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Stress Limiter Consideration for k-ω Turbulence Models in Shock Wave/Turbulent Boundary-Layer Interactions in

Supersonic and Hypersonic Flows

Jie TAN 1 and Jie JIN.2 School of Jet Propulsion, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

The shear stress transport (SST) turbulence model of Menter was proposed to predict an accurate separated flow in the adverse pressure gradient flows, but it has sometimes failed to predict the flowfield with strong shock wave/turbulent boundary-layer interaction. In an effort to improve the original k-ω model, Wilcox has recently proposed a modified k-ω model formulation (Wilcox08), which adopted two elements of the SST model, namely the cross diffusion term and SST limiter. However, the effects of freestream dependency have not been eliminated by the new model. In order to take full advantages of Wilcox08 model and SST model, in this paper, it is straightforward to apply Wilcox’s type of stress limiter to the SST model by adjusting the coefficient in the eddy viscosity computation. The new model with different stress limiter strength are evaluated for three cases involving strong shock wave/turbulent boundary-layer interactions at Mach 2.8 and Mach 5.0. It has been observed in all simulations that the stress limiter strength has significant effect on the features of flowfield and wall surface parameters while a constant value of stress limiter is not sufficient, and could be a function of the flow variables in some way or a specific scalar transport equation of stress limiter is necessary for better prediction of strong SWTBLI flows.

I. Introduction HE phenomenon of shock wave/turbulent boundary-layer interactions (SWTBLI) are the most interesting and challenging problems of modern aerothermodynamics and high speed fluid dynamics. Effective design of

control surfaces, high-speed inlets, or thermal protection systems requires accurate prediction of surface heat, friction drag and pressure loads.

Hence, modern numerical methods need to be validated to available experimental data of the aforementioned flows. Recently, there has been increased interest in applying high-fidelity models, such as large eddy simulation (LES) and direct numerical simulation (DNS) techniques to predict SWTBLI and have shown promising results,1-4

however, their use in production application on real geometry is years away. The Detached Eddy Simulation (DES)5 and its variants, as well as Scale Adaptive Simulation (SAS)6 have become increasingly popular in recent years, as they offer a reduced computational effort in comparison to LES while retaining much of the physical accuracy of the method, which show good potential to predict unsteady flows characterized by massive separation. However, the boundary layer separation emphasized in our work is, to a large extent, controlled by the RANS model, which means inapproriate. Engineering prediction of SWTBLI relies on Reynolds Average Navier-Stokes (RANS) simulations. The k-ω models are widely used for prediction of the SWTBLI. Two widely used k-ω models are due to Wilcox7-9 and Menter10. The Menter model is the shear stress transport version, referred to as SST model, which was proposed to predict an accurate separated flow in the adverse pressure gradient flows. The SST modification alters the turbulence frequency as used in the eddy viscosity definition in the vicinity of high strain rates. The general effect is to lower the eddy viscosity and thus to promote the growth of regions of separated flow, whereas it has sometimes failed to predict the flowfield with strong SWTBLI. Liou et al11 investigated the four popular turbulence models, including the SST model, for supersonic compression corner flow and oblique shock wave/turbulent boundary-layer interaction at Mach 2.87. His investigation showed that the SST model failure to predict the correct separation shock position and skin friction recovery distribution, and overpredicted the separation size by 2.0 times the experimental value at some cases. Oliver et al12 investigated the 2-D compression ramp flows with 2-D and 3-D computation

1 Ph. D. Candidate, School of Jet Propulsion, tanjie1985@gmail.com, AIAA Student Member. 2 Senior Engineer, School of Jet Propulsion, jinjie@buaa.edu.cn.

T

20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3980

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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using the OVERFLOW and DPLR codes including SST model, the results of which similar to that of Liou et al. Computational prediction show too much upstream influence of surface pressure for 2-D computation involving strong SWTBLI while the results were improved by running 3-D computation including the tunnel side walls.

