VOL. 22, NO. 7, JUL Y 1984 AIAA JOURNAL 921

M. Visbal* and D. KnighttRUlgers UniversilY, New Brunswick, New Jersey

The algebraic lurbulenl edd)' viscosily model of Baldwin and Lomax has been crilically examined for lhe caseof Iwo-dimensional (2-D) supersonic compression corner inleraclions. The flowfields are compuled usin)! lheNavier-Slokes equalions logelher with three differenl nrsions of lhe Baldwin-Lomax model, induding lheincorporation of a relaxation lechnique. The lurbulence models are nalualed b)' a delailed comparison wilhav'ailable expe~imental data for comprcssion ramp flows m'er a range of corner angle and Re)'nolds numher. TheBaldwin-Lomax outer formulalion is found 10 be unsuitable for separated 2-D supersonic inleractions due l;u;e

unphysical streamwise ,'ariation of lhe compuled len th scale in lhe v'iclnll)' orSelJaration. Minor modifications~p!QlJose to p..!Irtiall)'re~e~)' Ihi~ diff!cull)'. The use of relaxation prov'ides significant impro\ement in the

flowfield prediction upstream of the corner. However, the relaxalion length required is one-lenth of that em­ployed in a prnious computalional stud)'. Ali of lhe turbulence models lesled here Jail 10 simulale lhe rapid

recovery of lhe boundaf)'la)~ do_wn!~a~ r~t1achmenl.

,..yf

The Baldwin-Lomax Turbulence Modelfor Two-Dimensional Shock-Wave/Boundary-Layer Interactions ,

)v'\A R0'\1

~

I. Introduction

THE numerical solution of the Navier-Stokes equationsfor complex aerodynamic nows is now possible as a resultof increases in computer capability, the development of ef­ficient numericaJ algorithms,I.2 and the recent advances ingrid generation techniques.3 Praclical high-speed nows,however, are usually turbulent and thus a suitable empiricalturbulence model musl be selected. Algebraic eddy viscositymodels slill represent the most common choice for com­pressible Navier-Stokes codes since their implemel)tationresults in the minimum requirements of compuler time andstorage, which is particularly important in three-dimensional(3-D) compulalions.

Several Iwo-layer algebraic lurbulence models (such asCebeci-Smilh~) require, for their implemenlalion, lhedelerminalion of the boundary-layer Ihickness and edgevelocily. This constitules a praclical disadvantage in lhenumerical solution of the Navier-Slokes equalions.Specifically, IwO effects complicale any altempl 10 devise asuitable algorithm for determinalion of lhe boundary-Iayeredge, namely, I) lhe presence of nonuniform inviscid regionsin which lhe inviscid now varies in lhe direclion normal 10 lhe

boundary, and 2) lhe presence of small spurious oscillalions inlhe numerical solution. As discussed by Hung and Mac­Cormack5 for compression comer flows and Baldwin andLomax6 for Iransonic flow over an airfoil, large varialions inthe compuled OUler eddy viscosity can occur as a result of theuncertainlies in lhe boundary-Iayer thickness.

ln an altempt to overcome this difficulty, Baldwin andLomax6 recently proposed a new algebraic eddy viscositymodel, paltemed after thal of Cebeci and Smith. This newmodel does nOI require the delerminalion of the boundary­layer edge, and Iherefore eliminales a source of potenlial error

Submitled May 16, 1983; presented as Paper 83-1697 atthe AIAA16th Fluid and Plasma Dynamics Conference, Danvers. Mass, July1:!-14, 1983; rev'ision received 01.'1. 28, 1983. Cop)Tight © AmericanInstitute of Aeronautics and Astronautics, lnc., 1983. Ali rightsreserved.

'Graduate Student, Depl. af Mechanical and Aerospace Engineer­ing; presently, Visiting Scientist. Air Force Wright AcronaUlical Lab .•\\'right Pallerson AFB. Member AlAA.

tAssociate Professor. Depl. af Mechanical and AerospaceEngineering. Membcr AIAA.

in the compuled outer eddy viscosity. In addition, since itemploys the vorticily which is invariant under coordinaleIransformalions, the model may be applied to 3-D con­figuralions.

Oue to the above advantages and ease of implemenlalion,lhe Baldwin-Lomax model is a popular algebraic eddyviscosity model in compulationaI aerodynamics. lndecd, thismodel has becn applied (somelimes quite uncrilical!y) to avariely of 2-0 and 3-D flowfield calculalions (sec, forexample, Refs. 7-10). Baldwin and Lomax6 evaluated theirmodel in delail for the case of Iransonic now over an isolaledairfoil. Hungll employed lhe Baldwin-Lomax model (in itsoriginal form) for the simulation of several compressioncomer nows. Oegani and Schiffl2 recently applied theBaldwin-Lomax model in the computalion of supersonicnows around cones aI high incidence. They found the model10 be unsuitable for regions of cross-flow separation. due toambiguilies in lhe delerminalion of the length scale. Oespileits increasing use, no additional evaluations of the Baldwin­Lomax model have becn conducled and are therefore necded.

The presenl investigation is aimed at partial!y fulfilling thisneed by performing a critical examination of the Baldwin­Lomax model for the case of shock/boundary-Iayer in­leraction in a supersonic compression ramp (Fig. I). Thiswork represents a more detailed and eXlensive evaluation thanthat of Ref. 11. The major focus of this research is to identifythe merits and deficiencies of the Baldwin-Lomax mode! for

supersonic interactions, and to develop, whcn possible, simple'modifications (wilhin the limilations of the algebraic edd)'viscosity concept) that could improve the overal! nowfie!dprediction. The present test now case has becn selected fortwo main reasons: I) shock/boundary-laycr interaction is animportant phenomenon in many practical high-specd flows,l3and 2) sufficiently detailed experimental mcasurementsl~.16are available for lhe 2-0 compression ramp configuration.

