UMaine SECS Numerical Modeling Laboratory - User contributions [en]

Transition to Turbulence

2019-05-15T20:04:58Z

Ian:

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''
__NOTOC__

Welcome to our teaching module exploring the '''transition to turbulence''' through models of fluid flow around a cylinder!

Transitioning to turbulence involves decreasing the predictability of a system. Inertial forces begin to dominate over viscous forces, resulting in more chaotic particle motion in the system ''(take a quick look at the Hurricane Irene example below in the Complex Flow section)''. In this module, we will explore this transition and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. These naturally-occurring phenomena are fascinating and have many natural and engineering implications. By the end of the module, you should be able to describe what's occurring in each of the scenarios presented and have a good grasp on the dynamics involved in the complex flow examples. You will be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

Here is a satellite loop of a von Karman Vortex Street developing in the wake of Guadalupe Island off the coast of Baja California, Mexico. (source: twitter user [https://twitter.com/weatherdak/status/1128453166725353473 @weatherdak])

[[File:GuadalupeVonKarman.mp4|none|1000px]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

Mass * acceleration = Body forces + Shear forces - Pressure gradient

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

ρ (∂v / ∂t) = Δρg + µ∇2U - ∇P

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Shearthinthick.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid. (Image from https://neutrium.net/fluid_flow/viscosity/)]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will begin to transition towards turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described as '''Newtonian'''. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about paint - it runs readily when applied with a brush or roller, yet just moments after retains enough viscosity to stay in place with minimal running).

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

Re = ρVD/µ = VD/v

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved as a finite-element model with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. The SPH-generated models below (and all others in this module) show the motion of ~105 particles. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Below are velocity, shear rate, vorticity, Reynolds number, and pressure plots generated in COMSOL for the same fluid flowing around a teardrop-shaped object instead of the cylinder. Compare these to the plots shown above for the cylindrical impediment. You may still notice signs of oscillating flow resulting from the disturbance, but flow is ''much'' more laminar. Can you explain why this might be?

[[File:fairing_water_velocity.gif]]

Pay attention especially to how the velocity vectors change (or do not change) throughout time. Compare this to the velocity from the water model with the cylindrical impediment. What does this indicate about the differences in inertial forces between the two scenarios?

[[File:fairing_water_shear.gif]]

[[File:fairing_water_vorticity.gif]]

This looks vastly different from the vorticity plot for the cylinder model.

[[File:fairing_water_reynolds.gif]]

Note how much lower the cell Reynolds number is compared to the cylinder model!

[[File:fairing_water_pressure.gif]]

This also looks vastly different from the pressure plot for the cylinder model. It's amazing how such a simple design can result in a major shift in flow around the object.
How does the teardrop shape influence the pressure and vorticity?

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculated using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has walked you through various stages in the transition to turbulence. Can you describe this transition in your own words? We have shown you fluid systems with varying levels of predictability. In addition, this module introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in transitional flows can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|600px|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (gaging station is green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''105 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|400px|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

File:GuadalupeVonKarman.mp4

2019-05-15T19:55:23Z

Ian: Originally posted to Twitter by user @weatherdak (Dakota Smith)

Originally posted to Twitter by user @weatherdak (Dakota Smith)

University of Maine SECS Numerical Laboratory

2019-05-10T23:18:10Z

Ian:

__NOTOC__
{{DISPLAYTITLE:{{FULLPAGENAME}}}}


<h1 div id="mp-topbanner" style="clear:both; position:relative; box-sizing:border-box; width:100%; margin:1.2em 0 6px; min-width:47em; border:1px solid #ddd; background-color:#f3f7ff; color:#000; white-space:nowrap;">

<div style="margin:0.4em; width:22em; text-align:center;">
<div style="font-size:200%; padding:0.3em 1em;">'''Welcome to [https://umaine.edu/earthclimate/research/geodynamics/ UMaine Geodynamics]'''</div>
<div style="font-size:60%; padding:0.3em 1em; text-align:left;">
:: This wiki is a means for accessing research and educational modules produced by the geodynamics group at UMaine.
:: The left side panel provides links to current and past research. [[Special:Statistics|{{NUMBEROFARTICLES}}]] articles, in English.</div>
</div></h1>

{| role="presentation" id="mp-upper" style="width: 100%; margin-top:4px; border-spacing: 5px;"

| id="mp-right" class="MainPageBG" style="width:50%; border:1px solid #cedff2; padding:1; background:#f5faff; vertical-align:top;"|
<h2 id="mp-itn-h2" style="margin:0.5em; background:#cedff2; font-family:inherit; font-size:175%; font-weight:bold; border:1px solid #a3b0bf; color:#000; padding:0.2em 0.4em;">News</h2>
<div id="mp-itn" style="padding:0.1em 0.6em;">
*'''5/8/19:''' Dr. Koons' graduate level computational Fluid Dynamics class presented their coupled models evaluating the stability of Thwaites Glacier, West Antarctica for their [[Fluid Dynamics Course Projects#2019|final course project]] in a US Association of Polar Early Career Scientists (USAPECS) webinar.

*'''3/12/19:''' Dr. Koons' Fluid Dynamics class developed and published new teaching modules for [[Fluid Flow Past a Cylinder | fluid flow]] and [[Heat transfer module - Fluid Dynamics 2019|advective, and convective heat transfer]].

*'''4/20/18:''' New tutorials on [[Media Wiki Use| Wiki Use]] have come as well as a few [[Software| software tutorials]]!

*'''4/11/18:''' The "[[Current Research]]" and "[[Previous Research]]" sidebar navigations are receiving their own page and the long sidebar list will soon be removed.

*'''4/4/18:''' There are new sidebar entries for "[[Surface Hydrology |Watershed Hydrology]]" and "[[Sedimentation in Maine Lakes |Maine Lake Sedimentation]]," and the "[[Smoothed Particle Hydrodynamics]]" section has been updated to include our latest research efforts with SPH.

*'''3/30/18:''' The wiki has been fully updated to the latest version of Mediawiki (1.30). The [https://www.mediawiki.org/wiki/Extension:PDFEmbed PDFEmbed extension] has been installed allowing for easy embedding of PDFs to the wiki. 

*'''3/7/18:''' A Geodynamics Box has been created to add materials and files to in anticipation for future additions to this wiki. 

*'''2/17/15:''' Sam Roy made revisions to [[Tectonic-Geomorphic-Climatic Interaction (NSF-EAR-1324637, 1323137)]]

*'''10/28/15:''' An "Events" section has been added to the sidebar in anticipation of the upcoming GeoPRISMS collaborative session at UMaine. Participants can find session announcements and links to PDFs on the [[GeoPRISMS Workshop 2015]] page.

*'''10/1/15:''' Lynn Kaluzienski created "[[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamics]]," complete with model descriptions and animations.
:<h4 id="mp-itn-h4" style="margin:0.1em; width:400px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Periodic Boundary Conditions-1.gif|400px|link=Smoothed Particle Hydrodynamics]]</h4>

*'''9/17/15:''' Sam Roy updated text and model results in [[Subduction Zone Dynamics in the Mantle and at the Earth's Surface]]:
:<h4 id="mp-itn-h4" style="margin:0.1em; width:400px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Vel1.gif|400px]]</h4>

*'''5/26/15:''' fault motion and landscape evolution model results have been updated: [[The role of surface displacement in landscape evolution]]

*'''3/26/15:''' Sam Roy added modeling work on Mantle Wedge Hydration under "Previous Research": [[Mantle Wedge Hydration]]

*'''3/18/15:''' Sam Roy and Nick Richmond have been running coupled FLAC/CHILD models to investigate orographic precipitation. Stay tuned!
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:itsonlyamodel.png|300px]]</h4>

::<h5 style="font-size:80%>(http://www.funnyjunk.com/)</h5>
</div>


| id="mp-left" class="MainPageBG" style="width:50%; border:1px solid #cef2e0; padding:1; background:#f5fffa; vertical-align:top; color:#000;" |
<h2 id="mp-tfa-h2" style="margin:0.5em; background:#cef2e0; font-family:inherit; font-size:175%; font-weight:bold; border:1px solid #a3bfb1; color:#000; padding:0.2em 0.4em;">[[Current Research| Research]]</h2>
<div id="mp-tfa" style="padding:0.1em 0.6em;">
*;'''2018:''' [[Strain Partitioning and Fault Segmentation]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;"> [[File:Figure1 model setup.png|link=Strain Partitioning and Fault Segmentation|300px]]</h4>

*;'''2018:''' [[Smoothed Particle Hydrodynamics]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;"> [[File:KnickpointMeanderVel01.gif|link=Smoothed Particle Hydrodynamics|300px]]</h4>

*;'''2018:''' [[Microtopography and Surface Water Detention|Microtopography and Surface Water Detention]]

*;'''2018:''' [[Penobscot Watershed Hydrology and History|Penobscot Watershed Hydrology and History]]

*;'''2018:''' [[Sedimentation in Maine Lakes]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:RadargramFigure.png|link=Sedimentation in Maine Lakes|300px]]</h4>

*;'''2018:''' [[Glacier Edge Dynamics]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:600px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Jarvisvel2.png|link=Glacier Edge Dynamics|600px]]</h4>

</div>

| style="border:1px solid transparent;" |

|}

University of Maine SECS Numerical Laboratory

2019-05-10T23:14:53Z

Ian:

__NOTOC__
{{DISPLAYTITLE:{{FULLPAGENAME}}}}


<h1 div id="mp-topbanner" style="clear:both; position:relative; box-sizing:border-box; width:100%; margin:1.2em 0 6px; min-width:47em; border:1px solid #ddd; background-color:#f3f7ff; color:#000; white-space:nowrap;">

<div style="margin:0.4em; width:22em; text-align:center;">
<div style="font-size:200%; padding:0.3em 1em;">'''Welcome to [https://umaine.edu/earthclimate/research/geodynamics/ UMaine Geodynamics]'''</div>
<div style="font-size:60%; padding:0.3em 1em; text-align:left;">
:: This wiki is a means for accessing research and educational modules produced by the geodynamics group at UMaine.
:: The left side panel provides links to current and past research. [[Special:Statistics|{{NUMBEROFARTICLES}}]] articles, in English.</div>
</div></h1>

{| role="presentation" id="mp-upper" style="width: 100%; margin-top:4px; border-spacing: 5px;"

| id="mp-right" class="MainPageBG" style="width:50%; border:1px solid #cedff2; padding:1; background:#f5faff; vertical-align:top;"|
<h2 id="mp-itn-h2" style="margin:0.5em; background:#cedff2; font-family:inherit; font-size:175%; font-weight:bold; border:1px solid #a3b0bf; color:#000; padding:0.2em 0.4em;">News</h2>
<div id="mp-itn" style="padding:0.1em 0.6em;">
*'''5/8/19:''' Dr. Koons' graduate level computational Fluid Dynamics class evaluated the stability of Thwaites Glacier in their [[Fluid Dynamics Course Projects#2019|final course project]].

