; TeX output 2000.06.13:1138J5YO%2 &G cmb10Subgrid-ScaleLMozdelsforCompressibleLarge-Eddy~SimulationsDF cmmib10?Z5I & cmb10M.LPinoMartqn <o cmr9Departmen9t pofAerospaceEngineeringandMechanics,UniversityofMinnesota, {=}110 pUnionSt.SE,MinneapAolis,MN55455,USA /.pino@aem.umn.edu?5͹UgoLPiomelli [?aDepartmen9t pofMechanicalEngineering,UniversityofMaryland, College pP9ark,MD20742,USA ugo@eng.umd.edu>#5Β+GrahamLV.Candler  cmmi10"(Moin)': cmti10etal.,1991;El-Hadyetal.,1994)orfortheenthalpy perunitmassh(Specialeetal.,1988;Erlebacheretal.,1992).IntheseequationstheSGSheatfluxwasmoGdeledinamannersimilartothatusedfortheSGSstresses,whiletwoadditionalterms,theSGSpressure-dilatationٓRcmr7dilandLtheSGScontributiontotheviscousdissipation"v,wereneglected.V*reman)Tetal.(1995a,b)pGerformedaprioritestsusingDNS)KdataobtainedfromthecalculationofamixinglayeratMachnumbGersfrom0.2to0.6.TheyfoundthattheSGSpressure-dilatationdil UandSGSC>viscousC@dissipation"v]areofthesameorderasthedivergenceoftheSGSheatfluxQ 0ercmmi7j6,andthatmoGdelingO("vEimprovestheresults,espGeciallyatmoderateorhighMachnumbGers.Theyalsoproposedtheuse ofatranspGortequationforthetotalenergyofthefilteredfield,ratherthaneithertheenthalpy ortheinternalenergyequations;thesameunclosedtermsthatappGearintheinternalenergyandenthalpyequationsLarealsopresentinthisequation.V*ery^Tfewcalculationshave^TbGeencarriedoutusingthetransportequationforthetotalenergy*,despitetheidesirablefeaturethatitisaconservedquantity*,andthatalltheSGSHtermsinthisequationcanbGecast:inconservqativeform.Thisequationhasadi erentsetofunclosedterms,whosemoGdelingisnotveryadvqancedݜyet.NormandandLesieur(1992)pGerformedcalculationsofatransitionalboundarylayer,andemoGdeledonlytheSGSFheatflux,neglectingalltheotherterms.Knightetal.(1998)performedthe5LES ofisotropichomogeneousturbulenceonunstructuredgridsandcomparedtheresultsobtainedwiththeSmagorinsky(1963)moGdelwiththoseobtainedwhentheenergydissipationwasprovidedonlybyGthedissipationinherentinthenumericalalgorithm.TheymoGdeledtheSGSGheatfluxandanSGSturbulentRdi usionterm,andneglectedtheSGS=n(whereisthegridsizeandn=4,8,16,and32)wereused.NoteA thatthegridresolutionishighenoughthatnez=2A wouldcorrespGondtoaDNS.Thelo-cationjofthevqariousfiltercuto salongtheenergyspGectrumatt=t 8V=6:5areshowninFig-ureC1;theycoverCthedecayingrangeofthespGectrum(nD{=4),Ctheinertialrange(nD{=8and;16),andtheenergy-containingrange(n5=32).;Withthefiltersused,respGectively*,5%,15%,40%,and70%ofthetotalturbulentkineticenergyresidesintheSGSs.Thetwointer-mediatevqaluesarerepresentativeofactualLES=calculations,inwhichtheSGSkineticenergyisX}typicallybGetween15%and30%ofthetotalenergy*.AX8higherpGercentageofSGSX8energyingen-eralindicatesanunder-resolvedcalculation.Inthefollowing,resultswillbGeshownfor}feUW=8,exceptwhenLevqaluatingthee