The compression comer flows \Vere simulale:d using the fuI!2-0 mass-a\'eraged Navier-Stokes cquations 17 expressed inslrong conser\'ation form 18 and in general curvilinearcoordinates. Several versions of lhe Baldwin-Lomax algebraicturbulence modcl were invesligaled, induding lhe in­corporation of lhe relaxation tcchnique of Shang andHankey.19 The gowrning equations were: sol\'e:d in nc:arlyorthogonal body-fillcd grids~ e:mploying the implicil, ap­proximate-faclOrizalion algorithm of Bcam and \\'arming. I

922 M. VISBAL ANO O. KNIGHT AIAA JOURNAl

where k = 0.0168 is Clauser's constant and C,p is an ad­ditional constanl. The Olllcr function F •..a ••.• is

where T ••. is the waIl sheM stress. In the outer region, in orderto eliminate the need Df finding the boundary-Iayer edge,Baldwin and Lomax replaced Clauser's formulation by thefoIlowing relation

For equilibrium turbulent boundary layers, as weIl as fortransonic Oow over an airfoil,6 the outer function F typicaIlydisplays a single well-defined extremum and the deter­mination of F mIV. and Ymu is straight forward. For separatedsupersonic Oow over a compression ramp, however, thepresent research indicates that F displays two peaks in thevicinity of separation. Similar behavior was observed byBaldwin and Lomax6 for a 2-0 oblique shock/turbulentboundary-Iayer interaction. Since the values of Y m.u.

assoeiated with each one of thcse extrema may differ by onearder of magnitude, the sclection of the peak closer to the waIl(at the streamwise locations where it represents lhe absolutemaximum) results in an abrupt, unphysical reduction in thecompuled outer eddy viscosity. An additional problem in theBaldwin-Lomax model (for both the inrier and outer for­mulation) may be caused by the vanishing of the Van Oriestdamping factor D at the locations where ;T ••, approaches zero(i.e., near separation and reattachment in 2-0 Oows). ThesmaIl values of D result in an unphysical reduction of thecomputed eddy viscosity. This effeet is found to be morepronounced for the inner eddy viscosity in the vicinity ofreattachment, and in some cases (see Seco 111) can prevent theOowfield from achieving a fully steady state in this region.

In order to avoid the above difficulties, a second turbulencemodel, referred to subsequenlly as the modified Baldv.in­

Lomax model, was employed. This new verslOn J.!1corooratestwo modifications, namely, I) at the locations where Fdisplays two peaks, the values 01 F ma> and Y mu are obtainedfram the extremum farthest from the waIl (ou ter peak), and

2)jn the Van Oriest damping factor [Eq. (2)], the local \·alue~ the total shear stress (defined in terms of the velocitycomponent paraIlel to the wall) is used in place of T ••.•

The third turbulence model incorporates the relaxationtechnique of Shang and Hankeyl9 in an attempt to accountfor upstream turbulence history effects. The relaxation eddyviscosity E (for both the inner and the outer formulation) isgiven by

(3)

(I)

(2)

(4)

11. Mf:"lhod of Solution

GOH'rninl: Equations and T",rbulence Models

The governing equations are the fuIl mean compressibleNavier-Stokes equation~ in two dimensions using mass­averaged variables,17 strU1Jg conservation form,18 and generalcurvilinear coordinates. The Ouid is assumed thermaIly and

caloricaIly perfecl. The TT/olecular dynamic viscosity J1 is givenby Sutherland's law. Thc molecular Prandtl number Pr= 0.73(for air) and the turbulent Prandtl number Pr, =0.9.

Three different version~ of an algebraic two-Iayer turbulent

eddy viscosity rriodel w~re employed in the present com­putations. The first turbulence model is that proposed byBaldwin and Lomax.6 In lhe inner region the eddy viscosity is

given by the Prandtl- Van Driest formulation

where p is the density, rlw I the magnitude of the vorticity,K = 0.40 is von Kármáu's constant, and Y is the normaldistance from the wall. The Van Oriest damping factor D is

given by

where Fmu =max(YlwIlJ), and Ymax is the value Df Yatwhich Fmu oecurs. The Klebanoff intermittency correction isgiven by

(5)

where Ckleb is a constanl. In the original Baldwin-Lomaxmodel,6 an alternate formulation, applicable to wake Oows,was included in the expression for F"'rn- Although thisformulation has becn employed in previous computations6.7.1Iof shock/boundary-Iaycr interactions, it is the authors'opinion that its use is not justified for the present in­vestigation, and, therefore, was not considered. The turbulenteddy viscosity is switchcd fram the inner to the outer for­mulation at the location where Ei > EO' Baldwin and Lomax

suggested the values for Cep = 1.6 and Clleb = 0.3 based on acomparison with the' Cebeci and - S"mlth model4 forequilibrium boundary layers at transonic speeds. In thepresent research, these constants were found to be dependenLon the Mach number of the Oow .. 1I can oe Shown21 that for

- an equilibriumincomprcssible ~M"" = Ql..turbulent boundarylayer, which obeys the waIl/wa'e law,u the values Crp = 1.2and Clleb = 0.65 are reqllired in the Baldwin-Lomax model.1O

-c>rderro-matdí the Clauser-Klebanoff formulation. In ad­dition, a series of Oat plate near-adiabatic (Twl Tadiabatic

= 1.12) turbulent boundary-Ia)'er computations at Mach 3.0dictated the use of CC'p = 2.08 for the prescnt rampealculations.23

The elimination of the need to determine the boundary­

layer edge in the outer edd)' viseosity formulation eonstitutesa major advantage of lhe Baldwin-Lomax model over theCebeei-Smith model. This is onl)' true, howe\"er, when theouter funetion F= r \wlD pro\"ides an unambiguouse\"aluation of the veloeit)' seale F ma\ and the lengt h seale }'mo>'

(6)

where Ecq is obtained from the modified Baldwin-lomaxmodel and Eu!,,, denotes the value of the eddy viseosity at theupstream location Xo where the surface pressure rise begins.A relaxation length À equal to the incoming boundary-layerthickness 00 was employed in the present computations. Thedetermination of the value of the relaxation length isdiscussed in Seco 111.

