Models.cfd.Ahmed Body[1]

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A i r f l ow o v e r an Ahmed Body

Introduction

This model describes how to calculate the turbulent flow field around a simple car-like geometry using the CFD Module’s Turbulent Flow, k- interface. Detailed instructions guide you through the different steps of the modeling process in COMSOL Multiphysics.

Model Definition

The Ahmed body represents a simplified, ground vehicle geometry of a bluff body type. Its shape is simple enough to allow for accurate flow simulation but retains some important practical features relevant to automobile bodies. The geometry was first defined by Ahmed, who also measured its aerodynamic properties in wind-tunnel experiments (Ref. 1). Further experiments have also been performed by Lienhart and Becker (Ref. 2). The Ahmed body has become a popular benchmark case for RANS models (Ref. 3).

G E O M E T R Y

The Ahmed body is presented in Figure 1. The total length (L) of the body is 1.044 m from front to end. It is 0.288 m in height and 0.389 m in width. Cylindrical legs 0.05 m in length are attached to the bottom surface. The angle of the rear slanting

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surface is typically varied between 0 and 40 degrees. This particular geometry has a slant angle of 25 degrees which is the same slant angle used in Ref. 3.

Length

Width

Height

Figure 1: Ahmed body with 25 degree slant of the rear face.

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The body is placed in a flow domain that is 8L-by-2L-by-2L (length-by-width-by-height), with its front positioned 2L from the flow inlet face. Mirror symmetry reduces the computational domain by half, as shown in Figure 2.

8L

2L

L

2L

Inlet

Outlet

Wall function

Slip

Symmetry

Figure 2: The size of the computational domain is reduced by mirror symmetry.

TU R B U L E N C E M O D E L

The Reynolds number base on the length of the body, L, and the inlet velocity is 2.77·106 which means that the flow is turbulent. The k- turbulence model will be applied to account for the turbulence. The k- turbulence model is describe in Theory for the Turbulent Flow Interfaces in the CFD Module User’s Guide.

A common mesh size in Ref. 3 is half a million cells for simulations with wall functions. However, those simulations do not include the stilts (the legs that support the body), and the computational domains are smaller. Hence, you can expect to need an even larger mesh in this simulation to resolve the flow. How large is however difficult to know in advance. To avoid using a prohibitively large mesh, the modeling is carried out in a series of simulations were the Reynolds number is initially 25 times lower than the experiments and then gradually increased. After each simulation, the result is investigated to determine if it is likely that the mesh is able to sustain a higher Reynolds number. If not, the mesh must be refined before the Reynolds number is increased.

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The flow is considered to be incompressible. The temperature is assumed to be 293 K and the reference pressure is 1 atm.

B O U N D A R Y C O N D I T I O N S

Air enters the computational domain at a velocity of 40 m/s normal to the inlet surface. The turbulence intensity in the free stream is set to 0.5%. A turbulent length scale is also needed at the inlet. Upstream of the test section in a wind tunnel is equipment, for example honeycombs and screens, to reduce and homogenize the turbulence. Any turbulent structures that remain can therefore be expected to have a length scale in the same order of magnitude as the holes in the honeycombs and screens, that is in the order of a centimeter.

At the outlet, a Pressure condition is applied. The floor of the flow domain and surface of the Ahmed body are described by wall functions. Wall functions could also be applied to the outer wall and the ceiling of the wind tunnel. Their main effect on the flow around the body is however to keep the flow contained, and it will therefore suffice to model them as slip walls.

Results and Discussion

A key figure for the Ahmed body is the total drag coefficient, CD, which is defined as

(1)

where F is the total drag force on the body, Ap is area of the body projected on a plane perpendicular to the flow direction (that is the xz-plane), is the density and u is the freestream velocity. Evaluating the quantities in Equation 1 gives a drag coefficient equal to 0.279 which compares well to the experimental value of 0.285. The error is hence 2.2 and is fairly low since errors in order of 10 is not uncommon for simulations using wall functions (Ref. 3). A possible explanation to why CD is underestimated is that wall functions are not very good at predicting the transition that in the experiments takes place on the front of the body. This makes the turbulence levels too low which in turn results in a too low viscous drag (Ref. 4).

Figure 3 shows streamlines behind the ahmed body. The thickness of the lines is given by the turbulent kinetic energy. The most notable feature of the flow field is a large “empty” region behind the body. The streamlines on the edge of the region are thick but with low velocity magnitude. This region is a recirculation region. The low pressure in the recirculation region is the main contribution to the total drag on the

FAp------- CD

u2

2-----------=

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body. The region ends when vortices from the trailing edges of the body merge into two counter rotating vortices (only one vortex is visible since the other is on the other side of the symmetry plane).

Figure 3: Streamlines behind the Ahmed Body. The streamlines are colored by the velocity magnitude and their thickness is proportional to the turbulent kinetic energy.