ectoffilter-width.TheaccuracyofamoGdelisevqaluatedbycomputingtheexacttermR%|anditsmodelrepresentationMVand;comparingthetwo;usingthecorrelationcoGe icientC(RDz)andtheroot-mean-square(rms)amplitudesLθh(RL8hRǸi)^2|si^1=2ޘandh(MOhMi)^2|si^1=2 ʲ.ThecorrelationcoGe icientisgivenby\郵C(RDz)=<$h(RL8hRi)(MOhMi)iKwfeZ zM(㏸h(RL8hRi)r2|sih(MOhMi)r2i)|1=2.;8(15)hwhere?thebrackets?hidenoteaveraging?overthecomputationalvolume.A'\pGerfect"modelwouldgiveacorrelationcoGe icientof1.Inthefollowing,thequantitiesplottedaremadenondimensionalusingtheLinitialvqaluesof,u^09,and.'i\vʙ4.MoUVdelsLfortheMomentumEquation`ThebSGSbstresses(8)aretheonlyunclosedtermthatappGearsinthemomentumequation.V*arioustypGesof4moGdelshave4beendevisedtorepresenttheSGSstresses.Eddy-viscositymoGdelstrytoreproducetheBJ5䍍Subgrid-ScaleXMowiththeturbulenceenergycascade.Y*oshizawa>(1986)proposedaneddy-viscositymodelforweakly'compressibleturbulentflowsusingamultiscaledirect-interactionapproximationmethoGd.TheanisotropicpartoftheSGSstressesisparametrizedusingtheSmagorinsky(1963)moGdel,whiletheSGSLenergyk+Bk ismoGdeledseparately:f$Tij Cظ<$lijlwfe | (֍>_3k+Bk 8=C2፴s32}feUWUW2 ʟ~fe+gjx䍑heSTj^x䍑 pe Sijl<$lijlwfe | (֍>_3x䍑eSk+Bk^IQ=C2፴s3 ij ;k+Bk 8=CI2~fe+g7}feUW+2jx䍑heSTj2C=CI z;8(16)4׍withLεCs R=0:16,CI]=0:09,andjx䍑heSTj=(2x䍑heSijx䍑e ,SijY~)^1=2 ʲ.Moin*etal.(1991)propGosedamodificationoftheeddy-viscositymodel(16)inwhichthetwomoGdelpcoe icientsweredetermineddynamically*,ratherthaninputapriori,usingtheGermanoidentityLij =Tijϸ6c׵ij(Germano,1992),whichrelatestheSGSstressesij Ǡtothe\resolvedturbulentstresses"\rLij =`8d1ȍ~fe9Ygui~feguj'=~fe+g3`;ZrLcˠ~fe9Yguid~feguj Z=b~fe+g߲,andthesubteststressesTij=kb~fe+g8F1ȍ g GuiTLuj obˠ~fe+gT >ꬍ >e LuiTꬍeZIuǟj A7(whereqōs7;\qȫef ²=\xYRc fe $u$f =b~fe+g߲,and@thehatrepresentstheapplicationofthetestfilterx䍑TbG4ofcharacteristicwidthx䍑bͲ=,2}feUWUW)thatappGearif @thefilterx䍑CbG isappliedto(3).Moinetal.(1991)determinedthemoGdelcoe icientsbysubstituting(16)into theGermanoidentityandcontractingwithx䍑reSij6ɲ.InthepresentpapGerthecontractionpropGosedby Lilly(1992)tominimizetheerrorinaleast-squaressenseareusedinstead.Accordingly*,thetwomoGdelcoGe icientsLforthedynamiceddy-viscosity(DEV)moGdelwillbegivenby {C~4=C2፴s=<$ӚhLij MijiKwfe*( (֍hMk+Bl}XMk+Bli/Vj;CI]=<$GnhLk+Bk W iKwfe 5 (֍h ob8 i%˳;8(17)gwhereLε ij =2x䍑=b^2 uɫb ʟ~fe+gRj8ڪqȍx䍑heSTj(8ڪqȍx䍑heSij,8ij8墲qȍx䍑sܫe S Lk+Bkl=3),Mij = ij Cظ-c ij{,and N4=2x䍑=b^2 uɫb ʟ~fe+gRj8ڪqȍx䍑heS_j^2|s.Scale-similarmoGdelsarebasedontheassumptionthatthemostactiveSGSsarethoseclosertothecuto ,WandthatthescaleswithwhichtheyinteractarethoseimmediatelyabGovethecuto wavenumbGer(Bardinah#etal.,1980).Thus,scale-similarmoGdelsemploymultipleopGerationstoidentifythesmallestresolvedXscalesandusethesmallest\resolved"stressestorepresenttheSGSXstresses.AlthoughthesemoGdelsLaccountforthelocalenergyevents,Ltheyunderestimatethedissipation.SpGecialeetal.