Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1996A9618017, F49620-93-1-0005, DABT63-93-C-0064, AIAA Paper 96-0040

Computation of 3D asymmetric crossing shock wave/turbulent boundary layer interaction using a full Reynolds stress equation

turbulence model

Ge-Cheng ZhaRutgers Univ., Piscataway, NJ

Doyle KnightRutgers Univ., Piscataway, NJ

AIAA, Aerospace Sciences Meeting and Exhibit, 34th, Reno, NV, Jan. 15-18, 1996

The 3-D crossing shock wave-turbulent boundary layer interaction caused by an asymmetric 7 x 11 deg double fin was calculated by solving the Reynolds averaged Navier-Stokes equations with a full Reynolds stress equation turbulence model. An implicit approximate factorization method is used for the temporal integration. Roe’s scheme is used for evaluation of the convective terms of the mean flow and Reynolds stress equations with a third order MUSCL-type differencing. The computed surface pressure is in good agreement with the experiment. The computed heat transfer coefficient shows a modest improvement compared with the previous results obtained using the k-epsilon model with Chien’s low Reynolds number correction. Both computations of the heat transfer display significant deviations from the experiment. The tests of the grid refinement, different upstream boundary layer profiles, and different isothermal wall temperatures are also presented. (Author)

Page 1

AIAA-96-0040

Computation of 3D Asymmetric Crossing ShockWave/Turbulent Boundary Layer Interaction Using a Full

Reynolds Stress Equation Turbulence ModelGe-Cheng Zhaf and Doyle Knight*

Department of Mechanical and Aerospace EngineeringRutgers University - The State University of New Jersey

PO Box 909 • Piscataway NJ 08855-0909

Abstractt

The three dimensional crossing shock wave-turbulent boundary layer interaction caused byan asymmetric 7° x 11° double fin was calculatedby solving the Reynolds averaged Navier-Stokesequations with a full Reynolds Stress Equationturbulence model. An implicit approximate fac-torization method is used for the temporal in-tegration. Roe's scheme is used for evaluationof the convective terms of the mean flow and^Reynolds stress equations with a third orderMUSCL-type differencing. The computed sur-face pressure is in good agreement with the ex-periment. The computed heat transfer coefficientshows a modest improvement compared with theprevious results obtained using the k — e modelwith Chien's low Reynolds number correction.Both computations of the heat transfer displaysignificant deviations form the experiment. Thetests of the grid refinement, different upstreamboundary layer profiles and different isothermalwall temperature axe also presented.

1 Introduction

With the development of aerospace technology,the supersonic and hypersonic aerodynamic de-sign for aircraft attracts more attention. Even

fPostdoctoral Research Associate, Member AIAAJProfessor, Associate Fellow AIAA.Copyright ©1996 by G.-C. Zha and D. Knight. Pub-lished by the American Institute of Aeronautics andAstronautics, Inc. with permission.

though the Concorde supersonic transport andspace shuttle have been in use for more thana decade, many supersonic aerodynamic prob-lems are not well understood. An importantproblem is the shock wave/turbulent boundarylayer interaction which enhances the heat trans-fer, skin friction and reduces the energy of theflow. Good understanding and prediction of theshock wave/turbulent boundary layer interactionwill provide important information to aircraftdesigners. Knight recently reviewed the statusof the CFD research on three-dimensional shockwave/turbulent boundary layer interaction for afamily of geometries [1]. It is now generally un-derstood that the shock wave structure and pres-sure field can be accurately predicted in generalfor simple 3-D shock wave/turbulent boundarylayer interactions, e.g., sharp fin and swept com-pression corner. However, the heat transfer pre-diction remains a challenge. Typically, predictedheat transfer shows a large deviation from the ex-periment. Such disagreement is apparently dueto the inadequacy of the eddy viscosity conceptinherent in algebraic and k — c models. ReynoldsStress Equation (RSE) turbulence models arepromising candidates since they naturally incor-porate the non-local and history effects of theReynolds stress development. However, applica-tions of RSE models are mainly hi the researchfield of incompressible flow. RSE turbulent mod-els have begun to be used for compressible flowsrecently [2, 3,4, 5, 6]. However, most of the workis limited to 2D compressible flows and no workhas been done to apply an RSE model to a 3Dshock wave/turbulent boundary interaction.

Since the structure of shock wave-turbulent

boundary layer interactions for simple geometriesis generally well understood, attention has fo-cused in recent years on more complex 3-D shockwave-turbulent boundary layer interaction, in-cluding the crossing shock (double fin) interac-tion (Fig. 1). This configuration has been pro-posed as a possible hypersonic inlet [7, 8]. Vir-tually, all research to date has focused on thesymmetric crossing shock interaction [9, 10, 11,12, 13, 14, 15, 16]. Collaborative experimentaland computational research has elucidated thewave and streamline structure. Good agreementhas been observed between computed and ex-perimental surface pressure and flowfield profiles;however, computed heat transfer has been foundto typically overestimate the experimental data[1,10,17]. Various modifications to two equationturbulence models have been tested to improvethe prediction of heat transfer [16].

There are only a few studies of the asymmetriccrossing shock interaction.^ Garrison and Settles[18] [19] obtained surface flow visualization andplanar Laser Sheet images for a series of crossingshock configurations at Mach 3.9. Zheltovodovet al [19] obtained surface pressure, heat trans-fer and flow visualization for several asymmet-ric crossing shock interactions at Mach 3 and 4.Knight et al [20] computed two crossing shockinteractions at Mach 4 corresponding to exper-imental configurations of Garrison and Settles,and Zheltovodov et al, using Chien's k — € model.Good agreement was observed between computedand experimental surface pressure and flow visu-alization; however, surface heat transfer was overpredicted by 100% within the 3-D interaction re-gion.