*'''3/12/19:''' Dr. Koons' Fluid Dynamics class developed and published new teaching modules for [[Fluid Flow Past a Cylinder | fluid flow]] and [[Heat transfer module - Fluid Dynamics 2019|advective, and convective heat transfer]].

*'''4/20/18:''' New tutorials on [[Media Wiki Use| Wiki Use]] have come as well as a few [[Software| software tutorials]]!

*'''4/11/18:''' The "[[Current Research]]" and "[[Previous Research]]" sidebar navigations are receiving their own page and the long sidebar list will soon be removed.

*'''4/4/18:''' There are new sidebar entries for "[[Surface Hydrology |Watershed Hydrology]]" and "[[Sedimentation in Maine Lakes |Maine Lake Sedimentation]]," and the "[[Smoothed Particle Hydrodynamics]]" section has been updated to include our latest research efforts with SPH.

*'''3/30/18:''' The wiki has been fully updated to the latest version of Mediawiki (1.30). The [https://www.mediawiki.org/wiki/Extension:PDFEmbed PDFEmbed extension] has been installed allowing for easy embedding of PDFs to the wiki. 

*'''3/7/18:''' A Geodynamics Box has been created to add materials and files to in anticipation for future additions to this wiki. 

*'''2/17/15:''' Sam Roy made revisions to [[Tectonic-Geomorphic-Climatic Interaction (NSF-EAR-1324637, 1323137)]]

*'''10/28/15:''' An "Events" section has been added to the sidebar in anticipation of the upcoming GeoPRISMS collaborative session at UMaine. Participants can find session announcements and links to PDFs on the [[GeoPRISMS Workshop 2015]] page.

*'''10/1/15:''' Lynn Kaluzienski created "[[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamics]]," complete with model descriptions and animations.
:<h4 id="mp-itn-h4" style="margin:0.1em; width:400px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Periodic Boundary Conditions-1.gif|400px|link=Smoothed Particle Hydrodynamics]]</h4>

*'''9/17/15:''' Sam Roy updated text and model results in [[Subduction Zone Dynamics in the Mantle and at the Earth's Surface]]:
:<h4 id="mp-itn-h4" style="margin:0.1em; width:400px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Vel1.gif|400px]]</h4>

*'''5/26/15:''' fault motion and landscape evolution model results have been updated: [[The role of surface displacement in landscape evolution]]

*'''3/26/15:''' Sam Roy added modeling work on Mantle Wedge Hydration under "Previous Research": [[Mantle Wedge Hydration]]

*'''3/18/15:''' Sam Roy and Nick Richmond have been running coupled FLAC/CHILD models to investigate orographic precipitation. Stay tuned!
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:itsonlyamodel.png|300px]]</h4>

::<h5 style="font-size:80%>(http://www.funnyjunk.com/)</h5>
</div>


| id="mp-left" class="MainPageBG" style="width:50%; border:1px solid #cef2e0; padding:1; background:#f5fffa; vertical-align:top; color:#000;" |
<h2 id="mp-tfa-h2" style="margin:0.5em; background:#cef2e0; font-family:inherit; font-size:175%; font-weight:bold; border:1px solid #a3bfb1; color:#000; padding:0.2em 0.4em;">[[Current Research| Research]]</h2>
<div id="mp-tfa" style="padding:0.1em 0.6em;">
*;'''2018:''' [[Strain Partitioning and Fault Segmentation]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;"> [[File:Figure1 model setup.png|link=Strain Partitioning and Fault Segmentation|300px]]</h4>

*;'''2018:''' [[Smoothed Particle Hydrodynamics]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;"> [[File:KnickpointMeanderVel01.gif|link=Smoothed Particle Hydrodynamics|300px]]</h4>

*;'''2018:''' [[Microtopography and Surface Water Detention|Microtopography and Surface Water Detention]]

*;'''2018:''' [[Penobscot Watershed Hydrology and History|Penobscot Watershed Hydrology and History]]

*;'''2018:''' [[Sedimentation in Maine Lakes]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:RadargramFigure.png|link=Sedimentation in Maine Lakes|300px]]</h4>

*;'''2018:''' [[Glacier Edge Dynamics]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:600px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Jarvisvel2.png|link=Glacier Edge Dynamics|600px]]</h4>

</div>

| style="border:1px solid transparent;" |

|}

Fluid Dynamics Course Projects

2019-05-10T23:11:13Z

Ian: Created page with "= 2019 = Fluid Dynamics 2019 attempted a collaborative project focused on evaluating the stability of Thwaites Glacier, West Antarctica, over the next 200 years. This effort..."

= 2019 =

Fluid Dynamics 2019 attempted a collaborative project focused on evaluating the stability of Thwaites Glacier, West Antarctica, over the next 200 years. This effort involved creating cross-informed numerical models using [[Smoothed Particle Hydrodynamics]] (SPH; [[#References|1]]) and the [[ISSM Introductory Tutorials | Ice Sheet System Model]] (ISSM; [[#References|2]]). The results were presented in a webinar presentation format hosted and archived by the US Association of Polar Early Career Scientists ([https://usapecs.wixsite.com/usapecs USAPECS]).

[[User:Julialiu18 | Jukes Liu]] used observations by Pierre Dutrieux ([[#References|3]]) to calculate geostrophic flow near the margins of the Thwaites floating ice shelf, and input these calculations to thermodynamic equations to cross-evaluate with reported values of basal melt.

[[User:Ian | Ian Nesbitt]] created a coupled atmosphere ocean model in SPH, in an attempt to capture the local mixing processes that occur due to katabatic wind stress flowing off of the Thwaites ice sheet/ice shelf when there is no sea ice buffer protecting the ocean surface. Output from the SPH model was fed into a thermodynamic equation to determine melt rate potential, which was cross-checked with literature and observed melt rates, then used to parameterize basal melt on the floating ice shelf in the ISSM model.

[[User:Mariama | Mariama Dryak]] and [[User:Clara | Clara Deck]] created an ISSM solution designed to evaluate the stability of the Thwaites glacier system based on anticipated and experimental changes in surface mass balance (SMB) and basal melt. Surface mass balance values were derived from anticipated snow water equivalent expected in two [https://en.wikipedia.org/wiki/Representative_Concentration_Pathway IPCC Representative Concentration Pathways] (RCPs): RCP2.6 and RCP8.5 ([[#References|4]]). Basal melt values were derived from: 1) geothermal heat at the base, 2) measurements of large-scale melt holes described by Milillo (2019; [[#References|5]]), and 3) the outputs from the SPH simulation.

[[User:JackieFeng | Jackie Feng]] evaluated grounded ice rheology ([[#References|6]]), geothermal flux ([[#References|7]]), and basal friction ([[#References|7]], [[#References|8]]) in the ISSM model to ensure that these parameters were on par with values that other researchers had reported previously, and whether moderate changes to these values would cause changes to the stability of the system.

=== Model outputs ===

==== SPH model ====
{{#ev:vimeo|331417167}}

=== References ===

1. Crespo, A.J., Domínguez, J.M., Rogers, B.D., Gómez-Gesteira, M., Longshaw, S., Canelas, R., Vacondio, R., Barreiro, A. and García-Feal, O., 2015. DualSPHysics: Open-source parallel CFD solver based on Smoothed Particle Hydrodynamics (SPH). Computer Physics Communications, 187, pp.204-216.

2. Larour, E., Seroussi, H., Morlighem, M. and Rignot, E., 2012. Continental scale, high order, high spatial resolution, ice sheet modeling using the Ice Sheet System Model (ISSM). Journal of Geophysical Research: Earth Surface, 117(F1).

3. Dutrieux, P. Personal communication. NERC iSTAR program (UK), A. Jenkins and K. Heywood Principal Investigators.

4. IPCC, 2013: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change [Stocker, T.F., D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P.M. Midgley (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, pp. 1535.

5. Milillo, P., Rignot, E., Rizzoli, P., Scheuchl, B., Mouginot, J., Bueso-Bello, J. and Prats-Iraola, P., 2019. Heterogeneous retreat and ice melt of Thwaites Glacier, West Antarctica. Science advances, 5(1), p.eaau3433.

6. Suckale, J., Platt, J.D., Perol, T. and Rice, J.R., 2014. Deformation‐induced melting in the margins of the West Antarctic ice streams. Journal of Geophysical Research: Earth Surface, 119(5), pp.1004-1025.

7. Joughin, I., Tulaczyk, S., Bamber, J.L., Blankenship, D., Holt, J.W., Scambos, T. and Vaughan, D.G., 2009. Basal conditions for Pine Island and Thwaites Glaciers, West Antarctica, determined using satellite and airborne data. Journal of Glaciology, 55(190), pp.245-257.

8. Morlighem, M., Seroussi, H., Larour, E. and Rignot, E., 2013. Inversion of basal friction in Antarctica using exact and incomplete adjoints of a higher‐order model. Journal of Geophysical Research: Earth Surface, 118(3), pp.1746-1753.

