Tutorial: Turbulent Flow Through a Planar Asymmetric Diffuser

Introduction

The purpose of this tutorial is to provide guidelines and recommendations for solving a CFD problem which includes:

• Building the geometry and generating a mesh in GAMBIT.

• Setting up the CFD model in FLUENT.

• Solving the problem and comparing the results with the experimental data.

Prerequisites

This tutorial assumes that you are familiar with the FLUENT interface and you have a good understanding of the basic setup and solution procedures. Some steps will not be shown explicitly.

If you have not used FLUENT before, it would be helpful to first refer to FLUENT 13.0 User’s Guide and FLUENT 13.0 Tutorial Guide.

Problem Description

The geometrical description of the 2D asymmetric plane diffuser is shown in Figure 1. The origin of the x-axis is located at the intersection of the tangents to the straight and inclined walls at the beginning of the asymmetric expansion. The y-axis originates from the bottom wall of the downstream channel.

The problem is to simulate the flow through an asymmetric plane diffuser with a Reynolds number Re = 20000. The Reynolds number is based on the centerline velocity and the channel height at the inlet. The complete experimental results were obtained by Buice and Eaton [1]. This is a classical test case for flows dominated by adverse pressure gradient and boundary-layer separation.

Figure 1: Aysmmetric Planar Diffuser Geometry

Preparation

1. Copy the mesh file, asymmetric.msh and the profile file, channelu.prof to your working directory.

Fluent Case Setup and Solution

Step 1: Mesh

1. Start FLUENT 2DDP.

2. Read the mesh file, asymmetric.msh.

File −→ Read −→ Case...

3. Scale the mesh.

Mesh −→ Scale...

(a) Select Specify Scaling Factors, and specify values of 0.1 for both X

and Y

(b) Click Scale and close the panel.

4. Check the mesh.

Mesh −→Check

FLUENT will perform various checks on the mesh and will report the progress in the console window. If no error messages are reported in the FLUENT window, the mesh check was successful.

Step 2: Models

1. Keep the default General settings.

Define −→ General

2. Enable the realizable k-epsilon turbulence model.

Define −→ Models −→Viscous...

(a) Enable k-epsilon (2 eqn) under Model.

(b) Enable Realizable under k-epsilon Model and Enhanced Wall Treatment under Near- Wall Treatment.

Note: You have created a very fine near wall mesh in GAMBIT in anticipation of the use of Enhanced Wall Treatment with the turbulence models. A f t e r ca l cu la t i n g t he so lu t i o n , the xy plot tool in FLUENT can be used to verify the adequacy of the near wall mesh.

(c) Click OK to close the panel.

Step 3: Materials and Operating Conditions

The fluid is standard air with constant density so there is no need to visit the materials and operating conditions panel.

Step 4: Boundary Conditions

In order to obtain a fully-developed channel flow at the inlet, you can either extend the channel sufficiently long in the upstream direction, or separately compute a fully-developed channel flow using the same turbulence model for this problem (i.e., the same Reynolds number). Take the latter approach in this tutorial to minimize the size and CPU time required by the model. Profiles of u, v, k, and ε are stored in the file called channelu.prof. This fully-developed channel flow uses the inlet velocity (at the centerline) calculated as follows (the given Reynolds number Re = 20000 is based on the channel height and center- line velocity):

1. Read in the profiles.

Define −→Profiles...

(a) Click Read... and select the file channelu.prof.

(b) Close the panel.

2. Set the boundary conditions for velocity inlet (inlet v).

(a) Select Components as the Velocity Specification Method.

(b) Select inner x-velocity for X-Velocity (m/s) and inner y-velocity for Y-Velocity (m/s). (c)

Select K and Epsilon as Turbulence Specification Method.

(d) Select inner turb-kinetic-energy for Turb. Kinetic Energy and inner specific-diss-rate for Turb. Dissipation Rate.

The name ’inner’ refers to the to the zone where the profiles were exported from.

3. Use the default No-slip boundary conditions for both the walls.

4. Set the boundary conditions for pressure outlet (outlet).

(a) Select Intensity and Viscosity Ratio for Turbulence Specification Method. (b) Specify a value of 10 for both Backflow Turbulence Intensity and Backflow Turbu-

lence Viscosity Ratio.

