计算流体力学(CFD)文档――5. The momentum equation

CFD5–1DavidApsley5.THEMOMENTUMEQUATIONSPRING20115.1Scalar-transportequationsformomentum5.2Pressure-velocitycoupling5.3Pressure-correctionmethodsSummaryExamples5.1Scalar-TransportEquationsforMomentumEachmomentumcomponentsatisfiesitsownscalar-transportequation.Foronecell:sourcediffusionadvectionchangeofrateVSAnCVtfaces=∂ϕ∂-ϕ+ϕ∑)()(dd(1)whereCisthemassfluxthroughacellface.Formomentum:concentration,ϕvelocitycomponent(ϕ=u,vorw)diffusivity,viscosity,sourcenon-viscousforcesHowever,thescalar-transportequationsformomentumdifferinthreeimportantwaysfromthoseforpassivescalarsbecausetheyare:•non-linear;•coupled;•alsorequiredtosatisfyasecondcondition:massconservation.Forexample,thex-momentumfluxthroughanx-directedfaceisuuACu)(=MassfluxCisnotconstantbutchangeswithu.Themomentumequationisthereforenon-linearandmustbesolvediteratively.Similarly,they-momentumfluxthroughanx-directedfaceisvuACv)(=Thevequationdependsonthesolutionoftheuequation(andviceversa).Hence,themomentumequationsarecoupledandmustbesolvedtogether.Pressurealsoappearsineachmomentumequation.Thisfurthercouplestheequationsanddemandssomemeansofdeterminingpressure.Mathematically,thisislinkedtotherequirementofcontinuity(massconservation)•Incompressibleflow,continuityprovidesatransportequationfordensity().Pressureisobtainedbysolvinganenergyequationfortemperature(T)andthenusinganequationofstate(e.g.,p=RT).•Inincompressibleflow,densityvariations(ifthereareany)are,bydefinition,notdeterminedbypressure.Apressureequationarisesfromtherequirementthatthesolutionsofthemomentumequationsarealsomass-consistent.Inotherwords,massconservationactuallyleadstoapressureequation!uvmassfluxuAρCFD5–2DavidApsleyInmanyincompressible-flowcodesmassandmomentumequationsaresolvedsequentiallyanditerativelyaccordingtothefollowingpseudocode.Wecallthisasegregatedapproach.Incompressible-flowcodesthemainfluidvariablesareassembledtogetherandsolvedasavector;e.g.(,u,v,w,e).Thisiscalledacoupledapproach,butwon’tbefollowedhere.5.2Pressure-VelocityCouplingQuestion1.Howarevelocityandpressurelinked?Question2.Howdoesapressureequationarise?Question3.Shouldvelocityandpressurebeco-located(storedatthesamepositions)?5.2.1Pressure-VelocityLinkageInthemomentumequation,pressureforcesappearasasourceofmomentum;e.g.inthex-momentumequation:Appforcepressurenetew)(-=Thediscretisedmomentumequationisoftheformforcesother)(+-=-∑434214434421forcespressureewfluxnetFFFPPppAuaua(2)Hence,L+-=)(ewPPppduwherePPaAd=(3)Answer1(a)Themomentumequationlinksvelocityandpressure.(b)Velocitydependsonthepressuregradientor,whendiscretised,onthedifferencebetweenpressurevalues½celleitherside.momentumequation→L+-=)(ewPPppduorL+-=pduwhereΔindicatesacentreddifference(“rightminusleft”).DOWHILE(not_converged)CALLSCALAR_TRANSPORT(u)CALLSCALAR_TRANSPORT(v)CALLSCALAR_TRANSPORT(w)CALLMASS_CONSERVATION(p)ENDDOuPppweareaACFD5–3DavidApsleySubstitutingforvelocityinthecontinuityequation,LLL+-+-=+---=+-=EEppWWPWwEPewepapapappAdppAduAuA)()()()()()(0Thishasthesamealgebraicformasthescalar-transportequations.Answer2Themomentumequationgivesalinkbetweenvelocityandpressurewhich,whensubstitutedintothecontinuityequation,givesanequationforpressure.Theposhversionofthisis…Apressureequationarisesfromtherequirementthatsolutionsofthemomentumequationbemass-consistent.(***MSccourseonly***)TheNavier-Stokesequationforaconstant-densityfluidmaybewritteniijjiiuxpxuutu21)(∇+∂∂-=∂∂+∂∂Velocitydependsonthepressuregradient.Takingthedivergence(i.e./xi)andusingcontinuity(ui/xi=0)givespxxuujiji221)(∇-=∂∂∂orjijixxuup∂∂∂-=∇)(22Hence,analytically,thecombinationofmomentumandcontinuityequationsgivesaPoissonequationforpressure.EWNSPneswCFD5–4DavidApsley5.2.2Co-locatedStorageofVariablesInitially,supposethatpressureandvelocityareco-located(storedatthesamepositions)andthatadvectivevelocities(thecell-facevelocitiesusedtocalculatemassfluxes)arecalculatedbylinearinterpolation.Inthemomentumequationthenetpressureforceinvolves)()()(1121121121+-+--=+-+=-iiiiiiewppppppppHence,thediscretisedmomentumequationhastheform:L+-=+-)(1121iiiippduInthecontinuityequationthenetoutwardmassfluxdependsonL+---=-=+-+=---++-+-+)]()([)()()(2121411121121121iiiiiiiiiiiiweppdppduuuuuuuuThus,bothmassandmomentumequationsonlyproducelinksbetweenpressuresatalternatenodes,leadingtoodd-evendecoupling.Thus,thecombination:•co-locatedu,p;•linearinterpolationforadvectivevelocities;leadstodecouplingofoddnodalvaluesp1,p3,p5,…fromevennodalvaluesp2,p4,p6,….Thisodd-evendecouplingorcheckerboardeffectleadstoindeterminateoscillationsinthepressurefield.xp12345678910Therearetwocommonremedies:(1)useastaggeredgrid(velocityandpressurestoredatdifferentlocations);or(2)useaco-locatedgridbutRhie-ChowinterpolationfortheadvectivevelocitiesBothprovidealinkbetweenadjacentpressurenodes,preventingodd-evendecoupling.wei-1ii+1CFD5–5DavidApsley5.2.3StaggeredGrid(HarlowandWelch,1965)Inthestaggered-velocity-gridarrangement,velocitycomponentsarestoredhalf-waybetweenthepressurenodesthatdrivethem.Thisleadstodifferentsetsofcontrolvolumes.Theusualconventionisthateachvelocitynode