1

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 well-

known 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 its guide.

Results of the present study will be compared to the experimental data of the work of

Driver and Seegmiller [2] which is also provided in the aforementioned NASA website [1].

The scheme of experimental set-up done by Driver and Seegmiller [2] can be seen in

Figure 1. It should be mentioned that the present work only considers angle of the upper wall,

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

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Figure 1: Scheme of the experimental set-up and parameters

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,

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( ) . 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 re-

attachment 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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(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.

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

-2.5 -2.0 -1.5 -1.0 -0.5 0.0

y/H

[D

imen

sioen

less

]

avg(u'v')*1000/Uref2 [Dimensionless]

time-averaged u'v' at x/H = -4

experiment

no. of cells 14750

no. of cells 59000

no. of cells 132750

no. of cells 531000

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(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.

0.0

1.0

2.0

3.0

-15.0 -10.0 -5.0 0.0 5.0

y/H

[D

imen

sioen

less

]

avg(u'v')*1000/Uref2 [Dimensionless]

time-averaged u'v' at x/H = 1

experiment

no. of cells 14750

no. of cells 59000

no. of cells 132750

no. of cells 531000

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(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.

1

2

3

4

5

0.0 0.2 0.4 0.6 0.8 1.0

y/H

[D

imen

sioen

less

]

u/Uref [Dimensionless]

velocity profile at x/H = -4

experiment

no. of cells 14750

no. of cells 531000

10

(a)

(b)

Figure 8: 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. All are using SA turbulence model.

0

1

2

3

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

y/H

[D

imen

sioen

less

]

u/Uref [Dimensionless]

velocity profile at x/H = 1

experiment

no. of cells 14750

no. of cells 59000

no. of cells 132750

no. of cells 531000

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(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

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

-5 5 15 25

Cf[D

imen

sio

enle

ss]

x/H [Dimensionless]

Surface Skin Friction Coeff. (Cf) along Bottom-wall

experiment

no. of cells 14750

no. of cells 59000

no. of cells 132750

no. of cells 531000

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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.

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(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.

0.0

1.0

2.0

3.0

-15.0 -10.0 -5.0 0.0 5.0

y/H

[D

imen

sio

enle

ss]

avg(u'v')*1000/Uref2 [Dimensionless]

time-averaged u'v' at x/H = 1

experiment

SA, no. of cells 59000

SA, no. of cells 531000

SST, no. of cells 59000

SST, no. of cells 531000

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(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.

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

-5 5 15 25

Cf[D

imen

sioen

less

]

x/H [Dimensionless]

Surface Skin Friction Coeff. (Cf) along Bottom-wall

experiment

SA, no. of cells 59000

SA, no. of cells 531000

SST, no. of cells 59000

SST, no. of cells 531000

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(a)

(b)

Figure 12: Non-dimensional velocity profile along the wall for (a) the present study and (b)

NASA’s work using SST model.

-0.002

-0.001

0.000

0.001

0.002

0.003

0.004

-5 5 15 25

Cf[D

imen

sioen

less

]

x/H [Dimensionless]

Surface Skin Friction Coeff. (Cf) along Bottom-wall

experiment

SA, no. of cells 59000

SA, no. of cells 531000

SST, no. of cells 59000

SST, no. of cells 531000

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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 tested 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 Jan. 14, 2015,

http://cfd.mace.manchester.ac.uk/ercoftac/index.html