In an effort to improve the original k-ω model, Wilcox has recently proposed a modified k-ω model formulation, referred to as Wilcox08 model.9 It is interesting that Wilcox adopted two elements of the SST model, namely the cross diffusion term and SST limiter Clim. However, the effects of freestream dependency have also not been eliminated by the new model.10 In order to take full advantages of Wilcox08 model and SST model, in this paper, it is straightforward to apply Wilcox’s type of stress limiter to the SST model by adjusting the coefficient in the eddy viscosity computation. The value of Clim in the original SST model equals to 1.0, which is too large for some cases of strong SWTBLI.11-14 Wilcox8,9 recommends the value of Clim 0.875 for his model and Edwards et al15 found that a value of 0.90 agrees well with experiment of Mach 5 compression corner. As shown later, the predictions of SST model are sensitive to the value of the Clim, which ranges between zero (for no SST modification) to one (for the full SST contribution). Reducing the value of Clim, the turbulence kinetic energy level is increased thus to enhance the capacity of resistance to separation yielding to a smaller separated region.

The primary purpose of this work is to determine whether a constant value of stress limiter is sufficient or if should be a function of the flow variables in some way or requires a specific scalar transport equation of stress limiter for better prediction in strong SWTBLI. The computational cases studied in this paper include: 1) 24-deg compression corner at Mach2.84; 2) two-dimensional incident SWTBLI at Mach 5; 3) three-dimensional swept SWTBLI flow at Mach 5.

II. Governing Equations The Favre-average equations for conservation of mass, momentum, and energy are as follows:

0

jj

uxt

(1)

j

ji

iij

ji xx

puu

xu

t

ˆ

(2)

ˆPr Pr

L Tj i ji L k T

j j j L T j

k hE u H u

t x x x x

(3)

The total energy and enthalpy are 2iiuukeE and 2iiuukhH , respectively, with h e p and

1.0e p . ˆ ji is the sum of molecular and Reynolds stress, defined as

Fijij

k

kijLij x

uS

3

12ˆ (4)

ijijk

kijT

Fij k

x

uS 32

3

12

i

j

j

iij x

u

x

uS

2

1 (5)

The transport equations for the turbulence kinetic energy k and the specific dissipation rate ω for Menter’s SST model are given by

*j i

ij k tj j j j

u kk u kk

t x x x x

(6)

2

1 2

1 12 1

j iij t

j t j j j i i

u u kF

t x x x x x x

(7)

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Value for the constant k , , * , , , 2 and blending function 1F may be found in Ref. 10. The eddy

viscosity is defined as

1

1 lim 2,t t

a k

m ax a C SF

(8)

where 1 0.31a , 2 ij ijS S S and 2F is a second blending function define by

2

2 * 2

2 500tanh max ,

kF

d d

(9)

where d is the distance to the nearest wall and is the kinematic viscosity.

III. Numerical Methods The numerical simulations were carried out using an in-house CFD code. The computer code solves the three-

dimensional, time-dependent compressible RANS equations with an upwind finite volume formulation on block-structured grids. The inviscid fluxes are discretized using the Edwards’s16 low-diffusion flux-splitting scheme (LDFSS) together with a MUSCL variable extrapolation to provide second- or third-order accuracy in space. The minmod limiter is employed to ensure monotonic solutions around shock waves. Viscous terms are computed with standard central second-order differences. The linear system of the equations resulting from the first-order backward Euler time differencing is solved with a point-iterative Symmetric Gauss-Seidel (SGS)17,18 solver.

To increase the numerical stability of turbulent flow simulations, the turbulence models are decoupled from the mean flow equations. Their convective terms are evaluated by first order upwind scheme. The source terms of turbulence models are treated implicitly to avoid stiffness problems of the time-marching methods, and the turbulence equations are solved using a point-iterative SGS scheme. Clipping technique is applied to preserve the positivity of k and ω.

Because of the moderate test conditions during the wind-tunnel experiment, in this paper, the air is considered to be a calorically perfect gas with constant ratio of specific heats, γ=1.4. A constant turbulent Prandtl number of Prt=0.9 is assumed in the computation of turbulence contributions to thermal diffusion.

The computations reported here are performed on a parallel computer using 16 processors. The parallelization based on block-based message passing interface (MPI) formulation. Convergence is ascertained by monitoring several quantities including the global norm, surface pressure, and skin-friction coefficients.

