计算流体力学(CFD)文档――4. The scalar-transport equation

CFD4–1DavidApsley4.THESCALAR-TRANSPORTEQUATIONSPRING20114.1Thescalar-transportequation4.2Control-volumenotation4.3Thesteady-state1-dadvection-diffusionequation4.4Discretisingdiffusion4.5Discretisingthesourceterm4.6Assemblingthealgebraicequations4.7Extensionto2and3dimensions4.8Discretisingadvection(part1)4.9Discretisationproperties4.10Discretisingadvection(part2)4.11Implementationofadvancedadvectionschemes4.12Implementationofboundaryconditions4.13SolutionofthealgebraicequationsSummaryExamples4.1TheScalar-TransportEquationThisisagenericequationfortransportedphysicalquantities(momentum,energy,...)ForanarbitrarycontrolvolumeV:=+insideVyofboundaroutinsideVSOURCEFLUXCHANGEOFRATEThetotalfluxcomprisesadvection(transportwiththeflow)anddiffusion(molecular,orduetoturbulentfluctuations).Theresultingadvection-diffusionequationforconcentrationϕmaybewritten,foreachcomputationalcontrolvolume(or“cell”)as:sourcediffusionadvectionchangeofrateSVAnCVtfaces=∂ϕ∂-ϕ+ϕ∑)()(dd(Anynon-advectivefluxesnotdescribedbygradientdiffusionareassumedtobeincorporatedinthesourceterm–seeSection2.2.5.)4.2Control-VolumeNotationThecommonestconfigurationsarecell-centredstorageorcell-vertexstorage.cell-centredcell-vertexVAuunCFD4–2DavidApsleyThiscoursefocusesonstructuredmeshesusingcell-centredstorage.(Unstructuredmesheswillbediscussedbrieflyattheendofthecourse,butthe2ndeditionofVersteegandMalalasekera’sbookgivesamuchbetterdescription.)Atypical3-dcontrolvolumeisshownright.Relativetothecellcentre(pointP)thecoordinatedirectionsarecommonlydenotedwest,east,south,north,bottom,topwith:•lowercasew,e,s,n,b,tusedforcellfaces;•uppercaseW,E,S,N,B,Tforadjacentnodes.ForaCartesianmeshthesewouldusuallycorrespondto±x,±y,±zdirectionsrespectively.Cell-faceareaswillbedenotedAw,Ae,As,An,Ab,At.CellvolumeswillbedenotedV.In2dimensionsonecanthinkofasinglelayerofcellswithunitdepth.Whenreferringtotheentiresetofcontrolvolumes(asopposedtolookingatarepresentativeone)itiscommontoswitchbetweengeographicandijknotation,sothatϕP≡ϕijk,ϕE≡ϕi+1jk,etc.EWNSBTPewtbnsjikEWNSPneswEEWWNNSSi+1,ji-1,ji,ji,j-1i,j+1i+2,ji-2,ji,j-2i,j+2ijCFD4–3DavidApsley4.3TheSteady-State1-DAdvection-DiffusionEquationConsiderfirstthesteady-state,1-dadvection-diffusionequation.Thisisworthwhilebecause:•itgreatlysimplifiestheanalysis;•itpermitsahandsolutionofthediscretisedequations;•subsequentgeneralisationto2and3dimensionsisstraightforward;•inpractice,discretisationoffluxesisgenerallycarriedoutcoordinate-wise;•manyimportanttheoreticalproblemsare1-d.Integral(Control-Volume)FormConservationforonecontrolvolumegivessourcefluxfluxwe=-(1)where“flux”meanstherateatwhichsomethingpassesthroughagivenarea(here,acellface).Ifϕistheamountperunitmass,thenthetotalfluxhasadvectiveanddiffusiveparts:advection:ϕ)(uAdiffusion:xAddϕ-IfSisthesourceperunitlengththentheadvection-diffusionequationforϕisxSxAuAxAuAwedddd=ϕ-ϕ-ϕ-ϕ(2)ConservativeDifferentialFormDividingbyxandtakingthelimitasx→0givesacorrespondingdifferentialequation:SxAuAx=ϕ-ϕ)dd(dd(3)Non-ConservativeDifferentialFormMassconservationimpliesthatuA=constantandhence(3)canalsobewrittenSxAxxuA=ϕ-ϕ)dd(dddd(4)Note:•Thissystemisquasi-1-dinthesensethatthecross-sectionalareaAmayvary.Tosolveatruly1-dproblem,setA=1.ThedifferentialequationisthenSxux=ϕ-ϕ)dd(dd•Inthisinstance,,u,andSareassumedtobeknown.InthegeneralCFDproblem,uisitselfthesubjectofaseparatetransportequation.fluxefluxwxΔareaAsourceCFD4–4DavidApsley4.4DiscretisingDiffusionGradientdiffusionisusuallydiscretisedbycentraldifferencing:ϕ-ϕ-→ϕ-xAxAPEee)(dd(5)or)(ddPEeeDxAϕ-ϕ-→ϕ-wherexAD≡(6)DisadiffusivetransfercoefficientAsimilarexpressionisusedforthewestface:)(ddWPwwDxAϕ-ϕ-→ϕ-Note:•Inthefinite-volumemethod,fluxesarerequiredatcellfaces,notnodes.•Thisapproximationfor(dϕ/dx)eissecond-orderaccurateinx(seelater).•Ifthediffusivityvariesthenitscell-facevaluemustbeobtainedbyinterpolation.•Equalweightingisappliedtothetwonodesoneithersideofthecellface,consistentwithdiffusionactingequallyinalldirections.Lateron,weshallseethatthiscontrastswithadvection,whichhasadirectionalbias.ClassroomExample101T=100CT=500CooRodx=Aninsulatedrodoflength1mand1cm×1cmsquarecross-sectionhasitsendtemperaturesfixedat100ºCand500ºCasshown.TheheatfluxacrossanysectionofareaAisgivenbyxTkAdd-wheretheconductivityk=1000Wm–1K–1.(a)Dividetherodinto5controlsectionswithnodesatthecentreofeachsectionandcarryoutafinite-volumeanalysistofindthetemperaturealongtherod.(b)Writedownadifferentialequationforthetemperaturedistributionalongtherod.Solveanalyticallyandcomparewith(a).PEWweΔxCFD4–5DavidApsley4.5DiscretisingtheSourceTermClassroomExample201T=100CoRoddTdx=0T=20CoS=-c(T-T)ooooTherodconfigurationisnowchangedsuchthattheright-handtemperatureisnolongerfixedandtherodisallowedtocoolalongitslengthatarateproportionaltoitsdifferencefromtheambienttemperature(Newton’slawofcooling);i.e.theheatlossperunitlengthis:)(∞--=TTcSwheretheambienttemperatureT=20ºCandthecoefficientc=2.5Wm–1K–1.Repeatparts(a)and(b)ofExample1.Whenthesourceisproportionaltotheamountoffluid,thetotal