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More details are visible in Figure 4 and Figure 5 which show arrow plots of the velocity in the xz-plane 80 mm and 200 mm downstream of the body respectively.

Figure 4: Velocity in the xz-plane at yL0.08 m.

The flow pattern 80 mm downstream of the body shows two major vortices, one emanating from the outer edge of the slant and one emanating from the interaction between the floor and the stilts. The flow is qualitatively equal to the experimental results (Ref. 2). There are however quantitative differences. The upper vortex is smaller

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compared to experiments while the lower vortex is more pronounced than in the experiments.

Figure 5: Velocity in the xz-plane at yL0.20 m.

The flow pattern 200 mm downstream of the body shows that one major vortex is beginning to form but remains of the separate vortices can still be detected. The formation is however not proceeded as far as in the experiments where only one large vortex can be seen at this position.

In conclusion, the major features of the flow is well captured by the k- model, but there are details that deviate from experimental data. This finding is in agreement with other RANS simulations of the Ahmed body (Ref. 3).

References

1. S.R. Ahmed and G. Ramm, “Some Salient Features of the Time-Averaged Ground Vehicle Wake,” SAE-Paper 840300, 1984.

2. H. Lienhart and S. Becker, “Flow and Turbulence Structure in the Wake of a Simplified Car Model”, SAE 2003 World Congress, SAE Paper 2003-01-0656, Detroit, Michigan, USA, 2003.

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3. 9th ERCOFTAC/IAHR Workshop on Refined Turbulence Modelling, Darmstadt University of Technology, Germany, 2001.

4. A.C Benim, M.Cagan, A. Nahavandi, and E. Pasqualotto, “RANS Predictions of Turbulent Flow Past a Circular Cylinder over the Critical Regime”, Proceedings of the 5th IASME/WSEAS International Conference on Fluid Mechanics and Aerodynamics, Athens, Greece, 2007.

Model Library path: CFD_Module/Single-Phase_Benchmarks/ahmed_body

Modeling Instructions

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click Next.

3 In the Add physics tree, select Fluid Flow>Single-Phase Flow>Turbulent Flow>Turbulent

Flow, k- (spf).

4 Click Next.

5 In the Studies tree, select Preset Studies>Stationary.

6 Click Finish.

G L O B A L D E F I N I T I O N S

Parameters1 In the Model Builder window, right-click Global Definitions and choose Parameters.

2 Go to the Settings window for Parameters.

3 Locate the Parameters section. In the Parameters table, enter the following settings:

G E O M E T R Y 1

Import 11 In the Model Builder window, right-click Model 1>Geometry 1 and choose Import.

NAME EXPRESSION DESCRIPTION

L 1.044[m] Body length

u_in 40[m/s] Inflow velocity

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2 Go to the Settings window for Import.

3 Locate the Import section. Click the Browse button.

4 Browse to the model’s Model Library folder and double-click the file ahmed_body.mphbin.

5 Click the Import button.

Block 11 In the Model Builder window, right-click Geometry 1 and choose Block.

2 Go to the Settings window for Block.

3 Locate the Size and Shape section. In the Width edit field, type 2*L.

4 In the Depth edit field, type 8*L.

5 In the Height edit field, type 2*L.

6 Locate the Position section. In the x edit field, type -L.

7 In the y edit field, type -2*L.

8 Click the Build Selected button.

9 Click the Go to Default 3D View button on the Graphics toolbar.

Block 21 In the Model Builder window, right-click Geometry 1 and choose Block.

2 Go to the Settings window for Block.

3 Locate the Size and Shape section. In the Width edit field, type L.

4 In the Depth edit field, type 8*L.

5 In the Height edit field, type 2*L.

6 Locate the Position section. In the x edit field, type -L.

7 In the y edit field, type -2*L.

8 Click the Build Selected button.

Difference 11 In the Model Builder window, right-click Geometry 1 and choose Boolean

Operations>Difference.

2 Select the object blk1 only.

3 Go to the Settings window for Difference.

4 Locate the Difference section. Under Objects_to_subtract, click Activate Selection.

5 Select the objects imp1 and blk2 only.

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6 Click the Build Selected button.

7 In the Model Builder window, right-click Geometry 1 and choose Work Plane.

Rectangle 11 In the Model Builder window, right-click Geometry and choose Rectangle.

2 Go to the Settings window for Rectangle.

3 Locate the Size section. In the Width edit field, type L.

4 In the Height edit field, type 1.4*L.

5 Locate the Position section. In the y edit field, type 1.1*L.

6 Click the Build Selected button.

Union 11 In the Model Builder window, right-click Geometry 1 and choose Boolean

Operations>Union.

2 Select the objects dif1 and wp1 only.

3 Click the Build Selected button.

Block 31 In the Model Builder window, right-click Geometry 1 and choose Block.