Compulalionnl Domain and 80undnry Condilions

The shape of the compulationaI domain is sho\\'n in Fig. I.The inOow boundary was located ahead of the comer in aregion of no upstream inOuence. The outOow boundary "'-asplaced sufficiently far from the comer, in a region of smallstreamwise Oow gradients. The height of the computatio:Ja!domain (3-4 00) was chosen so as to obtain freestrcamconditions along lhe upper boundary and to ensure theemergence of the shock through the downstream boundar)·.

On the solid surface, the nonslip, isothermal condilio ..•su= v=O and T= T ••.were applicd along with a bound2..'}·condition for the pressure derived from the normal co:n­ponent of the momentum eqllation.:1 For the freestTc:2.."'I1boundary, a no-reOection condition, 2~suitable for superso:-icOow, was prescribed. Alon" the outOow boundar)·. :.'1econventional extrapolation eondition ala~ =0 was emp!Oy~j.The IIpstream boundary eondilions were obtained byca1culating the de\"clopment of a Oat plate lurbulent bou:-:d~'}'layer up to the locations where the computed mom~:1!::""'I1thickness O matched the experimental \"alue. At the S2::1~loeations, the computed and measured \"elocit)', skin fri~:x>nand displacement thiekness \\'ere also eompared and fOl:.:1':in\"er)' good agrecmenl. F0r inst:1I11:e, at the rnatching s.::::.:i-.'n

JUL Y 1984 SHOCK-\V A VE/BOUNDARY-LA YER INTERACTIONS 923

Fig. 1 Flow configuratlon and computatlonal domaln ..

1. O-2.0 -1.0 0.0 1.0 2.0

X/3oFig. 2 Surface pressure distrlbutlon for 16-deg ramp.

D EX1'ERt.e'lT- - - BAil>WN-LOMAX M::OEl.-- MOOtF1ED B-L M::OEl.- - •. - - RB.AXA TION M::OEl.4.'

~~ 2.0

Mo::> = 2.9b R b

·76x 1O( e8< 7.7x 10o

C~ ATIONA1. DOMAIN_~ .- /\r------III

experimental cases. For the first category (i.e., variable rampangle), the different versions of the algebraic turbulcncemode! were employed. For the second category (i.e., variableRe6 ), only the relaxation model was applied. A detailedco~parison of the computed flowfie!ds and the experimentaldata was performed; however, only the most significantresults are presented below. Reference 21 contains a moreextensive comparison along with the details of ali com­putations.

Results for Variable Ramp Angle

Resultsfor 16-deg Ramp

The 16-deg ramp flowfield was computed using each of theversions of the turbulence mode\. This compression comerflow eonstitutes, according to the experiments, an incipientlyseparated interaction. The computed and measured surfaeepressure is shown in Fig. 2. The results for the originalBaldwin-Lomax and re!axation models are in elose agreement

Numerical Algorithm

The governing equations were solved using the implicit,approximate-faetorization algorithm of Beam and \Varm­ing. I This seheme was formulated employing Euler implicittime-differencing and second-order, eentered approximationsfor the spatial derivatives. The boundary conditions wereimplemented in the explicit or lagged approaeh described bySteger. 26 Fourth-order explicit damping terms, required forthe smoothing of the embeddedshocks, were preseribedaecording to the procedure of Ref. 27. The developed Navier­Stokes eomputer code was extensively validated,21 with ex­cel1ent results, for several test cases including inviseid shockedflows, laminar and turbulent boundary layers, and laminarshoek/boundary-Iayer interactions.

Sinee the compression ramp flowfields are obtained by timeintegration of the unsteady Navier-Stokes equations until asteady state is reached, considerable care was exereised inensuring convergence. Separated interactions were run forphysieal times of up to 10le' where le is the time required for afluid parcel in the inviscid region to traveI from the upstreamto the downstream end of the mesh. The flowfields wereassumed eonverged when the maximum reIative variation ofthe flow variables over Ile were less than 1.0070.It should benoted that, while changes of order 1.0070 occur at a few meshpoints (in regions of shock smearing), the reIative ehangeswere much smal1er at most locations. The eorrespondingaverage variations over Ile were typical1y less than 0.05070.

where ReB =8.2 x 104, the computed and measured skin­friction coefficient cJ are 1.02 x 10- 3 and 1.00 x 10- 3,

respeetively. The value of cJ predieted by the Van Driest 11theory and the von Kármán-Schoenherr equation25 is1.04 X 10-3. At the same station, the eomputed displacementthickness ó· is essentially equal to the experimental value (0.66em). Exeellent agreement was also found21 between thecomputed and measured velocity profiles and the law of thewall. :u

The eom2utational grids were generated by the numericalpi"ocedure ofRêf-20:- A-n-onuniform mesh spacing was usedin both eoordinate direetions in order to provide sufficientresolution of the turbulent boundary layer and the interaetionregion. In the direction normal to the surfaee, the grid pointswere distributed using a eombination of geometrieaIlystretehed and uniform spacing. The normal spacing at thewall was ehosen in order to resolve the viseous sublayer, andsatisfied the requirement trY~in~ 2.5 at al1 loeations. Thetypical number of grid pomIS wlthm the boundary layer was25 to 30. lhe streamwise mesh spaeing in the interactionreglOn ranged from 0.027 Óo for the 8-deg ramp to 0.077 Óo

(for the 24-deg ramp). The maximum streamwise spaeing(outside the interaction region) was always less than 0.6 óo•

Ucfinltion of Intcractlon geometrlc dlstances.

