Flow Past a Cylinder

The flow of fluid behind a blunt body such as an automobile is difficult to compute due to the unsteady flows. The wake behind such a body consists of unordered eddies of all sizes that create large drag on the body. In contrast, the turbulence in the thin boundary layers next to the streamlined bodies of aircraft and fish create only weak disturbances of flow.

An exception to this occurs when you place a slender body at right angles to a slow flow because the eddies organize. A von Kármán vortex street appears with a predictable frequency and involves the shedding of eddies from alternating sides. Everyday examples of this phenomenon include singing telephone wires and an automobile radio antenna vibrating in an air stream.

From an engineering standpoint, it is important to predict the frequency of vibrations at various fluid speeds and thereby avoid undesirable resonances between the vibrations of the solid structures and the vortex shedding. To help reduce such effects, plant engineers put a spiral on the upper part of high smokestacks; the resulting variation in shape prohibits the constructive interference of the vortex elements that the structure sheds from different positions.

Model Definition

To illustrate how you can study such effects, the following model examines unsteady, incompressible flow past a long cylinder placed in a channel at right angle to the oncoming fluid. With a symmetric inlet velocity profile, the flow needs some kind of asymmetry to trigger the vortex production. This can be achieved by placing the cylinder with a small offset from the center of the flow.

The simulation time necessary for a periodic flow pattern to appear is difficult to predict. A key predictor is the Reynolds number, which is based on cylinder diameter. For low values (below 100) the flow is steady. In this simulation, the Reynolds number equals 100, which gives a developed von Kármán vortex street, but the flow still is not fully turbulent.

The frequency and amplitude of oscillations are stable features, but flow details are extremely sensitive to perturbations. To gain an appreciation for this sensitivity, you can compare flow images taken at the same time but with such minor differences as are created by different tolerances for the time stepping. It is important to note that this sensitivity is a physical reality and not simply a numerical artifact.

Before calculating the time-varying forces on the cylinder, you can validate the method of computation at a lower Reynolds number using the direct nonlinear solver. This saves time because you can find and correct simple errors and mistakes before the final time-dependent simulation, which requires considerable time.

The viscous forces on the cylinder are proportional to the gradient of the velocity field at the cylinder surface. Evaluating the velocity gradient on the boundary by directly differentiating the FEM solution is possible but not very accurate. The differentiation produces 1st-order polynomials when second-order elements are used for the velocity field. A far better approach is to use a pair of reaction force operators to compute the integrals of the viscous forces, comparable to second-order accurate integrals of the viscous forces. An alternative approach would be to use a pair of weak-constraint variables to enforce the no slip condition. Preferably use the reaction force operator instead of weak constraints when computing integrals of reaction forces or fluxes in postprocessing.

The drag and lift forces themselves are not as interesting as the dimensionless drag and lift coefficients. These depend only on the Reynolds number and an object’s shape, not its size. The coefficients are defined as

using the following parameters:

•

FD and FL are the drag and lift forces

•

ρ is the fluid’s density

•

Umean is the mean velocity

•

A is the projected area (product of thickness and diameter of cylinder)

Results and Discussion

Figure 1 shows the flow pattern resulting from the geometry.

Figure 1: A plot of the last time step clearly shows the von Kármán path.

The flow around a cylinder is a common benchmark test for CFD algorithms. Various research teams have tried their strengths on this problem using different techniques. Results from some of these experiments have been collected by Schäfer and Turek (Ref. 1), who also used them to compute a probable value for the “real” answer.

Figure 2 shows how the lift coefficient develops a periodic variation as the von Kármán vortex structure is formed.

Figure 2: Lift coefficient, CL, as a function of time.

Figure 3: Drag coefficient, CD, as a function of time.

Reference

1. M. Schäfer and S. Turek, “Benchmark Computations of Laminar Flow Around Cylinder”, E.H. Hirschel ed., Flow Simulation with High-Performance Computers II, Volume 52 of Notes on Numerical Fluid Mechanics, Vieweg, pp. 547–566, 1996.

COMSOL_Multiphysics/Fluid_Dynamics/cylinder_flow

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
U_mean	1[m/s]	1 m/s	Mean inflow velocity
H	0.41[m]	0.41 m	Height
W	2.2[m]	2.2 m	Width
R	0.05[m]	0.05 m	Cylinder radius

Next, create a smoothed step function feature that you will use for ramping up the inflow velocity.

Step 1 (step1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 0.05.

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type W.

In the text field, type H.

Click

Circle 1 (c1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.2.

Locate the section. In the text field, type R.

Click

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Click to select the

toggle button.

Select the object only.

In the toolbar, click

Click the

button in the toolbar.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	1	kg/m³	Basic
Dynamic viscosity	mu	1e-3	Pa·s	Basic

Laminar Flow (spf)

Inlet 1

In the window, under right-click and choose .

Select Boundary 1 only.

We define a parabolic velocity profile using the predefined parameters. Ramp up the velocity using the previously defined step function. Append the inverse unit bracket [1/s] to the time variable t because the step function expects a dimensionless argument.

In the window for , locate the section.

In the U0 text field, type 6*U_mean*y*(H-y)/H^2*step1(t[1/s]).

Outlet 1

In the toolbar, click

and choose .

Select Boundary 4 only.

Mesh 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Click

If you zoom in on the inlet and the cylinder you can see the boundary layers that the physics-controlled mesh gives.

Study 1

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,0.2,3.4) range(3.5,0.02,7).

Solution 1 (sol1)

In the toolbar, click

In the window, expand the node, then click .

In the window for , click to expand the section.

From the list, choose .

In the toolbar, click

Results

Velocity (spf)

Add a node to the first default plot group to reproduce the plot in Figure 1.

Particle Tracing with Mass 1

In the toolbar, click

and choose .

In the window for , click to expand the section.

In the text field, type 4*pi/3*1e-9.

Find the subsection. In the text field, type u.

In the text field, type v.

Locate the section. In the text field, type range(0.1,0.05,0.3).

In the text field, type 0.

Click to expand the section. From the list, choose .

Select the check box. In the associated text field, type 3.6.

In the text field, type 0.4.

Click to expand the section. Find the subsection. From the list, choose .

Find the subsection. From the list, choose .

In the toolbar, click

To reproduce Figure 2 and Figure 3 of the lift and drag coefficients, first add an dataset for computing the total reaction force on the cylinder.

Integral 1

In the toolbar, click

and choose .

Selection

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundaries 5–8 only.

1D Plot Group 3

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Point Graph 1

Right-click and choose .

In the window for , locate the section.

In the text field, type (-reacf(v)[N]*2/(spf.rho*U_mean^2*(2*R)[m^2]))[1/m].

Select the check box. In the associated text field, type Lift coefficient.

In the toolbar, click

Compare the graph with that shown in Figure 2.

1D Plot Group 4

Finally, visualize the drag coefficient using the following steps:

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Point Graph 1

Right-click and choose .

In the window for , locate the section.

In the text field, type (-reacf(u)[N]*2/(spf.rho*U_mean^2*(2*R)[m^2]))[1/m].

Select the check box. In the associated text field, type Drag coefficient.

In the toolbar, click

Compare with Figure 3.