2 Go to the Settings window for Block.

3 Locate the Size and Shape section. In the Width edit field, type L.

4 In the Height edit field, type 3*L.

5 Locate the Position section. In the y edit field, type 2.5*L.

6 Click the Build Selected button.

Rotate 11 In the Model Builder window, right-click Geometry 1 and choose Transforms>Rotate.

2 Select the object blk3 only.

3 Go to the Settings window for Rotate.

4 Locate the Rotation Angle section. In the Rotation edit field, type 45.

5 Locate the Point on Axis of Rotation section. In the y edit field, type 2.5*L.

6 Locate the Axis of Rotation section. In the z edit field, type 0.

7 In the x edit field, type 1.

8 Click the Build Selected button.

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Convert to Surface 11 In the Model Builder window, right-click Geometry 1 and choose Conversions>Convert

to Surface.

2 Select the object rot1 only.

3 Click the Build Selected button.

Split 11 In the Model Builder window, right-click Geometry 1 and choose Split.

2 Select the object csur1 only.

3 Click the Build Selected button.

Delete Entities 11 In the Model Builder window, right-click Geometry 1 and choose Delete Entities.

2 Click the Transparency button on the Graphics toolbar.

3 On the object spl1, select all faces except spl1(3).

4 Click the Build Selected button.

Union 21 In the Model Builder window, right-click Geometry 1 and choose Boolean

Operations>Union.

2 Select the objects uni1 (the block) and spl1(3) only.

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3 Click the Build Selected button.

Delete Entities 21 In the Model Builder window, right-click Geometry 1 and choose Delete Entities.

2 Select the boundary protruding from the top surface.

3 Click the Build Selected button.

Form Union1 In the Model Builder window, right-click Form Union and choose Build Selected.

2 Click the Go to Default 3D View button on the Graphics toolbar.

3 In the Model Builder window, collapse the Geometry 1 node.

4 Click the Transparency button on the Graphics toolbar to return to the default state.

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The model geometry is now complete.

M A T E R I A L S

1 In the Model Builder window, right-click Model 1>Materials and choose Open Material

Browser.

2 Go to the Material Browser window.

3 Locate the Materials section. In the Materials tree, select Built-In>Air.

4 Right-click and choose Add Material to Model from the menu.

5 In the Model Builder window’s toolbar, click the Show button and select Advanced

Physics Interface Options in the menu.

Turbulent Flow, k-

1 In the Model Builder window, click Model 1>Turbulent Flow, k-.

2 Go to the SettingsTurbulent Flow, k- window for .

3 Click to expand the Advanced Settings section.

4 From the CFL number expression list, select Manual.

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5 In the CFLloc edit field, type 1.3^min(niterCMP-1,9)+if(niterCMP>30,1.5*1.3^min(niterCMP-30,9),0

)+if(niterCMP>60,2.5*1.3^min(niterCMP-60,9),0).

The automatic formula for the local CFL number is too optimistic in this case. The delicate balance of complicated flow structures behind the body needs a more conservative ramping of the local CFL number to prevent the calculation from diverging.

Fluid Properties 11 In the Model Builder window, expand the Turbulent Flow, k- node, then click Fluid

Properties 1.

2 Go to the Settings window for Fluid Properties.

3 Locate the Fluid Properties section. From the list, select User defined. In the associated edit field, type 5e-4[Pa*s].

Wall 21 In the Model Builder window, right-click Turbulent Flow, k- and choose Wall.

2 Go to the Settings window for Wall.

3 Locate the Boundary Condition section. From the Boundary condition list, select Slip.

4 Select Boundaries 4, 12, 24, and 25 only.

Symmetry 11 In the Model Builder window, right-click Turbulent Flow, k- and choose Symmetry.

2 Select Boundaries 1 and 10 only.

Inlet 11 In the Model Builder window, right-click Turbulent Flow, k- and choose Inlet.

2 Select Boundary 2 only.

3 Go to the Settings window for Inlet.

4 Locate the Velocity section. In the U0 edit field, type u_in.

5 Locate the Boundary Condition section. In the IT edit field, type 0.005.

Outlet 11 In the Model Builder window, right-click Turbulent Flow, k- and choose Outlet.

2 Select Boundary 17 only.

3 Go to the Settings window for Outlet.

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4 Locate the Boundary Condition section. From the Boundary condition list, select Pressure.

M E S H 1

1 In the Model Builder window, click Model 1>Mesh 1.

2 Go to the Settings window for Mesh.

3 Locate the Mesh Settings section. From the Sequence type list, select User-controlled

mesh.

Size1 In the Model Builder window, click Size.

2 Go to the Settings window for Size.

3 Locate the Element Size section. Click the Custom button.

4 Locate the Element Size Parameters section. In the Maximum element size edit field, type 0.15.

5 In the Minimum element size edit field, type 0.005.

6 In the Resolution of curvature edit field, type 0.4.

7 In the Resolution of narrow regions edit field, type 0.5.

Size 11 In the Model Builder window, click 1.

