A Priori Analysis of Subgrid-Scale Models for Larg

APrioriAnalysisofSubgrid-ScaleModelsforLargeEddySimulationsofSupercriticalBinary-SpeciesMixingLayersNnranlmng70*23dJosef,te&f12=tA.ALCVL.LLJetPropulsionLaboratoryCaEiforniaInstituteofTechnologyPasadenaCA91109-8099Modelsforlargeeddysimulation(LES)areassessedonadatabaseobtainedfromdirectnumericalsimulations(DNS)ofsupercriticalbinary-speciestemporalmixinglayers.TheanalysisisperformedattheDNStransitionalstatesforheptane/nitrogen,oxygen/hydrogenandoxygen/heliummixinglayers.TheincorporationofsimplifyingassumptionsthatarevalidatedontheDNSdatabaseleadstoasetofLESequationsthatrequiresonlymodelsforthesubgridscale(SGS)fluxes,whicharisefromfilteringtheconvectivetermsintheDNSequations.Constant-coefficientversionsofthreedifferentmodelsfortheSGSfluxesareassessedandcalibrated.TheSmagorinskySGS-fluxmodelshowspoorcorrelationswiththeSGSfluxes,whiletheGradientandSimilaritymodelshavehighcorrelations,aswellasgoodquantitativeagreementwiththeSGSfluxeswhenthecalibratedcoefficientsareused.IntroductionSupercriticalfluidsareofgreatinterestinextractionprocessesaswellasinpropulsiondevicessuchasadvancedgas-turbineanddieselengines,andliquidrockets.Theperformanceofthesedevicesdependsontheefficiencyoffluiddisintegrationandturbulentmixing,whichmayoccurundersupercriticalconditions.Afluidisberedefinedtobeinasupercriticalstatewhenitisatathermodynamicpressure,p,ortemperature,T,exceedingitscritical(subscriptc)value,lp,orT,;therefore,inthesupercriticalregimethereisnolongerthepossibilityofatwophase(i.e.gas/liquid)region.2Formixtures,bothp,andT,dependonthecomposition.ThepresentinterestisinAuidmixtwesathighpressuresthataresupercriticalforthepurespecies.Pastthecriticalpointofthefluid(wherematerialsurfacesnolongerexist),thedisintegrationoffluidjetsdisplaysanaspectthatChehroudieta~.~call'fingers',or'comb-likestructures'attranscriticalconditions,whichhaveanincreasinglygaseousappearancewithincreasingp.SimilarexperimentalevidencewasproducedbyMayereta1.425-forOzdisintegration.ForsupercriticalfreeN2jets,theexperimentsofOschwaldandSchik6alsoshowedsharpdensityprofiles,indicatingtheoccurrenceofsharpdensitygradients.ResultsfiomDirectNumericalSimulations(DNS)showed,asintheexperiments,thatregionsofhighdensity-gradient-magnitudeexistinbothpre-transitiona17andtransitiona~'?temporalmixinglayers,arisingboth&omtheinitialdensitystrat5cationaswellasfrommixing.10TheseDNSwereconductedusingreal-gasequationsofstatefornon-idealmixturesinconjunctionwithrealistictransportpropertiesandthermal*Scientist,AIAAMember.'fSeniorResearchScientist,AIAAAssociateFellow(correspondingauthor,[e-mail]).1of15AmericanInstituteofAeronauticsandAstronauticsdiffusion(SoretandDufour)effects.Formodelingfullyturbulentsupercriticalflowsathighpressures,LargeEddySimulation(LES),whereinonlythelarge-scalesaresimulatedandthesubgridscales(SGS)aremodeled,presentlyseemsmorecomputationallyachievableforpracticalsystemsthanDNS,whichrequiresallturbulencescalestoberesolved.TheLESequationsarederivedbyapplyingaspatialfiltertotheDNSequations,leadingtovariousunclosedterms,includingtheSGSfluxes,whicharisefromfilteringtheconvectiveterms.Giventhedistinctivesupercriticalflowcharacteristics,itisofinteresttoinquirewhetherLESmodelsdevelopedforcompressibleperfect-gasandincompressibleflowscanbeextendedtoreal-gasnon-idealmixtures.Inthispaper,DNSdatabasesfortransitionalsupercriticaltemporalmixingareanalyzedonanaprioribasis.InSectionI,theLESgoverningequationsarepresented,inordertoderivetheunclosedtermsthatneedtobemodeled.SectionI1summarizestheDNSdatabase,followedbytheapriorianalysisofthedatabaseinSection111.TheanalysisincludesexplicitmodelingfortheSGSfluxesandsimplifyingassumptionsfortheremainingunclosedterms.Finally,SectionIVcontainstheconclusionsandareasforfuturework.I.GoverningEquationsforLargeEddySimulationsTheLESequationsarederivedfromtheDNSsetbyspatialfiltering.Thefilteringoperationisdefinedas:-(1)whereGisthefilterfunctionandVisthefilteringvolume;Ghasthepropertythatforaspatiallyconstantfunction,thefilteredfunctionisidentical.totheunfilteredone.Forcompressibleflow,weuseFavrefiltering,dehedas4=wherepisthedensity.Thegoverningequationsarewrittenfortheconservativevariables4={p,pui,pet,pya)whereuiisthevelocitycomponentinthexi-direction,etisthetotalenergyandY,isthemassfractionforspeciesa..A.DNSEquationsTheconservationequationsforamixtureofNspeciesare:wheretisthetime,aistheviscousstresstensor,qistheheatflux,e=et-ekistheinternalenergy,ek=uiui/2isthekineticenergy,andJ,andw,arethespecies-massfluxandreactionrateofspeciesa,respectively.Also,Lnthispaper,theEinsteinsummationisusedforromanindices(i,j,k),butnotforGreekindices(a,P).Thethermodynamicvariablesarefunctionsoftheflowfield4:AmericanInstitut,eofAeronauticsandAstronauticswherep,Tandtheenthalpyharecomputedfromtheequationofstate(EOS);likewise,thetransportquantitiesarefunctionsof4:ForaNewtonianfluid.wherepistheviscosityandSijistherate-of-straintensor.Thespecies-massandheatfluxesoriginateinKeizer'sl1fluctuation-dissipationtheorywhichisconsistentwithnon-equilibriumthermod