计算流体力学(CFD)文档――1. Introduction to CFD

CFD1–1DavidApsley1.INTRODUCTIONTOCFDSPRING20111.1Whatiscomputationalfluiddynamics?1.2BasicprinciplesofCFD1.3Formsofthegoverningfluid-flowequations1.4ThemaindiscretisationmethodsAppendixExamples1.1WhatisComputationalFluidDynamics?Computationalfluiddynamics(CFD)istheuseofcomputersandnumericaltechniquestosolveproblemsinvolvingfluidflow.CFDhasbeensuccessfullyappliedinmanyareasoffluidmechanics,including:aerodynamicsofcarsandaircraft;hydrodynamicsofships;pumpsandturbines;combustionandheattransfer;chemicalengineering.Applicationsincivilengineeringinclude:windloadinganddynamicresponseofstructures;wind,waveandtidalenergy;ventilation;fireandexplosionhazards;dispersionofpollutantsandeffluent;waveloadingoncoastalandoffshorestructures;hydraulicstructuressuchasweirsandspillways;sedimenttransport,hydrology.Morespecialistapplicationsincludeoceancurrents,weatherforecasting,plasmaphysics,bloodflow,heatdissipationfromelectroniccircuitry.ThisrangeofapplicationsisbroadandencompassesmanydifferentfluidphenomenaandCFDtechniques.Inparticular,CFDforhigh-speedaerodynamics(wherecompressibilityissignificantbutviscouseffectsareoftenunimportant)isverydifferentfromthatusedtosolvelow-speed,frictionalflowstypicalofmechanicalandcivilengineering.Althoughmanyelementsofthiscoursearegenerallyapplicable,thefocuswillbeonsimulatingviscous,incompressibleflowbythefinite-volumemethod.CFD1–2DavidApsley1.2BasicPrinciplesofCFDTheapproximationofacontinuously-varyingquantityintermsofvaluesatafinitenumberofpointsiscalleddiscretisation.ThefundamentalelementsofanyCFDsimulationare:(1)Theflowfieldisdiscretised;i.e.fieldvariables(,u,v,w,p,…)areapproximatedbytheirvaluesatafinitenumberofnodes.(2)Theequationsofmotionarediscretised(approximatedintermsofvaluesatnodes):control-volumeordifferentialequationsalgebraicequations(continuous)(discrete)(3)Theresultingsystemofalgebraicequationsissolvedtogivevaluesatthenodes.ThemainstagesinaCFDsimulationare:Pre-processing:–problemformulation(governingequations;boundaryconditions);–constructionofacomputationalmesh.Solving:–discretisationofthegoverningequations;–numericalsolutionofthegoverningequations.Post-processing:–visualisation;–analysisofresults.1.3FormsoftheGoverningFluid-FlowEquationsTheequationsgoverningfluidmotionarebasedonthefundamentalphysicalprinciples:•mass:changeofmass=0•momentum:changeofmomentum=force×time•energy:changeofenergy=work+heatAdditionalconservationequationsforindividualconstituentsmayapplyfornon-homogeneousfluids(e.g.containingdissolvedchemicalsorimbeddedparticles).Whenappliedtothefluidcontinuumthesemaybeexpressedmathematicallyaseither:•integral(i.e.control-volume)equations;•differentialequations.continuouscurvediscreteapproximationCFD1–3DavidApsley1.3FormsoftheGoverningFluid-FlowEquations1.3.1Integral(Control-Volume)ApproachThisconsiderschangestothetotalamountofsomephysicalquantity(mass,momentum,energy,…)withinaspecifiedregionofspace(controlvolume).Foranarbitrarycontrolvolumethebalanceofaphysicalquantityoveranintervaloftimeiscreatedamountoutamountinamountchange+-=Influidmechanicsthisisusuallyexpressedinrateformbydividingbythetimeinterval(andtransferringthenetamountpassingthroughtheboundarytotheLHSoftheequation):=+insideVboundaryofoutinsideVSOURCEFLUXNETCHANGEOFRATE(1)Therateoftransportacrossasurfaceorfluxiscomposedof:advection–movementwiththefluidflow;diffusion–nettransportbyrandom(molecularorturbulent)motion.=++insideVVofundarythroughboinsideVSOURCEDIFFUSIONADVECTIONCHANGEOFRATE(2)Theimportantpointisthatthisisasingle,genericscalar-transportequation,irrespectiveofwhetherthephysicalquantityconcernedismass,momentum,chemicalcontentetc.Thus,insteadofdealingwithlotsofdifferentequationswecanconsiderthenumericalsolutionofageneralscalar-transportequation(Section4).Thefinite-volumemethod,whichisthesubjectofthiscourse,isbasedonapproximatingthesecontrol-volumeequations.1.3.2DifferentialEquationsInregionswithoutshocks,interfacesorotherdiscontinuities,thefluid-flowequationscanalsobewritteninequivalentdifferentialforms.Thesedescribewhatisgoingonatapointratherthanoverawholecontrolvolume.Mathematically,theycanbederivedbymakingthecontrolvolumesinfinitesimallysmall.ThiswillbedemonstratedinSection2,whereitwillalsobeshownthatthereareseveraldifferentwaysofwritingthesedifferentialequations.Thefinite-differencemethodisbasedonthedirectapproximationofadifferentialformofthegoverningequations.VCFD1–4DavidApsley1.4TheMainDiscretisationMethods(i)Finite-DifferenceMethodDiscretisethegoverningdifferentialequationsdirectly;e.g.yvvxuuyvxujijijijiΔ-+Δ-≈∂∂+∂∂=-+-+2201,1,,1,1(ii)Finite-VolumeMethodDiscretisethegoverningcontrol-volumeequationsdirectly;e.g.0)()()()(=-+-=snwevAvAuAuAoutflowmassnet(iii)Finite-ElementMethodExpressthesolutionasaweightedsumofshapefunctionsS(x),substituteintosomeformofthegoverningequationsandsolveforthecoefficients(akadegreesoffreedomorweights).e.g.,forvelocity,∑=)()(xxSuuFinite-differenceandfinite-elementmethodsarecoveredinmoredetailintheComputationalMechanicscourse.Thiscou