The purpose of this paper is to study the asym-metric 3D crossing shock wave/turbulent bound-ary layer interaction with a full Reynolds StressEquation model of turbulence. The objectivesare twofold. First, we seek to evaluate the accu-racy of a full RSE model by comparison with theexperimental data of Zheltovodov et al [19] for aMaeh 3.95 asymmetric 7° x 11° crossing-stoek.The previous results of Knight et al using theChien's k — e model are also presented for com-parison with the full USE model. Second, weexamine the flowfield structure predicted by theRSE model, and offer some conjectures regard-

MtRes

6*9PrPrty+PuV

w

ing the enhancement of turbulence by the shockwave-boundary layer interaction.

2 Nomenclature

Symbol Definition__________________Moo Mach Number

Turbulent Mach NumberReynolds Number, Based onIncoming Boundary Layer ThicknessIncoming Boundary Layer ThicknessBoundary Layer Displacement ThicknessBoundary Layer Momentum ThicknessLaminar Prandtl NumberTurbulent Prandtl NumberDimensionless, Sublayer-Scaled, DistanceDensityX-velocityY-velocityZ-velocityPressureTemperatureTotal EnergyTotal EnthalpyEnthalpyTurbulence Kinetic EnergyDissipation Rate of Turbulence EnergyDilatation DissipationSolenoidal DissipationMolecular ViscosityViscous Stress TensorHeat-Flux VectorHeat Transfer CoefficientFriction VelocityAdiabatic Wall TemperatureTemperature at the WallDensity at the WallHeat Flux at the WallMolecular Viscosity at the WallShear Stress at the WallGrid Point Number in x-directionGrid Point Number in y-directionGrid Point Number in z-directionReynolds^ Stress'Tensor

PTeEhk

u*lawTwPw

NxNvN,

Governing EquationsTurbulence Modeling

and

The governing equations are the Reynolds av-eraged Navier- Stokes equations with the addi-tional equations for the Reynolds stresses, theturbulent dissipation rate and heat flux terms.The Reynolds-averaged Navier-Stokes equationsfor conservation of mass, momentum and energyare,

dtp + dkpiik = 0 (1)

(2)

+"— C'iP ~f* (sip { "~"/?tt* ttz. ""{~

t

dtpe+*dk(pe+p)iik =

"ufc -qk)

i Q I _/lf/"f/"'f*' -J- »/•'?•

/ 1——); ___\

where dt = d/dt, dk = d/dxk and the Einsteinsummation convention is employed. The overbardenotes a conventional Reynolds average, whilethe overtilde is used to denote the Favre massaverage. A double superscript represents fluc-tuations with respect to the Favre average, whilea single superscript ' stands for fluctuations re-spect to the Reynolds average.

In above equations, p is the mean density, Uiis the mass-averaged velocity, p is the mean pres-sure, and e is the mass-averaged total energy perunit mass. The following relations are employedto evaluate p and e:

e =

= pRf

+ 5 + k

(4)

(5)

where k is the mass-averaged turbulence kineticenergy

—r •*• n II /~ \pk = -pui ut (6)

The mean molecular viscous stress f is

23 n + + (7)

where ft = fi(f) is determined by the Sutherlandlaw. Similarly, the molecular heat flux is

(8)

where Pr is the molecular Prandtl number.

In energy equation, the triple velocity correla-tion v'Uu'k tai^- *ne velocity-molecular shearcorrelation u" TO, are considered small and are ne-glected [1].

To close the above equations, a turbulencemodel needs to be introduced to determine theReynolds stress —pu'-u'j and turbulent heat flux—CppT"u'l. A full Reynolds Stress Equation(RSE) turbulence model developed by Knight [21]is employed. The model is an extension of anincompressible flow RSE model to its compress-ible counterparts with some modification regard-ing the compressibility effects. The model is pre-sented below.

The equation for the Reynolds stress is,// // ,f Uj +

ij + Bij + Ca +

;/ /; _<*«,•«*

(9)

where the Aij, By, dj and Dij are the Reynoldsstress production term, diffusion term, pressure-rate of strain correlation and dissipation term.They are expressed as:

j II II n ~ II II n - ft n\ij = -pUjUkdkUi - pui ukdkuj (10)

o r= Ok{ -

n n // .U U +

.+

(11)

Dij = -

(12)

(13)

The production term of Reynolds stress requiresno further modeling since the Reynolds stress andmean velocity are dependent variables.

In the RSE model, the diffusion term is mod-eled as:

r\ f - Co0* { v [8

where v = fi/p and C^ is a constant. This modelis an extension of the incompressible flow modelof Launder, Reece and Rodi [22]

The correlation of the instantaneous pressureand fluctuating rate-of-strain is modeled as:

(15)

where CPI and C^ are constants. This is theextension of Rotta's model [23] for incompressibleflow.

An isotropic dissipation model with compress-ibility effect is used to determine the dissipationterm:

(16)2

'3'According to Sarkar et al [24] and Zeman [25]

pe = p(cg + ec)

where ec = CjfeeaMt2 and the turbulence Mach

number is Mt = \fzkf a. Even though this com-pressible flow effect for dissipation is incorpo-rated in the computer code, it is actually not usedand the Cfc = 0 because, for wall bounded flows,this effect is small and the above modification isalso questionable [26].

The conventional equation [27] is employed for

(17)

where Cei , Cg2 and Ct3 are constants.

The turbulent heat flux is modeled using a gra-dient diffusion hypothesis

(18)

where Ch is a constant.

Table 1 presents the values of the closure con-stants of the present RSE turbulence model [21].