Geodynamics Course Projects

2019-05-10T20:31:09Z

Ian: /* 2018 Geodynamics Course Projects */

== [http://umaine.edu/earthclimate/research/geodynamics/course-projects/ Past Geodynamics Course Projects] ==

[http://umaine.edu/earthclimate/research/geodynamics/course-projects/ Available here].

== 2018 Geodynamics Course Projects ==
'''Modeling Glacial erosion using ice velocity and rock strength in Coastal Alaska'''

''Will Kochtitzky and [[User:Ian|Ian Nesbitt]]''

This project uses a very basic, very generalized, "fudge factored" rock cohesion classification to estimate erosion in coastal Alaska at present and throughout the last 115,000 years. Modeled velocity and ice extent data provided graciously by Annie Boucher are part of the University of Maine Ice Sheet Model ([http://climatechange.umaine.edu/Research/Contrib/html/15.html UMISM]).

Seen here is a an estimate of glacial erosion based on the velocity-driven erosion model of Humphrey and Raymond ([https://doi.org/10.3189/S0022143000012429 1994]) since 115 thousand years before present.

[[File:Erosion_lgm.gif|800px]]

Below, the same model applied to a modern velocity dataset (Altena et al., [https://doi.org/10.5194/tc-13-795-2019 2019]) which uses a finer resolution.

[[File:Erosion_modern.gif|800px]]

<pdf>File:NesbittGeodynamicsFinal.pdf</pdf>

== References ==

* Altena, B., Scambos, T., Fahnestock, M., & Kääb, A. (2018). Extracting recent short-term glacier velocity evolution over Southern Alaska from a large collection of Landsat data. The Cryosphere Discussions, (May), 1–27. https://doi.org/10.5194/tc-2018-66

* Humphrey, N. F., & Raymond, C. F. (1994). Hydrology, erosion and sediment production in a surging glacier: Variegated Glacier, Alaska, 1982–83. Journal of Glaciology, 40(136), 539–552. https://doi.org/10.3189/S0022143000012429

Heat transfer module - Fluid Dynamics 2019

2019-03-12T18:59:28Z

Ian: Redirected page to Heat transfer

#REDIRECT [[Heat transfer]]

This page has moved [[Heat transfer|here]].

Heat transfer module - Fluid Dynamics 2019

2019-03-12T18:58:28Z

Ian:

This page has moved [[Heat transfer|here]].

Heat transfer module - Fluid Dynamics 2019

2019-03-12T18:58:19Z

Ian: Replaced content with "This page has moved [Heat transfer|here]."

This page has moved [Heat transfer|here].

Heat transfer

2019-03-12T18:55:18Z

Ian: Created page with " == '''Heat Transfer in a Lava Lamp''' == When you flip on the light in a lava lamp, heat from the bulb works slowly to warm up the wax (or ‘lava’). This kind of heat tra..."

== '''Heat Transfer in a Lava Lamp''' ==

When you flip on the light in a lava lamp, heat from the bulb works slowly to warm up the wax (or ‘lava’). This kind of heat transfer is called conduction, or movement of heat from one place to another. Heat energy flows from high heat (light bulb) to low heat (wax), and the gradient is the driver of conductive heat transfer.

[[File:lava1.png|200px|Image: 200 pixels]] [[File:lava2.png|200px|Image: 200 pixels]] [[File:lava3.png|185px|Image: 185 pixels]]

Until things start moving, conduction is the dominant heat transfer process in our lava lamp system.
*'''Conduction''', defined as '''κ∇²T''' depends on:
**thermal diffusivity of the material through which heat is moving (κ)
**thermal gradient (∇²T)

The density of the wax is inversely related to temperature, meaning that at higher temperatures, the wax is less dense. When the density becomes lower than the density of the liquid in the lava lamp, the wax will tend to float upward.

[[File:lava4.png|200px|Image: 200 pixels]] [[File:lava6.png|190px|Image: 190 pixels]]

Wax reaches critical density (less than liquid) and tends to float upward. That movement of wax, and thus heat, introduces velocity into the system and is a process that is referred to as advection.
*'''Advection''', defined as '''v∇T''' depends on:
**velocity of the fluid (v)
**thermal gradient (∇T)

''Heat Transfer Equation'':

Heat transfer within the lava lamp can be represented by an equation. The heat transfer equation brings these aforementioned processes together to describe how heat changes through time ('''∂T/∂t'''). It is defined as:

∂T/∂t = κ∇²T + v∇T + A

∂T/∂t = conduction + advection + production

The production term ('''A''') represents internally generated heat energy and we do not consider it in this lava lamp scenario. It becomes important in systems such as glaciers, when internal strain heating is an important source of heat.

''Peclét Number'':

The Peclét Number is a dimensionless number which indicates whether conduction or advection dominates the system.
Pe = uL/κ
where u = flow velocity, L = characteristic length Height of lava lamp, κ = thermal diffusivity
*Low peclet regime- dominated by heat transport by conduction
*High peclet regime- dominated by heat transport by advection

With the movement of wax in the upward direction away from the heat source at the bottom of the container and into a cooler material the wax density increases again (more than liquid). This wax density increase causes the wax to sink back down to the bottom of the container towards the heat source. This process is called convection.

[[File:lava5.png|200px|Image: 210 pixels]]

''Rayleigh Number'':

The Rayleigh Number is a dimensionless number used to describe whether convection grows or decays.
Ra ≡ (gρΔTαd³)/μκ

*'''Convection''' depends on driving forces and resisting forces.
**Driving forces (when these terms are dominant convection grows):
***Acceleration due to gravity (g)
***Density (ρ)
***The temperature difference between the bottom and top of the convection cell (ΔT)
***The volume coefficient of thermal expansion (α)
***The height of the convection cell (d)
**Resisting forces (when these terms are dominant convection decays):
***Viscosity (μ)
***Thermal diffusivity (κ)

At some critical Rayleigh (Racr) number (dependent on the system), the convection regime shifts.

=='''Heat Transfer in Ice Stream Shut Down and Start (based on Joughin and Alley, 2011)'''==

Stop and start of ice streams is dependent on:
*Thinning (steepens basal temperature gradients)
*Basal freezing (puts on the brakes for a system)
*Thickening (build up traps geothermal heat)
*Ice stream activation & the cycle is repeated

Watch the video below to walk through the steps of ice stream shut-down and speed-up, and see how this relates to the heat transfer equation introduced above.

{{#ev:youtube|"https://youtu.be/JZWqx4qiGJ0"}}

'''Further reading:'''

Different theories exist for ice stream shut down, and here we present varying hypothesis. Please note that the behavior we model in our video is based upon the hypothesis of '''Joughin and Alley (2011)'''.

*Joughin, I., and R. B. Alley (2011) Stability of the West Antarctic ice sheet in a warming world. Nature Geosci., 4, pp. 506–513. https://www.uib.no/sites/w3.uib.no/files/attachments/joughinnatgeoreview2011.pdf
*Anandakrishnan, S., Alley, R.B., Jacobel, R.W. and Conway, H. (2001) The flow regime of Ice Stream C and hypotheses concerning its recent stagnation. The West Antarctic Ice Sheet: Behavior and Environment, AGU Antarctic Research Series, 77, pp. 283-296. https://www.researchgate.net/profile/H_Conway/publication/266340045_The_Flow_Regime_of_Ice_Stream_C_and_Hypotheses_Concerning_Its_Recent_Stagnation/links/54f92db90cf210398e979b18.pdf
*Catania, G.A., Scambos, T.A., Conway, H. and Raymond, C.F. (2006) Sequential stagnation of Kamb ice stream, West Antarctica. Geophysical Research Letters, 33(14). https://doi.org/10.1029/2006GL026430
*Hulbe, C. & Fahnestock, M. (2007) Century-scale discharge stagnation and reactivation of the Ross ice streams, West Antarctica. J. Geophys. Res.-Earth 112, https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2006JF000603
*Lipovsky, B., and E. Dunham (2016) Tremor during ice-stream stick slip, Cryosphere, 10(1), pp. 385–399. https://www.the-cryosphere.net/10/385/2016/tc-10-385-2016.html
*Price, S.F., Bindschadler, R.A., Hulbe, C.L. and Joughin, I.R. (2001) Post-stagnation behavior in the upstream regions of Ice Stream C, West Antarctica. Journal of Glaciology, 47(157), pp. 283-294. https://doi.org/10.3189/172756501781832232

University of Maine SECS Numerical Laboratory

2019-03-12T17:40:06Z

Ian:

__NOTOC__
{{DISPLAYTITLE:{{FULLPAGENAME}}}}


<h1 div id="mp-topbanner" style="clear:both; position:relative; box-sizing:border-box; width:100%; margin:1.2em 0 6px; min-width:47em; border:1px solid #ddd; background-color:#f3f7ff; color:#000; white-space:nowrap;">

<div style="margin:0.4em; width:22em; text-align:center;">
<div style="font-size:200%; padding:0.3em 1em;">'''Welcome to [https://umaine.edu/earthclimate/research/geodynamics/ UMaine Geodynamics]'''</div>
<div style="font-size:60%; padding:0.3em 1em; text-align:left;">
:: This wiki is a means for accessing research and educational modules produced by the geodynamics group at UMaine.
:: The left side panel provides links to current and past research. [[Special:Statistics|{{NUMBEROFARTICLES}}]] articles, in English.</div>
</div></h1>

{| role="presentation" id="mp-upper" style="width: 100%; margin-top:4px; border-spacing: 5px;"

| id="mp-right" class="MainPageBG" style="width:50%; border:1px solid #cedff2; padding:1; background:#f5faff; vertical-align:top;"|
<h2 id="mp-itn-h2" style="margin:0.5em; background:#cedff2; font-family:inherit; font-size:175%; font-weight:bold; border:1px solid #a3b0bf; color:#000; padding:0.2em 0.4em;">News</h2>
<div id="mp-itn" style="padding:0.1em 0.6em;">
*'''3/12/19:''' Dr. Koons' graduate level computational Fluid Dynamics class developed and published new teaching modules for [[Fluid Flow Past a Cylinder | fluid flow]] and [[Heat transfer module - Fluid Dynamics 2019|advective, and convective heat transfer]].

