Investigation of Flow Dependent BlendingFunctions in Hybrid LES/RANS Simulations

X. Xiao∗, J. R. Edwards†, H. A. Hassan‡

North Carolina State University, Raleigh, NC

A blending function that depends on the ratio of the von Karman length scale andthe Taylor microscale in conjunction with k–ζ turbulence model is used to study twoMach 3 shock wave/boundary layer interactions on 20 degree compression ramp and 25degree compression-expansion ramp. The results are compared with two other blendingfunctions: one in which the von Karman length scale is replaced by the distance to thenearest wall while the other employs a grid dependent expression proportional to theratio of the RANS eddy viscosity to that of the LES. The results were further comparedwith a calculation where the underlying RANS model is that of Menter’s shear stress(SST) model with a blending function that depends on the distance to the wall. Allinflow boundaries employ the rescaling-reintroducing procedure developed by Xiao et al.for hybrid LES/RANS approaches. In general, the blending function based on the vonKarman length scale gives the best results when compared with available measurements.The skin friction measurements showed the highest sensitivity to the various blendingfunctions and the two RANS models employed.

Introduction

HYBRID Large-Eddy Simulation (LES) andReynolds-Averaged Navier-Stokes (RANS) simu-

lations have received increased attention lately becausethey provide a less expensive approach for studying un-steady turbulent flows where the geometry dependentlarge structures play a major role in dictating the be-havior of the flow. The bulk of the hybrid LES/RANSapproaches make use of two distinct concepts. Oneadvanced by Speziale1 in which the stress tensor is setas

τij = [1− exp (−β∆/Lk)]n τ(R)ij (1)

where Lk is the Kolmogorov’s length scale, ∆ is arepresentative grid size, τ

(R)ij is the Reynolds stress

provided by a RANS model, and β and n are modelconstants. The other concept is that of the Detached-Eddy Simulation (DES) of Spalart et al.2 In thisconcept, the distance to the nearest wall, d, in theSpalart-Allmaras3 one-equation model is replaced bya new length scale

d = min (d, CDES∆) (2)

where CDES is a model constant.Above approaches can lead to good results for flows

where there is a clear boundary between free shear re-gions and wall-bounded regions. In situations whereembedded circulation zones exist in the flow, such assupersonic ramp flows, the choice of the grid mayallow for RANS regions to be embedded into LES re-gions and vice-versa. To avoid such situations, our

∗Post-Doctorial Research Associate, Member AIAA.†Associate Professor, Senior Member AIAA.‡Professor, Associate Fellow AIAA.

approach emphasizes a zonal approach where certainregions of the flow are handled as either LES or RANSas dictated by the flow variables.4–7 In the earlierwork,4,5 a blending similar to that used in Menter’sshear stress (SST) model8 was employed. A successfulimplementation of this approach was demonstrated fora ramp-cavity configuration of Settles et al.9 Applica-tion of the same approach to a Mach 3 flow over 20 degramp compression corner10 resulted in poor agreementwith experiment. As is shown in Ref. 6, this turnedout to be an inflow boundary condition issue and not ablending issue, and when appropriate inflow boundaryconditions were employed, good results were obtainedin Ref. 6. Similar results were obtained for the 25 degcompression-expansion ramp of Zheltovodov et al.11

using the k–ζ RANS model and a different blendingfunction which depended on the grid but not explic-itly on the distance from the wall.

The results of any hybrid LES/RANS model are sen-sitive to the filter width and, as such, some dependenceon grid structure and refinement is unavoidable. Thus,there are two main objectives for this investigation.The first is to use a blending function that is explic-itly independent of the grid and geometry and comparethe results with blending functions that are either gridor geometry dependent. The second is to examine therole of the underlying RANS models. The bulk of thecalculations are carried out using the k–ζ turbulencemodel.12 These are compared with calculations usingMenter’s SST model with a blending function that isgeometry dependent. All calculations employ inflowconditions based on the rescaling-reintroducing proce-dure developed by Xiao et al.7

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ApproachBlending Functions

Three different hybrid blending functions are con-sidered in conjunction with the k–ζ RANS model. Allthree functions shift the closure from a RANS modelnext to the wall to a one-equation subgrid model awayfrom the wall. The first to be considered is the one thatis explicitly independent of the grid and the geometryand is written as