IV. Results and Discussion 24-deg Compression Corner

The freestream conditions for the experiment of Settles et al19 are: M∞=2.84, U∞=570m/s, T∞=100.3K, Re∞/L=6.8×107m-1, and Tw=276.1K. Three different grid refinement level: coarse (121×121), medium (241×121) and fine (481×241), the minimum normal spacing at the wall was 10-6m, yielding average y+ less than 0.6. Results employing the medium and fine grids gave grid-independent solutions. The boundary conditions are summarized as follows: on solid surface, the no-slip condition is enforced, the wall temperature is specified based on experiment and the normal pressure gradient and k are all assumed zero. The incoming boundary-layer profile is specified at a distance of 6δ (δ=21mm) upstream of the wedge apex by matching the momentum thickness (θ=0.106mm) with separated two-dimensional calculations. The downstream and top boundaries are assumed to be far enough away for the application of simple zero-gradient extrapolation boundary conditions.

Figure 1 shows the sensitivity of the wall pressure and skin friction coefficient distributions to the choice of the stress limiter Clim. As shown, the stress limiter has a barely noticeable effect on the computed surface pressure and skin friction coefficient, which can be traced to the different predicted increase in Reynolds shear stress caused by the vorticity in the vicinity of separation shock. The size of the separation region was overpredicted for all the stress limiters. The delayed reattachment appears to be a large source of the discrepancies with experimental data in the recovery regions. The separation point location -xs, reattachment point location xr, and separation bubble size increases monotonically as Clim increases. As it is noted, a value of Clim=0.0, corresponding to the Menter’s baseline model (BSL), provides the best overall agreement with the experimental data in the separation region, even though

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the separation shock location is underpredicted. Selecting Clim=1.0, corresponding to the original SST model, results in significant large separation bubble size and the predicted separation shock location is clearly further upstream than measured, which distorts the entire flowfield and the downstream skin friction coefficient is also too low. The Clim=0.8 calculation correctly predicts the upstream interaction length as well as the pressure recovery over the ramp, whereas appears to predict a boundary layer that separates earlier than what is apparent from the measured skin friction coefficient.

The comparison of measured and predicted flowfield structure is shown in Fig. 2. The schlieren image from the experiment illustrates the dramatic disagreement with the Clim=1.0 case. The experiment shows a separated region on the order of 2.0δ. In the case of the Clim=0.0, the predicted separation zone is slightly larger than experiment, whereras for the Clim=1.0 case, the separated region size is predicted to be more than twice as big as in the experiment. It should be noted here that the two-dimensionality of the flow may have deteriorated in this strong SWTBLI case, the evidence for the existence of three-dimensional large-scale streamwise structures, commonly referred to the presence of pairs of counter-rotating streamwise vortices,20 was found. Therefore, running 3-D computations allowing for some end effects might improve the accuracy for modeling the flow physics for strong SWTBLI cases.

-6 -4 -2 0 2 4 60

1

2

3

4

5

Settles et al. Dolling-Murphy C

lim=0.0

Clim

=0.80

Clim

=0.85

Clim

=0.90

Clim

=1.0

p/p

f

x/0

-6 -4 -2 0 2 4 6-1

0

1

2

3

Settles et al. Dolling-Murphy C

lim=0.0

Clim

=0.80

Clim

=0.85

Clim

=0.90

Clim

=1.0

Cfx1

03

x/0

Fig. 1 Comparison of wall surface parameters for 24-deg compression corner at Mach2.84. Left: normalized pressure, Right: Skin friction coefficient.

Fig. 2 Comparison of measured and predicted flowfield structure for 24-deg compression corner at Mach2.84. Left: Clim=0.0, Middle: Clim=1.0, Right: Experiment.

Two-Dimensional Incident SWTBLI at Mach 5 The experiment was performed at the Ludwig Tube Facility at DLR.21 The test model used for the investigation

of the incident shock wave interacting with the flat-plate turbulent boundary layer is shown in Fig. 3. Modern nonintrusive techniques are used to make skin friction and heat transfer measurements. The repeatability error in the skin friction varied from ±4% in undisturbed boundary layer up to ±10% at the highest wall shear-stress levels near the reattachment lines, and the maximum error in heat flux measurement was ±5%. The freestream conditions in the test section were M=5, T0=410K, P0=2.12Mpa, unit Reynolds number Re/L=3.7×107m-1, and Tw=(300±5)K.

Two shock generator angles β=10° and β=14° corresponding to flows with a moderate separation zone and a significant separation were investigated. The computational grid used in this study consists 241×121, providing in most flow regions well resolved results，the minimum normal spacing at the wall was 10-6m, yielding average y+ less than 0.1. The grid points are clustered around the incoming shock, in the boundary layer and around SWTBLI region. Similar computational domain and boundary conditions were used in Ref. 22.

