Revision as of 18:51, 25 February 2019

Created by Ian Nesbitt and Jukes Liu on 2019-02-19

Overview

In this module, we will examine the transition to turbulence and learn how viscosity affects the dynamics of a simple fluid system with a single cylindrical impediment placed in the flow path. Our tools in this endeavor are COMSOL Multiphysics and ParaView. 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!

We will move from looking at models with higher viscosity regimes and move to lower viscosity. Viscosity is 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. 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. 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.

Reynolds number

The flow regime of a fluid is defined by the Reynolds number:

Re = vD/µ

where
v	= fluid velocity
D	= linear dimension
µ	= viscosity

The 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 Reynold number increases while increasing viscosity lowers the Reynolds number). 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.

At low Reynolds numbers (Re < 100), viscous forces dominate and flow is laminar. As the Reynolds number increases, due to lower velocities or higher viscosities, inertial forces begin to play a greater role and flow becomes turbulent. In this transition between laminar and turbulent flow, we begin to observe some unique fluid phenomenon.

Navier-Stokes momentum equation

∂v / ∂t = ρg + µ∇²U - ∇P

To characterize the motion of particles in the fluid, we must account for the changes in momentum. This Navier-Stokes equation is the foundation of our models. It is derived based on the conservation of momentum in a system. This equation includes the influence from inertial forces, body forces, viscous forces, and the pressure gradient throughout the system.

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 (ρg), viscous forces ( µ∇²U), and the pressure gradient (∇P).

The cylinder models

Geometry

Below is the geometry of our simple COMSOL model.

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.

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.

Flow

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

Fluid	µ (viscosity at 20 ºC)
Honey	1 * 10^1.5 Pa s
Water	0.89 * 10^-3 Pa s
Air	1.81 * 10^-5 Pa s

We use an initial fluid velocity of 1 m/s in all of our models.

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?

Honey (μ = 10^1.5 Pa s)

In the first example, we have a fluid with the viscosity of about that of honey, μ = 10^1.5 Pascal seconds (Pa s). You might imagine honey to be pretty "sticky," 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 (x component)

Things are pretty orderly and logical here. Not a whole lot changes over time. You might describe this flow system as "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

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 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

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

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 stoss side and slightly lower on the 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.