E.GarnierandS.Deck
1Introduction
Thetransonicbuffetisanaerodynamicphenomenonthatresultsinalarge-scaleself-sustainedperiodicmotionoftheshockoverthesurfaceoftheairfoil.Thetimescaleassociatedtothismotionismuchslowerthantheoneofthewallboundedturbulence.ItisthenanappropriatecaseforURANSapproachesandfirstattemptswiththesemethodshavebeenreasonablysuccessfulinreproducingthemeanfea-turesofsuchflows.Nevertheless,asshownbyThieryandCoustols[2]resultsareverysensitivetotheturbulencemodel.Moreover,withsomemodels,itisneces-sarytoincreasetheangleofattackwithrespecttoexperimenttoobtainanunsteadyflow.Furthermore,theyhaveevidencedasignificantsensitivityoftheresultstotheconfinementduetothewindtunnelwalls.ThefirsthybridRANS/LEScomputationonthisconfigurationwasperformedbyDeck[1]whohasdemonstratedthatzonalDES(ZDES)generallyimprovestheresultswithrespecttoURANScomputationscarriedoutwiththeSpalart-Allmarasmodel.Inparticular,thespectralcontentofthepressurefluctuationsintheseparatedzoneismuchmoreclosertotheexperimentaldatawithZDESthanwithURANS.Inthislattercomputation,theshock/boundarylayerinteractionwastreatedinRANSmodeandoneofthepurposeofthepresentstudyistoassesstheimprovementthatmayresultfromafullyturbulenttreatmentoftheboundarylayeronthesuctionsideoftheairfoilbymeansofLES.
Moregenerally,themainobjectiveofthisstudyistoassessthecapabilitiesofLEStocapturethebuffetphenomenon.Thelargeamountofdataprovidedbythesesimulationscouldthensupporttheprogressinthephysicalunderstandingofsuch
E.Garnier
ONERA,AppliedAerodynamicsDepartment,8ruedesVertugadins,92190Meudon,France,e-mail:eric.garnier@onera.fr
S.Deck
ONERA,AppliedAerodynamicsDepartment,8ruedesVertugadins,92190Meudon,France,e-mail:sebastien.deck@onera.fr
1
2E.GarnierandS.Deck
flows.Thevalidationofthecomputationisperformedagainsttheverycomprehen-siveexperimentperformedatONERAbyJacquinetal.[3]whichwasalsousedbyThieryandCoustols[2]andDeck[1].
2Descriptionofthecomputation
ThesupercriticalOAT15AairfoilwascomputedinthesameflowconditionsthanintheexperimentbyJacquinetal.[3].Thisairfoilhasachordof230mmandarelativethicknessof12.3%.Itsangleofattackisequal3.5degrees.Thefree-streamMachnumberwassetto0.73andtheReynoldsnumberbasedonthechordlengthisequalto3106.
TheflowsolveristhestructuredmultiblockcodeFLU3MdevelopedatONERA.Itissecond-orderaccurateinspaceandtime.ThenumericalschemededicatedtothecomputationoftheconvectivefluxesisbasedonaRoeschemewhichwasmodifiedtoadaptlocallyitsdissipationusingtheDucrosetal.sensor[4].Thedissipationleveloftheschemecanbetunedbyanadditionalmodificationofthesensor[5].TheSelectiveMixedScalesModelhasbeenchosenforthisstudy[6].Thetimestephasbeenimposedto3.10−7sinordertoensuretheconvergenceofthesubiterativeprocessofthetemporalimplicitschemeusing5subiterations.
Inordertolimittherequiredcomputationaleffort,theflowiscomputedinRANSmodeonthepressuresideoftheairfoilandinLESmodeonthesuctionsideandinthewake.Moreover,RANSzonesaretreatedin2D.ThegridrefinementcriteriacommonlyusedinLESofattachedflowsaresatisfied(∆x+≈50inthelongitudinaldirection,∆z+≈20inthespanwisedirectionand∆y+min≈1inthewall-normaldi-rection).Despitethezonaltreatmentoftheflow,20.8millionsofcellsarenecessarytocomputeadomainwidthofonly3.65%ofchordinthegridA(Nz=140).ThespanandconsequentlythenumberofpointsweredoubledtoconstructthegridB(Nz=280).Thismaybeinsufficientbutthegridsizeresultsfromacompromisewiththelongintegrationtimerequiredtocapturefewbuffetingperiods.Twocom-putationsdenotedB1andB2havebeencarriedoutwiththegridB.ThefirstoneisbasedonahighdissipationsetofparametersdefiningthemodifiedDucrossensorwhereasthesecondonecorrespondstolowdissipationparameters.