IH. Results and Discussion

Figure I iIlustrates the eompression comer geometryemployed in the present evaluation of the Baldwin-Lomaxmode\. An extensive experimental study of this flow con­figuration, for a nominal Mach number AI"" = 2.9, has becoconducted in recent years.14.16 The available experimentaldate basel4 may be divided into two major categories:1) surface and mean flowfie!d data for four comer angles(a=B, 16, 20, and 24 deg) at a fixed Reynolds number

Rel>o= 1.6 x 106, and 2) surface data (including wal1 pressureano separation and reattachment locations) for a fixed rampangle of 20 deg at four different Reynolds numbers(Rel>o=0.76x 106, 3.4x 106, 5.6x 106, and 7.7 x 106). Thefirst category ineludes measurements of surface pressure; skinfriction; and velocity, f\lach number, and static pressureprofiles at nine streamwise stations for each ramp angle.Thesc flowfields encompass nominal1y attaehed (B-deg ramp),as wcll as fully separated (20- and 24-deg ramps) interactions.Computations were performcd for al1 cight of the above Flg.3

I' 10.p

x

• 924 M. VISBAL AND D. KNIGHT AIAA JOURNAL

with each other and with the experimental data. However, thecomputed surface pressure obtained with the modified modeldisplays an insufticient upstream propagation ~p (see Fig, 3for the definition of geometric distances), similar to theresults of Shang and Hankeyl9 and Horstman et al.,28 whoemployed the Cebeci-Smith turbulence mode!.

The skin-friction results are presented in Fig. 4. Alicalculations predict the existence of a separated region, whichis not displayed by the experiments. The original Baldwin­Lomax and relaxation models are in reasonably goodagreement with the expe:riment in the region of sharplydecreasing cf' Howe:ver, both models seriously underpredictthe skin-friction values in this recovery region (Le:., down­stream of reattachment). Also, the compute:d separation-to­reattachment lengths are toa large. The modified mode1provides substantial improvement in this regard. Downstreamof attachment, however, ali models approach the same skin­friction levei which is significantly below the measured value.This behavior is similar to previous computations usingalgebraic eddy viscosity models. 28

The computed and expe:rimental velocity profiles at threestations (upstream, at the comer, and downstream of theinteraction) are shown in Fig. 5. The original and relaxationmodels give slightly bette:r results upstream and at the comer(first two profiles). Downstream of reattachment, ali threemode1s result in a ve]ocity profile that displays an insufficientrecovery or "filling out" near the wall. This observation isconsistent with the underprediction of skin friction diseussedabove.

Figure 6 shows the: evolution of the outer funetionF= Y IwlD across the interaction, for the ealculation with theoriginal Baldwin-Lomax model. The corresponding value ofYmax' and the: maximum value of fO at each streamwise: stationare given in Figs. 7 and 8. Upstream of the interaction

(X /Óo = - 1. I), F displays a single:, well-define:d peak. lm­mediately before the: separation point (X/óo = - 0.27), Fexhibits two distinct extrema (re:ferred to subse:quently as the:inner and outer peak). At this loeation the: outer peak. whichrepresents the absolute maximum. is still chosen by the modelto compule F ma> and Yma>' Downstream of separation(X/óo= -0.19). the inner peak, whieh is very dose to thewall, exceeds the outer one, and Ymax abruptly decreases byone order of magnitude: (Fig. 7). Despite the increase in Fma>.'

a net sudden drop in F•.•kt occurs. This reduetion in F•.akt'eombined with the effeet of Ymu in the: Klebanoff in­termittency correction [Eq. (5)], produces a sharp decrease inthe computed outer eddy viseosity (Fig. 8). As Fig. 6 in­dicates, lhe outer peak disappears further downstream, andthe inner peak moves away from the: surfaee to a newequilibrium position. These large streamwise variations in theeomputed length scale: Yma> are unphysical and constitute amajor deficieney of the original Baldwin-Lomax model for

.5E.o1

F/ Uc>

Fig. 6 E"olution of lhe Baldwin-Lomax OUler edd)' viscosil)' fuoe­lion F(Y) across 16-dcg ramp inleraclion.

1.0 t a

--,,­,"I," - - - BAl.DWN-lOMAX '-COEI.

, I

:1' -- MCOFED B-l '-COEI.II

0.2

0.3

.1msu.~o

O. t

a

--::-:....., .', .­, ., .'" .'I' •.i

D

D EXPERr.e;T- -- BAl.DWlN-LOMAX MODEl.-- MXJFED B-l MODEl.- - - - - RELAXATION MODEl.---'- CQNSTANT YMAX ••

0.0

2.0

0.0 2.0

X/SoFig.4 Skin·friclion coeffieienl for 16-deg ramp.

2.0

xjSo

Fig. 7 Variation of }'max across 16-deg ramp inleraeeion.

-1.0-2.0 4.0

0.0-2.0 0.0 4.0 6.0

Fi~. 5 "doeil)" profiles aI sneral slalions (a = 16 deg. Rt'ho =\.6 X 106).

