Comparison of Turbulence Models on 2D Backward Facing Step

Comparison of Turbulence Models Applied on Two-dimensional Backward-Facing-Step Flow

Eldwin Djajadiwinata1 Student ID: 434107763 1

 Graduate Studies, Mechanical Engineering Department, College of Engineering, King Saud University, Kingdom of Saudi Arabia.

ABSTRACT

This report is part of Advanced CFD course, PhD program, at M.E. Dept., College of Engineering, King Saud University. It focuses on comparing two turbulence models applied on two-dimensional backward-facing-step flow for educational purpose, i.e., to learn how to  perform the steps on solving flow problem through CFD commercial package. The turbulence models compared are Spalart-Allmaras model and         model. The geometry and the grid generated are explained. The results are compared with other wellknown simulation results and experimental results. Comparison between both turbulence models is discussed. Finally, conclusion is presented in the end.

1. METHODOLOGY Two turbulence models, i.e., Spalart-Allmaras Turbulence Model and       , are compared by applying them on two-dimensional backward facing step flow. This study is mainly referring to the NASA Turbulence Modeling Resource website [1] as [1] as its guide. Results of the present study will be compared to the experimental data of the work of Driver and Seegmiller [2] which [2] which is also provided in the aforementioned NASA website [1]. [1]. The scheme of experimental set-up done by Driver and Seegmiller [2]  [2]  can be seen in Figure 1. It 1. It should be mentioned that the present work only considers angle of the upper wall,

, (in Figure (in Figure 1) to 1) to be equal zero due to time limitation. That is, flat-horizontal upper wall.

1

Figure 1: Scheme of the experimental set-up and parameter s conducted by Driver and Seegmiller [2]

The flow is an incompressible turbulent flow over a backward-facing step in a channel. The mean-velocity and turbulence parameters of the flow were measured using a laser Doppler velocimeter (LDV). Eddy viscosities, production, convection, turbulent diffusion, and dissipation (balance of kinetic energy equation) terms are extracted from the measured data. A more detail explanation can be read in ERCOFTAC "Classic Collection" Database website [3]. The grid/mesh was generated using ANSYS version 14. The ANSYS DesignModeler was used to generate the geometry and, afterwards, it is exported to ANSYS Meshing to generate the mesh. There were four meshes created with each of them having different number of cells, i.e., 14750, 59000, 132750, and 531000. These meshes were used for grid independent study. Basically, the finer mesh was obtained by increasing the number of points on every boundary uniformly by a certain factor. For example, the 59000-mesh was obtained by increasing the number of points on the boundary by a factor of two on, both, x and y directions. Thus, 2

  (  )  . Quadrilateral-shape cells were used throughout the mesh as can  be seen in Figure 2.

Figure 2: Typical quadrilateral-shape cells used on the mesh

The dimension of the geometry as well as the boundary conditions was exactly following those in NASA website [1] except at the inlet and outlet (Figure 3). In the present simulation, the inlet and outlet boundary conditions were ‘velocity inlet’ and ‘outflow’  boundary conditions, respectively, whereas NASA was using pressure boundary conditions. It is stated in the website that any type of boundary conditions at the inlet and outlet will work as long as they result in the same boundary layer thickness and local-center Mach number of around    and 0.128, respectively, at non-dimensional -location of  

 . Here, ‘H’ is the step height which value is  .

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Figure 3: Geometry and boundary condition used in the simulation (source: NASA [1]) The Uref   is the reference velocity at the center-channel near x/H=-4, used to nondimensionalize velocity and turbulent shear stress profiles. The skin friction coefficient was also calculated with respect to conditions near this location. The simulation was done in double precision mode. The pressure-velocity coupling scheme used was SIMPLE. The spatial discretization schemes were ‘least squares cell based’ and PRESTO! for the gradient and pressure. PRESTO! scheme was intentionally used based on the information stated in ANSYS FLUENT 14 user’s guide (page 1316) as quoted below: “For most cases the “standard” scheme is acceptable, but some types of models may benefit from one of the other schemes: • For problems involving large body forces, the body-force-weighted scheme is recommended. • For flows with high swirl numbers, high -Rayleigh-number natural convection, high speed rotating flows, flows involving porous media, and flows in strongly curved domains, use the PRESTO! scheme.”

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The spatial discretization for the remaining parameters was set to second order upwind. Example of Fluent GUI for the aforementioned solution methods can be seen in Figure 4.  These settings were used for both, Spalart-Allmaras and         turbulence models simulation.