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The prediction and the measurement of the surface parameters for β=10° are shown in Fig. 4. As expected, the stress limiter has a relatively great effect on surface parameters. However, the stress limiters Clim less than 1.0 have a detrimental effects in this case. The best result is provided by the Clim=1.0 calculation, which predicts the upstream interaction length and separation point correctly, even though overpredicts the downstream reattachment point and return somewhat lower skin friction values in the recovery region. Obviously, the Clim=0.85 and Clim=0.90 separate too far downstream and predict a excessively smaller separation than the measured. The Clim=0.0 calculation significantly underpredicts the upstream interaction length and gives a too steep pressure rise, and worse than that, no separation zone was predicted.

As shown, the agreement between experimental data and computed Stanton number is fair in the undisturbed turbulent region whereas deteriorates dramatically in the interaction region. The initial rise in heat transfer occurs at approximately x=325mm in the experiment, and all models predicte that rise too far downstream. Moreover, both the gradient and the magnitude of that rise are dramatically overpredicted for all stress limiters. Farther downstream of the reattachment, the computed heat transfer decrease to the levels measured in the experiment. With the increasing of the strength of the stress limiter by using a larger value of Clim, which results in the decrease of peaking heating. For the Clim=0.0 case, the peak Stanton number is as much as one and three quarters times the measured value. Even with the Clim increases to 1.0, the peak Stanton number is 50% higher than measured. It should be noted here that the assumption of constant turbulent Prandtl number has been recently questioned, relative to its effect on heat flux for SWTBLI.23 A variable turbulent Prandtl number model was shown to improve heat flux near reattachment. Moreover, it was show that there may be a significant influence of the unsteadiness of the shock system on the heat flux distribution.24

275 300 325 350 375 400 4250

3

6

9

Exp. C

lim=0.0

Clim

=0.85

Clim

=0.90

Clim

=0.925

Clim

=1.0

pw/p

f

X(mm) 275 300 325 350 375 400 4250

1

2

3

4

Exp. C

lim=0.0

Clim

=0.85

Clim

=0.90

Clim

=0.925

Clim

=1.0

St

X(mm) 275 300 325 350 375 400 425

-2

0

2

4

6

8

Exp. C

lim=0.0

Clim

=0.85

Clim

=0.90

Clim

=0.925

Clim

=1.0

Cfx

10

3

X(mm) Fig. 4 Surface properties for β=10 deg case: Left: wall pressure, Middle: Stanton number, Right: skin friction coefficient.

For the β=14 deg case, analogous conclusions can be drawn, although the difference between Clim =0.0 and Clim

=0.85 is more noticeable than the β=10 deg. There is an interesting phenomenon that increasing the Clim up to 1.05, the predicted upstream influence and separation point show a great improvement over Clim=1.0 case and agree well with the experiment, whereas the downstream pressure recovery and reattachment point make little changes.

275 300 325 350 375 400 4250

3

6

9

12

15

Exp. C

lim=0.0

Clim

=0.85

Clim

=0.90

Clim

=1.0

Clim

=1.05

p w/p

f

X(mm) 275 300 325 350 375 400 4250

2

4

6

8

Exp. C

lim=0.0

Clim

=0.85

Clim

=0.90

Clim

=1.0

Clim

=1.05

St

X(mm) 275 300 325 350 375 400 425

-2

0

2

4

6

8

10

12

Exp. C

lim=0.0

Clim

=0.85

Clim

=0.90

Clim

=1.0

Clim

=1.05

CfX

10

3

X(mm) Fig. 5 Surface properties for β=14 deg case: Left: wall pressure, Middle: Stanton number, Right: skin friction coefficient.

Figure 3. Test model for two-dimensional incident SWTBLI at Mach 5.

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Three-Dimensional Swept SWTBLI Flow at Mach 5 1. Experiment and numerical method

The experiment was also performed at the Ludwig Tube Facility at DLR21 and the freestream condition is same as the two-dimensional incident SWTBLI at Mach 5 as aforementioned. The test model used for the investigation of three-dimensional swept shock wave/boundary-layer interaction flows is shown in Fig. 6(a). It consists of an unswept sharp fin that acts as the shock generator, mounted on a flat plate. The fin may be mounted at different deflection angles β in the range from 2 to 27 deg. The β=23 deg was selected in this paper, which flowfield characterized with the secondary separation.