(1988)proposedtheadditionofascale-similarparttotheeddy-viscositymodelofY*oshizawau(1986)introGducingthemixedmodel.Inthisway*,utheeddy-viscositycontributionprovidesthe-dissipationthatisunderestimatedbypurelyscale-similarmoGdels.ThismixedmodelwasalsousedbyLErlebacheretal.(1992)andZangetal.(1992),andisgivenby*r[-ij Cظ<$lijlwfe | (֍>_3k+Bk 8=CsF: ij+8Aij<$lijlwfe | (֍>_3Ak+Bk W ;k+Bk 8=CI BZ+8Ak+Bk;8(18);whereLεAij =~fe+g O(T*gꬍ\eui je ʵuHj6ԸTNeꬍe8u^iT \Leꬍ ie Fu(j).Erlebacheretal.(1992)testedtheconstantcoGe icientmoGdelaprioribycomparingDNSandLESresultsCDofcompressibleisotropicturbulenceandfoundgoGodCDagreementinthedilatationalstatisticsofthelflow,aswellashighcorrelationbGetweentheexactandthemoGdeledstresses.Zangetal.(1992)comparedtheDNS{andLESresultsofisotropicturbulencewithvqariousinitialratiosofcompressibletototalikineticenergy*.TheyobtainedgoGodiagreementfortheevolutionofquantitiessuchascompressiblekineticLenergyandfluctuationsofthethermoGdynamicvqariables.Dynamic^moGdeladjustmentcanbealsoappliedtothemixedmodel(18),toyieldthedynamicmixedmoGdelL(DMM)nhC~4=<$KhLij Miji8hNij MijiKwfeZH (֍hMl `k}XMl `ki_v+;CI]=<$KhLk+Bk 8Nk+Bk W iKwfe5џ (֍ Nh ob8 i:O;8(19)BwithLεBij =kb~fe+g G(?qōFg;T\ꬍ\eu~iT jꬍ je ʵuHj6Ըx獟qōNe;Tꬍe8u^i ix獟qō \Le;T iꬍ ie Fu(j),andNij =Bij Cظx䍑Exd8Aijٲ.An"oissuethatrequiressomeattentionisthenecessitytomoGdelseparatelythetraceoftheSGSstressesk+Bk W .Y*oshizawa(1986),Moinetal.(1991),andSpGecialeetal.(1988)proposedaseparatemodelfor{pthisterm.Erlebacheretal.(1992)conjecturedthat,forturbulentMachnumbGersMtLn<0:4thistermisnegligible;theirDNSofisotropicturbulenceconfirmthisconjecture.Zangetal.(1992)confirmedTJ5䍍6״M.PJ.XMart8n,U.Piomelli,andG.V.CandlerYthesexresultsap}'osteriori:theyrancalculationswith0CI]0:066x(thelattervqalueistentimeshigher thanLthatpredictedbythetheory)andobservedlittledi erenceintheresults.Comte andLesieur(1998)propGosedincorporatingthistermintoamodifiedpressureP}.ThisleadstoLthepresenceofanadditionaltermintheequationofstate,whichtakestheform͍P=~fe+g ORx䍑ɫeǵT 6+<$l3 85lwfe| (֍ z>6 µk+Bk W ;8(20)(8for) UP= K5K&fes3 theadditionaltermiszero,andfor = K7K&fes5 itmightbGenegligible,unlessMtBisverylarge.ThisobservqationccanbGeusedtoexplainap}'osterioritheinsensitivityoftheLEScresultstothevalueofCIdiscussedbyZangetal.(1992):theSGSstresstracecanbGeapproximatelyincorpGoratedinthepressurewithnomoGdificationtotheequationofstate.AnotherfactormaybethatboththecalculationsbyErlebacher>etal.(1992)andthosebyZangetal.