Table 1: Turbulence Model Constants

QuantitycdlCP1OptCciCe3CC3

ckch

Value0.0864.3250.1791.011.800.100.00.0857

There are two ways to treat the flow field in thevicinity of the walls: one is to use wall functionsand the other is to use the low Reynolds numbercorrection to resolve the viscous sublayer. Thewall function method is chosen in this paper tosave computational tune. The asymptotic solu-tions of the above governing equations in the fullyturbulent region of a 2-D boundary layer are ob-tained in [28]. The asymptotic solutions (wallfunctions) are:

———r-r^-B

where

sin «* (logy+ + 5'K)

-arcsn ( . B „}] 1=0,W52+4AVJJ

(19)

A2 =

B = -Prt

B' = 5.0,

ic = 0.41,

. _ ! _ B + A^i = 0P «oo V«00/

(20)'00 /

p = Rpwfw (21)

kiso (fw - fwaiij + (1 - kiso) (qw - qwaii) = 0.(22)

1.5

y P* - OtlfVi* = 0

- a2fM = 0

= 0

= 0, (23)

(24)

(25)

(26)(27)

v = w = 0 (28)

where Prt = 0.9, «oo and T^ are reference veloc-ity and temperature, kiso = 1 means the isother-mal wall and the wall temperature is given whilekiso = 0 the wall heat flux is given. The con-

stantslowing:

, 0:2 , 0:3 and 0:4 are denned as the fol-

a3 = 0:2

•(<7 p l+2) (29)

(30)

(31)

— (32)

where a = |(ai + a2 + as).These wall functions are extended in a straight-

forward manner to 3-D flow under the assump-tion that the turbulent shear stress is locallyaligned with the direction of the mean flow ve-locity at the point of application of the wall func-tions. The correction to the wall functions dueto the pressure gradient [29] is omitted. The rel-ative contribution of the pressure gradient termsto the Reynolds shear stress at the point of appli-cation of the wall functions [29] is proportional toy and thus decreases with decreasing y. As indi-cated in Section 4.2, the computed flowfield wasfound to be insensitive to the y-value of the pointof application of the wall functions, and thus it isconcluded that the pressure gradient correctionto the wall functions would have a small effecton the computed flowfield.

4 Numerical Algorithm

An implicit finite volume approximate factoriza-tion method is used to solve the discretized equa-tions. The individual block matrix size is 12 x 12since all the governing equations are fully cou-pled. The implicit solver is slightly different fromthe standard AF method. The method suggestedby Vandromme [4] with the source term Jacobiansplit has been employed. This method was alsoused by Morrison et al.[5]. The formulations canbe written as:

Our numerical experiments have shown that theconvergence rates using the standard AF schemeand the source term Jacobian splitting schemeare almost exactly the same. But the source termsplitting scheme only calculates the source Jaco-bian S once and the CPU time per time step istherefore a little less than the one used by thestandard AF scheme. For the source term Ja-cobian, we only keep the diagonal elements ofthe original source term Jacobian and set the off-diagonal elements to be zero. This treatment en-hances the diagonal dominance and makes thecomputation more stable. To efficiently vectorizethis implicit solver, a matrix inversion computercode has been specially written for the 12 x 12matrix.

Roe's flux difference splitting scheme is em-ployed for the mean flow and Reynolds stressequation convective terms due to its ability toobtain high resolution shock waves and low nu-merical diffusion for viscous flows. The 3rd orderMUSCL-type differencing is used for these con-vective terms [30]. However, because central dif-ferencing is used for the viscous and turbulencediffusion and the source terms, the general accu-racy of the solution is second order. The tempo-ral discretization is first order to save CPU timebecause only the steady state solutions are of in-terest. First order accuracy is also used for theLHS implicit operators to save CPU time and toobtain diagonal dominance.

At the turbulent/non-turbulent interface, theturbulence variables have discontinuous deriva-tives as observed by other researchers [31] [29].The flux limiters, designed to achieve monotoneshock wave profiles for the higher order upwindschemes, play a very important role in captur-ing the turbulent/non-turbulent interface. The3rd order MUSCL scheme generates oscillationsat the interface and drives the turbulent kineticenergy and dissipation rate to be negative. Thecomputations have no way to be stable unlesssmooth limiters are used even though there is noshock wave in the flow fields [32]. A smooth lim-iter switches the higher order scheme to first or-der when the turbulent/non-turbulent interfaceis encountered and1 can nicely capture the crispprofiles. Two smooth limiters, namely Roe's Su-per Bee limiter and Minmod Limiter were testedwith the MUSCL-type reconstruction [33] [30][34], The numerical tests indicated that theMinmod limiter was better able to treat theturbulent/non-turbulent interface and thereforeis used in the present work. The computationusing Super Bee limiter was able to reduce theresidual one order of magnitude while the Min-mod was able to reduce the residual to machinezero. Having benefited from the TVD schemeused in the present computations, the variableprofiles captured at the turbulent/non-turbulentinterface are as sharp as those resolved by theboundary layer code computation which used amuch finer adaptive mesh [32].

5 Results and Discussion

To ensure the accuracy of the 3D computer code,several validation computations were performed.Computed supersonic laminar flat plate resultshave no distinguishable difference from the an-alytical Blasius solution [35]. This ensures thatthe code has small numerical dissipation. Forthe supersonic turbulent flat plate flow, excellentagreement was also obtained by comparing theresults with those from a separate computation ofthe compressible boundary layer equations withthe same ESE model [32]. This ensures that theturbulence model is correctly implemented.

