计算流体力学(CFD)文档――7. Introduction to turbulence

CFD7–1DavidApsley7.TURBULENCESPRING20117.1Whatisturbulence?7.2Momentumtransferinlaminarandturbulentflow7.3Turbulencenotation7.4Effectofturbulenceonthemeanflow7.5Turbulencegenerationandtransport7.6ImportantshearflowsSummaryExamplesPART(a)–THENATUREOFTURBULENCE7.1WhatisTurbulence?InstantaneousMean•A“random”,3-d,time-dependenteddyingmotionwithmanyscales,superposedonanoftendrasticallysimplermeanflow.•AsolutionoftheNavier-Stokesequations.•ThenaturalstateathighReynoldsnumbers.•Anefficienttransporterandmixer...ofmomentum,energy,constituents.•Amajorsourceofenergyloss.•Asignificantinfluenceondragandboundary-layerseparation.•“Thelastgreatunsolvedproblemofclassicalphysics.”(variouslyattributedtoSommerfeld,EinsteinandFeynman)CFD7–2DavidApsley7.2MomentumTransferinLaminarandTurbulentFlowInlaminarflowadjacentlayersoffluidslidepasteachotherwithoutmixing.Transferofmomentumoccursbetweenlayersmovingatdifferentspeedsbecauseofviscousstresses.Inturbulentflowadjacentlayerscontinuallymix.Anettransferofmomentumoccursbecauseofthemixingoffluidelementsfromlayerswithdifferentmeanvelocity.Thismixingisafarmoreeffectivemeansoftransferringmomentumthanviscousstresses.Consequently,themean-velocityprofiletendstobemoreuniforminturbulentflow.7.3TurbulenceNotationTheinstantaneousvalueofanyflowvariablecanbedecomposedintomean+fluctuation.isdecomposedintomean+fluctuationMeanandfluctuatingpartsaredenotedbyeither:•anoverbarandprime:uuu′+=or•uppercaseandlowercase:uU+Thefirstisusefulinderivingtheoreticalresultsbutbecomescumbersomeingeneraluse.Thenotationbeingusedis,hopefully,obviousfromthecontext.Bydefinition,theaveragefluctuationiszero:0=′uInexperimentalworkandinsteadyflowthe“mean”isusuallyatimemean,whilstintheoreticalworkitistheprobabilistic(or“ensemble”)mean.TheprocessoftakingthemeanofaturbulentquantityoraproductofturbulentquantitiesiscalledReynoldsaveraging.Thenormalaveragingrulesforproductsapply:222uuu′+=(variance)vuvuuv′′+=(covariance)Thus,inturbulentflowthe“meanofaproduct”isnotequaltothe“productofthemeans”butincludesan(oftensignificant)contributionfromtheneteffectofturbulentfluctuations.vulaminarturbulentCFD7–3DavidApsley7.4EffectofTurbulenceontheMeanFlowEngineersareusuallyonlyinterestedinthemeanflow.However,turbulencemuststillbeconsideredbecause,althoughtheaveragesofindividualfluctuations(e.g.u′orv′)arezero,theaverageofaproduct(e.g.vu′′)isnotandmayleadtoasignificantnetflux.ConsidermassandmomentumfluxesintheydirectionacrosssurfaceA.Forsimplicity,assumeconstantdensity.7.4.1ContinuityMassflux:vAAveragemassflux:AvTheonlychangeisthattheinstantaneousvelocityisreplacedbythemeanvelocity.Themeanvelocitysatisfiesthesamecontinuityequationastheinstantaneousvelocity.7.4.2Momentumx-momentumflux:AuvuvA)()(=Averagex-momentumflux:AvuvuuvA)()(′′+=TheaveragemomentumfluxhasthesameformastheinstantaneousmomentumfluxexceptforadditionalfluxesAvu′′duetotheneteffectofturbulentfluctuations.Theseadditionaltermsarisebecauseoftheaveragingofaproductoffluctuatingquantities.AnetrateoftransportofmomentumAvu′′fromlowertouppersideofaninterface...•isequivalenttoanetrateoftransportofmomentumAvu′′-fromuppertolower;•hasthesamedynamiceffect(i.e.samerateoftransferofmomentum)asastress(i.e.forceperunitarea)ofvu′′-.ThisapparentstressiscalledaReynoldsstress.Inafully-turbulentflowitisusuallymuchlargerthantheviscousstress.OtherReynoldsstresses(uu′′-,vv′′-,etc.)emergewhenconsideringthefluxofthedifferentmomentumcomponentsindifferentdirections.Themeanvelocitysatisfiesthesamemomentumequationastheinstantaneousvelocity,exceptforadditionalapparentstresses:theReynoldsstressesjiuu′′-vAρuvAρCFD7–4DavidApsleyInasimpleshearflowthetotalstressis{321stressturbulentstressviscousvuyu′′-∂∂=(1)Infully-turbulentflowturbulentstressisusuallysubstantiallybiggerthanviscousstress.canbeinterpretedaseither:•theapparentforce(perunitarea)exertedbytheupperfluidonthelower,or•therateoftransportofmomentum(perunitarea)fromupperfluidtolower.Thedynamiceffect–atransferofmomentum–isthesame.Thenatureoftheturbulentstresscanbeillustratedbyconsideringthemotionofparticleswhosefluctuatingvelocitiesallowthemtocrossaninterface.IfparticleAmigratesupward(v′0)thenittendstoretainitsoriginalmomentum,whichisnowlowerthanitssurrounds(u′0).IfparticleBmigratesdownward(v′0)ittendstoretainitsoriginalmomentumwhichisnowhigherthanitssurrounds(u′0).Inbothcases,vu′′-ispositiveand,onaverage,tendstoreducethemomentumintheupperfluidorincreasethemomentuminthelowerfluid.Hencethereisanettransferofmomentumfromuppertolowerfluid,equivalenttotheeffectofanadditionalmeanstress.VelocityFluctuationsNormalstresses:222,,wvu′′′Shearstresses:vuuwwv′′′′′′,,(Inslightlycareless,butextremelycommon,usagebothvu′′-andvu′′arereferredtoas“stresses”.)Mostturbulentflowsareanisotropic;i.e.222,,wvu′′′aredifferent.Turbulentkineticenergy:)(22221wvuk′+′+′=Turbulenceintensity:UkUuitymeanvelocctuationsquarefluroot-mean-rms32=′==yUv'BAvuyUττCFD7–5DavidApsley7.4.3GeneralScalarIngeneral,theadvectionofanyscalarquantityϕgivesrisetoanadditionalscalarfluxinthemean-flowequations;e.g.321fluxadd