计算流体力学(CFD)文档――10. Advanced turbulence modelling

Advancedturbulencemodelling1DavidApsley10.ADVANCEDTURBULENCEMODELLINGSPRING20111.Turbulencemodelsforgeneral-purposeCFD2.Lineareddy-viscositymodels3.Non-lineareddy-viscositymodels4.Differentialstressmodels5.ImplementationofturbulencemodelsinCFDReferencesExamples1.TurbulenceModelsForGeneral-PurposeCFDTurbulencemodelsforgeneral-purposeCFDmustbeframe-invariant–i.e.independentofanyparticularcoordinatesystem–andhencemustbeexpressedintensorform.Thisrulesoutsimplermodelsofboundary-layertype.TurbulentflowsarecomputedeitherbysolvingtheReynolds-averagedNavier-Stokesequationswithsuitablemodelsforturbulentfluxesorbycomputingthefluctuatingquantitiesdirectly.Themaintypesaresummarisedbelow.Reynolds-AveragedNavier-Stokes(RANS)Models•Lineareddy-viscositymodels(EVM)–(deviatoric)turbulentstressproportionaltomeanstrain;–eddyviscositybasedonturbulencescalars(usuallyk+oneother),determinedbysolvingtransportequations.•Non-lineareddy-viscositymodels(NLEVM)–turbulentstressisanon-linearfunctionofmeanstrainandvorticity;–coefficientsdependonturbulencescalars(usuallyk+oneother),determinedbysolvingtransportequations;–mimicresponseofturbulencetocertainimportanttypesofstrain.•Differentialstressmodels(DSM)–akaReynolds-stresstransportmodels(RSTM)orsecond-orderclosure(SOC);–solve(modelled)transportequationsforallturbulentfluxes.ModelsThatComputeFluctuatingQuantities•Large-eddysimulation(LES)–computetime-varyingfluctuations,butmodelsub-grid-scalemotion.•Directnumericalsimulation(DNS)–nomodelling;resolvethesmallestscalesoftheflow.Advancedturbulencemodelling2DavidApsley2.LinearEddy-ViscosityModels2.1GeneralFormStress-strainconstitutiverelation:ijijjitjikxUxUuu32-∂∂+∂∂=-,tt=(1)Theeddyviscositytisderivedfromturbulentquantitiessuchastheturbulentkineticenergykanditsdissipationrate.Thesequantitiesarethemselvesdeterminedbysolvingscalar-transportequations(seebelow).Atypicalshearstressandnormalstressaregivenby∂∂+∂∂=-xVyUuvtkxUut3222-∂∂=-Fromthesetheotherstresscomponentsareeasilydeducedbyinspection/cyclicpermutation.Mosteddy-viscositymodelsingeneral-purposeCFDcodesareofthe2-equationtype;(i.e.scalar-transportequationsaresolvedfor2turbulentscales).Thecommonesttypesarek-andk-models,forwhichspecificationsaregivenbelow.2.2k-ModelsEddyviscosity:2kCt=(2)Scalar-transportequations(non-conservativeform):ndissipatioproductiondiffusionadvectionchangeofratekCPCxxtPxkxtkkiikiki+-+∂∂∂∂=-+∂∂∂∂=)()(DD)()(DD2)(1)()()((3)Thediffusivitiesofkandarerelatedtotheeddy-viscosity:)()()()(,tktk+=+=Therateofproductionofturbulentkineticenergy(perunitmass)isjijikxUuuP∂∂-≡)((4)Inthestandardk-model(LaunderandSpalding,1974)thecoefficientstakethevaluesC=0.09,C1=1.92,C2=1.44,(k)=1.0,()=1.3(5)Advancedturbulencemodelling3DavidApsleyOtherimportantvariantsincludeRNGk-(Yakhotetal.,1992)andlow-RemodelssuchasLaunderandSharma(1974),LamandBremhorst(1981),andLienandLeschziner(1993).Modificationsareemployedinlow-Remodels(seelater)toincorporateeffectsofmolecularviscosity.Specifically,C,C1andC2aremultipliedbyviscosity-dependentfactorsf,f1andf2respectively,andanadditionalsourcetermS()mayberequiredintheequation.Thedampingfactorfisnecessarybecauset∝y3asy→0,butk∝y2and~constant,sothatk2/yieldsthewrongpowerofy.Somemodels(notablyLaunderandSharma,1974)solveforthehomogeneousdissipationrate~whichvanishesatsolidboundariesandisrelatedtoby22/1)(2,~kDD∇=+=(6)Thisreflectsthetheoreticalnear-wallbehaviourof(i.e.2/2yk~)inaformwhichavoidsusingageometricdistanceyexplicitly.2.3k-Models(nominallyequaltokC)issometimesknownasthespecificdissipationrateandhasdimensionsoffrequencyor(time)–1.Eddyviscosity:kt=(7)Scalar-transportequations:)()(DD)()(DD2)()()()(*-+∂∂∂∂=-+∂∂∂∂=ktiikikiPxxtkPxkxtk(8)Again,thediffusivitiesofkandarerelatedtotheeddy-viscosity:)()()()(,tktk+=+=Theoriginalk-modelwasthatofWilcox(1988a)wherethecoefficientstakethevalues1009*=,95=,403=,(k)=2.0,()=2.0(9)butinlaterversionsofthemodelthecoefficientsbecomefunctionsof/2k(seeWilcox,1998).Menter(1994)devisedashear-stress-transport(SST)model.Themodelblendsthek-model(whichis–allegedly–superiorinthenear-wallregion),withthek-model(whichislesssensitivetothelevelofturbulenceinthefreestream)usingwall-distance-dependentblendingfunctions.Transportequationsaresolvedforkand,butthisisanoddchoiceAdvancedturbulencemodelling4DavidApsleybecauseinafreeflowwithnowallboundaries(e.g.ajet)themodelissimplyatransformedk-model.Allmodelsofk-typesufferfromaproblematicwallboundarycondition(asy→0)whichisroutinelyfudged!2.4BehaviourofLinearEddy-ViscosityModelsinSimpleShearInsimpleshearflowtheshearstressisyUuvt∂∂=-Thethreenormalstressesarepredictedtobeequal:kwvu32222===whereas,inpractice,thereisconsiderableanisotropy;e.g.inthelog-lawregion:6.0:4.0:0.1::222≈wvuActually,insimpleshearflows,thisisnotaproblem,sinceonlythegradientoftheshearstressuv-playsadynamically-significantroleinthemeanmomentumbalance.However,itisawarningofmoreseriousproblemsincomplexflows.uvU(y)yAdvancedturbulencemodelling5DavidApsley3.Non-LinearEddy-ViscosityModels3.1GeneralFormThestress-strainre