Step 5: Solution

1. Set the solution methods

Solve −→ Methods...

(a) Select SIMPLEC for the pressure-velocity coupling scheme

(b) Click OK to close the panel

2. Set the solution controls

Solve�Controls

Change the Momentum under-relaxation factor from 0.7 to 0.3. Otherwise keep the default

values for the other entries.

2. Initialize the solution.

Solve −→ Initialization…

(a) Select all-zones in the Compute From drop-down list. (b)

Click Initialize

3. Use default convergence criteria for all residuals.

4. Set up a monitor for wall-shear stress on the wall.

Solve −→ Monitors… then click Create… below Surface Monitors

(a) Enable Plot and Print

(b) Enable Plot and Print.

(c) Select Area-Weighted Average in the Report Type drop-down list

(d) Select Wall Fluxes... and Wall Shear Stress in the Report Of drop-down lists.

(e) Select wall bottom and wall top under Surfaces.

(f) Click OK to close the panel.

Step 6: Define Custom Field Functions

Define −→Custom Field Functions...

1. Select Mesh... and X-Coordinate in the Field Functions drop-down list, and click Select.

2. Click the buttons /, ., and 1 in a sequence in the Custom Field Function Calculator Pad

3. Specify x-by-h as the New Function Name and click Define.

4. Close the panel.

Step 7: Iterations and Convergence

1. Start the calculations by requesting 1000 iterations.

Solve −→Run Calculation...

Click the Calculate button. Due to the default convergence criteria based on the reduction of the level of the residuals, the solution will converge after just over 300 iterations. Though the calculations have proceeded smoothly so far, two things need to be noted:

(1) The monitor plot shows that the average surface shear stress (on the walls) has not yet reached a constant value.

(2) You have used the first-order upwind scheme for the convective terms of the governing equations. This scheme is numerically diffusive. Hence it should not be used for obtaining the final results.

Switch the discretization scheme for convective terms for the momentum and turbulence equations to second-order upwind.

2. Save the case and data files (asdn3L-initial.cas.gz).

3. Change the reference values.

Report −→Reference Values...

in

(a) Change the Velocity value to 2.921469. (b) Change the Length value to 0.1.

(c) Click OK to close the panel.

4. Plot the initial results.

Display --> Plots… --> XY Plot

(a) Deselect Node Values and Position on X Axis under Options.

(b) Select Wall Fluxes... and Skin Friction Coefficient under Y Axis Function. (c)

Select Custom Field Functions... and x-by-h under X Axis Function.

(d) Click Load File... and select the cf top.xy file and click OK.

Experimental data for skin friction coefficient (Cf = τw /0.5ρU 2 ) for the top wall and bottom wall are stored in cf top.xy and cf bot.xy respectively.

(e) Change the line and symbol style for Curve 0.

i. Click on Curves... in the Solution XY Plot panel.

ii. Make the changes as shown in the panel. iii.

Click Apply and close the panel.

(f ) Select wall top under Surfaces and click Plot.

(g) Repeat the same procedure for bottom wall by loading file cf bot and selecting wall bottom under Surfaces.

Figure 3: Skin Friction Coefficient Vs x/h (rke - Unconverged Solution) for Top Wall

Figure 4: Skin Friction Coefficient Vs x/h (rke - Unconverged Solution) for Bottom Wall

5. Change the Discretization scheme to Second Order Upwind for all equations.

6. Disable convergence check for all residuals.

Solve −→ Monitors −→Residual...

7. Increase the number of iterations to 4000 and continue the calculation until the monitored quantity becomes a constant value.

You can see the residuals of all the equations also have dropped below 5 orders of magnitude, so the solution can be taken as converged.

8. Save the case and data files (asdn3L-rke.gz).

9. Change the viscous model to SST k-omega.

Define −→ Models −→Viscous...

10. Set the boundary conditions for inlet_v

(a) Set the Spec. Dissipation Rate to inner specific-diss-rate.

11. Continue the calculation with more iterations until the monitored quantity becomes a constant value.

12. Save the case and data files (asdn3L-sst.gz).

Step 8: Postprocessing

Results and Discussion Define a new custom field function as shown below. This will make comparison of the skin friction on the lower wall more convenient.