*'''4/20/18:''' New tutorials on [[Media Wiki Use| Wiki Use]] have come as well as a few [[Software| software tutorials]]!

*'''4/11/18:''' The "[[Current Research]]" and "[[Previous Research]]" sidebar navigations are receiving their own page and the long sidebar list will soon be removed.

*'''4/4/18:''' There are new sidebar entries for "[[Surface Hydrology |Watershed Hydrology]]" and "[[Sedimentation in Maine Lakes |Maine Lake Sedimentation]]," and the "[[Smoothed Particle Hydrodynamics]]" section has been updated to include our latest research efforts with SPH.

*'''3/30/18:''' The wiki has been fully updated to the latest version of Mediawiki (1.30). The [https://www.mediawiki.org/wiki/Extension:PDFEmbed PDFEmbed extension] has been installed allowing for easy embedding of PDFs to the wiki. 

*'''3/7/18:''' A Geodynamics Box has been created to add materials and files to in anticipation for future additions to this wiki. 

*'''2/17/15:''' Sam Roy made revisions to [[Tectonic-Geomorphic-Climatic Interaction (NSF-EAR-1324637, 1323137)]]

*'''10/28/15:''' An "Events" section has been added to the sidebar in anticipation of the upcoming GeoPRISMS collaborative session at UMaine. Participants can find session announcements and links to PDFs on the [[GeoPRISMS Workshop 2015]] page.

*'''10/1/15:''' Lynn Kaluzienski created "[[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamics]]," complete with model descriptions and animations.
:<h4 id="mp-itn-h4" style="margin:0.1em; width:400px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Periodic Boundary Conditions-1.gif|400px|link=Smoothed Particle Hydrodynamics]]</h4>

*'''9/17/15:''' Sam Roy updated text and model results in [[Subduction Zone Dynamics in the Mantle and at the Earth's Surface]]:
:<h4 id="mp-itn-h4" style="margin:0.1em; width:400px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Vel1.gif|400px]]</h4>

*'''5/26/15:''' fault motion and landscape evolution model results have been updated: [[The role of surface displacement in landscape evolution]]

*'''3/26/15:''' Sam Roy added modeling work on Mantle Wedge Hydration under "Previous Research": [[Mantle Wedge Hydration]]

*'''3/18/15:''' Sam Roy and Nick Richmond have been running coupled FLAC/CHILD models to investigate orographic precipitation. Stay tuned!
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:itsonlyamodel.png|300px]]</h4>

::<h5 style="font-size:80%>(http://www.funnyjunk.com/)</h5>
</div>


| id="mp-left" class="MainPageBG" style="width:50%; border:1px solid #cef2e0; padding:1; background:#f5fffa; vertical-align:top; color:#000;" |
<h2 id="mp-tfa-h2" style="margin:0.5em; background:#cef2e0; font-family:inherit; font-size:175%; font-weight:bold; border:1px solid #a3bfb1; color:#000; padding:0.2em 0.4em;">[[Current Research| Research]]</h2>
<div id="mp-tfa" style="padding:0.1em 0.6em;">
*;'''2018:''' [[Strain Partitioning and Fault Segmentation]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;"> [[File:Figure1 model setup.png|link=Strain Partitioning and Fault Segmentation|300px]]</h4>

*;'''2018:''' [[Smoothed Particle Hydrodynamics]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;"> [[File:KnickpointMeanderVel01.gif|link=Smoothed Particle Hydrodynamics|300px]]</h4>

*;'''2018:''' [[Microtopography and Surface Water Detention|Microtopography and Surface Water Detention]]

*;'''2018:''' [[Penobscot Watershed Hydrology and History|Penobscot Watershed Hydrology and History]]

*;'''2018:''' [[Sedimentation in Maine Lakes]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:RadargramFigure.png|link=Sedimentation in Maine Lakes|300px]]</h4>

*;'''2018:''' [[Glacier Edge Dynamics]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:600px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Jarvisvel2.png|link=Glacier Edge Dynamics|600px]]</h4>

</div>

| style="border:1px solid transparent;" |

|}

University of Maine SECS Numerical Laboratory

2019-03-12T17:39:02Z

Ian:

__NOTOC__
{{DISPLAYTITLE:{{FULLPAGENAME}}}}


<h1 div id="mp-topbanner" style="clear:both; position:relative; box-sizing:border-box; width:100%; margin:1.2em 0 6px; min-width:47em; border:1px solid #ddd; background-color:#f3f7ff; color:#000; white-space:nowrap;">

<div style="margin:0.4em; width:22em; text-align:center;">
<div style="font-size:200%; padding:0.3em 1em;">'''Welcome to [https://umaine.edu/earthclimate/research/geodynamics/ UMaine Geodynamics]'''</div>
<div style="font-size:60%; padding:0.3em 1em; text-align:left;">
:: This wiki is a means for accessing research and educational modules produced by the geodynamics group at UMaine.
:: The left side panel provides links to current and past research. [[Special:Statistics|{{NUMBEROFARTICLES}}]] articles, in English.</div>
</div></h1>

{| role="presentation" id="mp-upper" style="width: 100%; margin-top:4px; border-spacing: 5px;"

| id="mp-right" class="MainPageBG" style="width:50%; border:1px solid #cedff2; padding:1; background:#f5faff; vertical-align:top;"|
<h2 id="mp-itn-h2" style="margin:0.5em; background:#cedff2; font-family:inherit; font-size:175%; font-weight:bold; border:1px solid #a3b0bf; color:#000; padding:0.2em 0.4em;">News</h2>
<div id="mp-itn" style="padding:0.1em 0.6em;">
*'''3/12/19:''' Dr. Koons' Fluid Dynamics class published new teaching modules for [[Fluid Flow Past a Cylinder | fluid flow]] and [[Heat transfer module - Fluid Dynamics 2019|advective, and convective heat transfer]].

*'''4/20/18:''' New tutorials on [[Media Wiki Use| Wiki Use]] have come as well as a few [[Software| software tutorials]]!

*'''4/11/18:''' The "[[Current Research]]" and "[[Previous Research]]" sidebar navigations are receiving their own page and the long sidebar list will soon be removed.

*'''4/4/18:''' There are new sidebar entries for "[[Surface Hydrology |Watershed Hydrology]]" and "[[Sedimentation in Maine Lakes |Maine Lake Sedimentation]]," and the "[[Smoothed Particle Hydrodynamics]]" section has been updated to include our latest research efforts with SPH.

*'''3/30/18:''' The wiki has been fully updated to the latest version of Mediawiki (1.30). The [https://www.mediawiki.org/wiki/Extension:PDFEmbed PDFEmbed extension] has been installed allowing for easy embedding of PDFs to the wiki. 

*'''3/7/18:''' A Geodynamics Box has been created to add materials and files to in anticipation for future additions to this wiki. 

*'''2/17/15:''' Sam Roy made revisions to [[Tectonic-Geomorphic-Climatic Interaction (NSF-EAR-1324637, 1323137)]]

*'''10/28/15:''' An "Events" section has been added to the sidebar in anticipation of the upcoming GeoPRISMS collaborative session at UMaine. Participants can find session announcements and links to PDFs on the [[GeoPRISMS Workshop 2015]] page.

*'''10/1/15:''' Lynn Kaluzienski created "[[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamics]]," complete with model descriptions and animations.
:<h4 id="mp-itn-h4" style="margin:0.1em; width:400px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Periodic Boundary Conditions-1.gif|400px|link=Smoothed Particle Hydrodynamics]]</h4>

*'''9/17/15:''' Sam Roy updated text and model results in [[Subduction Zone Dynamics in the Mantle and at the Earth's Surface]]:
:<h4 id="mp-itn-h4" style="margin:0.1em; width:400px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Vel1.gif|400px]]</h4>

*'''5/26/15:''' fault motion and landscape evolution model results have been updated: [[The role of surface displacement in landscape evolution]]

*'''3/26/15:''' Sam Roy added modeling work on Mantle Wedge Hydration under "Previous Research": [[Mantle Wedge Hydration]]

*'''3/18/15:''' Sam Roy and Nick Richmond have been running coupled FLAC/CHILD models to investigate orographic precipitation. Stay tuned!
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:itsonlyamodel.png|300px]]</h4>

::<h5 style="font-size:80%>(http://www.funnyjunk.com/)</h5>
</div>


| id="mp-left" class="MainPageBG" style="width:50%; border:1px solid #cef2e0; padding:1; background:#f5fffa; vertical-align:top; color:#000;" |
<h2 id="mp-tfa-h2" style="margin:0.5em; background:#cef2e0; font-family:inherit; font-size:175%; font-weight:bold; border:1px solid #a3bfb1; color:#000; padding:0.2em 0.4em;">[[Current Research| Research]]</h2>
<div id="mp-tfa" style="padding:0.1em 0.6em;">
*;'''2018:''' [[Strain Partitioning and Fault Segmentation]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;"> [[File:Figure1 model setup.png|link=Strain Partitioning and Fault Segmentation|300px]]</h4>

*;'''2018:''' [[Smoothed Particle Hydrodynamics]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;"> [[File:KnickpointMeanderVel01.gif|link=Smoothed Particle Hydrodynamics|300px]]</h4>

*;'''2018:''' [[Microtopography and Surface Water Detention|Microtopography and Surface Water Detention]]

*;'''2018:''' [[Penobscot Watershed Hydrology and History|Penobscot Watershed Hydrology and History]]

*;'''2018:''' [[Sedimentation in Maine Lakes]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:300px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:RadargramFigure.png|link=Sedimentation in Maine Lakes|300px]]</h4>

*;'''2018:''' [[Glacier Edge Dynamics]]
:<h4 id="mp-itn-h4" style="margin:0.1em; width:600px; background:#ffffff; border:1px solid #a3b0bf; color:#000; padding:0.1em 0.2em;">[[File:Jarvisvel2.png|link=Glacier Edge Dynamics|600px]]</h4>