2 Go to the Settings window for Size.

3 Locate the Geometric Entity Selection section. Click Clear Selection.

4 Select Boundaries 13 and 14 only.

5 Go to the Settings window for Size.

6 Locate the Element Size section. Click the Custom button.

7 Locate the Element Size Parameters section. Select the Maximum element size check box.

8 In the associated edit field, type 0.01.

9 Click the Build Selected button.

Size 21 In the Model Builder window, right-click Mesh 1 and choose Size.

2 Go to the Settings window for Size.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Boundary.

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4 Select Boundaries 7 and 9 only.

5 Locate the Element Size section. Click the Custom button.

6 Locate the Element Size Parameters section. Select the Maximum element size check box.

7 In the associated edit field, type 0.025.

Size 31 In the Model Builder window, right-click Mesh 1 and choose Size.

2 Go to the Settings window for Size.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Edge.

4 Select Edges 19, 21, 22, 48, and 49 only.

5 Locate the Element Size section. Click the Custom button.

6 Locate the Element Size Parameters section. Select the Maximum element size check box.

7 In the associated edit field, type 0.005.

Size 41 In the Model Builder window, right-click Mesh 1 and choose Size.

2 Go to the Settings window for Size.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Boundary.

4 Select Boundaries 3 and 15 only.

5 Locate the Element Size section. Click the Custom button.

6 Locate the Element Size Parameters section. Select the Maximum element size check box.

7 In the associated edit field, type 0.05.

Distribution 11 In the Model Builder window, right-click Mesh 1 and choose Distribution.

2 Go to the Settings window for Distribution.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Edge.

4 Select Edge 23 only.

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5 Locate the Distribution section. From the Distribution properties list, select Predefined distribution type.

6 In the Number of elements edit field, type 50.

7 In the Element ratio edit field, type 3.

Free Tetrahedral 11 In the Model Builder window, click Free Tetrahedral 1.

2 Select Domain 1 only.

3 Click the Build Selected button.

Swept 1In the Model Builder window, right-click Mesh 1 and choose Swept.

Distribution 11 In the Model Builder window, right-click Swept 1 and choose Distribution.

2 Select Domain 2 only.

3 Go to the Settings window for Distribution.

4 Locate the Distribution section. From the Distribution properties list, select Predefined distribution type.

5 In the Number of elements edit field, type 17.

6 In the Element ratio edit field, type 10.

7 From the Distribution method list, select Geometric sequence.

Boundary Layers 11 In the Model Builder window, click Boundary Layers 1.

2 Go to the Settings window for Boundary Layers.

3 Click to expand the Advanced Settings section.

4 In the Maximum angle per split edit field, type 120.

Boundary Layer Properties 11 In the Model Builder window, expand the Boundary Layers 1 node, then click

Boundary Layer Properties 1.

2 Go to the Settings window for Boundary Layer Properties.

3 Locate the Boundary Layer Properties section. In the Thickness adjustment factor edit field, type 2.

4 In the Model Builder window, right-click Mesh 1 and choose Build All.

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M E S H 2

In the Model Builder window, right-click Model 1 and choose Mesh.

Reference 11 In the Model Builder window, right-click Model 1>Meshes>Mesh 2 and choose More

Operations>Reference.

2 Go to the Settings window for Reference.

3 Locate the Reference section. From the Mesh list, select Mesh 1.

Scale 11 Right-click Reference 1 and choose Scale.

2 Go to the Settings window for Scale.

3 Locate the Scale section. In the Element size scale edit field, type 2.

Size1 In the Model Builder window, right-click Reference 1 and choose Expand.

2 In the Model Builder window, click Size.

3 Go to the Settings window for Size.

4 Locate the Element Size Parameters section. In the Maximum element growth rate edit field, type 1.2.

Boundary Layer Properties 11 In the Model Builder window, expand the Boundary Layers 1 node, then click

Boundary Layer Properties 1.

2 Go to the Settings window for Boundary Layer Properties.

3 Locate the Boundary Layer Properties section. In the Number of boundary layers edit field, type 4.

4 In the Boundary layer stretching factor edit field, type 1.25.

5 In the Model Builder window, right-click Mesh 2 and choose Build All.

M E S H 3

In the Model Builder window, right-click Model 1>Meshes and choose Mesh.

Reference 11 In the Model Builder window, right-click Model 1>Meshes>Mesh 3 and choose More

Operations>Reference.

2 Go to the Settings window for Reference.

3 Locate the Reference section. From the Mesh list, select Mesh 2.

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Scale 11 Right-click Reference 1 and choose Scale.

2 Go to the Settings window for Scale.

3 Locate the Scale section. In the Element size scale edit field, type 2.

Size1 In the Model Builder window, right-click Reference 1 and choose Expand.