Plot Cf for the top and the bottom walls versus data as explained in this tutorial, (Figures 5 and 6). Compare the results with those obtained from the unconverged solution. There is a very substantial difference.

The predictions for Cf along the top wall are substantially different (lower) than the experimental data by the realizable k-ε model. The main reason for the failure is due the fact that it does not correctly predict the size of the separation/recirculation zone along the inclined wall.

On the other hand, SST k-ω is the only turbulence model among all the two-equation turbu- lence models which can successfully capture the recirculation zone. SST k-ω model’s predic- tion of Cf on the top wall is good (see Figure 7), but along the bottom wall it predicts the flow separates slightly upstream of the actual separation point (see Figure 8).

Figure 5: Skin Friction Coefficient Vs x/h (rke - Converged Solution) for Top Wall

Figure 6: Skin Friction Coefficient Vs x/h (rke - Converged Solution) for Bottom Wall

Figure 7: Skin Friction Coefficient Vs x/h (sstkw) for Top Wall

Figure 8: Skin Friction Coefficient Vs x/h (sstkw) for Bottom Wall

Grid Independence Study

Test whether the converged results (from the SST k-ω model: asdn3L-sst.cas.gz, asdn3L-sst.dat.gz) obtained so far are independent of the grid resolution, you can either uniformly double the total cell count, or use the grid adaption feature of the solver to achieve the objective more efficiently.

Grid independence is attained when further mesh refinement yields only small and insignificant changes in the solution fields. You can use many possible criteria to adapt the mesh. Here we choose the pressure gradient. The separation/recirculation zone should be sensitive to the computed pressure gradient of the flow. If you proceed from your own calculation, first save the case and data before attempting any adaption since any change is irreversible.

1. Open the Gradient Adaption panel.

Adapt −→Gradient...

(a) Ensure Pressure... and Static Pressure is selected in the Gradients Of drop-

down list.

(b) Click Compute. This will list the current Max and Min gradients in the boxes.

You can use the so-called “10-percent rule” to determine the adaption threshold: to refine the mesh wherever the gradient exceeds 10% of the maximum level.

(c) Enter a value of 8.7e-07 for the Refine Threshold and click Mark. (d) Click Manage...

button to open t h e Manage Adaption Registers panel.

i. Plot the adaptively refined mesh by clicking Display.

In general, it is desirable to have the marked cells clustered in a contiguous manner. (If they are not, delete the register and reduce the Refinement Threshold and do it again.) For the the current problem we have increased about 10000 cells (about 17% more) and they are mainly concentrated around the inclined section of the channel. We consider it to be satisfactory and proceed.

ii. Click Adapt and click Yes when prompted Hanging-node mode: Ok to adapt grid?.

(e) Continue the iterations until the case is converged. (f ) Save the case and data files (asdn3L-sst-adapt.gz).

(g) Plot the results, (Figures 9 and 10).

It can be seen that there are no detectable changes from the previously obtained results (except some small improvement over the range of 10 < x/h < 20), s o now you can say that the converged solution for this case is grid-independent.

Figure 9: Skin Friction Coefficient Vs x/h (sstkw) After Grid Adaption for Top Wall

Figure 10: Skin Friction Coefficient Vs x/h (sstkw) After Grid Adaption for Bottom Wall

Summary

In this tutorial, you performed a simulation of steady-state turbulent flow through an asymmetric, planar diffuser by using the popular realizable k-ε model. The calculated skin friction coefficients (Cf ) at the top and bottom of the diffuser walls were compared with experimental data reported by Buice and Eaton. Between the two-equation turbulence models, only the SST k-ω model gives reasonable predictions of the skin friction and the recirculation zone.

You have also learned how to use FLUENT́s grid adaption feature to test whether or not the calculation is grid independent, without having to uniformly double up the cell count in the whole flow domain.

References

[1] C.U. Bruice and J.K. Eaton. Experimental investigation of flow through an asymmetric plane diffuser. Technical Report No. TSD-107, Thermosciences Division, Dept. of Mechanical Engineering, Stanford University, Stanford, CA, USA, August 1997.