The asymmetric double fin channel was calcu-

lated using the aforementioned algorithm. Thetwo fins have angles of 7° and 11°, respectively.The Mach number of the upstream flow is 3.95.The thin boundary layers on the fin surfaceswere not resolved, and slip boundary conditionswere applied at the fin surfaces. Previous stud-ies of the single fin interactions [36] have demon-strated that the bottom wall shock wave - tur-bulent boundary layer interaction is essentiallyunaffected by the boundary layer on the fin. Ofcourse, sufficiently far downstream of the inter-section of the crossing shocks, the shocks will in-teract with the turbulent boundary layers on thesidewall fin surfaces. However, in the presentstudy, nearly all of the experimental data isobtained upstream of the sidewall shock wave-turbulent boundary layer interactions, and con-sequently the omission of the fin boundary layersdoes not affect comparison with the experimentin this regard. Also, Bardina and Coakley [17]observed that the treatment of the fin boundarylayers as laminar/turbulent affected the flowfieldin the immediate vicinity of the fin/plate junctionfor a Mach 8.3 crossing shock interaction. How-ever, in the present study, experimental data isavailable only in the central portion of the flow,and thus our comparison with experiment is ex-pected to be unaffected by the use of slip bound-ary conditions on the fins.

A boundary layer equation code with the sameRSE turbulence model [21] was used to generatethe upstream variable profiles. Since it is gener-ally not possible for the computed upstream pro-files to exactly match all the experimental param-eters upstream of the intersection, the highestpriority was matching the experimental displace-ment thickness which is considered to have thebest measurement accuracy. A computationaltest is conducted to study the influence of thevariation of upstream profiles and the results aregiven in the last section. Table 2 presents theflow parameters for the computations.

5.1 Baseline Case

As the baseline case, the first computation forthis asymmetric double fin flow used a mesh sizeequal to 101x69x49 in the x-, y- and z-directions.The y+ of the first grid point adjacent to the wall

Table 2: Flow Conditions

Parameter Experiment ComputationMc

ReK>

Soo

3.953.033 xlO5

O"0 1.12 x 10~3

000

chlPta

Tto

(m)

^MPa)0°K

1.3 x7.9 x1.492260.4

io-4

io-4

3.953.033 x IO5

1.12 x 10~3

1.246 x IO-4

5.91 x IO-4

1.492260.4

is equal to 28.96 at the inflow boundary. A seriesof computations were carried out to test the de-pendence of the present RSE model on the gridrefinement, upstream profiles and wall tempera-ture. Those results are compared with this base-line case in the following sub-section. Table 3provides the mesh size information for the base-line case and grid refinement cases.

t

Table 3: Grid Sizes for Mesh Refinement Tests

Parameter Baseline in x iny in zNxNvNz

yfAj/iyi/Sooy 'max /&o<

Ay in/'

Aymai/Ase/tfooAzmjn/(

1016949296.21.0032615.51

Joo 0.007£00 1-16

0.5Joo 0.2

2016949296.21.0032615.510.0071.160.250.2

1011374920.92.64.0023715.510.0030.6210.50.2

1016997296.21.0032615.510.0071.160.50.1

0.4762 0.4762 0.4762 0.2881

Fig. 2 shows the experimental measurementlocations on the bottom surface of the channel.The location 1 is the throat middle line whichis along the streamwise direction. Location 2, 3,and 4 are in spanwise direction with the stream-wise position equal to 46 mm, 79 mm and 112mm. Fig. 3 to Fig. 6 are the pressure distribu-tions compared with the experiment [20] and theresults using Chien's k — e model [20] at those lo-cations. In Fig. 3, the computed pressures using

both the RSE and k — e model along the throatmiddle line agree well with the experiment exceptat the very end where there is a strong shock wavereflection from the sidewalls. Since the side wallis treated as inviscid, the computed reflected side-wall shock is displaced downstream of the exper-imental shock. This is the cause of the deviationbetween the computed and experimental pressurefor x > 140 mm. The sections at x=46 mm andx= 79 mm are located upstream of the inviscidintersection of the initial waves. The computedpressures agree closely with the experiment asshown in Fig. 4 and 5. At x = 112 mm which islocated after the shock wave intersection, thereis a pressure plateau in the experiment which isnot accurately represented in the present compu-tation (see Fig. 6). The interaction of the shockwaves with the turbulent boundary layer gener-ates two counter rotating vortices. The computedpressure "valley" in Fig. 6 corresponds to thevortex generated by the 11° fin. At the vortexpressure "valley" region, the k — e model pre-dicted the pressure a little better than the RSEmodel. The computed and experimental resultsshow that the sharp pressure gradient generatedby the shock waves in the inviscid flow field aregreatly eased through the shock wave/turbulentboundary layer interaction.

Fig. 7 shows the mean streamline structurewith the computed turbulent kinetic energy con-tours developing from the upstream turbulentboundary layer. The turbulent kinetic energyis normalized by the free stream mean flow ki-netic energy u^. The two bulges on the firstturbulent kinetic energy contours represent theinitial formation of the individual vortices gener-ated by the oblique shock wave/turbulent bound-ary layer interaction. These vortices converge toform a counter-rotating vortex pair. The shockwave/turbulent boundary layer interaction en-hances the computed turbulence intensity by in-creasing the maximum value of k about 3 timesfrom the channel entrance to exit. The turbulentregion is also enlarged from the boundary layerat the entrance to about 3.5 times higher at theexit. Such a flowfield structure may possibly beused for the design of supersonic ramjet enginefuel mixture system. The fuel may be injectedfrom the surface at the entrance and may be well

mixed for combustion at the exit due to the highturbulent intensity. These results are conjecturesand are not verified by the experiment. The totalpressure contours looks almost the same as Fig.7 but with the value decreasing from the entranceto exit [20].

Fig. 8 is an enlargement at a streamwise lo-cation after the intersection of the shock waves(x=112 mm). It demonstrates the detailed tur-bulent kinetic energy contours and the vortexlocations shown by the " streamlines"1 Thereare two counter-rotating vortices generated bythe shock wave/turbulent boundary layer inter-action. The right vortex generated by the 11° finis larger than the left one generated by the 7° fin.The counter rotating vortex pair has entrainedmost of the incoming wall boundary layer. Theturbulent region has a mushroom like shape andthe maximum turbulent intensity is predicted tobe midway between the two vortices. A sharpturbulent/non-turbulent interface is clearly seen.Detailed turbulence measurements are not avail-able to assess the accuracy of this prediction.