2 In the Model Builder window, click Size.

3 Go to the Settings window for Size.

4 Locate the Element Size Parameters section. In the Maximum element growth rate edit field, type 1.25.

Boundary Layer Properties 11 In the Model Builder window, expand the Boundary Layers 1 node, then click

Boundary Layer Properties 1.

2 Go to the Settings window for Boundary Layer Properties.

3 Locate the Boundary Layer Properties section. In the Number of boundary layers edit field, type 3.

4 In the Boundary layer stretching factor edit field, type 1.3.

5 In the Model Builder window, right-click Mesh 3 and choose Build All.

M E S H 1

In the Model Builder window, collapse the Model 1>Meshes>Mesh 1 node.

M E S H 2

In the Model Builder window, collapse the Model 1>Meshes>Mesh 2 node.

M E S H 3

1 In the Model Builder window, collapse the Model 1>Meshes>Mesh 3 node.

2 In the Model Builder window’s toolbar, click the Show button and select Advanced

Study Options in the menu.

S T U D Y 1

Step 1: Stationary1 In the Model Builder window, expand the Study 1 node.

2 Right-click Study 1>Step 1: Stationary and choose Multigrid Level.

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3 Go to the Settings window for Multigrid Level.

4 Locate the Mesh Selection section. In the associated table, enter the following settings:

5 In the Model Builder window, right-click Step 1: Stationary and choose Multigrid

Level.

6 Right-click Study 1 and choose Show Default Solver.

7 In the Model Builder window, expand the Study 1>Solver Configurations node.

Solver 11 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1

node.

2 In the Model Builder window, expand the Stationary Solver 1>Iterative 1 node, then click Multigrid 1.

3 Go to the Settings window for Multigrid.

4 Find the subsection. From the Hierarchy generation method list, select Manual.

5 In the Model Builder window, expand the Stationary Solver 1>Iterative 2 node, then click Multigrid 1.

6 Go to the Settings window for Multigrid.

7 Find the subsection. From the Hierarchy generation method list, select Manual.

8 In the Model Builder window, collapse the Study 1 node.

Rename the study as a reminder of the viscosity used.

9 In the Model Builder window, right-click Study 1 and choose Rename.

10 Go to the Rename Study dialog box and type mu=5e-4 in the New name edit field.

11 Click OK.

12 Right-click Study 1 and choose Compute.

It is advisable to disable the automatic plot update when working with large 3D models.

1 In the Options menu, choose Preferences.

2 Go to the Preferences dialog box.

3 Click the Results tab.

GEOMETRY MESH

Geometry 1 {geom1} mesh2

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4 Clear the Update plot when selected check box.

5 Click the OK button.

R E S U L T S

Investigate the lift-off in viscous units to verify that the wall resolution is sufficient.

Wall Resolution (spf)1 In the Model Builder window, click Results>Wall Resolution (spf).

2 Go to the Settings window for 3D Plot Group.

3 Click the Plot button.

There is no need to refine the surface mesh since the wall lift-off is 11.06 almost everywhere (the exact result depends on the computational platform).

Figure 6: Wall lift-off in viscous units for 5e4 Pa·s.

Velocity (spf)1 In the Model Builder window, expand the Results>Velocity (spf) node, then click Slice

1.

2 Go to the Settings window for Slice.

3 Locate the Plane Data section. From the Entry method list, select Coordinates.

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4 In the x-coordinate edit field, type 0.15.

5 Click the Plot button.

The slice plot of the velocity shows that the flow for this Reynolds number is well resolved except perhaps in the wake. The main Reynolds number effect in the wake is expected to be the thickness of the shear layers and those seem well resolved all together. It is therefore probable that the resolution in the wake will suffice to converge a simulation at a higher Reynolds number.

Figure 7: Slice plot at x=0.15 m of the velocity magnitude for =5e4 Pa·s.

Turbulent Flow, k-

Fluid Properties 11 In the Model Builder window, click Model 1>Turbulent Flow, k->Fluid Properties 1.

2 Go to the Settings window for Fluid Properties.

3 Locate the Fluid Properties section. In the edit field, type 1e-4[Pa*s].

M O D E L W I Z A R D

1 In the Model Builder window, right-click Untitled.mph and choose Add Study.

2 Go to the Model Wizard window.

3 In the Studies tree, select Preset Studies>Stationary.

4 Click Finish.

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S T U D Y 2

Step 1: Stationary1 In the Model Builder window, click Study 2>Step 1: Stationary.

2 Go to the Settings window for Stationary.

3 Locate the Mesh Selection section. In the associated table, enter the following settings:

4 Right-click Study 2>Step 1: Stationary and choose Multigrid Level.

5 Go to the Settings window for Multigrid Level.

6 Locate the Mesh Selection section. In the associated table, enter the following settings:

7 In the Model Builder window, right-click Step 1: Stationary and choose Multigrid

Level.