</div>

| style="border:1px solid transparent;" |

|}

Transition to Turbulence

2019-03-11T00:16:32Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. These naturally-occurring phenomena are fascinating and have many natural and engineering implications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid. (Image licensed under Creative Commons 4.0.)]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will begin to transition towards turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described as '''Newtonian'''. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|600px|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (gaging station is green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|400px|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-11T00:15:21Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. These naturally-occurring phenomena are fascinating and have many natural and engineering implications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid. Image licensed under Creative Commons 4.0.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will begin to transition towards turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described as '''Newtonian'''. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|600px|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (gaging station is green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|400px|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-11T00:13:48Z

Ian:

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. These naturally-occurring phenomena are fascinating and have many natural and engineering implications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will begin to transition towards turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described as '''Newtonian'''. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|600px|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (gaging station is green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|400px|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-11T00:10:27Z

Ian: /* Fully turbulent flows */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will begin to transition towards turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described as '''Newtonian'''. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|600px|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (gaging station is green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|400px|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:55:50Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will begin to transition towards turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described as '''Newtonian'''. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:50:53Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will begin to transition towards turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described as '''Newtonian'''. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:50:07Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will begin to transition towards turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:49:51Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. Molasses and toothpaste are more viscous than water, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulence.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:39:43Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://youtu.be/BN2D5y-AxIY?t=8 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:38:26Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://www.youtube.com/watch?v=BN2D5y-AxIY?t=7 here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:37:06Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British demonstration of this behavior [https://www.youtube.com/watch?v=BN2D5y-AxIY here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:36:29Z

Ian: /* Viscosity */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch a very British example [https://www.youtube.com/watch?v=BN2D5y-AxIY here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:34:04Z

Ian: /* Reynolds number and things that affect it */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch example [https://www.youtube.com/watch?v=BN2D5y-AxIY here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:33:26Z

Ian: /* Characterizing the fluid dynamics */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number and things that affect it ===

The Reynolds number, ''Re'', is a dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

==== Viscosity ====

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch example [https://www.youtube.com/watch?v=BN2D5y-AxIY here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

==== Velocity ====

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T23:30:55Z

Ian: /* Reynolds number */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

[[File:Viscous regimes.png|thumb|400px|Viscous regimes. Newtonian fluids exhibit a nearly linear relationship between stress and strain rate. This means that deformation occurs proportional to the forces acting on the fluid.]]

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

The conceptual diagram at right shows viscous regimes. A fluid that exhibits a linear relationship between stress and strain is described Newtonian. Any force that is applied to Newtonian fluids results in a proportional amount of deformation. '''Shear thickening''' is a term used to describe a nonlinear regime in which greater force can result in less deformation (corn starch and water, for example...watch example [https://www.youtube.com/watch?v=BN2D5y-AxIY here]). Conversely, '''shear thinning''' describes a fluid that deforms more easily with increased force (think about quicksand, which behaves like a shear-thinning fluid—the more you struggle, the more you sink!)

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

File:Viscous regimes.png

2019-03-10T23:11:45Z

Ian: Ian uploaded a new version of File:Viscous regimes.png

Graphic by Ian Nesbitt, shared under Creative Commons 4.0

File:Viscous regimes.png

2019-03-10T23:07:46Z

Ian: Ian uploaded a new version of File:Viscous regimes.png

Graphic by Ian Nesbitt, shared under Creative Commons 4.0

File:Viscous regimes.png

2019-03-10T23:06:49Z

Ian: Graphic by Ian Nesbitt, shared under Creative Commons 4.0

Graphic by Ian Nesbitt, shared under Creative Commons 4.0

Transition to Turbulence

2019-03-10T22:26:48Z

Ian: /* Navier-Stokes momentum equation */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Mass * acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T22:26:03Z

Ian: /* Geometry */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient (shear) between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T22:24:14Z

Ian: /* Geometry */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model for fluid flow described by Navier Stokes equation (above) and solved with help from numerical methods of COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T22:22:54Z

Ian: /* Navier-Stokes momentum equation */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ρ (∂v / ∂t) = Δρg + µΔ2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''Δ2U'''
|= curvature of velocity field (2nd derivative)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T22:15:45Z

Ian: /* Characterizing the fluid dynamics */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 (approximately that of freshwater) to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T22:12:15Z

Ian: /* Pressure 3 */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency] for this channel.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T22:11:53Z

Ian: /* Pressure 3 */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a [https://en.wikipedia.org/wiki/Resonance resonant frequency].

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T22:09:49Z

Ian: /* Pressure 3 */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth upstream of the cylinder, it seems to be oscillating in the channel behind it as well. Pressure oscillations that occur under these fluid conditions may be occurring near what's called a resonant frequency.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T22:07:42Z

Ian: /* Pressure 3 */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

In this plot, pressure is not only oscillating back and forth, it seems to be oscillating at a rate that causes resonance in the rest of the channel. This is called a resonant frequency.

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T21:44:38Z

Ian: /* Shear rate 3 */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

Compare this shear rate plot with [[Fluid Flow Past a Cylinder#Shear rate 2|the one above]]. Why is this one "sharper," i.e. the velocity gradient is more severe across the shear zone? What about air in comparison to water allows that to happen?

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T21:37:06Z

Ian: /* Truck tail fairings */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances. Have you thought about why this works? What about the truck and survey pole fairings decreases that instability and allows less vortex shedding?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T21:29:22Z

Ian: /* Real-world applications 2 */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on mechanical joints, which can suffer brittle failure.

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

The photos above show a weld joint that has failed due to shear forces caused by vortex shedding. At the time of failure, the vessel was traveling about 2-2.5 m/s. Luckily, the mount was secured with rope and the instrument was not lost.

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing normal (perpendicular) to the direction of travel. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped mount would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances.

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-10T21:18:10Z

Ian:

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our computational models of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity after the fluid passes the cylinder. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on the mechanical joints of the survey mount, which can cause brittle failure (see below).

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing. Think back to the vorticity and pressure plots from the flow past a cylinder model above. Can you explain why a teardrop-shaped "cylinder" would result in more stable flow?

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

Compare this velocity plot to the velocity from scenario 2 (Water). The initial velocity of the fluid is the same for both scenarios. Why might the maximum velocity be higher for one versus the other?

Note: The scale bar is different between the two.

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

These cell Reynolds numbers are several orders of magnitude higher than those calculated for the water scenario. In this case, the Reynolds number is above the critical number of 100, which makes it a "transitional" flow. In the range of transitional Reynolds numbers, 100 to 105, eddy-like vortices begin to form and propagate downstream. This phenomenon is much easier to visualize using a 3D particle solution. Let's look at the SPH models for this scenario.

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

If you've ever seen a tractor trailer with panels hanging off its tail (see image below), you've also seen strategies to counter the effects of vortex shedding firsthand. Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances.

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-08T04:35:28Z

Ian: /* Truck tail fairings */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our generated model of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on the mechanical joints of the survey mount, which can cause brittle failure (see below).

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing.

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Question #1: Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

Question #2: Have you ever seen a tractor trailer with panels hanging off its tail (see image below) and wondered what they were? Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances.

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-08T04:34:40Z

Ian: /* Truck tail fairings */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our generated model of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on the mechanical joints of the survey mount, which can cause brittle failure (see below).

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing.

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Question #1: Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

Question #2: Have you ever seen a tractor trailer with panels hanging off its tail and wondered what they were? Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Tail fairings.jpg|Fluid dynamics being sold to consumers!]]
[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances.

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

File:Tail fairings.jpg

2019-03-08T04:34:11Z

Ian: Photo: Nathan Hamilton https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html (Accessed 2019-03-07)

Photo: Nathan Hamilton
https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html (Accessed 2019-03-07)

Transition to Turbulence

2019-03-08T04:31:45Z

Ian: /* Real-world applications 3 */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our generated model of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on the mechanical joints of the survey mount, which can cause brittle failure (see below).

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing.

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

==== Atmospheric instability ====

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

==== Truck tail fairings ====

Question #1: Have you ever gotten a little too close to the back of a tractor trailer on the highway and noticed that your car was being buffetted around side-to-side in the wind? If so, you have experienced the effects of von Kármán vortex streets firsthand.

Question #2: Have you ever seen a tractor trailer with panels hanging off its tail and wondered what they were? Remember the marine survey mount fairings from [[Fluid Flow Past a Cylinder#Real-world applications 2|the previous Real-world applications section]]? These trailer fairings serve essentially the same purpose: reducing drag and causing a more aerodynamic slipstream. The following image is an example of a commercial application of fluid dynamics.

[[File:Atdynamics trailertail.jpg|Fluid dynamics being sold to consumers!]]

This image shows the effects of creating a more efficient slipstream which reduces vortex shedding and improves fuel efficiency. One estimate claims [https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html] that these tail fairings can increase fuel efficiency by approximately 10% over hauling distances.

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

File:Atdynamics trailertail.jpg

2019-03-08T04:22:42Z

Ian: Copyright ATDynamics 2013/STEMCO Aerodynamics 2019. Retrieved from https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html on 2019-03-07.

Copyright ATDynamics 2013/STEMCO Aerodynamics 2019. Retrieved from https://slate.com/culture/2013/04/truck-panels-what-do-they-do-explained-photos.html on 2019-03-07.

Transition to Turbulence

2019-03-08T04:08:25Z

Ian:

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our generated model of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications 2 ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on the mechanical joints of the survey mount, which can cause brittle failure (see below).

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing.

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world applications 3 ===

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

von Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>

Transition to Turbulence

2019-03-08T03:58:14Z

Ian: /* Real-world Applications */

''Created by [[User:Ian|Ian Nesbitt]] and [[User:Julialiu18|Jukes Liu]] on 2019-02-19. Because this page has some high-resolution animations, it is best viewed on a reliable internet connection.''

Welcome to our teaching module on fluid flow past a cylinder!

In this module, we will explore the transition to turbulence and learn how viscosity affects the dynamics of a fluid flowing around a cylindrical impediment. We can observe an interesting phenomenon called Von Kármán Vortex Streets at this simulated transition to turbulence. Their naturally-occurring counterparts are fascinating and have many environmental and engineering applications. Below is a snapshot of one of our generated model of water flowing past a cylinder. In this early time-step, we can see an instability begin to develop in the wake of the cylinder.