Fig. 9 and 10 are the computed heat transfercoefficient compared with the experiment and thecomputational results using Chien's k - e model.The heat transfer coefficient Ch is defined as [20]:

oCp (TW(X, Z) - Taw(x,Ch =

where qw(x,z) = —KwdT/dy is the wall heatflux. In the experiment, the Ch has a measure-ment accuracy uncertainty of ±10% to 15%. Twocomputations are needed to determine Ch, i.e.,an isothermal case and an adiabatic case. Forthe isothermal case, the wall temperature is setto 265°K and the local heat transfer qw(x,z) iscomputed. For the adiabatic case, the zero walltemperature gradient dT/dy = 0 is used as theboundary condition and the local adiabatic walltemperature Taw(x,z) is computed.

The Ch measurement locations are the same asthose for the pressure. Results at two represen-

1They are not, of course, streamlines in the proper sense.Rather, they represent the curves which are everywheretangent to the component of the velocity in the y-z plane.The purpose of presenting these curves is to emphasize thelocation of the vortices.

tative locations are presented. Along the throatmiddle line shown in Fig. 9, the computationalresults using k - e model and RSE model agreequite well with the experiment before the inter-section of the shock waves. Both of the com-putational results are substantially higher thanthe experiment after the intersection of the shockwave interacting with the turbulent boundarylayer. The maximum deviation of the RSE modelfrom the experiment is 80% and the k — € model is100%. The RSE model therefore has a 20% accu-racy improvement. Fig. 10 shows the heat trans-fer coefficient at the location x=112 mm whichis downstream of the intersection of the invis-cid shock waves. The whole profile computed byRSE model is lower than that computed by thek — e model and closer to the experiment. TheCh computed by RSE model has a maximum de-viation 39% from the experiment, which is 28%more accurate than the one computed by Jb — emodel. On the basis of these results, we may con-clude that the RSE turbulence model improvesthe prediction of the heat transfer for the in-tersecting shock waves/turbulent boundary layerinteraction. However, the deviation from the ex-periment remains significant.

Fig. 11 and 12 are the adiabatic wall temper-ature results. Both the RSE model and k — emodel predict the adiabatic wall temperaturewithin 5% deviation. Before the shock waveintersection,'the RSE model predicts an adia-batic wall temperature which is in closer agree-ment with the experiment. In the region of theshock wave/turbulent boundary layer interactionas shown in Fig. 12, the RSE model predicts anadiabatic wall temperature variation which is op-posite to the one obtained by k — e model.

The CPU time per time step for this baselinecase with the mesh size 101 X 69 x 49 is 113.8s onthe Cray C-90. Approximately 2000 time stepsare needed to get the converged solution. Thetotal CPU time for a converged solution is there-fore approximately 63 hours. The computer codeperformance is 160 MFlops.

5.2 Solution Convergence Tests

A series of computations axe conducted to inves-tigate the solution convergence of the baselinecase. There are five tests including grid refine-ment tests, different isothermal wall temperaturetest and different upstream boundary layer thick-ness test.

Because there are 12 equations to be solved forRSE turbulence model, the CPU time is fairlyintensive. The computation for the completelydoubled size mesh in every direction would bevery time consuming. The mesh refinement istherefore implemented one by one in each coor-dinate direction. Table 3 shows the mesh sizefor each refinement test. The following is the de-scription of each test:

1) Refinement test in x- direction: Thistest halves the mesh spacing in the streamwisedirection while the mesh sizes in the other direc-tions are kept the same. This test will show if thebaseline streamwise mesh resolution is adequateto capture the boundary layer .development.

2) Refinement test in y- direction: Thistest halves the mesh spacing in y direction. Theyf at the channel entrance is reduced from thebaseline case value yf = 29 to yf = 20.9. Thistest will show if the solution is sensitive to thedifferent yf values used.

3) Refinement test in z- direction: Thistest halves the mesh spacing in spanwise direc-tion. The refinements in y- and z- direction willshow the influence of the resolution of the shockwaves and vortices on the computational results.

4) Different upstream boundary layerprofiles test: As mentioned in the baseline case,it is difficult to match all the upstream exper-imental parameters simultaneously. Therefore,this test investigates the influence of the up-stream boundary layer. In this test case, theupstream boundary layer displacement thicknesswas reduced 45% from the baseline case, whichresulted in an increase in the upstream hsattransfer coefficient of 10%. This test will showif the variations in the upstream boundary layerprofiles significantly affect the solution.

5) Different isothermal wall temperature

test: In the experiment, the temperature on thebottom wall was not spatially uniform. Thereare not sufficient experimental data which canbe used to specify the wall temperature for thecomputation. The isothermal wall temperatureis therefore treated as constant in the computa-tion. This is based on the hypothesis that theheat transfer coefficient is independent of thewall temperature when the isothermal wall tem-perature is close to the adiabatic wall temper-ature. This test is to verify this hypothesis byusing a wall temperature which is different fromthe baseline case. The wall temperature is setto 270°.K" which is 5°K higher than the baselineone.

Fig. 13 displays representative results of theconvergence tests for the surface pressure alongthe throat middle line. The grid refinement inthe x-, y- and z-directions does not perceptivelychange the pressure distribution. Also, the gridrefinement changes the pressure distribution atx = 49, 79 and 112 mm (not shown) by lessthan 1%. The disagreement between the com-puted and experimental pressure at x = 112 mm(Fig. 6) may be attributable to differences in thestrength of the vortex generated by the 11° fin;however, measurements of the vortex strengthare unavailable.