Γvk = tanh(

lvk

α1λ

)2

(3)

where

lvk =S

|∇S|, λ =

√k/ζ,

S =√

SijSij , Sij =12

(∂ui

∂xj+

∂uj

∂xi

), (4)

k is the turbulence kinetic energy per unit mass or thevariance of velocity, ζ the enstrophy or the variance ofvorticity and α1 a model constant. The length scalelvk is referred to as the von Karman length scale and isequal to the distance normal to the wall in the log-lawregion. The other length scale, λ, is proportional tothe Taylor microscale which is the characteristic eddysize in the inertial subrange. The inertial subrangeis the region that lies between the energy containingrange representative of the large scale and the viscousrange representative of the fine scales

The second blending function is a variation on theexpression given in Eq.(3) where lvk is replaced by d,the distance from the nearest wall. Thus

Γd = tanh(

d

α1λ

)2

(5)

Note that lvk is quite different from d outside the loglayer.

The third blending function is the grid dependentfunction used in Ref. 7. Thus

Γ∆ = tanh(

lεα2∆

)2

(6)

where

lε =k3/2

νζ,∆ = (Vcell)

1/3, (7)

ν is the kinematic viscosity and α2 is a model constant.The RANS and LES eddy viscosities are chosen as

νt,R = Cµk2

νζ , νt,L = Cs

√k∆, (8)

Cµ = 0.09, Cs = 0.01

Using Eq.(8), it can be seen that

lε/∆ ∼ νt,R/νt,L (9)

When using Menter’s RANS model, the followingblending function is employed6

Γω = 1− tanh η4

η = max(τ1,τ2)ω , τ1 = 500ν

d2 , τ2 =√

kCµd (10)

It was found in Ref. 7 that Γ must be a non-decreasing function as the distance from the nearestwall increases. This requirement was implemented forΓvk and Γ∆ by choosing

Γj = max(Γj ,Γj−1) (11)

where j is the index of the grid cell in the normal direc-tion. For complex shapes, it may be necessary to keeptrack of the distance from the nearest wall. This re-quirement introduce a degree of geometry dependenceon the formulation. For the other two cases consid-ered here, where d appears explicitly in the blendingfunction, the resulting Γ’s are non-decreasing.

The k-equation

For the k-ζ model, the k-equation for the hybridLES/RANS scheme can be written as7

∂ρk

∂t+

∂ρujk

∂xj=

∂

∂xj

[(µ

3+

µt

σk

)∂k

∂xj

]+ τij

∂ui

∂xj

−(1− Γ)(

1Ck

µt

ρ2

∂ρ

∂xi

∂P

∂xi+ C1

ρk

τρ+ µζ

)−ΓCdρ

k3/2

∆(12)

whereµt = (1− Γ)µt,R + Γµt,L (13)

For the Menter’s k–ω model, k-equation is6

∂ρk

∂t+

∂ρujk

∂xj= Pk − (1− Γ)β∗ρωk +

ΓCdρk3/2/∆ +∂

∂xj

[(µ + σkµt)

∂k

∂xj

](14)

with Pk being the production term

Pk = µt

[(∂ui

∂xj+

∂uj

∂xi

)∂ui

∂xj− 2

3

(∂uk

∂xk

)2]

(15)

When production balances dissipation in the LESregion, Eqs.(12) and (14) yield a Smagorinsky-typesubgrid eddy viscosity

νt = Cs

√Cs

CdS∆2 (16)

Both Cs and Cd are chosen as 0.01.Note that when using k–ζ model, the ζ-equation

plays no role in the LES region. On the other hand,the ω-equation will have a role. However, the influenceof eddy viscosity on ω is limited to the diffusion term.Thus, ω is weakly influenced by the eddy viscosity asthe model shifts from RANS to LES.

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Model Constants

The model constants α1 and α2 were chosen so asΓ achieves a value of 0.5 somewhere in the log-law re-gion. To meet this requirement the following valueswere adopted: α1 = 25 and α2 = 5. In this regard, theraging controversy concerning the validity of the clas-sical log-law that describes the overlap region and theneed to replace it by a power law can be put to gooduse. As has been indicated by Buschmann and Gad-el-Hak,13 part of the overlap region is well represented byeither a log-law or a power-law (see Fig. 7 of Ref. 13).This common region, however, is Reynolds number de-pendent. Thus, if the blending function is distancedependent, it is recommended that y+ be chosen sothat Γ = 1/2 in this region. The choice of α1 = 25 isconsistent with this suggestion.