[[File:Water Vort VelMagVectors.0134.png|none|800px|Early-timestep vorticity map of a numerical model of water flowing past a cylinder, showing an instability developing in the wake of the cylinder.]]

= Overview =

In order to model the transition to turbulence and the effects of changing viscosity on the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path, we use several modeling tools: [https://www.comsol.com/ COMSOL Multiphysics], [https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics Smoothed-Particle Hydrodynamics] (SPH), and [https://www.paraview.org/ ParaView]. We will describe the model set-up and its physical basis, the differences between the three scenarios with fluids of different viscosities, and several real-world implications. We will move from looking at the behavior of more viscous fluids and move towards fluids with lower viscosities. Note: [https://en.wikipedia.org/wiki/Viscosity Viscosity] is defined as the measure of a fluid's resistance to deformation.

At the end of the module, you should be able to describe the initial conditions of a system, the transition from laminar to turbulent flow, and the effect viscosity has on the inertia, complexity, and predictability of a system!

First, let's start by learning how to talk about the various components of flow.

= Characterizing the fluid dynamics =

When we talk about fluid flow, we need to first describe the fluid. Some of the fundamental characteristics of fluid are density, viscosity, compressibility, heat capacity, and thermal conductivity. In this exercise we will ignore the thermal properties entirely, define a constant density and compressibility, and focus most of our attention on how viscosity affects the kinematics of a fluid system.

Compressibility and density are fairly easy to define. [https://en.wikipedia.org/wiki/Compressibility Compressibility] is the tendency of a fluid to change its volume due to changes in pressure. Since we are not dealing with immense pressure, we assume that compressibility is zero across all models. [https://en.wikipedia.org/wiki/Density Density] is the quantity of mass per unit volume. For the purposes of this model, we will hold density constant at 1 kg m-3 to try and isolate the effects of viscosity. These parameters will be fed into the Navier-Stokes fluid momentum equation (below) for each location in the model at each time step.

=== Navier-Stokes momentum equation ===

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is derived based on the conservation of momentum in a system. It is the foundation of our models. This equation can be broken down into its different terms, which represent the effects of the inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

This is the basic structure of the Navier-Stokes momentum equation:

'''Acceleration''' = '''Body forces''' + '''Shear forces''' - '''Pressure gradient'''

These qualitative terms are represented by some greek symbols which may or may not mean anything to you right now, but we'll break them down for you.

==== ∂v / ∂t = ρg + µ∇2U - ∇P ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''∂v'''
|= change in velocity
|-
|'''∂t'''
|= change in time
|-
|'''ρ'''
|= density
|-
|'''g'''
|= acceleration due to gravity
|-
|'''µ'''
|= viscosity
|-
|'''∇2U'''
|= divergence of velocity (gradient)
|-
|'''∇P'''
|= divergence of pressure (gradient)
|}

The inertial forces (acceleration) is represented by '''∂v / ∂t'''.

On the right hand side of the equation, we have the body forces from density and gravity ('''ρg'''), viscous forces from the viscosity property of the fluid ('''µ∇2U'''), and the pressure gradient throughout the system ('''∇P'''). Each of these forces will influence the acceleration or motion of particles. Can you visualize how these different forces on the right-hand side of the equation might influence a particle's motion?

Solving this equation for each grid point or particle at each time step will define how our fluid will behave in a computational sense. Once our fluid starts to move, we need ways to describe how it is flowing. Fluid flow can be characterized as either laminar or turbulent, which has to do with the predictability of its motion throughout the system. The parameter that is used to describe this characteristic is called the Reynolds number.

=== Reynolds number ===

The Reynolds number, ''Re'', is a simple, dimensionless number that represents the flow regime of a fluid. The Reynolds number is a way of representing whether a flow is laminar, transitional, or fully turbulent using numerical values. When flow is laminar, the fluid motion is more uni-directional, smooth, and more predictable. Imagine the flow of molasses being poured out onto a table. Another example would be toothpaste being squeezed out of the tube. Laminar flow is represented by lower Reynolds numbers. When flow is turbulent, fluid motion is more irregular. A gust of wind flowing around a flag-pole causing the flag to flap behind it is an every-day example of a more turbulent flow. These flows are characterized by higher Reynolds numbers.

What are some other real-world examples of fluid flow that you can think of? Would you characterize the flow as more laminar or turbulent?

Keeping those real-world examples in mind, we can brainstorm about the differences in those fluid systems that might affect the flow pattern. There are differences in the fluid properties between the molasses, toothpaste, and atmospheric air. Once of the main differences, which we examine in depth in this module, is their difference in viscosity. "Viscosity" is a term you might be familiar with in terms of describing a fluid. The technical definition of [https://en.wikipedia.org/wiki/Viscosity viscosity] is a fluid's capability of resisting deformation. The molasses and toothpaste are more viscous, which means that their flow is more difficult to deform or disturb. Meanwhile, air has a much lower viscosity and is quite easily deformed! We deform air every time we move. Therefore, if everything else is held constant, increasing viscosity decreases the Reynolds number which means that fluid flow becomes more laminar. If we decrease viscosity, fluid flow will transition to turbulent flow.

Another main parameter that influences the flow regime is the velocity of the fluid. Faster-flowing fluids tend to result in more turbulent responses. In the air around a flagpole example, higher wind velocity will result in a more turbulent response which one would observe as more violent "flapping".

Therefore, Reynolds number is a function of viscosity and velocity: f(µ, v). Velocity and viscosity have inverse influences on the Reynolds number (i.e. as velocity increases, the Reynolds number increases while increasing viscosity lowers the Reynolds number). This module will walk you through the transitions that occur when we decrease '''viscosity''' while holding everything else, including velocity, constant. We are choosing to focus on examining the effects of lowering viscosity by examining fluid flow fluids with different viscosities (honey, water, air) while holding the velocity constant. This would have a similar effect as increasing velocity and holding viscosity constant. Using our models, we examine the effects of changing viscosity, effectively changing the Reynolds number, on fluid flow.

As we have introduced above, the Reynolds number is an extremely useful way to characterize flow conditions. The ''Re'' number is equal to the '''inertial forces''' (velocity times characteristic length) over the '''resisting forces''' (viscosity).

==== Re = ρVD/µ = VD/v ====

{| class="wikitable" style=""
!colspan=2|where
|-
|'''V'''
|= fluid velocity (m s-1)
|-
|'''D'''
|= linear dimension (m)
|-
|'''µ'''
|= dynamic viscosity (Pa s)
|-
|'''v'''
|= kinematic viscosity (Pa s)
|-
|'''ρ'''
|= density (kg/m3)
|}

Note: ''Re'' can also be thought of as inversely related to how well flow conditions can be described mathematically across a given dimension (''x'', ''y'', ''z'', or ''time'').

The table below lists the qualitative descriptions for each quantitative ''Re'' range.

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105< ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar," while those from 100 to 105 are "transitional," and above 105 are "turbulent." At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocity or higher viscosity, inertial forces begin to play a greater role and flow becomes increasingly turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomena. Let's first describe our model and then take a look at these phenomena!

= Model setup =
=== Geometry ===

Below is the geometry of our simple model in COMSOL.

[[File:Geometry of cylinder model.png|none|COMSOL model geometry]]

This is a two-dimensional COMSOL model, which assumes that the channel is of infinite depth and thus does not produce drag from the bottom of the flow in any way. Both the horizontal (long) walls and the walls of the cylinder have a "no slip" condition. This means that as the flow gets closer to the wall, velocity goes to zero. This condition sets up a velocity gradient between the walls at zero and the center of the flow. The wall on the far left side has an inflow condition of 1 meter per second, and the wall on the right side has a zero-pressure outlet condition, which lets flow pass out of the model unhindered.

COMSOL solves a finite element model, which means that it performs calculations for a mesh grid at each step. We use a relatively small mesh cell size ("Extra Fine" setting in COMSOL) on the order of ~1 cm.

[[File: meshdetail.png|600px|COMSOL model mesh]]

When we run the model, we are executing a variation of the Navier-Stokes momentum equation for each mesh element of the model at each time step. This equation, shown above, is fundamental to describing the behavior of fluids in continuum mechanics.

=== Experimental variable: Viscosity ===

Let's look at the dynamic viscosities of fluids we'll be working with.

{| class="wikitable" style="text-align:center"
!Fluid
!µ (viscosity at 20 ºC)
|-
|'''1. Honey'''
|1 * 101.5 Pa s
|-
|'''2. Water'''
|0.89 * 10-3 Pa s
|-
|'''3. Air'''
|1.81 * 10-5 Pa s
|}

As mentioned above, we use an initial fluid velocity of 1 m/s in all of our models.

=== Questions to consider ===

From the diagram of the model geometry, what is the linear dimension (D)? In this model, it is equivalent to the diameter of the cylinder.

Before we go through the model results, try calculating the initial Reynolds numbers for each fluid viscosity using the linear dimension and the fluid velocity. Based on the Reynolds number calculated for the honey, do you think the fluid flow will be more laminar or turbulent? What about for water? Air?

= Models =

== 1) Honey (''μ'' = 101.5 Pa s) ==

In the first example, we have a fluid with the viscosity of about that of honey, ''μ'' = 101.5 Pascal seconds (Pa s). You might imagine honey to be a pretty "stiff" liquid, which has something to do with its viscosity. What do you think will happen? Let's look at a few parameters of flow in the honey model.

=== Velocity 1 (x-component) ===
[[File:Mu101.5_ux1_velocity.gif]]

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "'''[https://en.wikipedia.org/wiki/Laminar_flow laminar]'''," since you can describe what will happen at any given point in the model space at any given time fairly easily. Note that velocity is highest near the cylinder where the honey must "squeeze by" the cylinder in order to keep up with the 1 m s-1 inflow rate. This sets up a steep velocity gradient near the no-slip boundary of the cylinder, which we can see in the shear rate:

=== Shear rate 1 ===
[[File:Mu101.5 ux1 shear.gif]]

[https://en.wikipedia.org/wiki/Shear_stress Shear rate] is by definition the derivative of the velocity field, so it is no surprise that the highest shear values are in the areas next to the cylinder where the steep velocity gradient exists. If we break shear down into its component parts, we get [https://en.wikipedia.org/wiki/Vorticity vorticity]. Vorticity is the curl of the velocity field, or in other words, the rate of rotation of the angle between two particles as they slip past each other.