Clustering was applied close to the plate and fin for adequate resolution of the viscous effects. In Fig. 6(b) the mesh which was used is shown. In each crossflow plane (z, y directions) 121×101 points are used, while in the streamwise x direction there are 161 grid planes uniformly spaced, with Δx=0.389δ (δ=3.8mm). The inflow plane is located at a distance equal to 6δ upstream of the leading edge of the fin , and the downstream boundary at x=56.3δ. The height of the computational field is 26.3δ, and the width is large enough ensuring the computational domain laterally extended more than the interaction domain.

The walls are assumed impermeable, and no-slip boundary conditions are applied. The first point off the surface have a distance from the plate Δy =2.63e-4δ, and from the fin Δz =2.63e-3δ. The boundary layer profile upstream of the interaction region is used as boundary condition on the inflow plane and as initial condition for the whole flowfield. The inflow velocity profile is specified 6δ upstream of the fin leading edge and is determined from separated two dimensional calculations so that the momentum thickness (θ=0.157mm) of the incoming equilibrium boundary layer matches that reported in the experimen. At the outflow and top boundaries simple zero-gradient extrapolation boundary conditions are applied. The pressure gradient normal to the wall is taken to be zero. The temperature of the wall is assumed constant (isothermal condition) with Tw=295K.

X(mm)

0

50

100

150

200

Y(m

m)

0

50

100

Z(mm

)

0

50

100

150

200

250

300

X

Y

Z

3D Swept SWTBLI Grid1 cell out of 3

(a) Test model (b) Computational grid

Fig. 6 Test model and computational grid for three-dimensional swept SWTBLI flow at Mach 5. 2. Comparison with the experiment

The surface pressure distributions at several measurement planes are shown in Fig. 7. The local dip in pressure obtained by experiment indicate that there exits secondary flow separation, whereas which is not captured for all stress limiters and also does not predict the secondary shock. The pressure peak location coindent with the primary reattachment R1 was well predicted for all Clim selection, whereras the peak value of the pressure is underestimated by about 20%. It is interesting that the pressure distribution of different measurement planes between he primary attachment line R1 and peak valley location nearly collapse into one line for the Clim=0.0, Clim=0.85, and Clim=0.9, whereras the Clim=1.0 calculation show that the pressure distribution change a little. In general, the agreement with experiment deteriorates as the value of Clim decreases. With the Clim=0.0, the overall distribution of pressure is quite far from the measurements, and the Clim=1.0 calculation provides the best overall agreement with the experiment.

And it's worth pointing out that Thivet25 successfully predicted the secondary separation in the single sharp fin plate configurations using non-linear Eddy-Viscosity Models (EVM), whereras the linear EVM failed in this case. Menter also indicated that the inclusion of anisotropic effects for EVM should offer some potential for increased accuracy for secondary flows, as the transfer of momentum into the corner by secondary flows might be driven by the anisotropy of turbulence.10 The non-linear EVM could be considered for a better prediction of secondary flows.

1 cell out of 3

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0.4 0.6 0.8 1.0 1.2 1.40

3

6

9

12

X= 82mm Exp. X= 92mm Exp. X=152mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm Computation

pw/p

f

Z/X 0.4 0.6 0.8 1.0 1.2 1.4

0

3

6

9

12

X= 82mm Exp. X= 92mm Exp. X=152mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm Computation

p w/p

f

Z/X (a) Clim=0.0 (b) Clim=0.85

0.4 0.6 0.8 1.0 1.2 1.40

3

6

9

12

X= 82mm Exp. X= 92mm Exp. X=152mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm Computation

p w/p

f

Z/X 0.4 0.6 0.8 1.0 1.2 1.4

0

3

6

9

12

X= 82mm Exp. X= 92mm Exp. X=152mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm Computation

p w/p

f

Z/X (c) Clim=0.90 (d) Clim=1.0

Fig. 7 Surface pressure distribution for 23-deg single fin at different x positions

Figures 8a-8d show the skin friction coefficient Cf distribution at several measurement planes. In accordance with the surface pressure distribution, the surface skin friction prediction show that secondary separation did not exist in the computation. It is note that the maxima in the surface pressure, skin friction coefficient, and Stanton number distribution are located along the primary reattachment line R1 in all measurement planes. All stress limiters yield an underestimated by at least 50% of the maximum skin-friction coefficient whereas the location of peak was well predicted. As shown, for the Clim=1.0 case, the predicted Cf agree fairly well with the measured in the region between upstream influence line and secondary reattachment line R2 but the agreement deteriorates as Clim decreases. There is a discrepancy between the prediction and experiment in the region between the R1 and R2 for all stress limiter selection. As a whole, the best overall agreement with experiment was achieved by the Clim=1.0 calculation.