Figure 4: Example of Fluent GUI of ‘solution methods’ menu

The default values were taken for all under-relaxation factors. The globally-scaled residual was set to reach      throughout the simulations in this study.

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2. RESULTS AND DISCUSSION This section is divided into two. The first one focuses on discussing the results of simulation using Spalart-Allmaras one-equation model (SA) and the other part focuses on discussing the results when k-omega-SST model is used.

2.1. Spalart-Allmaras One-equation Model The meshes used in the grid independence study have been explained previously in the methodology section. The results show that the variables considered approach certain values as the number of cells increased. This can be seen clearly in Figure 5(a) to Figure 9(a) (letter (a) only). Thus, it can be concluded that the mesh generated is acceptable. It is known that the smaller the cell size, the lower the truncation error which, then, will increase the accuracy of the simulation. Consequently, it is desirable to have small cell size. Computing time, however, should be considered because the smaller the cell size, i.e., higher number of cells, will result in longer computation time. Therefore, as a rule of thumb, usually one would choose a mesh with larger cell size which simulation result does not differ significantly than that mesh with smaller cell size. Based on this, in the present study, mesh with number of cells of 59000 can be chosen  because its simulation results are close to those of mesh with higher number of cells, i.e., 132750 and 531000 as shown in Figure 5(a) to Figure 9(a) (letter (a) only). In Figure 9(a) it can be seen that the re-attachment point obtained (i.e., at shear stress equals to zero or, alternatively, surface skin friction coefficient equals to zero) for these three meshes are close which is around x/H = 6 with difference between them around 1%. The experimental reattachment point is

 

    whereas that predicted by NASA’s simulation via SA

turbulence model is close to

 

 . On Figure 6 it can be seen that maximum difference

 between the 59000-cell mesh and the 531000-cell mesh is only less than or equal to 5% which happens on the left peak of the curve. Of course, if the recourses are sufficient and  better accuracy is highly needed, mesh with higher number of cells is preferable. Figure 5 to Figure 9 also show comparison between the present simulation results,  NASA simulation results, and experimental results from the work of Driver and Seegmiller [2]. All of these simulations are using SA turbulence model.

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5.0

4.5 experiment 4.0

no. of cells 14750

   ]   s   s 3.5   e    l   n   e   o    i   s   n 3.0   e   m    i    D    [    H    / 2.5   y

no. of cells 59000 no. of cells 132750 no. of cells 531000

2.0

1.5

1.0 -2.5

-2.0

-1.5

-1.0

-0.5

0.0

avg(u'v')*1000/Uref 2 [Dimensionless]

(a)

(b) Figure 5: Non-dimensional turbulent shear stress along y direction at x/H =-4 for (a) the  present study and (b) NASA’s work. All are using SA turbulence model.

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3.0

experiment no. of cells 14750 2.0

no. of cells 59000 no. of cells 132750

   ]   s   s   e    l   n   e   o    i   s   n   e   m    i 1.0    D    [

no. of cells 531000

   H    /   y

0.0 -15.0

-10.0

-5.0

0.0

5.0

avg(u'v')*1000/Uref 2 [Dimensionless]

(a)

(b) Figure 6: Non-dimensional turbulent shear stress along y direction at (a) x/H=1 for the  present study and (b) at various x/H locations for NASA’s work. All are using SA turbulence model.

8

5

experiment 4

no. of cells 14750 no. of cells 531000

   ]   s   s   e    l   n   e   o    i   s   n   e   m    i    D    [    H    /   y

3

2

1 0.0

0.2

0.4

0.6

0.8

1.0

u/Uref  [Dimensionless]

(a)

(b) Figure 7: Non-dimensional velocity profile along y direction at x/H=-4 for (a) the present study and (b) NASA’s work. All using SA turbulence model.

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3

experiment no. of cells 14750 no. of cells 59000 2

no. of cells 132750 no. of cells 531000

   ]   s   s   e    l   n   e   o    i   s   n   e   m    i    D    [    H    /   y

1

0 -0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

u/Uref  [Dimensionless]

(a)

(b) Figure 8: Non-dimensional velocity profile along y direction at (a) x/H=1 of the present stud y and (b) at various x/H locations of NASA’s work. All are using SA turbulence model. 10

0.004

experiment no. of cells 14750 no. of cells 59000

0.003

no. of cells 132750 no. of cells 531000 0.002      ]     s     s     e      l     n     e     o      i     s 0.001     n     e     m      i      D      [      f

     C

0.000 -5

5

15

25

x/H [Dimensionless] -0.001

-0.002

(a)

(b) Figure 9: Non-dimensional velocity profile along the wall for (a) the present study and (b)  NASA’s work. All are using SA turbulence model

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It can be seen clearly that the results from this work are not only qualitatively agree with the NASA’s work but also quantitatively. If the 59000 -cell mesh is taken as the base, then the maximum difference between these two CFD works are 10% which is shown in Figure 6. The left peak of the present simulation reaches -10 while the NASA’s work reaches -9. However, if we compare with the 531000-cell mesh, the difference reduces to around 5% only. This proves that the meshes and boundary condition of the present study are acceptable.