3Meanfieldanalysis
Afteratransientof2periods,theflowhasbeenaveragedoveronlyoneperiodofthebuffetphenomenonforthecaseA.ThespanwasthendoubledtogeneratethegridBand,afteratransientofoneperiod,thestatisticswerecollectedoversevenadditionalperiodstogeneratetheresultsdenotedB1.Thedissipationlevelwasthendiminished(B2case)andafteraoneperiodtransient,statisticswerecollectedoveroneperiod.Figure1presentsanisovalueoftheQcriterioncoloredbythelongitudi-
Large-eddysimulationoftransonicbuffetoverasupercriticalairfoil3
nalvelocity.Theseparationoccursafterthelocationoftheshockidentifiedbyoneisovalueofthepressure(inpurple).Onthissnapshotwhichcorrespondstoasitu-ationwheretheshockmovesdownstream,theflowisseparatedunderthelambdashockandnearthetrailingedge.
Fig.1AnisovalueoftheQcriteriacoloredbythelongitudinalvelocityandoneisovalueofthepressure(purple)tomarktheshocklocation(B1case)
Figure2(left)showstheaveragedpressuredistributionontheairfoil.ForbothcasesAandB1,thebuffetzoneisshifteddownstreamby7%ofchordwithre-specttotheexperimentwhereastheagreementissligtlybetterinthecaseB2.Thisdemonstratessomeinfluenceofthenumericaldissipationonthemeanshockposi-tion.InpreambletothediscussionconcerningthepressurefluctuationspresentedinFig.2(right),itisworthnoticingthatintheexperimentthesignaliscollectedover2300periodswhereasatbest7periodsareavailableinthecomputation.Inordertopermitfaircomparisonsbetweencomputationandexperiment,thevariability(min-imumandmaximumvalues)oftheexperimentalpressuredistributionsaveragedover7periodsisalsoreportedinFig.2(right).Itisobservedthattheforwardexcur-sionoftheshockcansignificantlydifferfromitslongtermaveragedvalue,itsaftexcursionbeingmorerepeatable.Theshockmovementisfarfrombeingexactlype-riodicintheexperimentandpressurefluctuationsdistributionaveragedoversuchashorttimeinthesimulationcanonlybeexpectedtoliebetweentheupperandlowerboundsofthe7periodsaveragedexperimentaldistribution.SomeimprovementisregisteredinthecaseB2withrespecttoothercases.Acorrelationbetweentheup-streamexcursionoftheshockandthemaximumleveloffluctuationisevidenced,aupstreammovementoftheshockgivingasmallerlevelofpressurefluctuations.Asmallpartofthiserror(about7%atx/c=0.9)canbeattributedtothefactthatpressuresignalswereacquireduptoa5kHzcut-offfrequencybuttherestoftheoverestimationremainstobeexplained.Theuseofadoubledspan(gridB)signifi-cantlyreducesthefluctuationsnearthetrailingedgewithrespecttotheoriginalgrid.TheanalysisoftheinstantaneousfieldsobtainedongridAhasevidencedthatthisoverestimationwasduetothepresenceofintensetwo-dimensionalcoherentstruc-turesdevelopingwhentheflowseparatesfromtheshockuptothetrailingedge.
4E.GarnierandS.Deck
ThespanofthegridBallowsthethree-dimensionalisationofthesestructureswhichlimitstheirintensityandsubsequentlythewallpressurefluctuations.
Fig.2Averagedpressurecoefficientdistribution(left)andrmspressuredistribution(right)
Theprofilesofaveragedandfluctuatinglongitudinalvelocityatx/c=0.35areplottedinFig.3forcasesB1andB2.Thesedataevidencethatupstreamoftheinteractionthevelocityfieldiswellestimated,theagreementwiththeLDA(LaserDopplerAnenometry)measurementsbeingalmostperfectfortheB1case.Thisre-sultisfarfrombeingtrivialsincetheflowundergoesanumericallyforcedtransitionatthesamestationthanintheexperiment(x/c=0.07).IntheB2case,thetransitionoccursinstantaneouslyatx/c=0.07whereasittakessomedistanceintheB1case(inFig.2(right),thelevelofpressurefluctuationsoftheB1casereachestheoneoftheB2caseatx/c=0.12).ThisleadstoafullervelocityprofileintheB1casethanintheB2case,theformercasebeinginfairagreementwiththeexperimentbothforthemeanandfluctuatingvelocityfield.ThisfullervelocityprofileisconsistentwithadownstreamshiftedshockpositionforcaseB1evidencedinFig.2.ThissuggeststhatapossibleimprovementontheshocklocationdeducedfrompressuresensorsmightbedoneattheexpenseoftheagreementwiththeLDAmeasurementsinthevelocityprofilesupstreamoftheshock.
Fig.3Meanlongitudinalvelocity(left)andlongitudinalvelocityfluctuations(right)profilesatx/c=0.35(GridB).
Large-eddysimulationoftransonicbuffetoverasupercriticalairfoil5
Downstreamfromtheinteraction(atx/c=0.75),onecanobserveinFig.4thattheagreementofthesimulationswithbothaveragedandfluctuatinglongitudinalvelocityprofilesismorethansatisfactory.AbetteragreementwithLDAdataisregisteredinthefirstthirdoftheboundarylayerfortheB2casethanfortheB1case.Thecontraryisobservedabovethislimit.Itishoweverworthwhiletonoticethatbetweenx/c=0.4andx/c=0.6,experimentalandnumericalvelocityprofilesdiffersignificantlysincetheshockisnotlocatedatthecorrectmeanposition.