0.4 0.6 0.8 1.0 1.2 1.40.000

0.005

0.010

0.015

0.020

0.025

X=162mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm Computation

Cf

Z/X 0.4 0.6 0.8 1.0 1.2 1.4

0.000

0.005

0.010

0.015

0.020

0.025

X=162mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm Computation

Cf

Z/X (a) Clim=0.0 (b) Clim=0.85

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0.4 0.6 0.8 1.0 1.2 1.40.000

0.005

0.010

0.015

0.020

0.025

X=162mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm Computation

Cf

Z/X 0.4 0.6 0.8 1.0 1.2 1.4

0.000

0.005

0.010

0.015

0.020

0.025

X=162mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm Computation

Cf

Z/X (c) Clim=0.90 (d) Clim=1.0

Fig. 8 Surface skin drag coefficient distribution for 23-deg single fin at different x positions Figures 9a-9d show the Stanton number St distribution at several measurements planes. Both the experimental

and computational findings show that the surface pressure and skin friction distribution changes little at different measurement planes, whereas the St measured in the experiment decrease noticeably farther downstream of the leading edge. By contrast, the predicted St of different measurement planes nearly collapse into one line. Besides, in contrast with the skin friction coefficient prediction, all stress limiters selection yield an overestimated by at least 40% of the maximum Stanton number. For strong SWTBLI flows of high Mach numbers, temperature fluctuations are as important as velocity fluctuations, play a major role in determining the wall heat flux, and their effects must explicitly be taken into consideration for better prediction.23

0.4 0.6 0.8 1.0 1.2 1.40.000

0.002

0.004

0.006

X= 82mm Exp. X=102mm Exp. X=142mm Exp. X=162mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm ComputationS

t

Z/X 0.4 0.6 0.8 1.0 1.2 1.4

0.000

0.002

0.004

0.006

X= 82mm Exp. X=102mm Exp. X=142mm Exp. X=162mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm ComputationS

t

Z/X (a) Clim=0.0 (b) Clim=0.85

0.4 0.6 0.8 1.0 1.2 1.40.000

0.002

0.004

0.006

X= 82mm Exp. X=102mm Exp. X=142mm Exp. X=162mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm ComputationS

t

Z/X 0.4 0.6 0.8 1.0 1.2 1.4

0.000

0.002

0.004

0.006

X= 82mm Exp. X=102mm Exp. X=142mm Exp. X=162mm Exp. X=182mm Exp. X= 92mm Computation X=122mm Computation X=152mm Computation X=182mm ComputationS

t

Z/X (c) Clim=0.90 (d) Clim=1.0

Fig. 9 Surface Stanton number distribution for 23-deg single fin at different x positions

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V. Conclusion and Perspectives The goal of this work was to explore the impact of stress limiter strength for k-ω turbulence models in shock-

wave/turbulent boundary-layer interactions in supersonic and hypersonic flows. It has been observed in all simulations that the stress limiter strength has significant effect on the features of flowfield and wall surface parameters. However, there is no optimum stress limiter strength for covering the supersonic and hypersonic regimes. In other words, a constant value of stress limiter is not sufficient, and could be a function of the flow variables in some way or a specific scalar transport equation of stress limiter is necessary for better prediction of strong SWTBLI flows. A clear limitation of the present work is the assessment of the stress limiter with not enough validation cases. Further work is necessary to extend our work to more suitable cases for the validation.

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High-Order Method,” AIAA Journal, Vol. 39, No. 12, 2002, pp. 2283, 2292. 3Loginov, M. S., Adams, N. A., and Zheltovodov, A. A., “Large Eddy Simulation of Shock-Wave/Turbulent-Boundary-Layer

Interaction,” Journal of Fluid Mechanics, Vol. 565, Oct. 2006, pp. 135, 169. 4Wu, M., and Martin, M. P., “Direct Numerical Simulation of Supersonic Turbulent Boundary Layer over a Compression

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