Starting Solution

It has not been possible to start from a RANS so-lution based on the k–ζ RANS model to arrive at anLES/RANS solution for the two cases under consider-ation. The reduced eddy viscosity resulting from theLES component results in an unrestricted growth ofthe separated region that will eventually contaminatethe recycle plane and the inflow boundary condition.To avoid this problem, the starting procedure devel-oped in Ref. 7 is used for all calculations involving thek–ζ model. In this procedure, a hybrid LES/RANSsolution for the flat plate is generated separately whilethe region downstream of the corner is initialized bythe RANS solution. The flat plate solution does notneed to have reached a statistically-steady solutionsince the flat plate solution will be evolved further aspart of the computational domain. All that is neces-sary is to generate structural content so as to preventthe contamination of the recycle plane.

Numerical Procedure

A second order ENO (Essentially Non-Oscillatory)upwinding method based on the low-diffusion flux-splitting scheme of Edwards14 is used to discretizethe inviscid fluxes while central differences are em-ployed for the viscous and diffusion terms. A pla-nar relaxation subiteration procedure, which results insecond-order temporal accuracy, is used to integratethe system. The code is parallelized using domaindecomposition and message passing interface (MPI)strategies and is optimized for the IBM SP3 architec-ture.

Results and Discussion20 Degree Compression Ramp

A schematic of the experimental setup is shown inFig.1, which depicts a 20 cm flat plate connected toa 20 deg ramp. The initial conditions are given atx = −3.81 cm and are summarized in Table.1. Therecycle plane is located at x = −10.3 cm. To facilitate

comparisons with the results of Ref. 6, an identical gridconsisting of 449× 129× 69 nodes in the flow, normaland span directions is employed. The span size is set as2.9δ, where δ is the boundary layer thickness at inflowboundary. Equal mesh spacing is used in x- and z-directions, while the mesh spacing in the y directionis refined near the wall with y+ < 1 at the first cellcenter off the wall. The computational domain hasbeen decomposed into 14× 3× 2 blocks.

Table 1 20 Ramp Inflow Condition @ x=-3.81 cm

M∞ 2.79P∞(Pa) 26001T∞(K) 100.8

ρ∞(kg/m3) 0.77δ(cm) 2.7

Adiabatic and non-slip boundary conditions are em-ployed at the wall and extrapolation conditions areused at the outflow boundaries. Periodic boundaryconditions are employed in the z-direction. As indi-cated earlier, the recycling-reintroducing procedure ofRef. 7 was implemented at the inflow boundary for allcases considered.

The time step employed is 1.2 × 10−7 seconds.The flow reaches a statistically-steady condition af-ter about 7 characteristic times. Additional runs forabout 4.6 characteristic times were carried out to col-lect statistics. The initial solution for all cases is shownin Fig. 2.

A typical velocity contour is shown in Fig.3. Fig-ures 4 to 6 compare the wall pressure, skin frictionand velocity profiles for blendings Γvk, Γd, k–ζ and ex-periment. Figure 4 shows that two blending functionsyield identical results for wall pressure. Both yield alower pressure in the separated region, indicating thatthe initial displacement effect of the separation bubbleis not as pronunced as that in the RANS calcula-tion. Overall, however, the RANS solution is in betteragreement with experiment when comparing wall pres-sure and skin-friction. The Γvk blending results inlower overall skin friction values than the Γd blending.This leads to better predictions downstream of reat-tachment but worse upstream of separation. Theseconclusions are reversed when comparing velocity pro-files. Both hybrid schemes are in better agreementwith experiment, especially in the recovery region.

Figures 7–9 compare results of Γvk and Γ∆ blend-ing with k–ζ and experiment. As may be seen fromFigs. 7 and 8, Γvk blending gives better agreement withexperiment than Γ∆ does. Figure 9 shows a similarconclusion in the separated region. However, equallygood agreement is indicated in the recovery region.Thus, Γvk blending appears to yield the best hybridscheme.