Fig.4Meanlongitudinalvelocity(left)andlongitudinalvelocityfluctuations(right)profilesatx/c=0.75(GridB).
ThesignaldurationbeingtooshorttoundertakeaphysicalanalysisoftheflowwiththeB2caseresults,thefollowingdiscussionisbasedontheresultsoftheB1case.
4spectralanalysis
DuetotheshortdurationoftheLESsimulations,anauto-regressive(AR)modelmethodhasbeenusedtocomputethePowerSpectralDensityofthepressure.In-deed,thismethodiswelladaptedtostudyshortdatathatareknowntoconsistofsi-nusoidsinwhitenoise[9].TheARparametersareobtainedwithBurg’smethod[10].Thepressurespectrumforx/c=0.9iscomparedtoexperimentinFig.5.Theoc-currenceofstrongharmonicpeakshighlightstheperiodicnatureofthemotion.Ontheexperimentalside,themainpeakat69Hzrepresentsthefrequencyoftheself-sustainedmotionoftheshockovertheairfoil.Aslightlyhigherfrequencynear72Hzisfoundinthecomputation.Thetwofirstharmonicsofthemainpeakarealsocorrectlycapturednumericallybutadditionalharmonicsarealsoobservedinthecomputation.Thisillustratesthefactthattheperiodicityofthecomputationisstrongerthanintheexperiment.
6E.GarnierandS.Deck
Fig.5PSDofpressurefluctuations.
5Spaceandtimescales
Once,themainstatisticalandspectralfeaturesoftheflowhavebeenfound,itisworthwhiletostudythekinematicsofthesepressurewaves.Tothisend,letusconsiderthefluctuatingpressureatdifferentstations.Thetwo-pointtwo-timecor-relationcoefficient:R(∆ξ,τ)=√P(x1,t)P(x1+∆ξ,t−τ)(P2(x1))
√(P2(x1+∆ξ))
establishesthecorrelation
betweentwosignalslocatedatabscissax1etx1+∆ξandseparatedbyatimedelayτ.Theconvectionvelocitycanbeobtainedastheslopeofthelinearfittingofthe∆ξversusτmax(τmaxrepresentsthedelaywherethecorrelationcoefficientreacheditsmaximum),asillustratedinFig.6.
Fig.6Propagationvelocitiesobtainedbyaleastsquarefittingofthelinearrelationbetweentheseparationdistance∆ξandtimedelayτ(filledsymbol:exp,solidline:uppersideoftheairfoil,dashedline:uppersideoftheairfoil).
Ontheupper-sideoftheairfoil,adownstreampropagationvelocityequal6.1110−3U∞isclearlyidentifiedfortheLESandappearstobeslightlylowerthanintheexperi-ment.Onthelowersideoftheairfoil,aforwardmotionatvelocity0.34110−3U∞isevidenced.Thelattervelocityisclosetotheupstreamtravellingacousticwavesonthelowersideoftheairfoil.
Large-eddysimulationoftransonicbuffetoverasupercriticalairfoil7
6Discussion
Toassessthefrequencyofthemotion,Lee[7]proposedthattheperiodoftheshockoscillationshouldagreewiththetimeittakesforadisturbancetopropagatefromtheshocktothetrailingedgeaddedtothetimeneededforanupstreammovingwavetoreachtheshockfromthetrailingedge.Asimplifiedmodelhasbeenusedinreference[1]toassessthetotaldurationtocompletesuchaloop:T=vc−xs+
downstream
c−xs|vupstream|wherecisthechordandxsisthemeanlocationoftheshockwave.xScanbeobtainedbynotingthefirstabscissawheretheskewnessofpressurefluctuationsiszero.Onegets(xs/c)LES=0.52while(xs/c)exp=0.45.Thevelocityofupstream-travellingacousticwavesisvupstream=a(M−1)whereaisthelocalspeedofsoundinthefieldoutsidetheseparatedarea.WithM=0.8anda=330m/s,theLee’sequationgivesf=1/T≈110HzwhichishigherthanthefrequencyFLES≈70Hz.Morerecently,Crouchetal.[8]advocatedthattransonicbuffetresultsfromglobalinstabilitywheretheunsteadinessischaracterizedbyphase-lockedoscillationsoftheshockandtheseparatedshearlayer.Withinthisscenario,theregiondownstreamfromtheshockisnottheonlyregioncontributingtothefeedbackloop.Indeed,anupstreamtravellingacousticmotionhasbeenhighlightedonthelowersurfaceoftheairfoil(seeFig.6).Adeeperinvestigationofthesephenomenawillfollowthepresentwork.
AcknowledgementsThisworkhasbeenpartlysponsoredbytheFrenchNationalResearchAgency(projectANR-07-CIS7-009-04).
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