The computed surface pressure is insensitiveto the upstream profile. The 45% reduction inthe upstream displacement thickness causes a7% change in the maximum, surface pressure onthe throat middle line. Also, the change in theisothermal wall temperature has a negligible ef-fect on the surface pressure.

Fig. 14 displays representative results of theconvergence tests for the heat transfer coefficientalong the throat middle line. The maximumchange in Ch is 4% for all convergence studies,which is less than the experimental uncertaintyof ±10% to 15%. In particular, it is noted that Ch.is insensistive to the location y\ at which the wallfunction boundary conditions are applied, and tothe incoming boundary layer dispiacment thick-ness 8*. Also, the Ch distribution changes by lessthan 2.5% at x = 46, 79 and 112 mm. Moreover,the computed Taw changes by less than 1.5% forall convergence studies on the throat middle line

and at x = 46, 79 and 112 mm.

To see the influence of y^ location on the tur-bulence variables, Fig. 15 and 16 display the pro-files of the dissipation rate per unit mass of turbu-lent kinetic energy € and a Reynolds stress tensorcomponent Rxx at two locations for the refine-ment test in y-direction. The two locations havethe same x and z coordinates which are equal to(x=42.875 mm, z=-0.182 mm) and (x=130.375mm and z=1.471 mm ) and close to the throatmiddle line (z=0.0). The two computations havethe same symbol style at the same location, onehollow and one solid. The hollow symbol repre-sents the refined mesh result and the solid onerepresents the baseline result. The first locationis upstream of the intersection of the two shockwaves and the second location is after the inter-section where the turbulence intensity is high.Fig. 15 shows excellent asymptotic agreement forthe € profiles for the two computations at eachlocation. In Fig. 16, RXXl shows the same be-havior at location one before the intersection ofthe two shock waves. At location two, the pro-files of the two computation differ by no morethan 11%. The profile difference is mainly in theouter part where there are two counter-rotatingvortices shown in Fig. 8. This disagreement maybe mainly because of the different mesh reso-lutions for the two vortices and turbulent/non-turbulent interface. Near the wall, the two pro-files still show close agreement. Fig. 15 and16 also predict that, at the two locations, theshock wave/turbulent boundary layer interactionenhances the maximum e value by the factor of3.5 and Rxx value by the factor of 7.5.

The computational cases above have similarconvergence rates even though the CFL numberfor each case is slightly different. Fig. 17 showstwo representative convergence histories, one forthe baseline case and the other for the mesh re-finement case in y- direction. The average L2norm residual are reduced three orders of magni-tude with 2000 steps and all the variables havelittle change after that. A local_CEL_numberequal to 1.5 is used for the baseline case and 2.0for the refined mesh case.

6 Conclusions

The three-dimensional shock wave-turbulentboundary layer interaction caused by an asym-metric 7° X 11° double fin was calculated us-ing the Reynolds averaged Navier-Stokes equa-tions with a full Reynolds Stress Equation model.An implicit approximate factorization method isemployed for the temporal integration. Roe'sscheme is used for evaluation of the convectiveterms with a third order MUSCL-type differenc-ing. Min-Mod TVD limiter is used and is es-sential to capture the irregular turbulent/non-turbulent interface to make the computation sta-ble. The computed surface pressure is in goodagreement with the experiment. The computedheat transfer coefficient achieved 20% improve-ment compared with the results from the k — emodel with Chien's low Reynolds number cor-rection. However, the maximum heat transfercoefficient still has a 80% deviation from theexperiment. The computation has resolved thecounter-rotating vortex pair generated by thecross shock wave/turbulent boundary layer inter-action, and predicts an enhanced turbulence in-tensity between the vortices. The test of mesh re-finements indicates that the solution is grid con-verged based on the mesh size. It also showsthat, within a proper range of y f , the flow solu-tions are basically insensitive to the location ofy f . Wall functions are used as boundary con-ditions to treat the solid walls. The test withreduced upstream boundary layer thickness indi-cates that the heat transfer coefficient is generallynot sensitive to the Reynolds number. The testwith different isothermal wall temperature showsthat the computed heat transfer is independent ofthe isothermal wall temperature when it is closeto the adiabatic wall temperature. The computercode with the RSE turbulence model is quite ro-bust.

7 Acknowledgments

This research was sponsored by AFOSR GrantF49620-93-1-0005 monitored by Dr. Len Sakell,and ARPA Contract DABT-63-93-C-0064 mon-itored by Dr. Bob Lucas (Hypercomputing and

10

Design (HPCD) Project). The content of the in-formation herein does not necessarily reflect theposition of the government and official endorse-ment should not be inferred. The authors ac-knowledge the beneficial collaboration with Dr.Alexander Zheltovodov and his Acolleagues atthe Institute of Theoretical and Applied Mechan-ics, Russian Academy of Sciences, Novosibirsk,Russia. The authors appreciate the helpful dis-cussion with Dr. Joseph Morrison in Analyt-ical Services & Materials, Inc.. The authorswould also like to thank Ms. Marianna Gnedinfor her help in generating the upstream turbu-lent boundary layer profiles using the bound-ary layer code and Mr. Ken Tsang and Mr.Patrick McPartland for their work for visualiz-ing the computed flow field. The computationshave been performed at the DoD Shared Re-source Center: Naval Oceanographic Office atStennis Space Center and at the DOD High Per-formance Computing Center USAE WaterwaysExperiment Station. Postprocessing has beenperformed at the Rutgers University Supercom-puter Remote Access Center.

References

[1] D. Knight, "Numerical Simulation of 3-d Shock Wave Turbulent Boundary LayerInteractions," Special Course on Shock-Wave/Boundary Layer Interactions in Su-personic and Hypersonic Flows, AGARDReport 192, pp. 3-1 to 3-32. Editor: G. De-grez, Aug. 1993.