8 Right-click Study 2 and choose Show Default Solver.

9 In the Model Builder window, expand the Study 2>Solver Configurations node.

Solver 21 In the Model Builder window, expand the Study 2>Solver Configurations>Solver 2

node, then click Dependent Variables 1.

2 Go to the Settings window for Dependent Variables.

3 Locate the Initial Values of Variables Solved For section. From the Method list, select Solution.

4 From the Solution list, select Solver 1.

5 In the Model Builder window, expand the Stationary Solver 1>Segregated 1 node.

6 In the Model Builder window, expand the Stationary Solver 1>Iterative 1 node, then click Multigrid 1.

7 Go to the Settings window for Multigrid.

8 Find the subsection. From the Hierarchy generation method list, select Manual.

MESH

mesh1

MESH

mesh2

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9 In the Model Builder window, expand the Stationary Solver 1>Iterative 2 node, then click Multigrid 1.

10 Go to the Settings window for Multigrid.

11 Find the subsection. From the Hierarchy generation method list, select Manual.

12 In the Model Builder window, collapse the Study 2 node.

13 In the Model Builder window, right-click Study 2 and choose Rename.

14 Go to the Rename Study dialog box and type mu=1e-4 in the New name edit field.

15 Click OK.

16 Right-click Study 2 and choose Compute.

R E S U L T S

Wall Resolution (spf) 11 Go to the Settings window for 3D Plot Group.

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2 Click the Plot button.

The wall lift-off is now larger than 11.06 at several locations. It is hence not likely that the current mesh will suffice to converge a simulation with five times higher Reynolds number.

Figure 8: Wall lift-off in viscous units for =1e4 Pa·s.

M E S H 4

In the Model Builder window, right-click Model 1>Meshes and choose Mesh.

Reference 11 In the Model Builder window, right-click Model 1>Meshes>Mesh 4 and choose More

Operations>Reference.

2 Go to the Settings window for Reference.

3 Locate the Reference section. From the Mesh list, select Mesh 1.

Size1 Right-click Reference 1 and choose Expand.

2 In the Model Builder window, click Size.

3 Go to the Settings window for Size.

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4 Locate the Element Size Parameters section. In the Minimum element size edit field, type 0.0025.

5 In the Maximum element growth rate edit field, type 1.1.

Size 21 In the Model Builder window, click 2.

2 Go to the Settings window for Size.

3 Locate the Element Size Parameters section. In the Maximum element size edit field, type 0.02.

Size 31 In the Model Builder window, click Size 3.

2 Go to the Settings window for Size.

3 Locate the Element Size Parameters section. In the Maximum element size edit field, type 0.0025.

Distribution 11 In the Model Builder window, click Distribution 1.

2 Go to the Settings window for Distribution.

3 Locate the Distribution section. In the Number of elements edit field, type 60.

4 In the Element ratio edit field, type 3.5.

Distribution 11 In the Model Builder window, expand the Swept 1 node, then click Distribution 1.

2 Go to the Settings window for Distribution.

3 Locate the Distribution section. In the Number of elements edit field, type 23.

Boundary Layer Properties 11 In the Model Builder window, expand the Boundary Layers 1 node, then click

Boundary Layer Properties 1.

2 Go to the Settings window for Boundary Layer Properties.

3 Locate the Boundary Layer Properties section. In the Thickness adjustment factor edit field, type 1.

4 In the Number of boundary layers edit field, type 6.

5 In the Model Builder window, right-click Mesh 4 and choose Build All.

6 In the Model Builder window, collapse the Mesh 4 node.

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M E S H 5

In the Model Builder window, right-click Model 1>Meshes and choose Mesh.

Reference 11 In the Model Builder window, right-click Model 1>Meshes>Mesh 5 and choose More

Operations>Reference.

2 Go to the Settings window for Reference.

3 Locate the Reference section. From the Mesh list, select Mesh 4.

Scale 11 Right-click Reference 1 and choose Scale.

2 Go to the Settings window for Scale.

3 Locate the Scale section. In the Element size scale edit field, type 2.

Size1 In the Model Builder window, right-click Reference 1 and choose Expand.

2 In the Model Builder window, click Size.

3 Go to the Settings window for Size.

4 Locate the Element Size Parameters section. In the Maximum element growth rate edit field, type 1.15.

Boundary Layer Properties 11 In the Model Builder window, expand the Boundary Layers 1 node, then click

Boundary Layer Properties 1.

2 Go to the Settings window for Boundary Layer Properties.

3 Locate the Boundary Layer Properties section. In the Number of boundary layers edit field, type 5.

4 In the Boundary layer stretching factor edit field, type 1.3.

5 In the Model Builder window, right-click Mesh 5 and choose Build All.

6 In the Model Builder window, collapse the Mesh 5 node.

M E S H 6

In the Model Builder window, right-click Model 1>Meshes and choose Mesh.

Reference 11 In the Model Builder window, right-click Model 1>Meshes>Mesh 6 and choose More

Operations>Reference.