(1992)usedmixedmoGdels,inwhichthescale-similarparthgaveacontributiontothenormalSGSLstresses.Thus,k+Bk 8istakenintoaccount,atleastpartially*,byLthescale-similarcontribution.If themixedmoGdelisused,thetraceoftheSGSstressescanbeparameterizedwithoutrequiringaseparateLterm.Aone-coGe icientdynamicmixedmodel(DMM-1)wouldbeoftheformc!ij =C ij Cز+8Aij ;8(21)withC܍+۵C~4=<$KhLij Miji8hNij MijiKwfeZH (֍hMl `k}XMl `ki_v+:8(22)qTheLmoGdelsDEV(16){(17),DMM(18){(19),andDMM-1(21){(22)areevqaluatedinFigures2{4. Figurep2(a)showsthattheDMM-1moGdelgivesthehighestcorrelationforthediagonalcompGonentsofthe SGS stresstensor;Figure2(b)showsthatneithertheeddy-viscositymoGdelnorthetwo-coGe icientmixedmoGdelDMMpredictthermsoftheSGSstressesaccurately*.TheDMM-1moGdelgivesthemostaccurateLpredictionamongthosetested.Figure63(a)showsthecorrelationcoGe icientfortheo -diagonalcompGonentsoftheSGS6stress.Asinincompressibleflows,theeddy-viscositymoGdelgivesverypGoorcorrelation(near0.2),whilemuchimproved"resultsareobtainedwiththemixedmoGdels.Notethatthecorrelationcoe icientforDMM"Qand'OeFFigure: 2.A9prioricomparisonofthenormalSGSstressesq11N.(a)Correlationcomixinglayer>studiedbyV*remanetal.(1995b),inthisflowthepressuredilatationdil isnegligible,andTtheviscousdissipation"v#ϲisoneorderofmagnitudesmallerthanthedivergenceoftheSGSTheatflux.vInthetotalenergyequation(6),theSGSLturbulentdi usion@8Jj6=@xj&"isvcomparablewiththedivergence#oftheSGS#heatfluxandtheSGSviscousdi usionisoneorderofmagnitudesmallerthantheLotherterms.InthissectionseveralmoGdelsforthemoresignificanttermsareexamined.!+5.1.jJSGSLHeatFluxVTheʊsimplestapproachtomoGdelingtheSGSjheatfluxQj6istouseaneddy-di usivitymoGdeloftheformPAQjIJ=<$Y~fe+g/T33wfeHߟ (֍Pr T<$;@x䍑::e8Txwfe (֍@8xj)=C<$O}feUW ?2~fe+gPjx䍑heSTjOwfe돟 (֍XPrT<$%a@x䍑::e8T$e8ui F!Dz(uj6;uk됲)8euj ) (uiTL;uk됲)euk (uiTL;uj6)~fe+g k֫e uiyeujiʫe uk&:8(39)UsingLthisnotationtheturbulentdi usiontermcanbGewrittenas2JjIJ=!Dz(uj6;uk됵;uk)8+2\euk !Dz(uj6;uk);8(40)since?7!Dz(uj6;uk됲)=jgk<.Usingthisformalism,scale-similarmoGdelscanbederivedbyapproximatingthequadratic(1termsusingthefilteredveloGcitieseuj@toreplacetheveloGcitiesuj6;forinstance,onecanwritewR!Dz(uiTL;uj6)(\eui ʵ;geuj Ҳ) )~fe+gG(qg+uiTLuj.b[e8ui ie Fuj6Բ)R ~fe+g G`Tgꬍe@۵uideNuj&wTNeꬍe8u^iT \Leꬍ ie Fu(j6ԟ`K-:8(41),If}thepropGortionalityconstantin(41)issettoone,thescale-similarpartofthemixedmoGdel(18)isobtained.LAnalogously*,thetripleproGductcanbewrittenasW1]2Jjj/(=tF!Dz(uj6;uk됵;uk)8+2\euk !Dz(uj6;uk)j/('tFCJ!Dz(\euj *;geuk O;geuk)8+2\euk (uj6;uk됲)\uj/(=tFCJ@X`h <~fe+gT3gꬍ,2esuj\euk'je'duk4B8~fe+gT $eꬍ ke uȕjTeꬍ\eAukTeꬍeOu#]͟k*=TNeꬍe8u^j ) Ak+Bk 82Teꬍ\eu~k Ajgk<`iE++82\euk jgk<;8(42)ύthe-lasttermisparametrizedbythesamemoGdelusedinthemomentumequation.ThecoGe icientCJcanLbGesetusingtheidentity~uTsdꬍ~fe+g&ePuj9ezukGe^ukꬍU>eu$kT)Lꬍ)Le)u/W k6{{߫b8~fe+g ~x獟qō ae;T ~ꬍ ~e us=jx獟qōe;Tꬍeucgkx獟qōde;TꬍeNu%uk,,x獟qōNe;Tꬍe8u^j ) Bk+Bk 82\x獟qōe;T\ꬍ\eu~k Bjgk<^}+82T\ꬍ\eu~k Tjgk<;1ҍS3Bjgkfkֲ=qbq2~fe+g{^{^?qō̧g;TꬍeutmjTزꬍثeudkx獟qōNe;Tꬍe8u^j ɲx獟qō >e;T ɲꬍ ɫe ) u∟k^%Y;8(44)эtoLyield6|{CJ\Ȳ=K`D `Tdꬍ~fe+gBeصuj!2e uk+ϫe+{uk8X߫b8~fe+gT ~ꬍ ~e us=jTꬍeucgkTꬍeNu%uk)`qPjo8Qj6Pj`EKfeʥ (֍AhPk됸Pki#;8(45)ҍwheresgPjʴ=ҟ^|b~fe+g?vqōg;TꬍeYϵuMjTꬍeIuwkTKƲꬍKƫeuk߫b8~fe+g ~x獟qō ae;T ~ꬍ ~e us=jx獟qōe;Tꬍeucgkx獟qōde;TꬍeNu%uk,,x獟qōNe;Tꬍe8u^j ) Bk+Bk 82\x獟qōe;T\ꬍ\eu~k Bjgk<^ʴҟ^㌍Jdt~fe+gTfgꬍ e'ujeQukeD_uk"M*:dX8~fe+gT $eꬍ ke uȕjTeꬍ\eAukTeꬍeOu#]͟k*=*cdXTNeꬍe8u^j ) Ak+Bk9 82**dXTeꬍ\eu~k AjgkK^D$;8(46) ri[Qjʴ=2`TKꬍKeu ] kHTjgk T/dꬍe8uk jgkG`LE:8(47),Figure10(a)showsthecorrelationcoGe icientforthetwomoGdels(37)and(42)andusing(21){(22)to=SGS!viscousdi usion@8Dj6=@xjis>thesmallestofthetermsinthetotalenergyequation,andis abGoutU5%ofthedivergenceofQj6.Nomodelforthistermhasbeenproposedintheliteraturetodate.OneLpGossibilityistoparametrizeitusingascale-similarmodeloftheformqJ5䍍14״M.PJ.XMart8n,U.Piomelli,andG.V.Candler荟 Figure412.AprioricomparisonofthesumoftheSGStermsinthetotalenergyequation(6).(a)Correlationcoheatfluxalone,Figure6(b),butmoreaccuratethanfortheSGSturbulentdi usion alone,LFigure10(b).Figure12showsthattheoverallpGerformanceisverygoGod.)D?6.řConclusions`Severalmixedandeddy-viscositymoGdelsforthemomentumandenergyequationshavebGeentested.TheGveloGcity*,pressure,density*,andtempGeraturefieldsobtainedfromtheDNSGgofhomogeneousisotropicturbulenceatRe =S50,Mt'=0:52werefilteredandtheunclosedtermsinthemomentum,internalenergy*,Landtotalenergyequationswerecomputed.Inthemomentumequation,mixedmoGdelswerefoundtogivebGetterprediction,intermsofbothcorrelationmbandrmsamplitude,thanthepureeddy-viscositymoGdels.ThedynamicadjustmentofthemoGdelLcoe icientwasbGeneficial,asalreadyobservedbyMoinetal.(1991).IntheinternalenergyandenthalpyequationsonlythedivergenceoftheSGSheatfluxwassignificantin thisflow;theSGS pressuredilatationdil andviscousdissipation"v,whichweresignificantinthemixing̀layerstudiedbyV*remanetal.(1995b),werefoundtobGenegligiblehere.Onceagain,mixeddynamicLmoGdelsgaveLthemostaccurateresults.Inthetotalenergyequationtwoadditionaltermsarepresent,oneofwhich,theturbulentdi usion@8Jj6=@xj,issignificant.ThemoGdelproposedbyKnightetal.(1998)andanewscale-similarmoGdelpropGosedherecorrelatewellwiththeactualSGSDturbulentdi usion,andpredictthecorrectrmsamplitude.~However,thenewscale-similarmoGdelwasfoundtobGemoreaccurate.A~mixedmodelfortheqSGSqviscousdi usionwasalsopropGosedandtested,althoughthistermismuchsmallerthantheothers.TheaccuracyofthemoGdelsforthesumofthetermswasalsoevqaluated,anditwasfoundthatthe'moGdelsproposedstillpredictnearlythecorrectrmsamplitude,andanacceptablevqalueofthecorrelationLcoGe icient.TheװresultsobtainedinthisinvestigationװarepromisingandindicatethatitispGossibletomodelaccuratelythetermsintheenergyequations.F*urtherworkmayextendtheseresultstocasesinwhichtheqpressure-dilatationissignificant,aswellastoinhomogeneousflows,andevqaluatethesemoGdelsap}'osteriori.ReferencesUdBardina,NJ.,FJerziger,J.H.,andReynolds,W.C.(1980).ImproÎvedNsubgrid-scalemoX.,andLesieur,M.(1992).Directandlarge-eddysimÎulationoflaminarbreakdowninhigh-sp cmmi10 0ercmmi7K`y cmr10ٓRcmr7u cmex10