[2] M. A. Leschziner, "Computation of Aero-dynamic Flows with Turbulence-TransportModel Based on Second-Moment Closure,"International Journal of Computer & Flu-ids, vol. 24, pp. 377-392,1995.

[3] F. Ladeinde, "Supersonic Flux-Split Proce-dure for Second Moments of Turbulence,"AIAA Journal, vol. 33, pp. 1185-1195,1995.

[4] D. Vandromme and H. H. Minh, "Aboutthe Coupling of Turbulence Cclosure Modelswith Averaged Navier-Stokes Equations,"JournaZ of Computational Physics, vol. 65,pp. 386^09,1986.

[5] J. H. Morrison, T. B. Gatski, T. P. Som-mer, H. S. Zhang, and R. So, "Evaluation ofa Near-Wall Turbulent Closure in Predict-ing Compressible Ramp Flows." Near WallTurbulent Flows, edited by R.M.C. So, C.G.Speziale and B. E. Launder, Elsevier, Ams-terdam, 1993.

[6] R. Abid, T. B. Gatski, and J. H. Morrison,"Assessment of Pressure-Strain Models inPredicting Compressible, Turbulent RampFlows," AIAA Journal, vol. 33, pp. 156-159,1995.

[7] C. Edward, "A Forebody Design Techniquefor Highly Integrated Bottom-MountedScramjets with Application to a HypersonicResearch Airplane." Tech. Rep. NASA TND-8369, Dec. 1976.

[8] L. Sakell, D. Knight, and A. Zheltovodov,"Proceedings of the AFOSR Workshop onFluid Dynamics of High Speed Inlets." Dept.of Mechanical and Aerospace Engineering,Rutgers University, May 1994.

[9] D. Gaitonde, J. Shang, and M. Visbal,"Structure of a Double-Fin Turbulent In-teraction at High Speed," AIAA Journal,vol. 33, pp. 193-200, 1995.

[10] N. Narayanswami, C. C. Horstman, andD. Knight, "Computation of Crossing ShockTurbulent Boundary Layer Interaction atMach 8.3," AIAA Journal, vol. 31, pp. 1369-1376,1993.

[11] N. Narayanswami, D. Knight, S. BogdonofF,and C. C. Horstman, "Interaction betweenCrossing Oblique Shocks and a TurbulentBoundary Layer," AIAA Journal, vol. 30,pp. 1945-1952,1992.

[12] N. Narayanswami, D. Knight, and C. C.Horstman, "Investigation of a HypersonicCrossing Shock Wave/Turbulent BoundaryLayer Interaction," Shock Waves, vol. 3,pp. 35-48,1993.

[13] T. Garrison, G. Settles, N. Narayanswami,and D. Knight, "Structure of Crossing-Shock wave/Turbulent Boundary Layer In-teraction." AIAA Paper 92-3670,1992.

11

[14] T. Garrison and G. Settles, "Interac-tion Strength and Model Geometry Ef-fects on the Structure of Crossing - ShockWave/Turbulent Boundary Layer Interac-tion." AIAA Paper 93-0780,1993.

[15] T. Garrison, G. Settles, N. Narayanswami,and D. Knight, "Comparison of FlowfieldSurveys and Computation of a Crossing -Shock Wave/Turbulent Boundary Layer In-teraction." AIAA Paper 94-2273,1994.

[16] J. E. Bardina and T. J. Coakley, "The Struc-ture of Intersecting Shock-Waves/TurbulentBoundary Layer Interaction flow." AIAAPaper 95-221$, 1995.

s

[17] J. E. Bardina and T. J. Coakley, "ThreeDimensional Navier-Stokes Simulations withTwo Equation Turbulence Models of Inter-secting Shock Waves/Turbulent BoundaryLayer at Mach 8.3." AIAA Paper 94-1905,1994.

[18] T. Garrison, "The Interaction betweenCrossing - Shock Waves and a TurbulentBoundary Layer." PhD Thesis, Dept. of Me-chanical Engineering, Penn State University,1994.

[19] A. - • - Zheltovodov,A. Maksimov, A. Shevchenko, A. Vorontsov,and D. Knight, "Experimental Study andComputational Comparison of Crossing •Shock Wave - Turbulent Boundary Layer In-teraction." Proceedings of the InternationalConference on Method of Aerophysical Re-search -Part 1, Russian Academy of Sci-ences, Siberian Division, Aug. 1994.

[20] D. Knight, T. Garrison, G. Settles, A. Zhel-tovodov, A. Maksimov, A. Shevchenko, andS. Vorontsov, "Asymmetric Crossing ShockWave-Turbulent Boundary Layer Interac-tion." AIAA Paper 95-0231,1995.

[21] M. Gnedin and D. Knight, "A ReynoldsStress-Equation Tmbulenee-Model for Com-pressible Flows. Part I: Flat Plate BoundaryLayers." AIAA Paper 95-0860,1995.

[22] B. Launder, G. Reece, and W. Rodi,"Progress in the Development of a Reynolds

Stress Turbulence Closure," J. Fluid Me-chanics, vol. 68, pp. 537-566, 1975.

[23] J. Rotta, "Recent Attempts to Develop aGenerally Applicable Calculation Methodfor Turbulence Shear Flows." AGARD CP-93, 1972.

[24] S. Sarkar, G. Erlebacher, M. Hussaini, andH. Kreiss, "The Analysis and Modellingof Dilatational Terms in Compressible Tur-bulence," J. Fluid Mechanics, vol. 227,pp. 473-493, 1991.

[25] 0. Zeman, "Dilatational Dissipation: TheConcept and Application in Modelling Com-pressible Mixing Layers," Physics of FluidA, pp.178-188,1990.

[26] P. Huang, G. N. Coleman, and P. Bradshaw,"Compressible Turbulent Channel Flow- AClose Look Uuing DNS Data." AIAA Paper95-0584, 1995.