=== Vorticity 1 ===
[[File:Mu101.5 ux1 vorticity.gif]]

Again, unsurprisingly, the curl is highest where the gradient of velocity is high. This makes sense because as two particles slip past each other at different velocities, their orientation changes (rotates) with regards to one another. Vorticity will be highest near the no-slip boundaries in a simple laminar system like this.

=== Pressure 1 ===
[[File:Mu101.5 ux1 pressure.gif]]

The pressure field should also make sense, because our one obstacle in this model sets up the overall gradient from inflow to outflow pretty nicely. Notice that the pressure values are very slightly higher on the [https://en.wiktionary.org/wiki/stoss stoss side] and slightly lower on the [https://en.wiktionary.org/wiki/lee lee side] of the cylinder. Note also that pressure gradients define the shortening and stretching of a system. Flow is shortened as it is forced towards the front of the cylinder and stretched as it goes past the cylinder.

=== Reynolds number 1 ===
[[File:Mu101.5 ux1 reynolds.gif]]

[https://en.wikipedia.org/wiki/Reynolds_number Reynolds number] describes how chaotic a system is. Since our system has almost no chaos, this plot is pretty boring at this viscosity.

You may have noticed that the Reynolds numbers shown here are different orders of magnitude than those you calculated earlier. That's because the Reynolds numbers calculated via COMSOL are cell Reynolds numbers. This means that the Reynolds number is calculated using the characteristic length scale of the mesh grid size rather than the whole system. Despite this distinction, examining relative Reynolds numbers throughout the model will be useful for qualitatively describing what's happening with the flow.

=== Discussion 1 - Honey ===

[[File:Honey-miel.jpg|thumb|"The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. ... when the flow meets resistance it slows and [shortens by folding] upon itself."[https://en.wikipedia.org/wiki/Reynolds_number#Object_in_a_fluid] (used under Creative Commons 2.0)]]

Okay, so what's happening here? Things look pretty "boring" so far. With the exception of the very first couple of frames, in which the fluid has not achieved its full velocity yet, all of the frames are visually nearly the same. As you can tell by the color scale on the velocity plot, the fluid is moving as fast as 2 m s-1 past the cylinder, but there seems to be no instability here. This system is the result of '''kinematic''' forces at work, but the system seems to be stable, i.e. there is very little '''dynamic''' element to this system. Another way to say this is that there is not much variance in any given cell through time.

Why is this? Well, we said before that honey is "sticky." In more technical terminology, higher viscosity means higher internal friction and resistance to deformation. This has the effect of dampening the inertial term in Navier-Stokes, and thus causes the flow to gain stability and to behave in a "boring," predictable manner. Take another look at the Reynolds number plot. All areas on the plot are '''''Cell Re'' < 1'''. The overall ''Re'' is on the order of 100.5 (around 3.2) since velocity is 1.0 m/s, the linear dimension or cylinder width is 10-1 m, and viscosity is 101.5 Pa s. This indicates a very stable and predictable flow, in which the kinematics of any one particle anywhere in the continuum can be described quite well mathematically. Let's take another look at the table we saw above:

{| class="wikitable" style="text-align:center"
!colspan=3|Quantifying qualitative descriptions of Reynolds number
|-
|Laminar
|Transitional
|Turbulent
|-
|''Re'' < 100
|100 < ''Re'' < 105
|105 < ''Re''
|}

Flows with Reynolds numbers of less than 100 are considered "laminar". This model shows laminar fluid flow around a cylinder.

== 2) Water (''μ'' = 10-3 Pa s )==

Now let's look at how flow changes when viscosity is lowered more than four orders of magnitude to that of water, ''μ'' = 10-3 Pa s (remember, we're holding density constant). What do you think will be different? What do you think the system will look like?

=== Velocity 2 (x-component) ===
[[File:Mu10-3_ux1_velocity.gif]]

Whoa! That's quite different. Notice that things initially begin laminar but start to decay into a regular pattern of "wobbling." This process of vortex formation in the wake of the flow around the cylinder is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Notice also how violently the velocity vectors swing around as the vortices begin forming and shedding and the flow field oscillates from side to side. Flow is actually going backwards in some places! Why?

There's no question this flow is more dynamic than the first one. Technically, as you'll see just below, this is still a laminar flow. Why? Can you still predict what's happening in certain places or at certain times? Why or why not? Keep these questions in mind as you scroll through the next few plots.

=== Shear rate 2 ===
[[File:Mu10-3 ux1 shear.gif]]

Here we see the shear plot, which still tracks the derivative of the velocity field, but in this case it's much more dynamic as well. We can see the formation of vortices as the dominant shear jumps back and forth between the two sides of the cylinder. Why do you think this is happening?

=== Vorticity magnitude 2 ===
[[File:Mu10-3 ux1 vorticity.gif]]

What a gorgeous vorticity plot. After you've admired it for a bit, can you describe what's happening here? Look back at the shear plot. When the newest vortex is about to detach from the [https://en.wikipedia.org/wiki/Windward_and_leeward lee side] of the cylinder, where is the shear rate highest?

=== Pressure 2 ===
[[File:Mu10-3 ux1 pressure.gif]]

Look at the similarities between this pressure plot and the vorticity plot, and you'll notice that pressure is more or less inversely related to vorticity. Why? Can you think of real-life scenarios (other than flow past a cylinder in a flume) where you can observe this phenomenon? Recall the flow vectors going backwards. Given this pressure plot, can you give an explanation for this phenomenon?

=== Reynolds number 2 ===
[[File:Mu10-3 ux1 reynolds.gif]]

Notice the change in magnitude of the cell Reynolds numbers calculated in this model. Recall that higher Reynolds numbers suggest more turbulent flow while lower Reynolds numbers indicate more laminar flow. Where do you see the most turbulent flow in the model? The least? Why do you think this is happening?

The overall Reynolds number for the system is 89. The Reynolds number doesn't quite make it above the critical number of 100 in order to be considered a "transitional" flow. What is happening in this flow that keeps it relatively "predictable?"

''Note: The stippling is an artifact of the meshing process in COMSOL.''

=== Discussion 2 - Water ===

The vortex shedding pattern shown in these examples develops after a couple of brief moments in which the flow is fairly laminar and there are few dynamics acting on the system. Then the system begins to develop an instability. In real life this instability is the result of tiny perturbations on the surface of the cylinder that cause differences in the boundary layer thickness as flow passes, causing one side to briefly slip faster than the other. The momentum of the vortex continues over to the other side of the cylinder, which increases the fluid pressure on that side and thus slows flow. As pressure always wants to equilibrate, this process reverses and a competing vortex is sent spinning to the opposite side. If flow conditions do not change, this process will continue indefinitely.

In mathematical terms, fluid particles will accelerate over time—'''∂v / ∂t'''—towards areas of low pressure—'''- ∇P''' (or decelerate if they are already traveling away from an area of low pressure). In order to figure out whether a particular particle will accelerate (or decelerate) towards a certain area of pressure, you'd have to examine the direction magnitude component partial differentials of the velocity and term, '''∇2U'''. In other words, the ''grad'' ('''∇''') symbol is a shorthand for the fact that you're looking at something that changes in more than one linear dimension (in the case of this simple 2D model, '''∇''' denotes that the variable will change in both ''x'' and ''y'').

In a 3D model, the same patterns hold true. Let's look at a particle model of the same process. Particles in the first plot here are given colors based on the ''x'' component of their velocity.

[[File:2019-03-01-Water_VelX.gif]]

In the second plot, particles are given colors based on their vorticity.

[[File:2019-03-01-Water Vort VelMagVectors.gif]]

As a fluid becomes less viscous, the momentum term will begin to dominate the Navier-Stokes solution, meaning that the internal shear strength of the fluid is less able to keep it from deforming. As we said before, in this ''flow past a cylinder'' model, that means vortex shedding will become more and more viable at either lower viscosities or higher velocities.

=== Real-world applications ===

[[File:Transitional.jpg|thumb|Quasi-laminar flow past a cylinder (a GPS survey staff) on Third River in Montclair, NJ. (Photo: Ian Nesbitt, licensed under Creative Commons 4.0)]]

Pictured at right, a GPS survey staff causes a slight flow instability downstream.

As beautiful as they are, von Kármán vortex streets are an engineering problem in many real-world applications. Vortex shedding can put enormous shear forces on the mechanical joints of the survey mount, which can cause brittle failure (see below).

[[File:Mount-shear.jpg|200px|A survey mount with the Z-joint shorn off.]]
[[File:Mount-shear-zjoint.jpg|632px|A survey mount with the Z-joint shorn off.]]

Low pressure behind a cylindrical marine survey mount pole can cause approximately an order of magnitude more drag than a hydrodynamic shape like a teardrop, even at low speeds. Stainless steel "fairings" are sometimes used to reduce this drag and stop the pole from flexing.

[[File:ADCPfairing.JPG|A fairing sheath used to reduce drag on a survey mount pole.]]
[[File:ADCPfairing_operation.JPG|Fairing sheath in action. Note teardrop shape, meant to reduce pressure instability behind survey mount pole.]]

Now, let's take a look at this behavior in a fluid with two orders of magnitude less viscosity than water: air.

==3) Air (''μ'' = 10-5 Pa s)==

Air has a viscosity of ''μ'' = 10-5 Pa s (still holding density and velocity constant). How do you think this will affect the deformation of the fluid?