6.04.0

-----

2.0

X/~o

­,,-I

II"

, - - - BAl.Dwr./-LOMAXI

I I -- MODFED B-l MODEl.,':(

I

Variall"n "f IOmn Der"" t6-dcg ramp Inlcn~ell"n.

5.0

4.0

'o '"~3.0

~ 8••• :s.. 2.0 IIIt.O ~

III"i0.0 I

.-2.0

0.0

Fig.8

--

5.6 I"li--

lili

0.0 I-"li

x _

&0-

0.0

0.5

1.0

t

JULYl984 SHOCK-WA VE/BOUNDARY-LA YER INTERACTIONS 925

2.0

X/t;oFig. 10 Sldo·frielino eoeffieieol for :!O.degramp.

Downstream of reattachment, ali models seriously un­derpredict the reeovery of the boundary layer. The computedseparation-to-reattachment length is toa large, although lhemodified model does slightly better in this respeet.

Figure I I shows the computed and measured velocityprofiles at several streamwise loeations. The relaxation modelgives some improvement in the computcd vcloeity for the firstpart of the interaction. This is consistent with lhe betterprediction of upstream propagation discussed above.Downstream of reattaehment, ali models fail to predict therapid recovery of the velocity near the wall. The results for thestalie pressure are given in Fig. 12. ln the interaclion region,the relaxation model displays the best comparison with lheexperiment. The measured static pressure profiles down­stream of reattaehment exhibit a normal gradient near lhe

-1.0-2.0 6.0

••

4.00.0

o EXPER~- - -SALOWIN-LOMAX MODEl.--- MODIFIED S-L MODEl. a-. - - •• RELAXA TION MODEl."

1.0

2.0

0.0

the predietion of supersonie shoek/boundary-layer in­teraetions. Similar problems were eneountercd by Baldwinand Lomax6 in the computation of a 2-D oblique shoek in­teraction, and by Knight23 in a 3-D interaetion ..

The behavior of the outer funetion F in the interaetionregion for the modified Baldwin-Lomax model is similar tothat deseribed above. The corresponding value of Yma.. and€Om" are also shown in Figs. 7 and 8. Sinee in the modifiedmodel the outer peak is always selected, an abrupt deerease inYma' and €Omax still oeeurs at the streamwise loeation wherethe outer peak in F disappears.

For the 16-deg ramp eomputation using the originalBaldwin-Lomax model, a fully steady-state solution could notbe aehieved in a small region close to the wall in the immediatevicinity of reattaehment. In this region the flow variablesexhibited bounded oscillations in time. This problem is ap­parently assoeiated with the very low values of the eomputedinner eddy viseosity, eased by the Van Driest damping factorD [Eq. (2)] approaching zero in the reattachment region .This diffieulty was overcome in the modified version of themodel by employing the local total shear stress in theevaluation of D.

In order to investigate the effects of the length seale Ymax'

an additional computation was performed for the 16-degramp utilizing the modified Baldwin-Lomax model with aconstant or "frozen" value of Ymax throughout the flowfield.The eomputed surface pressure was essentially identical to theresults for the modified model. The eomputed skin-frictioncoefficient, shown in Fig. 4, gave only a slight improvement inthe recovery region. The velocity profile at the downstreamstation XI Óo = 5.4 (not shown) was very close to the previousresults (Fig. 5), and again failed to predict the rapid recoveryof the boundary layer downstream. This caIculation indicatesthat the computed flowfield in the recovery region is not verysensitive to changes in the ou ter eddy viscosity.

o EXI'<'RMNf--~-LCM.tJ< MCXJa.--MOOFEO 91. MCXJa.•••• - RElAXA T10N MCXJa.

Fig. 11 "elodl~' profiles aI senral slallo05

R~60 = t.6x 106).

Resu/rsfor 20-deg Ramp

The 20-deg compression ramp was simulated using the Ihreedifferent versions of the turbulence model. This caserepresents, according to the experiments, a fully separatedinteraetion. The computed and measured surface pressure isshown in Fig. 9. The results for the modified Baldwin-Lomaxmodel significantly underprediet lhe extent of upstreampropagation and do not display the pressure "plateau" ob­served in the experiments. The use of relaxation, with arelaxation length À = óo. substantially improves the wallpressure predietion and gives the correct upstream influence.However, the computed pressure plateau is more pronouneedthan in the experiments. The computed pressure for theoriginal model falls between the results for the modified andrelaxation models. The skin-friction results are shown in Fig.10. The relaxation and original models provide a betteragreement with the experiment upstream of separation.

\.0

0.5

0.0

x _ -0.44~-

o.

0.0 1.0

(a = :!O de\:,

'\.0

Surfare prt"SSul"l'dislribulioo for :!O-de):ramp.

: )\ 4-6

:1111• I.1••aa""""•

a',eI., ,...•

.... '.\ '", .•.• ot...... ,,."

. p/Pco

OE~- - -BAlDWIV-LOMA.X "*DEI.-- MOOFEO B1. "*DEI.•••••• RElAXA 110'< "*DEI.

0.0 1 0.16

2.

•..,I.

0.5

0.0

Fig. n Slalic presslIl'l' prtlriks aI st'\er.ll sl:llinos Ia =:!O de\:.

R~6o = 1.6 X 106).

6.04.02.0

X/t;o

() EXPffi~- - - BAl.DW1N-LOMAX MODa.-- ~\X)F18) S-L MODa.- - - - - - RELAXA TION MOOEl.