2.2.        Two-equation Model The grid independence study for simulation using k-omega-SST model was not done as thorough as for that using SA model because the time available is limited. There are only two meshes tested, i.e., the 59000-cell mesh and the 531000-cell mesh. Therefore, this is actually not a proper grid independence study but it is only to get the sense of the effect of changing cell size on the results. Regarding grid independency, if one observes Figure 10 to  Figure 12, it is clear that the two meshes tested (59000-cell and 531000-cell) for grid independence give close results except those showed in Figure 11. They are far away to each other. On the other hand, the results of these two meshes are close to each other in all figures when SA model is used instead of k-omega-SST model. Thus, it can be concluded that grid independent mesh for a certain turbulence model cannot be generalized for another model. The second thing that one can conclude is that parameters/variables to be looked at during grid independence test must be those which become the main concern of the research. This means that, a certain parameter/variable might be independent of such mesh size while other parameters might not. Hence, one needs to have a grid independent mesh for the  parameter of interest.

12

3.0

experiment SA, no. of cells 59000 SA, no. of cells 531000 2.0

SST, no. of cells 59000 SST, no. of cells 531000

   ]   s   s   e    l   n   e   o    i   s   n   e   m    i 1.0    D    [    H    /   y

0.0 -15.0

-10.0

-5.0

avg(u'v')*1000/Uref 2

0.0

5.0

[Dimensionless]

(a)

(b) Figure 10: Non-dimensional turbulent shear stress along y direction at (a) x/H=1 for the  present study and (b) at various x/H locations for NASA’s work  using SST model.

13

0.004

experiment SA, no. of cells 59000 SA, no. of cells 531000

0.003

SST, no. of cells 59000 SST, no. of cells 531000

0.002      ]     s     s     e      l     n     e     o      i     s 0.001     n     e     m      i      D      [      f      C

0.000 -5

5

15

25

x/H [Dimensionless] -0.001

-0.002

(a)

(b) Figure 11: Non-dimensional velocity profile along y direction at (a) x/H=1 of the present study and (b) at various x/H locations of NASA’s work  using SST model.

14

0.004

experiment SA, no. of cells 59000 SA, no. of cells 531000

0.003

SST, no. of cells 59000 SST, no. of cells 531000 0.002      ]     s     s     e      l     n     e     o      i     s 0.001     n     e     m      i      D      [      f      C

0.000 -5

5

15

25

x/H [Dimensionless] -0.001

-0.002

(a)

(b) Figure 12: Non-dimensional velocity profile along the wall for (a) the present study and (b)  NASA’s work  using SST model.

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The same as the results obtained when using SA model, the results of this work utilizing k-omega-SST model show a close agreement with the results obtained by NASA using the same model (k-omega-SST). Moreover, using this model, the simulation results match closer to the experimental data compared to that when using SA model.

3. CONCLUSION Comparison study between Spalart-Allmaras and k-omega-SST turbulence model applied on backward-facing step flow has been conducted. The conclusion can be summarized as follows: 1. Grid independent mesh for a certain turbulence model cannot be generalized for another model. 2. In backward-facing step flow, k-omega-SST model can predict the flow better compared to the Spalart-Allmaras model. 3. The commercial code used in this study (ANSYS FLUENT 14) is able to predict the flow as good as the prediction of other code te sted by NASA. 4. The prediction of CFD on turbulent flow is good enough qualitatively but, quantitatively, still needs more improvement.

REFERENCES [1] Nasa Turbulence Modeling Resource: 2d Backward Facing Step, Last accessed Jan. 14, 2015, http://turbmodels.larc.nasa.gov/backstep_val.html [2] Driver, D. M., and Seegmiller, H. L., Feb 1985, "Features of Reattaching Turbulent Shear Layer in Divergent Channel Flow," AIAA Journal, 23(2), pp. 163-171. [3] E.R.C.O.F.T.A.C, Last accessed http://cfd.mace.manchester.ac.uk/ercoftac/index.html

16

Jan.

14,

2015,