FLUENT 高手进阶―Numerical Implentation of Low-Reynolds

AppendixANumericalImplementationofLow-Reynolds-Number(LRN)ModelinFLUENTPage210_________________________________AppendixANumericalImplementationofLow-Reynolds-Number(LRN)ModelinFLUENTA.1IntroductionFLEUNTVersion4.32hasthreebuilt-inturbulencemodels.Theyarethek-εmodel,theReynoldsStressmodelandtheRNGk-εmodel.Thek-εmodelisthestandardkε−model,whichneedsspecialtreatmenttoitsnear-wallflowcomputationasdiscussedinChapter2.FLUENTsuppliesthreeoptionsforthesespecialtreatments:thestandardwallfunction,thenon-equilibriumwallfunction,orthetwo-layerzonalmodel.Thereisnodirectmodelsuppliedtoallowthestraightforwarduseofagenerallow-Reynolds-number(LRN)k-εmodelwithitsownturbulentviscositydampingfunctionandspecificwallconditions.Fortunately,FLUENT’sUser-DefinedSubroutine(UDS)optionallowstheusertocustomisetheturbulencemodels.Inthisappendix,theimplementationoftheLRNk-εmodel,usingFLUENT’sUDS,isdiscussedindetail.A.2ImplementationofLRNk-εmodelinFLUENTLetusfirstcomparetheequationsofthestandardk-εmodelandthegeneralequationsoftheLRNk-εmodel.AppendixANumericalImplementationofLow-Reynolds-Number(LRN)ModelinFLUENTPage211_________________________________Thestandardk-εequations:Turbulentviscosity:ερµµ2tkC=Continuity:0xUii=∂∂Momentum:()+=++−=+ijjijtijijixUxUxxPxUUtU∂∂∂∂∂∂µµ∂∂∂∂ρ∂∂ρTurbulencekineticenergy(k):()+−∂∂+∂∂=+jktjjiijjitjjxk/xxUxUxUxkUtk∂∂σµ∂∂ρε∂∂µ∂∂ρ∂∂ρDissipationrate(ε):()+−=+jtj22jiij1jjx/xkCxUCxUt∂∂εσµ∂∂ερ∂∂τ∂∂ερ∂∂ερεεεWallboundaries:Wallfunction()5.5yln44.2U+=++(A-1)ThegeneralLRNk-εequationsare:Turbulentviscosity:ερµµµ2tkCf=Continuity:0xUii=∂∂Momentum:()+++−=+ijjijtijijixUxUxxPxUUtU∂∂∂∂∂∂µµ∂∂∂∂ρ∂∂ρTurbulencekineticenergy(k):AppendixANumericalImplementationofLow-Reynolds-Number(LRN)ModelinFLUENTPage212_________________________________()kjktjjiijjitjjDxk)/xxUxUxUxkUtkΠ++++−+=+∂∂σµµ∂∂ρε∂∂∂∂∂∂µ∂∂ρ∂∂ρDissipationrate(ε):()εεε∂∂εσµµ∂∂ερ∂∂∂∂∂∂µε∂∂ερ∂∂ερΠ++++−+=+Ex/xkCfxUxUxUkCfxUtjktj222jiijjit11jjWallboundaries:22wyk,0k,0V,0U∂∂====νε(A-2)WenoticethatsomeextratermsoccurinLRNequations-----thedampingfunctionsand,themoleculardiffusiontermsinkandεequationsandsourcetermsD,,Eand1f,fµkΠ2fεΠ.Also,wallboundaryconditionsforthetwogroupsofequationsarequitedifferent.BecausewewanttouseFLUENT’sbuilt-ink-εmodelasafoundationimplementingtheseextratermsisessential.TheUser-DefinedSubroutineUSRMUTprovidestheuserwiththeoptionofdefiningtheeffectiveviscosity.Theturbulentviscositydampingfunctionfµisimplementedinthissubroutine.ThesourcetermsD,,EandkΠεΠ,moleculardiffusiontermsanddampingfunctionsintheLRNkandεequationsarespecifiedusingthesourcetermUser-DefinedSubroutineURSTRM.AppendixANumericalImplementationofLow-Reynolds-Number(LRN)ModelinFLUENTPage213_________________________________ApplicationoftheLRNwallboundaryconditionsisnotsostraightforward.Ifawallboundaryisspecified,FLUENTimposesthestandardboundaryconditionsinawaywhichrendersmodificationsbytheUDSinactive.A.2.1TurbulentviscositydampingfunctionfµOneofFLUENT’sphysicalpropertyUser-DefinedsubroutinesisUSRMUT.WhenturbulenceisactiveinFLUENT,theuserhastheoptionofdefiningtheeffectiveviscosityusingthisfunction.Theeffectiveviscosityµeisdefinedastmeµµµ+=,(A-3)wheremµisthemolecularviscosityandtµistheturbulentviscosity.ThemainprogramlineofFLUENTdefaultUSRMUTis,USRMUT=MUMOLE+CMU*DENSTY*KAY**2/MAX(SMALL,EPSILN)Herethedefaultprogramcalculatestheeffectiveviscosityusingthestandardk-εmodelformula,ερµµµ2mekC+=(A-4)However,theeffectiveviscosityofthegeneralLRNmodelintroducesadampingfunctionfµ,suchthatερµµµµ2mekfC+=.(A-5)AppendixANumericalImplementationofLow-Reynolds-Number(LRN)ModelinFLUENTPage214_________________________________Forexample,MKmodel’s(MyongandKasagi,1990)dampingfunctiontakestheformas−−+=+70yexp1R45.31fTµ.(A-6)ThustheprogramismodifiedasUSRMUT=MUMOLE+CMU*fmu*DENSTY*KAY**2/MAX(SMALL,EPSILN)Inordertoobtain(fmuinprogram)beforethisline,thefollowingadditionalcalculationstogetherwithEquation(A-6)shouldbeincludedintheprogram,µfνε2TkR=,,dydUwµτ=ρττwu=,uyy+=τν(A-7)Also,topassrelateddatacalculatedbyFLUENTtotheaboveequations,someFLUENT“include”functionsmustbedeclaredintheprogram.FinalsampleprogramtoimplementtheturbulentviscositycomputationfortheMKmodelisshownin.SimilarprocesscanbeappliedtootherLRNmodels.1-AAppendixA.2.2SourcetermsD,EandkΠ,dampingfunctions,andmoleculardiffusionterms1f2fThekandεtransportequationsforthestandardk-εmodelorgeneralLRNk-εmodelcanbewritteninthefollowinggeneralform,Convection–Diffusion=ΣSources.(A-8)AppendixANumericalImplementationofLow-Reynolds-Number(LRN)ModelinFLUENTPage215_________________________________FLUENT’suser-definedsubroutineURSTRMallowstheusertocustomiseuser-specifiedsourcetermsontherighthandsideofEquationA-8usingthisfunction.AccordingtoEquationA-8,theadditionaltermsofLRNk-εmodels,D,EandkΠ,dampingfunctionsand1f2fcanbeclassifiedintothesourcetermsoftherighthandsideofEquationA-8.Forthemoleculardiffusiontermsofkandε,ifwemovethemtotherighthandsideofEquationA-8assourceterms,itwillnotbringtoomuchtroublebeca