计算流体力学(CFD)文档――3. Common approximations

CFD3–1DavidApsley3.APPROXIMATIONSANDSIMPLIFIEDEQUATIONSSPRING20113.1Steady-statevstime-dependentflow3.2Two-dimensionalvsthree-dimensionalflow3.3Incompressiblevscompressibleflow3.4Inviscidvsviscousflow3.5Hydrostaticvsnon-hydrostaticflow3.6Boussinesqapproximationfordensity(buoyancy-affectedflow)3.7Depth-averaged/shallow-waterequations3.8Reynolds-averagedequations(turbulentflow)ExamplesFluiddynamicsisgovernedbyequationsformass,momentumandenergy.ThemomentumequationforaviscousfluidiscalledtheNavier-Stokesequation;itisbasedupon:•continuummechanics;•themomentumprinciple;•alinearstress-strainrelationship(yU∂∂∝/).AfluidforwhichthelastistrueiscalledaNewtonianfluid;thisisthecaseforalmostallfluidsincivilengineering.However,thereareimportantnon-Newtonianfluids;e.g.mud,blood,paint,polymersolutions.SpecialisedCFDtechniquesexistforthese.Thefullequationsaretime-dependent,3-dimensional,viscous,compressible,non-linearandhighlycoupled(i.e.thesolutionofoneequationdependsonthesolutionofothers).However,inmostcasesitispossibletosimplifytheanalysisbyadoptingareducedequationset.Somecommonapproximationsarelistedbelow.Reductionofdimension:•steady-state;•2-d(oraxisymmetric).Neglectofsomephysicalfeature:•incompressible;•inviscid.Simplifiedforces:•hydrostatic;•Boussinesqdensity.Approximationsbaseduponaveraging:•Reynolds-averaging(turbulentflows);•depth-averaging(shallow-waterequations).CFD3–2DavidApsley3.1Steady-StatevsTime-DependentFlowManyflowsarenaturallytime-dependent.Examplesincludewaves,tides,reciprocatingpumpsandinternalcombustionengines.Otherflowshavestationaryboundariesbutbecometime-dependentbecauseofaninstability.Animportantexampleisvortexsheddingaroundcylindricalobjects.Somecomputationalsolutionproceduresrelyonatime-steppingmethodtomarchtosteadystate;importantexamplesaretransonicflow(becausethemathematicalnatureofthegoverningequationsisdifferentforMa1andMa1)andopen-channelflows(e.g.,weirs)Thus,therearethreemajorreasonsforusingthefulltime-dependentequations:•time-dependentproblem;•time-dependentinstability;•time-marchingtosteadystate.NumericalConsequences.•Time-dependentequationsareparabolic;theyare1st-orderintimeandaresolvedbytime-marching,onlystoringvaluesatoneortwotimelevelsatanyinstant.•Steady-stateequationsareelliptic(inincompressibleflow)andaresolvedbyimplicit,iterativemethodsforallpointssimultaneously.3.2Two-DimensionalvsThree-DimensionalFlowGeometryandboundaryconditionsmaydictatethattheflowistwo-dimensional(atleastinthemean).Two-dimensionalcalculationsrequireconsiderablylesscomputerresources.“Two-dimensional”maybeextendedtoinclude“axisymmetric”.Thisisactuallyeasiertoachieveinthelaboratory,whereside-wallboundarylayerspreventtrue2-dimensionality.3.3IncompressiblevsCompressibleFlowCompressibilityisimportantwhenflow-inducedvariationsinpressureortemperaturecausesignificantchangesindensity.Thiscanarise:(a)inhigh-speedflow;(b)wherethereissignificantheatinput.Aflow(notafluid,note)issaidtobeincompressibleifflow-inducedpressureortemperaturechangesdonotcausesignificantdensitychanges.Liquidflowsareusuallytreatedasincompressible(waterhammerbeinganimportantexception),butgasflowscanalsoberegardedasincompressibleatspeedsmuchlessthanthespeedofsound(acommonruleofthumbbeingMa0.3).Densityvariationswithinthefluidcanoccurforotherreasons,notablyfromsalinityvariations(oceans)andtemperaturevariations(atmosphere).Theseleadtobuoyancyforces.Becausethedensityvariationsarenotflow-inducedtheseflowscanstillbetreatedasincompressible,eventhoughthedensityvariesfromplacetoplace.Thus,“incompressible”doesnotnecessarilymean“uniformdensity”.CFD3–3DavidApsleyCompressibleFlowFirstLawofThermodynamics:fluidondoneworkinputheatenergyofchange+=Atransportequationhastobesolvedforanenergy-relatedvariable(e.g.,internalenergyeorenthalpy/peh+=)inordertoobtaintheabsolutetemperatureT:Tcev=orTchp=cvandcparespecificheatcapacitiesatconstantvolumeandconstantpressurerespectively.Massconservationprovidesatransportequationfor,whilstpressureisderivedfromanequationofstate;e.g.,theideal-gaslaw:RTp=Forcompressibleflowitisnecessarytosolveanenergyequation.Formally,density-basedmethodsforcompressibleCFD:•massequation;•energyequationT;•equationofstatep.IncompressibleFlowInincompressibleflowpressurechanges(bydefinition)causenegligiblechangestodensity.Temperatureisnotinvolvedandsoaseparateenergyequationisnotnecessarytocomputetheflow.TheMechanicalEnergyPrinciple:changeofkineticenergy=workdoneisequivalentto,andreadilyderivedfrom,themomentumequation(seetheExamples,Q2).Incompressibilityimpliesthatdensityisconstantalongastreamline:0DD=tbutmayvarybetweenstreamlines(e.g.,duetosalinitydifferences).Conservationofmassisthenreplacedbyconservationofvolume:0)(=∑facesfluxvolumeor0=∂∂+∂∂+∂∂zwyvxuPressureisnotderivedfromathermodynamicrelationbutfromtherequirementthatsolutionsofthemomentumequationbemass-consistent.InSection5weshallseethatthisleadstothecontinuityequationbeingrephrasedasapressureequation.Inincompressibleflowitisnotnecessarytosolveaseparateenergyequation.Formally,•incompressibilityisconstan