Reynolds_Transport

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1、TheReynoldsTransportTheoremThequestionarisesastohowagivenquantitychangesΨ,(whereΨ(x,t)issomefluidpropertyperunitvolume)withrespecttotimeasamaterialvolumedeforms.Wedefinethematerialvolumeas:Anarbitrarychosencontrolvolumeoffluidwhosesurfacemovesattheparticlevelocity.Thesignifi

2、canceofthevolumemovingatthesamerateastheparticlevelocityisthatnomassistransportedacrossthechosencontrolsurfacethatenclosesthecontrolvolume.Inaddition,westatebydefinitionthatthematerialcontrolvolumedeformswiththebodymotion.nSm(t)vV(t)mMaterialControlVolumeConsideramaterialvol

3、umeVm(t)withsurfaceareaSm(t).Theunitnormaltothesurfaceisdenotedbyn,andthesurfacevelocityisdenotedbyv.WithinthiscontrolvolumeisapropertyofthecontinuumΨthatisofinterest.WewouldliketodeterminehowΨchangeswithtime.ThetotalamountofΨpresentinthevolumeVm(t)canbeexpressedas:F()t=∫Ψ(x

4、,t)dV(1)Vm(t)Thetimerateofchangeifthefluidpropertyisdefinedas:dFd=∫Ψ(x,t)dV(2)dtdtVm(t)Sincethelimitsofintegrationareafunctionoftime(inourcasethisisthematerialvolumeVm(t)),thetimederivativecannotbetakeninsidetheintegraldirectly.Inordertoovercomethisproblem,wetransformthevolu

5、me,andmaterialpropertyfromaspatialrepresentation,toareference/materialrepresentation..Recallthatanydifferentialvolumeelementattimetisrelatedtothevolumeattinet=0by:dV=JdVowheretheJacobianJisdefinedas:∂x1∂x2∂x3∂X1∂X1∂X1∂x1∂x2∂x3J=detF=givenx=χ(X,t)(3)∂X2∂X2∂X2∂x1∂x2∂x3∂X3∂X3∂X

6、3also,thequantityΨ(x,t)canalwaysbeexpressedintermsofthematerialcoordinatesbyemployingtheresultsx=χ(X,t).Therefore,wecanexpressΨ(x,t)as:Ψ(x,t)=Ψ[χ(X,t),t]≡Ψ(X,t)(4)Thereforewecanexpresstheintegralinequation2intermsofthematerialcoordinates:dFd=∫Ψ(X,t)JdVo(5)dtdtVoSinceVoisinde

7、pendentoftime,theorderofintegrationanddifferentiationcanbeswitched.NotethatsinceXisheldconstant,weareineffecttakingthematerialderivativeoftheintegrand.dFd(ΨX,t)dJdt=∫dtJ+Ψ(X,t)dtdVo(6)XVoXButrecallthatdΨΨD∂Ψ==+v•∇ΨdtXDt∂txdJDJand==J()∇•vdtDtXThereforewecanrea

8、rrangeequation6tohavetheformdF∂Ψ=∫+v•∇∇ΨΨ+()•vJdVo(7)dt∂tVoorintermso