2 Go to the Settings window for Reference.

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3 Locate the Reference section. From the Mesh list, select Mesh 5.

Scale 11 Right-click Reference 1 and choose Scale.

2 Go to the Settings window for Scale.

3 Locate the Scale section. In the Element size scale edit field, type 2.

Size1 In the Model Builder window, right-click Reference 1 and choose Expand.

2 In the Model Builder window, click Size.

3 Go to the Settings window for Size.

4 Locate the Element Size Parameters section. In the Maximum element growth rate edit field, type 1.2.

Boundary Layer Properties 11 In the Model Builder window, expand the Boundary Layers 1 node, then click

Boundary Layer Properties 1.

2 Go to the Settings window for Boundary Layer Properties.

3 Locate the Boundary Layer Properties section. In the Number of boundary layers edit field, type 4.

4 In the Boundary layer stretching factor edit field, type 1.4.

5 In the Model Builder window, right-click Mesh 6 and choose Build All.

6 In the Model Builder window, collapse the Mesh 6 node.

Turbulent Flow, k-

Fluid Properties 11 In the Model Builder window, expand the Model 1>Turbulent Flow, k- node, then

click Fluid Properties 1.

2 Go to the Settings window for Fluid Properties.

3 Locate the Fluid Properties section. From the list, select From material.

M O D E L W I Z A R D

1 In the Model Builder window, right-click Untitled.mph and choose Add Study.

2 Go to the Model Wizard window.

3 In the Studies tree, select Preset Studies>Stationary.

4 Click Finish.

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S T U D Y 3

Step 1: Stationary1 In the Model Builder window, click Study 3>Step 1: Stationary.

2 Go to the Settings window for Stationary.

3 Locate the Mesh Selection section. In the associated table, enter the following settings:

4 Right-click Study 3>Step 1: Stationary and choose Multigrid Level.

5 Go to the Settings window for Multigrid Level.

6 Locate the Mesh Selection section. In the associated table, enter the following settings:

7 In the Model Builder window, right-click Step 1: Stationary and choose Multigrid

Level.

8 Right-click Study 3 and choose Show Default Solver.

9 In the Model Builder window, expand the Study 3>Solver Configurations node.

Solver 31 In the Model Builder window, expand the Study 3>Solver Configurations>Solver 3

node, then click Dependent Variables 1.

2 Go to the Settings window for Dependent Variables.

3 Locate the Initial Values of Variables Solved For section. From the Method list, select Solution.

4 From the Solution list, select Solver 2.

5 In the Model Builder window, expand the Stationary Solver 1>Iterative 1 node, then click Multigrid 1.

6 Go to the Settings window for Multigrid.

7 Find the subsection. From the Hierarchy generation method list, select Manual.

8 In the Model Builder window, expand the Stationary Solver 1>Iterative 2 node, then click Multigrid 1.

MESH

mesh4

MESH

mesh5

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9 Go to the Settings window for Multigrid.

10 Find the subsection. From the Hierarchy generation method list, select Manual.

11 In the Model Builder window, collapse the Study 3 node.

12 In the Model Builder window, right-click Study 3 and choose Rename.

13 Go to the Rename Study dialog box and type mu from material in the New name edit field.

14 Click OK.

15 Right-click Study 3 and choose Compute.

R E S U L T S

Wall Resolution (spf) 21 In the Model Builder window, right-click Results>Wall Resolution (spf) 2 and choose

Plot.

The wall lift-off is larger than 11.06 at some locations, but it is close to 11.06 on most of the body and can hence be considered to be acceptable.

Figure 9: Wall lift-off in viscous units for from material.

To evaluate F and Ap in Equation 1, perform the following steps:

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Explicit 11 In the Model Builder window, right-click Definitions and choose Selections>Explicit.

2 Go to the Settings window for Explicit.

3 Locate the Input Entities section. From the Geometric entity level list, select Boundary.

4 Click the Select Box button on the Graphics toolbar.

5 Select Boundaries 5–9, 13, 14, and 18–23 only.

6 Right-click Explicit 1 and choose Rename.

7 Go to the Rename Explicit dialog box and type Body in the New name edit field.

8 Click OK.

R E S U L T S

Derived Values1 In the Model Builder window, right-click Results>Derived Values and choose

Integration>Surface Integration.

2 Go to the Settings window for Surface Integration.

3 Locate the Data section. From the Data set list, select Solution 3.

4 Locate the Selection section. From the Selection list, select Body.

5 In the upper-right corner of the Expression section, click Replace Expression.

6 From the menu, choose Turbulent Flow, k->Total stress>Total stress, y component

(spf.T_stressy).