0.01.0-2.0

2.0

3.0

Fi!:. 9

•

, 926 M. VISBALAND D. KNIGHT AIAA JOURNAL

wall, which is not duplicated in the computations. This ob·servation also applies to the 16- and 24-deg ramp flows.

Resu/tsfor 140deg Ramp

This fully separated compression comer interaction wassimulated using the modified BaldwinoLomax and relaxationturbulence models. The comparison of computed and ex­perimental results, contained in Figs. \3-15, exhibits the samecharacteristics observed for the 20-deg ramp. As comparedwith the modified model, the relaxation model (with À = óo)

provides a marked improvement in the prediction of surfacepressure distribution (Fig. \3), ineluding the pressure plateauleveI and the extent of upstream influence. This agreementdictated the use of À=óo in the present research. This value ofÀ was also found by Horstman et aI. 28 to predict ÁXreasonably well. On the other hand, Shang and Hankeyl~required a value of À = JOóo in their compression comercalculations. This will be~discussed in more detail below. Theuse of relaxation also results in a eloser agreement with themeasured skin friction ahead of separation (Fig. 14). Bothmodels, however, overpredict the length of the separationregion, and do not provide the correct skin-friction leveIdownstream of reattachment. The higher values of c, ob­tained by Baldwin and Lomax6 for the 24-deg ra~ case are

~in_contraâjCtion with the presçnt results and those of Ref. 28.Upstream of the corner, the relaxatian model praduces

again an improvement in the predicted velacity prafiles (Fig.15). Downstream of reattachment, bath models fail tosimulate the rapid recovery of the baundary layer near thewall.

ln arder to examine the effects af the relaxatian length anthe computed flawfield, the 24-deg ramp flaw was alsocalculated using a "frozen" eddy viscosity madeI. The term"frazen" denotes that the eddy viscosity profile upstream af

the interaction was cmploycd at ali strcamwisc locations [i.e.,À = 00 in Eq. (6)]. Thc computalion was run for ap­proximately three characteristic times, aI which point theextent af upstream propagatian D.Xp had increased by almast100070, as comparcd with the previaus results for therelaxation mode! with À = óo. Results by Harstman ct aI. 28 farthe same flow conditions indicated an increase of up to 500;0in ÁXp when À was changed from I Óo to 5 óo. On lhe otherhand, camputations by Shang and Hankeyl9 for a 25-degramp (with M ••-2.96 and Re6(J = 1.4 x 105) employing a frozenand a relaxation (À = IOóo) model, did nOI exhibit this drasticdifference in ÁXp. The use of À= 10óo is then expected toproduce a grass overpredictian af the upstream influence farthe present compression comer flow. Since esscntially thesame baseline turbulence model (namely CebecioSmith4) wasused in Refs. 19 and 28, the discrepancy in the value of À,

required ta match the experimental pressure, is probably dueto the difference in the flaw Reynalds number Reho' In fact,the measured separatian-to-reattachment length for thepresent 24-deg ramp flow is 1.7óo• while the correspondinglength for the 25-deg ramp cansidered in Ref. 19 is ap­proximately 8 óo. This could be interpreted as a dependence ofÀ on the extent of the interaction, which far a given geometryis a function of the flow parameters (Reho and M •• ). Thisvariatian of the relaxation length À (which is intended torepresent the lagged response of the turbulent stress to suddenmean flow gradients) is perhaps reasonable since forseparated shock/boundary-Iayer interactians large increasesin the Reynolds stress are observed29 before the reattachmentlocation.

Results for Variable Re)'nolds Number

Since the previous results far variable ramp angle indicatedthat the use of relaxation pravides some improvement in thepredictian of the interaction upstream influence, it is of in-

5.0

Fig. 13 Surface pressure distribution for 24-deg ramp.

C EX.~-- M(X)FID Sol ..x>El.- .•..•. - RElAXA ~ MCX:El

slations (a = 24 de'l.U/Uco

promes at se\"eral

-'.2t=

I.D

D.O o.

D.S

Fig. IS Velocit)·

R~iO = 1.6 X 106).

6.04.0

D.

1.0-4.0 -2.0 0.0 2.0

X/tJo

2.0

4.0

~Poo

3.0

2.0

.SE+07

I x,. COM'.[;)

O-t- ---

.•.•. -----"' "' "' "' "'

.•..••..

••..

11

0.0 I ,.SI:':.06

0.5

><1 O<Jlc.<) 1.5

2.5

.....1..8 2.0

RecS o

Fil:. 16 ~rrl'<"1of Rt'ho on .:lXp'~Xs' nnd [os (cr= 20 dc!:).

6.04.0

li.o

[;) E~ 11-- MOOFlED S-L MOOel.- - - - - RELAXAT10N MOOEI. li

1.0

0.0

-1.0-4.0 -2.0 0.0 2.0

X/tJo

Fig. 14 Sldn·friclion coeHicient for 24-del: ramp .

•.. - --

c.

•JULY 1984 SHOCK-WA VE/BOUNDARY-LA YER INTERACTIONS 927

terest to evaluale lhe relaxation model for the case of variableReynolds number. For this purpose, several calculations wereperformed for a 20-deg ramp over the Reynolds number range0.76 x 106 sReho s7.75 X 106, using the relaxation turbulencemodelonly.

In order to illustrate the effects of Reynolds number on theinteraction, the values of t:.Xp, t:.X, (see Fig. 3 for definitions)and the separation-to-reauachment length L, are presented asfunctions of Reho in Fig. 16. Although the computed resultsfor t:.Xp, t:.X,. and L, exhibit the correct Reynolds numbertrend, only the upstream pressure innuence t:.Xp is predictedwith reasonable accuracy (see Refs. 16 and 28 for the scaUerof the experimental data). The distance from the separationlocation to the comer t:.X, as well as the overall separationlength L, are consistently overpredicted. The above ob­servations are in agreement with the results of Ref. 28. Therelaxation model (with a constant relaxation length À = 00) iscapable of predicting the upstream pressure innuence withreasonable accuracy over the entire Reynolds number rangeinvestigated.