In the above comparisons, non-decreasing blendingfunctions, one explicitly independent of both geome-try and grid, the other grid-dependent, were compared

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Table 2 25 Ramp Inflow Condition @ x=-3.3 cm

M∞ 2.88P∞(Pa) 11956T∞(K) 114.8

ρ∞(kg/m3) 0.36Re/meter 3.24× 107

θ(mm) 0.3δ(mm) 4.1

with a blending that is geometry-dependent and in-creases with the distance from the nearest wall. Thenext comparison involves two calculations that usedifferent geometry-dependent blending functions anddifferent RANS models. Figures 10–12 compare hy-brid results using Γd and Γω blending of Fan et al.6

with experiment. It is seen that the Γd blending is inbetter agreement with experiment for all flow proper-ties indicated in the figures. This may be a result ofthe fact that Γω blending results in a wider separationbubble and a higher pressire in the separated region.It is difficult to pinpoint the exact reason why the Γd

blending is in better agreement with the Γω scheme. Inall probablity, this is a result of the underlying RANSschemes as the Menter’s SST method used in Ref. 6has a tendency in overpredicting axial separation.

25 Degree Compression-Expansion Ramp

The schematic of the experiment together with therelevant dimensions is shown in Fig.13. The inflowconditions at x = −3.3 cm are presented in Table 2.To facilitate comparison with Ref. 7, the grid employedis the fine grid used there, which consists of 481×145×65 nodes in the flow, normal and spanwise directionswith the grid in the z-direction being 2.9δ. The entiredomain is decomposed into 12× 4× 2 blocks.

The time steps employed is 6 × 10−8 seconds. Theflow reached statistically steady conditions after 9.2characteristic times. Additional runs of about 5 char-acteristic times were carried out to collect statistics.Boundary and initial conditions were handled in amanner similar to that of the 20 degree compressionramp.

Because Γvk and Γd blendings gave similar resultsfor the 20 degree ramp, only comparisons involvingΓvk and Γ∆ are shown in Figs. 14–16. Both hybridschemes indicate a lower pressure in the separated re-gion similar to the 20 degree ramp. The two schemesand RANS predict the extent of upstream separationwell. In general, the Γvk scheme yields pressure andskin friction distributions that are in better agreementwith experiment than the Γ∆ scheme. However, nei-ther scheme gives as good agreement with experimentas the RANS scheme.

Similar conclusions hold when considering velocityprofiles in the separated region, Fig.12(a). Note thatthis is different from the 20 degree compression rampresults. In Fig.12, y′ = y − ywall and x′ is measured

from the compression corner. The experimental datado not indicate negative axial profiles but the “cusp”may indicate the edge of the backflow region. The Γvk

scheme gives better agreement with experiment thanthe Γ∆ scheme, but the k–ζ scheme gave better predic-tions than both. The situation is, however, reversed inthe recovery region, where both hybrid schemes giveexcellent agreement with experiment as is indicated inFig.12(b).

Concluding RemarksResults of three blending schemes and two RANS

models are compared with experiments for a 20 de-gree compression ramp and a 25 degree compression-expansion ramp. In general, the scheme that is ex-plicitly independent of both geometry and grid sizeperformed the best while the scheme in which the vonKarman length scale is replaced by the closest distanceto the wall performed nearly as well. Although the k–ζscheme compared better with experiment in predict-ing wall pressure, skin friction coefficient and velocityin the separated region for the 25 degree compression-expansion ramp, it did not do as well for the 20 degreecompression ramp. In addition, the Γd blending usingthe k–ζ model performed better than the Γω blend-ing using Menter’s SST model. Since the two blendingfunctions are somewhat comparable, the difference, inall probability, is a result of differences in the twoRANS models.

A number of lessons can be drawn from this inves-tigation. It is not clear why a certain hybrid modelperforms better when compared with one set of exper-iment than the other. This suggests that the quality ofexperimental data and the manner in which the datawas reduced were not the same. The fact that dif-ferent blending functions and different RANS modelsgave different results suggests that additional work isneeded to sort out the differences.

The fact remains, however, that without a coordi-nated computational and experimental effort, many ofthese issues will remain unresolved.

AcknowledgmentIBM SP3 computer resources is provided by a grant

from the North Carolina Supercomputing Center.