[27] W. Jones and B. Launder, "The Predic-tion of Laminarization with a Two-EquationModel of Turbulence," Int. J. Heat and MassTransfer, vol. 15, pp. 301-304, 1972.

[28] M. Gnedin and D. Knight, "Compress-ible Turbulence Wall Layer Reynolds StressEquation Model Version No. 4." InternalReport No. 17, Dept. of Mechanical andAerospace Engineering, Rutgers University,1993.

[29] D. Wilcox, Turbulence Modelling for CFD.DCW Industries, Inc., 1993.

[30] W. Anderson, J. Thomas, and B. V. Leer,"Comparison of Finite Volume Flux VectorSplittings for the Euler Equations," AIAAJournal, vol. 24, pp. 1453-1460,1986.

[31] P. Safrman, "A Model for InhomogeneousTurbulent Flow." Proc. Roy. Soc., Lond.,Vol. A317, pp. 417-433,1970.

[32jG.-C.Sia and D. Knight, "SupersonicTurbulent Flat Plate Flow Calculation bySolving 3D Navier-Stokes Equations witha Reynolds Stress Equation Turbulence

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Model," Tech. Rep. 22, Department of Me-chanical and Aerospace Engineering, Rut-gers University, December 1994.

[33] P. Roe, "Some Contributions to the Mod-elling of Discontinuous Flows," Lectures inApplied Mathematics, vol. 22, pp. 163-93,1985.

[34] C. Hirsch, Numerical Computation of Inter-nal and External Flows, Volume 2. John Wi-ley & Sons Ltd., 1990.

[35] G.-C. Zha and D. Knight, "Supersonic FlatPlate Flow Tests for 3D RSE Code," Tech.Rep. 21, Department of Mechanical andAerospace Engineering, Rutgers University,June 1994.

[36] D. Knight, D. Badekas, C. Horstman, amG. Settles, "Quasiconical Flowfield Struc?ture of the Three-Dimensional Single Fin In-teraction," AIAA Journal, vol. 30, pp. 2809-2816,1992.

Fin Fin

Figure 1: The sketch of a double fin

Figure 2: 7° x 11° double fin measurement loca-tions: 1, 2, 3 and 4

13

6

5

4P / P~3

2

1

0

Figuremiddle

7

6

5

4P/P"3

2

1

°3

Figurethe cro

7

6

5

4P / P~3

2

1

0

-— ------- k-e Computation---- - - - - - - - Inviscid

Experiment f

^^0 50 100 150

X(mm)

3: Pressure distribution along theline

..................... ],,_£ Computation- - - - - - - - - - - Inviscid ,

Experiment

^

^"~1~r"~-*--!' '^^"^

200

throat

0 -20 -10 0 10 20 30Z - Z7ML (mm)

4: Bottom wall pressure distribution atss section x — 46 mm

- —————— RSE Computation.................... k.g Computation- - - - - - - - - - - Inviscid

• Experiment

S^~*~^^ y******^-^^J^

30 -20 -10 0 10 20 30

——————RSE Compulation——————k-e Computation— - - - - - - - - - Inviscid

Experiment

-20 -10 0 10 20 3CZ-Z-,., (mm)

Figure 6: Bottom wall pressure distribution atthe cross section x = 112 mm

C 312C 0.0150 0 0180 C 0210 0 Oi*

Figure 7: Turbulent kinetic energy contours withstreamlines at different locations

Figure 5: Bottom wall pressure distribution atthe cross section x = 79 mm

Figure 8: Turbulent kinetic energy contours withstreamlines at x=112 mm

14

0.005

0.004

0.003h

0.002

0.001

0.000

RSE computationk-e computationExperiment

50 100X(mm)

150

Figure 9: Heat transfer coefficient distributionalong the throat middle line

4.2 r

4.1 -4.0 -3.9^3.8:

3.73.63.5 -3.4:

3.3-3-?:o-

RSE Computationk-e ComputationExperiment

-20 -10 10 20(mm)

30

Figure 12: Adiabatic wall temperature distribu-tion at the cross section x = 112 mm

0.005

0.004

0.003

C*0.002

0.001

o.oo

— RSE computation•-- Experiment

k-e computation

%0 -20 -10 0 10 20 30

Figure 10: Heat transfer coefficient distributionat the cross section x = 112 mm

P/P

BaselineRefined in xRefined in yRefined in zReduced 8*Increased T.

50 100X(mm)

150

Figure 13: Pressure distribution along the throatmiddle line at the bottom wall

4.24.14.03.93 Q

t-1 37£-* o a

3.53.43.3

r

- ————— k-e computationExperiment

: ' ' " \'''7X___" " " "../ \s

i0 50 100 150

X (mm)

Figure 11: Adiabatic wall temperature distribu-tion along the throat middle line

0.005

0.004

0.003

0.002

0.001

0.000

BaselineRefined in xRefined in yRefined in zReduced 8*Increased T

50 100X(mm)

150

Figure 14: Heat transfer coefficient distributionalong the throat middle line

15

xltf

. y;-20.9, x-12.0 mm. Z-0.1S2 mm- y>20.t.ta130.4 mm. 1-1.471 mm

y!«2S,x«t30.4 mm, 2—1.471 mm

0.0 OK 1.0 1.1 2.0 IS**_

Figure 15: Turbulent dissipation rate profiles atdifferent locations.

R.X101

0.0 0.5 1.0 1.5 2.0

Figure 16: Reynolds stress tensor component Rxxprofiles at different locations.

10

10"4

a:

Grid: 101x68x49,Grid: 101x137x49,

500 1000 1500 2000 2500Time Step

Figure 17: Convergence histories of the baselinecase and refined mesh case

16

Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc. Fig. 7

Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc. Fig. 8