7 Click the Evaluate button.

8 In the Model Builder window, right-click Derived Values and choose Integration>Surface Integration.

9 Go to the Settings window for Surface Integration.

10 Locate the Data section. From the Data set list, select Solution 3.

11 Click the Wireframe Rendering button on the Graphics toolbar.

12 Select Boundaries 5–7, 18, and 19 only.

13 Locate the Expression section. In the Expression edit field, type ny.

14 Click the Evaluate button.

The following steps reproduce Figure 3:

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Data Sets1 In the Model Builder window, right-click Results>Data Sets and choose Solution.

2 Go to the Settings window for Solution.

3 Locate the Solution section. From the Solution list, select Solver 3.

4 Right-click Solution 4 and choose Add Selection.

5 Go to the Settings window for Selection.

6 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Boundary.

7 From the Selection list, select Body.

8 Select Boundaries 3, 5–9, 13–16, and 18–23 only.

Velocity (spf) 21 In the Model Builder window, expand the Results>Velocity (spf) 2 node.

2 Right-click Slice 1 and choose Delete.

3 Click Yes to confirm.

4 Right-click Velocity (spf) 2 and choose Surface.

5 Go to the Settings window for Surface.

6 Locate the Data section. From the Data set list, select Solution 4.

7 Locate the Expression section. In the Expression edit field, type 1.

8 Locate the Coloring and Style section. Clear the Color legend check box.

9 From the Coloring list, select Uniform.

10 From the Color list, select Gray.

11 Click the Plot button.

12 In the Model Builder window, click Velocity (spf) 2.

13 Clear the Plot data set edges check box.

14 Right-click Velocity (spf) 2 and choose Streamline.

15 Go to the Settings window for Streamline.

16 Locate the Streamline Positioning section. From the Positioning list, select Start point

controlled.

17 From the Entry method list, select Coordinates.

18 In the x edit field, type range(0.01,0.03,0.16) range(0.01,0.03,0.16) range(0.01,0.03,0.16) range(0.01,0.03,0.16) range(0.01,0.03,0.16).

19 In the y edit field, type -0.5*L.

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20 In the z edit field, type 0.02*1^range(1,6) 0.08*1^range(1,6) 0.14*1^range(1,6) 0.2*1^range(1,6) 0.26*1^range(1,6).

21 Locate the Coloring and Style section. From the Line type list, select Tube.

22 In the Tube radius expression edit field, type k*1[s^2/m].

23 Select the Radius scale factor check box.

24 In the associated edit field, type 3e-4.

25 Right-click Streamline 1 and choose Color Expression.

26 In the Settings window, click Plot.

The following steps will reproduce Figure 4 and Figure 5:

Data Sets1 In the Model Builder window, right-click Results>Data Sets and choose Cut Plane.

2 Go to the Settings window for Cut Plane.

3 Locate the Data section. From the Data set list, select Solution 3.

4 Locate the Plane Data section. From the Plane list, select zx-planes.

5 In the y-coordinate edit field, type L+0.08.

6 In the Model Builder window, right-click Data Sets and choose Cut Plane.

7 Go to the Settings window for Cut Plane.

8 Locate the Data section. From the Data set list, select Solution 3.

9 Locate the Plane Data section. From the Plane list, select zx-planes.

10 In the y-coordinate edit field, type L+0.2.

3D Plot Group 101 In the Model Builder window, right-click Results and choose 3D Plot Group.

2 Go to the Settings window for 3D Plot Group.

3 Locate the Data section. From the Data set list, select Solution 3.

4 Locate the Plot Settings section. Clear the Plot data set edges check box.

3D Plot Group 101 In the Model Builder window, right-click Results>Velocity (spf) 2>Surface 1 and

choose Copy.

2 Right-click Results>3D Plot Group 10 and choose Paste Surface.

3 Right-click 3D Plot Group 10 and choose Arrow Surface.

4 Go to the Settings window for Arrow Surface.

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5 Locate the Data section. From the Data set list, select Cut Plane 1.

6 Locate the Expression section. In the y component edit field, type 0.

7 Locate the Coloring and Style section. In the Scale factor edit field, type 2.4.

8 In the Number of arrows edit field, type 2500.

9 From the Color list, select Black.

10 In the Model Builder window, right-click Arrow Surface 1 and choose Filter.

11 Go to the Settings window for Filter.

12 Locate the Element Selection section. In the Logical expression for inclusion edit field, type (x<0.35)*(z<0.45).

13 In the Model Builder window, click 3D Plot Group 10.

14 Go to the Settings window for 3D Plot Group.

15 Locate the Plot Settings section. Select the Title check box.

16 In the associated edit field, type Arrow: Velocity in xz-plane.

17 Click the Plot button.

18 Right-click 3D Plot Group 10 and choose Duplicate.

3D Plot Group 111 In the Model Builder window, expand the 3D Plot Group 11 node, then click Arrow

Surface 1.

2 Go to the Settings window for Arrow Surface.

3 Locate the Data section. From the Data set list, select Cut Plane 2.

4 Click the Plot button.

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