References1Speziale, C. G., “Turbulence Modeling for Time-Dependent

RANS and VLES: A Review,” AIAA Journal , Vol. 36, No. 2,February 1998, pp. 173–184.

2Spalart, P. R., Jou, W.-H., Strelets, M., and Allmaras,S., “Comments on the Feasibility of LES for Wings and on aHybrid RANS/LES Approach,” Advances in DNS/LES , editedby C. Liu and Z. Liu, First AFOSR Int. Conf. on DNS/LES,Aug 4-8, Ruston, LA., Greyden Press, Columbus, OH, 1997,pp. 137–147.

3Spalart, P. R. and Allmaras, S. R., “A One-EquationTurbulence Model for Aerodynamic Flows,” La RechercheAerospatiale, , No. 1, 1994, pp. 5–21.

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4Fan, T. C., Tian, M., Edwards, J. R., Hassan, H. A.,and Baurle, R. A., “Validation of a Hybrid Reynolds-Averaged/Large-Eddy Simulation Method for Simulating Cav-ity Flameholder Configuration,” AIAA Paper 2001-2929, June2001.

5Fan, T. C., Xiao, X., Edwards, J. R., Hassan, H. A., andBaurle, R. A., “Hybrid LES/RANS Simulation of a shock Wave/ Boundary Layer Interaction,” AIAA Paper 2002-0431, Jan2001.

6Fan, T., Xiao, X., Edwards, J., Hassan, H., and Baurle,R., “Hybrid LES/RANS Simulation of a Mach 3 Shock Wave /Boundary Layer Interaction,” AIAA Paper 2003-0080, Jan 2003.

7Xiao, X., Edwards, J. R., and Hassan, H. A., “InflowBoundary Conditions for LES/RANS Simulations with Appli-cation to Shock Wave/ Boundary Layer Interactions,” AIAAPaper 2003-0079, Jan 2003.

8Mentor, F. R., “Two Equation Eddy Viscosity TurbulenceModels for Engineering Applications,” AIAA Journal , Vol. 32,No. 8, 1994, pp. 1598–1605.

9Settles, G. S., Williams, D. R., Baca, B. K., and Bogdonoff,S. M., “Reattachment of a Compressible Turbulent Free ShearLayer,” AIAA Journal , Vol. 20, No. 1, Jan 1982, pp. 60–67.

10Settles, G. S. and Dodson, L. J., “HypersonicShock/Boundary-Layer Interaction Database,” NASA Contrac-tor Report 177577, April 1991.

11Zheltovodov, A. A., Zaulichnii, E. G., Trofimov, V. M., andYakolev, V. N., “Heat Transfer and Turbulence Study in Com-pressible Separated Flows,” Preprint 22-87, Inst. Theoreticaland Applied Mechanics, Russian Acad. of Sciences, Novosobirsk,1987.

12Robinson, D. F. and Hassan, H. A., “Further Developmentof the k-ζ(Enstrophy) Turbulence Closure Model,” AIAA Jour-nal , Vol. 36, No. 10, 1998, pp. 1825–1833.

13Buschmann, M. H. and Gad-el Hak, M., “Debate Concern-ing the Mean-Velocity Profile of a Turbulent Boundary Layer,”AIAA Journal , Vol. 41, No. 4, 2003, pp. 5650–572.

14Edwards, J. R., “A Low Diffusion Flux Splitting Schemefor Navier-Stokes Calculations,” Computers & Fluids, Vol. 26,No. 6, 1997, pp. 635–659.

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x, m

y,m

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15

0

0.05

0.1

0.15

0.2

outflow boundary

compression ramp

inflow boundary

flat plate

recycle-plane

Fig. 1 20 deg compression ramp

X (m)

y(m

)

-0.2 -0.1 0 0.10

0.05

0.1

0.15

0.2

LES/RANS

RANS

Fig. 2 A combined initial solution: contour plot of T

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x, m

y,m

-0.2 -0.1 0 0.1

0.05

0.1

0.15

0.2

u: -120 -35 50 135 220 305 390 475 560

Fig. 3 Instantaneous streamwise velocity (u) contours

x’, cm

p/p ∞

-8 -4 0 4 8 12 160.5

1

1.5

2

2.5

3

3.5

4

EXPRANS (k-ζ)LES/RANS (Γd)LES/RANS (Γvk)