IV. Conciusions

A critical evaluation of the algebraic turbulence model ofBaldwin and Lomax was performed for the case of 2-Dshock/boundary-Iayer interactions induced by compressioncomers. Three different versions of this algebniic eddyviscosity mode! were investigated, including the incorporationof relaxation.19 A detailed compai-ison of the computednowfields with the available experimental data 14 was per­formed, and the capabilities and deficiencies of the turbulencemodels were identified.

Regarding the characteristics of the Baldwin-Lomax modelfor 2-D supersonic interactions, the following specificconclusions can be made:

I) The constants Ccp and Clltb, appearing in the Baldwin­Lomax outer formulation, were found to be dependent on thenow Mach number. These constants vary by a factor of twoover the Mach number range OsM '"s3.0, and therefore needto be adjusted accordingly.

2) The Baldwin-Lomax outer function (F= Y Iw ID) is notsuitable for the determination of the length scale in theseparation region of the interactions investigated. This is dueto the appearance, near separation, of a double peak in F( Y)which results in an abrupt (unphysical) decrease in thecomputed length scale. This behavior constitutes a majordeficiency of the model for supersonic interactions, and couldperhaps be eliminated by the use of a different ou ter function.In addition, for an incipiently separated interaction, the smallvalues of the eddy viscosity near the reattachment location(caused by the vanishing of the Van Driest damping factor)can prevent the solution from achieving a fully steady state.

The above difficulties can be panially overcome by usingthe local total shear stress in the Van Driest damping term andby the selection of the outermost peak of F( Y) in the com­putation of the length scale. These modifications provide abetter now prediction near reattachment.

3) Computations with the original and modified Baldwin­Lomax models exhibit an insufficient upstream propagation,caused by the inability of the models to reproduce the laggedresponse of the turbulence structure to the sudden adversepressure gradient. Significant improvement in lhe nowfieldprediction upstrcam of the comer can be obtained with the useof relaxalion. A relaxalion length equal 10 the incomingboundary-Iayer Ihickness was found 10 be suilable for therange of Reynolds numbers and ramp angles considered. Thisvalue, however, is one-tenth of thal suggesled by Shang andHankeyl9 and is expecled to depend on the extent of the in­teraction.

4) Ali of lhe turbulence modcls lested here fail to predicIthe_~a[1id recovery of=lhe boundary layer downstr~í

- 7' reattachment. This is due 10 lhe inability of lhe models 10simulatc the observed:~33 amplification of lhe turbulence

nuctualions across a shock/boundary-Iaycr interaction. This,deficiency would lead to a ralher poor prediction of nowswilh multiple interactions, and means of improving thepresent results have nOI yct been found. The fact Ihatdownslream of the interaction, the mean velocity profilerapidly approaches its equilibrium shape, while the cnhancedlurbulence nucluations relax very slowly loward equili­brium,3c}'J2points OUI lhe inadequacy of the algebraic eddyviscosilY concept for these complex nowfields.

AcknowlcdgmcntsThe authors gratefully acknowledgc many helpful and

stimulating discussions with S. Bogdonoff, D. Dolling, G.Settles, and L. Smits. This research was sUPP0rled by lhe AirForce Office of Scienlific Research under AFOSR Grant 82­0040, monilored by J. Wilson. Inilial development of lhenumerical algorithm was supported by lhe Air Force FlightDynamics Laboralory-Air Force Wright AeronaulicalLaboratories under AFOSR Grant 80-0072, monilored by J.Mace.

Refcrenccs

1 Beam, R. and Warming. R., "An Implicit Factored Scheme forthe Compressible Navier-Stokes Equations," AIAA Journal, Vol. 16,April 1978, pp. 393-402.

2MacCormaek, R. \V., "A Numerical Method for Solving theEquations of Compressible Viscous Flow," AIAA Paper 81-0110.1981.

3Thompson, l. F., Warsi, Z. U. A., and Mastin, C. \V.• "Bound­ary Fitled Coordinate Systems for Numerical Solution of PartialDifferentiaI Equations-A Review," Journal 01 CompUlarional

Phrics. Vo\. 47,1982, pp. 1-108.Cebeci, T. and Smith, A. M. O., Analysis 01 Turbulenr Boundary

La{,ers, Academic Press, New York; 1974.Hung, C. M. and MacCormack, R.W., "Numerical Simulation ofSupersonic and Hypersonic Turbulent Compression Comer Flows."AIAA Journa/, Vol. 15, March 1977, pp. 410-416.

6BaIdwin, B. and Lomax, H., "Thin Layer Approximation andAlgebraic Model for Separated Turbulenl Flows," AIAA Paper 78­257,1978.

7 Hung, C.M. and Chausee. D. S., "Comput3tion of SupersonicTurbulent Flows Over an Inclined Ogi\'~Cylinder-Flare," AIAA

par,:,r 80-1410. 1980.Deiwert, G. S.• "Numerical Simulation of Three-DimensionaIBoattail Afterbody Flow Fie1d," AIAA Paper 80-1347, 1980.

9Kutler, P., Chakravarthy, S. R., and Lombard. C. K., "Super­sonic Flow Over Ablated Nosetips Using an Unsteady, ImplicitNumerical Procedure," AIAA Paper 78-213, 1978.