=== Velocity (x-component) 3 ===
[[File:Mu10-5_ux1_velocity.gif]]

=== Shear rate 3 ===
[[File:Mu10-5 ux1 shear.gif]]

=== Vorticity magnitude 3 ===
[[File:Mu10-5 ux1 vorticity.gif]]

What is happening to the vorticity as the fluid moves further past the cylinder? Energy from the disturbance propagates forward and eventually dissipates. Can you see where this is happening from each of other variable plots as well?

=== Pressure 3 ===
[[File:Mu10-5 ux1 pressure.gif]]

=== Reynolds number 3 ===
[[File:Mu10-5 ux1 reynolds.gif]]

Note: The stippling is an artifact of the meshing process in COMSOL.

=== Discussion 3 - Air ===

This specific type of oscillating flow is called a [https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street von Kármán vortex street] (VKVS), named after [https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n Theodore von Kármán], the Hungarian-American physicist who first described it in detail. The process that leads to this type of flow is called [https://en.wikipedia.org/wiki/Vortex_shedding vortex shedding]. Let's take a deeper look at how these vortices are forming by looking at the particle solution generated in SPH.

[[File:2019-03-02-Air_VelX.gif]]

[[File:2019-03-02-Air Vort VelMagVectors.gif]]

These von Kármán vortex streets refer to the double row of vortices being shed from either side of the cylinder. The vortices formed from one side rotate in the opposite direction as the vortices formed from the other side. Each respective vortex row is referred to as a vortex street. Notice that angles at which the vortexes are moving relative to each other.

Compare the vortices here to those generated in the water flow around a cylinder model. What stands out to you about these vortices and what do you think is causing those differences? The ability of these vortices to maintain their form and stability through the entire simulation is remarkable. What do you notice about how they change over time and space? You may notice that these vortices are spaced differently and that they are different sizes than those from the water flow model. These VKVS can be described using several parameters:

The geometry of a VKVS can be defined by two dimensionless ratios, their '''aspect ratio''' and their '''dimensionless width'''. The aspect ratio is equal to '''h/a''' where h is equal to the separation distance between 2 the counter-rotating vortex streets and a is the the distance between 2 vortices in the same vortex street. What would you estimate this ratio to be for the vortices above? The dimensionless width of the VKVS is '''h/D''' where D is the diameter of the obstacle (cylinder in this case). In other words, the dimensionless width is just the ratio of the vortex street width to the width of the obstacle. What would you estimate the dimensionless width to be for the vortex street above? These parameters allow us to compare the geometry of vortex streets that vary vastly in size scale. The stable aspect ratio and dimensionless widths observed in lab settings are 0.28 and 1.2, respectively. How do these compare to those you calculated for this VKVS?

The vortex shedding frequency of a VKVS can be calculated from its [https://en.wikipedia.org/wiki/Strouhal_number Strouhal number], which describes the periodicity of oscillating flow mechanisms. The Strouhal number (St) is calculating using the frequency of vortex shedding (f), the obstacle/object diameter (D), and the flow velocity (u): '''St = f*D/u'''. Knowing the cylinder diameter (0.1 m) and the flow velocity (1.0 m/s), can you count the vortex shedding frequency from the animation to calculate the Strouhal number of this VKVS?

These von Kármán vortex streets are well-studied fluid phenomena that occur during the transition to turbulence (when 100 < Re < 105). Vortex shedding has many real-world engineering and design applications, some of which were described in the water flow discussion. Next, let's learn about their atmospheric counterparts!

=== Real-world Applications ===

[[File:canaryIslands_tmo_2015140.jpg|thumb|900px|right|View of atmospheric VKVS off of Madeira and the Canary Islands on May 20, 2015 from the NASA TerraX Satellite.]]

von Kármán vortex streets have been thoroughly-modeled in lab settings, but they occur naturally more often than you might think. In fact, the first VKVS described by Theodore von Kármán were observed in the sky. He noticed remarkable swirling patterns in the clouds forming on the lee side of an island. These VKVS occur on a much larger scale than the VKVS that are modeled in lab settings. In the case of these atmospheric vortex streets, there are vertical forces, atmospheric pressure and temperature gradients, and a whole slough of other factors that influence the geometry and stability of the vortices. Many studies have modeled atmospheric VKVS such as the ones shown in the image to the right.

In the case of these atmospheric VKVS, the island acts as the cylinder in our fluid flow models, creating the flow disturbance. As air flows around the island obstacle, vortex shedding begins and these counter-rotating vortices propagate downwind. Do you notice any other differences between these atmospheric VKVS and the simulated VKVS from our air flow around a cylinder model?

Using the scale bar in the image, measure the linear dimension (D) or width of Madeira. Using a wind speed of 10 m/s and the viscosity of air, can you calculate the general Reynolds number for flow around Madeira using the same steps as you previously did in the module?

Now use the same scale bar to measure and characterize the geometry of these atmospheric VKVS. What's the aspect ratio of the VKVS created by Madeira? The dimensionless width? How do these compare to those values observed in lab settings?

= Conclusions =

This module has introduced you to some of the tools that you can use to qualitatively and quantitatively characterize fluid flow as well as the physics behind it. The Navier-Stokes momentum equation governs the fluid particle motion while the Reynolds number describes the flow regime. The von Kármán vortex streets generated in the transition to turbulence can also be characterized by their aspect ratio, dimensionless width, and vortex shedding frequency (Strouhal number). Now that you have explored and described the transition to turbulence through models of fluid flow around a cylinder, you can use your knowledge to examine turbulence and flow behavior in other fluid systems!

= Bonus: Complex flows =

Although our module focused on models of simple fluid flow, more complex flows make up the vast majority of flow behavior you'll encounter in the real world. It's important to note that the systems modeled above above are highly simplified in order to highlight the physical forces acting on fluids. Below are some examples of more complex flow systems that one might encounter in the real world.

== Fully turbulent flows ==

[[File:Turbulent.jpg|thumb|Even at low flow, a US Army Corps of Engineers flood control project on the Hoosac River in North Adams, Massachusetts causes a brief transition from laminar to turbulent flow. Photo was taken four years to the day after Hurricane Irene (see GIF, left), the flood of record at USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] (green shed with solar panel in this photo). (Photo: Ian Nesbitt, August 28, 2015, licensed under Creative Commons 4.0)]]

Fully turbulent flow, where '''2000 < Re''' and predictability over a scale larger than a couple of particles or beyond a couple of time steps approaches zero, is very difficult to model. In some cases, flows of this nature are easier to compute and the solutions are better represented by [[Smoothed Particle Hydrodynamics|Smoothed Particle Hydrodynamic]] models, in which individual particle smoothing kernels, rather than mesh elements, define the behavior of the fluid. However, flows in high Reynolds cases are are difficult to describe mathematically, and thus solutions that approach full turbulence in continuum or smoothed particle models can take orders of magnitude longer to compute.

[[File:Fullturbulence.gif|(Hurricane Irene flood GIF: Ian Nesbitt, licensed under Creative Commons 4.0)]]

The gif above was filmed on August 28, 2011, near peak flow during Hurricane Irene, the current flood of record for USGS gaging station [https://waterdata.usgs.gov/nwis/uv/?site_no=01332500 01332500] near Williamstown, MA. The video shows flow over a US Army Corps of Engineers (USACE) flood control structure (also pictured at baseflow on right). The concrete-bedded flood control flume has low bed roughness and thus high velocity, and comparatively lower ''Re''. Flows like this can be very erosive when they encounter a jump in bed roughness due to a transition from a flood-control flume with a concrete bed to natural (rougher) bed conditions. The purpose of induced turbulence—from USACE's perspective—is to limit scour caused by a sudden increase in bed roughness. By increasing turbulence and thereby reducing friction at the boundary layer, the flow will not be quite as erosive in its transition between bed conditions. The flow captured in this video is at or near the flood control project's design capacity to limit the destructive nature of the flow, and thus this event still eroded cobble- and boulder-sized particles from the natural bed.

== "Multiphase" flows ==

Using a particle physics solution, one can model fluid particle behavior without the need for computationally remeshing, which can have substantial processing power overhead in a continuum model. In this case we use the word "multiphase" to describe the use of two fluids, each with different properties, to initialize the model. The following GIF shows a DualSPHysics two-phase example model of a dam break, where a parcel of water flows over and deforms denser substrate (sediment) material. This type of model can be used to examine physical processes of erosion at a small scale, and potentially could be used to reproduce and expound on results of physical modeling studies of these phenomena.

[[File:MultiphaseSPH_01.gif]]

As you can see here, water flowing on the surface of the sediment creates shear stress at the bed, which both deforms the sediment to certain depths, and entrains grains from the surface. You'll notice that as the front of the flow takes on sediment it behaves more like a debris flow than a water wave. This has the effect of buoying dense sediment up to the surface, which then remains entrained as the flow hits the wall and begins to curl around. From the DualSPHysics documentation:

<blockquote>These multi-phase sediment scouring phenomena are induced by rapid flows creating shear forces at the surface of the sediment which causes the surface to yield and produce a shear layer of sediment suspended particles at the interface and finally sediment suspension in the fluid. Applications include scouring in industrial tanks, port hydrodynamics, wave breaking in coastal applications and scour around structures in civil and environmental engineering flows among others.</blockquote>

Some of the more relevant physical properties of this model are shown in the following table.

{| class="wikitable" style="text-align:center"
!colspan=6|Fluid properties of multiphase model
|-
|''Fluid''
|''Density (kg m-3)''
|''Viscosity (Pa s)''
|''Speed of sound (m s-1)''
|''Cohesion coeff.''
|''Angle of internal friction (deg)''
|-
|'''Water'''
|1000
|0.001
|80
|0
|0
|-
|'''Saturated sediment'''
|1500
|0.002
|81
|1
|35
|}

A physical multiphase model typically involves a flume, a pump, and a specific particle size or range of sizes to test the effects that flow-induced boundary stress will have on particle saltation, sliding, and entrainment. A simple sediment flume demonstration conducted by Dr. Ronadh Cox (Williams College) is shown below. Apart from the geometry, one major difference between this physical model and the numerical one is that the grain density is that of quartz (2650 kg m-3) in the physical model.

<youtube>XAyA_p99njw</youtube>