Fig. 4 Wall pressure distribution

x’, cm

Cf

-10 0 10

-0.0005

0

0.0005

0.001

0.0015

0.002

EXPRANS (k-ζ)LES/RANS (Γd)LES/RANS (Γvk)

Fig. 5 Skin friction distribution

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U/Ue

y’,c

m

0

0.5

1

1.5

2

2.5

3

3.5

EXPRANS (k-ζ)LES/RANS (Γd)LES/RANS (Γvk)

x’, cm=-3.81 -1.11 0.0 0.397

1 11 10

a) Separated region

U/Ue

y’,c

m

0

0.5

1

1.5

2

2.5

3

3.5

4x’, cm= 1.27 2.54 4.13 7.62

111 1

11.43

10

b) Recovery region

Fig. 6 Comparison of velocity profile, Γvk vs Γd

x’, cm

p/p ∞

-8 -4 0 4 8 12 160.5

1

1.5

2

2.5

3

3.5

4

EXPRANS (k-ζ)LES/RANS (Γ∆)LES/RANS (Γvk)

Fig. 7 Wall pressure distribution

x’, cm

Cf

-10 0 10

-0.0005

0

0.0005

0.001

0.0015

0.002

EXPRANS (k-ζ)LES/RANS (Γ∆)LES/RANS (Γvk)

Fig. 8 Skin friction distribution

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U/Ue

y’,c

m

0

0.5

1

1.5

2

2.5

3

3.5

EXPRANS (k-ζ)LES/RANS (Γ∆)LES/RANS (Γvk)x’, cm=-3.81 -1.11 0.0 0.397

1 11 10

a) Separated region

U/Ue

y’,c

m

0

0.5

1

1.5

2

2.5

3

3.5

4x’, cm= 1.27 2.54 4.13 7.62

111 1

11.43

10

b) Recovery region

Fig. 9 Comparison of velocity profile , Γvk vs Γ∆

x’, cm

p/p ∞

-8 -4 0 4 8 12 160.5

1

1.5

2

2.5

3

3.5

4EXPLES/RANS (Γω, Menter’s SST)LES/RANS (Γd, k-ζ)

Fig. 10 Wall pressure distribution

x’, cm

Cf

-10 0 10

-0.0005

0

0.0005

0.001

0.0015

0.002

EXPLES/RANS (Γω, Menter’s SST)LES/RANS (Γd, k-ζ)

Fig. 11 Skin friction distribution

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U/Ue

y’,c

m

0

0.5

1

1.5

2

2.5

3

3.5

EXPLES/RANS (Γω, Menter’s SST)LES/RANS (Γd, k-ζ)

x’, cm=-3.81 -1.11 0.0 0.397

1 11 10

a) Separated region

U/Ue

y’,c

m

0

0.5

1

1.5

2

2.5

3

3.5

4x’, cm= 1.27 2.54 4.13 7.62

111 1

11.43

10

b) Recovery region

Fig. 12 Comparison of velocity profile, Γd vs Γω

x, m

y,m

-0.05 0 0.050

0.01

0.02

0.03

0.04

0.05

0.06

0.07

outflow boundary

compression ramp

inflow boundary

flat plate

recycle-plane

15m

m

25o

Fig. 13 Zheltovodov 25 deg compression-expansion ramp

x’, cm

p w/p

∞

-5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

EXPRANS (k-ζ)LES/RANS (Γ∆)LES/RANS (Γvk)

Fig. 14 Wall pressure distribution, 25 deg ramp

x’, cm

Cf

-5 0 5-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008EXPRANS (k-ζ)LES/RANS (Γ∆)LES/RANS (Γvk)

Fig. 15 Skin friction distribution, 25 deg ramp

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u/ue

y’,c

m

0

0.25

0.5

0.75

ExpRANS (k-ζ)LES/RANS (Γ∆)LES/RANS (Γvk)

x’ =-0.8 cm -0.5 cm 0.5 cm

1 1 10

a) Separated region

u/ue

y’,c

m

0

0.1

0.2

0.3

0.4

0.5

0.6

x’ = 1.25 cm 2.35 cm 3.10 cm

1 1 10

b) Recovery region

Fig. 16 Comparison of velocity profile, Γvk vs Γ∆, 25 deg ramp

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