IOSteger, l., Pulliam, T., and Chima, R., "An Implicit FiniteDifference Code for Inviscid and Viscous Cascade Flows," AlAA

par.;r 80-1427,1980.IHung, C. M.• 1980-198/ AFOSR-HTTM·STANFORD Con­lerence on Comp/ex Turbulenr F/ows. Vol. 111.edited by S. l. Kline etai., 1982, Thermoscienccs Div., Mechanical Engineering Depanment,Stanford Univ., pp. 1283-84 and 1372-74.

12 Degani, D. and Schiff, L. B., "Computation of SupersonicViscous Flows Around Pointed Bodies 8t Large Incidence," AIAA

PaRer 83-0034, 1983.3Korkegi, R. H., "Survey of Viscous Interactions Associated withHigh Mach Number Flight," AIAA Joumal, Vol. 9, May 1971,pp.771-784.

14Settles, G., Gilbert, R., and Bogdonoff, A., "Data Compilationfor Shock \\'aveITurbulent Boundary Layer Intcraction Experimentson Two-DimensionaI Compression Comers," Dep!. of Aerospacc andMechanical Engineering, Princcton University, Princeton, N. l.,Re~t. MAE-1489, 1980.~Settles, G., Fitzpatrick, T., and Bogdonoff, S., "Dctailed Studyof Attached and Separated Compression Comer Flo\\'fields in HighReynolds Number Supersonic Flow," AIAA JOllrnal, Vol. 17, 1979,pp. 579-585.

16Settles. G., Bogdonoff, S., and Vaso I.E., "Incipient Separalionof a Supersonic Turbulcnt Boundary L1ycr aI High ReynoldsNumbers," AIAA JOllrnal, Vol. 14, Jan. 1976, pp. 50-56.

17 Rubesin, M. and Rose. W., "The Turbulcnt /llean-Flow,Reynolds-Stress and Heat·l-lux 'Equations in Mass A\'eragedDependent Variables," NAS.-\ TMX-6:!:!48. M:udI1973.

928 M. VISBAL AND D. KNIGHT AIAA JOURNAL

18Pulliam, T. and Steger, J., "Implicit Finite-DifferenceSimulations of Three-Dimensional Compressible Flow," AJAAJournal, Vol. 18, Feb. 1980, pp. 159-167.

19Shang, J. and Hankey, W. L., Jr., "Numerical Solution forSupersonic Turbulent Flow Over a Compression Ramp," AJAAJOllrnal, Vol. 13, Oc\. 1975, pp. 1368-1374.

2oVisbal, M. and Knight, D. D., "Generation of Orthogonal andNearly Orthogonal Coordinates with Grid Control Near Boundaries,"AJAA Journal, Vol. 20, March 1982, pp. 305-306.

21Visbal, M., "Numerical Simulation of Shock/TurbulentBoundary Layer Interactions Over 2-D Compression Corners," Ph.D.Thesis, Dep\. of Mechanical and Aerospace Engineering, RutgersUni"ersity, New Jersey, Oc\. 1983.

22Sun, c.-c. and Childs, M. E., "Wall-Wake Velocity Profile forCompressible Nonadiabatic Flows," AIAA JOllrnal, Vol. 14, June1976, pp. 820-822.

23Knight, D. D., "A Hybrid Explicit-Implicit Numerical Algorithmfor the Three-Dimensional Compressible Navier-Stokes Equations,"AIAA Paper 83-0223,1983

24Knight, D. D., "Calcul~lion of High Speed Inlet Flows Using theNa~;er-Stokes Equations," AFFDL- TR-79-3130, Vol. I, 1980.

2..' Hopkins, E. and Inouye, M., "An Evaluation of Theories forPredicting Turbulent Skin Friction and Heat Transfer on Flat PlatesaI Supersonic and Hypersonic Mach Numbers," AIAA Journal, Vol.9, June 1971, pp. 993-1003.

26Steger, J. L., "Implicil Finite-Diffcrcnce Simulation of FlowAbout Arbitrary Geometries with Application to Airfoils," AIAA

PaRer77-665,1977.7Thomas, P. O., "Boundary Conditions and ConservativeSmoothing Operators for Implicit Numerical Solutions to theCompressible Navier-Stokes Equations," LMSC-D630729, 1978.

28Horstman, c., Hung, c., Settles, G., Vas, 1., and Bogdonoff, S.,"Reynolds Number Effects on Shock- Wavc Turbulent Boundary­Layer Intcraction-A Comparison of Numerical and ExperimentalResults," A1AA Paper77-42, 1977.

29Delery, J. M., "Experimental In"estigation of TurbulcnceProperties in Transonic Shock/Boundary-Layer Interactions," AJAAJournal, Vol. 21, Feb. 1983, pp. 180-185.

30Settles, G. S., Baca, B. K., Williams, O. R., and Bogdonoff, S.M., "A Study of Rcattachment of a Free Shc.1r Layer in CompressibleTurbulent Flow," AIAA Paper 80-1408,1980.

31Rose, W. c., "The Behavior of a Compressiblc TurbulcnlBoundary Layer in a Shock· Wave-Induced Adverse PressureGradient," NASA TN D-7092, 1973.

32Hayakawa, K., Smits, A. J., and Bogdonoff, S. M., "TurbulenceMeasurements in a Compressible Reattaching Shear Layer," AIAA

Pater 83-0?99, 1983.. " ,.Hayakawa, K., Smlts, A.J., and Bogdonoff, S. M., Hot-\\IreInvestigation of an Unseparated Shock- Wave/Turbulent BoundaryLayer Interaction," AIAA Paper 82-0985,1982.

Fro/n the AIAA Progress in Astronautics and Aeronautics Series

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