流体力学特论Ch8课件

IntroductiontoFluidMechanicsChapter8InternalIncompressibleViscousFlow1MainTopicsEntranceRegionFullyDevelopedLaminarFlowinaPipeFullyDevelopedLaminarFlow

BetweenInfiniteParallelPlatesTurbulent

VelocityProfilesin

FullyDevelopedPipeFlowEnergyConsiderationsinPipeFlowCalculationofHeadLossSolutionofPipeFlowProblemsFlowMeasurement2INTRODUCTIONFlowscompletelyboundedbysolidsurfacesarecalledINTERNALFLOWSwhichincludeflowsthroughpipes

(Roundcrosssection),ducts(NOTRoundcrosssection),nozzles,diffusers,suddencontractionsandexpansions,valves,andfittings.Thebasicprinciplesinvolvedareindependentofthecross-sectionalshape,althoughthedetailsoftheflowmaybedependentonit.Theflowregime(laminarorturbulent)ofinternalflowsisprimarilyafunctionoftheReynoldsnumber.Laminarflow:Canbesolvedanalytically.Turbulentflow:RelyHeavilyonsemi-empiricaltheoriesandexperimentaldata.3DuctFlowandPipeFlowandOpen-channelFlowAlthoughnotallconduitsusedtotransportfluid(liquidorgas)fromonelocationtoanotherareroundincrosssection,mostofthecommononesare.FlowscompletelyfillingthepipearecalledPipeflows.Flowswithoutcompletelyfillingthepipearecalledopen-channelflows.4LaminarorTurbulentFlowOsborneReynolds,aBritishscientistandmathematician,wasthefirsttodistinguishthedifferencebetweentheseclassificationofflowbyusingasimpleapparatusasshown.For“smallenoughflowrate”thedyestreakwillremainasawell-definedlineasitflowsalong,withonlyslightblurringduetomoleculardiffusionofthedyeintothesurroundingwater.Forasomewhatlarger“intermediateflowrate”thedyefluctuatesintimeandspace,andintermittentburstsofirregularbehaviorappearalongthestreak.For“largeenoughflowrate”thedyestreakalmostimmediatelybecomeblurredandspreadsacrosstheentirepipeinarandomfashion.5當流率很小時，染劑的軌跡依然隨著水流保持明顯的細條，染劑分子會稍微擴散到周圍水流，並呈現輕微混濁現象。當流率增加時，染劑軌跡會隨時間與位置振動，並呈現間歇性的不規則現象。當流率足夠大時，染劑軌跡幾乎是迅速迸裂開來，並以隨機方式在管中擴散開來。

LaminarorTurbulentFlow6TransitionfromLaminartoTurbulentFlowinaPipeApipeisinitiallyfilledwithafluidatrest.Asthevalveisopenedtostarttheflow,theflowvelocityand,hence,theReynoldsnumberincreasefromzero(noflow)totheirmaximumsteadyflowvalues.7TimeDependenceofFluidVelocityataPoint8Indicationof

LaminarorTurbulentFlowThetermflowrateshouldbereplacedbyReynoldsnumber,,whereVistheaveragevelocityinthepipe.Itisnotonlythefluidvelocitythatdeterminesthecharacteroftheflow–itsdensity,viscosity,andthepipesizeareofequalimportance.Forgeneralengineeringpurpose,theflowinaroundpipeLaminarTurbulent

9EntranceRegionand

FullyDevelopedFlow(I)Anyfluid

flowinginapipe

hadtoenterthepipeatsomelocation.Theregionofflownearwherethefluidentersthepipeistermedtheentranceregion.10EntranceRegionand

FullyDevelopedFlow(II)Thefluidtypicallyentersthepipewithanearlyuniformvelocityprofileatsection(1).Asthefluidmovesthroughthepipe,viscouseffectscauseittosticktothepipewall(thenoslipboundarycondition).Theboundarylayerinwhichviscouseffectsareimportantisproducedalongthepipewallsuchthattheinitialvelocityprofilechangeswithdistancealongthepipe,x,untilthefluidreachestheendoftheentrancelength.flownearwherethefluidentersthepipeistermedtheentranceregion,section(2),beyondwhichthevelocityprofiledoesnotvarywithx.ForlaminarflowForturbulentflow11EntranceRegionand

FullyDevelopedFlow(III)Oncethefluidreachestheendoftheentranceregion,section(2),theflowissimplertodescribebecausethevelocityisafunctionofonlythedistancefromthepipecenterline,r,andindependentofx.Theflowbetween(2)and(3)istermedfullydeveloped.任何流體欲在圓管中流動，則必須在某一位置導入圓管；其中，流體在進入圓管的區域，稱之為入口區（entranceregion）；若從流體沿著pipe前進的發展過程來看，在截面（1）處的速度接近均勻分佈，再往下游，則由於黏性效應使得boundarylayer逐漸成長，並使速度曲線隨著X軸逐漸改變，直到流體達到『入口長度』的尾端，也就是說，在截面（2）處以後，速度分佈曲線就不再變化了！至於管流中的速度曲線？則依流動為laminarflow，或turbulentflow以及入口長度le

而定！12FULLYDEVELOPEDLAMINARFLOWINAPIPE

VELOCITYDISTRUBUTION(I)Consideringafullydevelopedaxisymmetriclaminarflowinapipe.Thecontrolvolumeisadifferentialannulus.Thecontrolvolumelengthisdxanditsthicknessisdr.13FULLYDEVELOPEDLAMINARFLOWINAPIPE

VELOCITYDISTRUBUTION(II)UsingtheMomentumequationsTosumtheforcesactingonthecontrolvolumeinthex-direction.14FULLYDEVELOPEDLAMINARFLOWINAPIPE

VELOCITYDISTRUBUTION(III)ThepressureattheleftfaceoftheCVThepressureattherightfaceoftheCVTheshearforceattheinnersurfaceoftheCVTheshearforceattheoutersurfaceoftheCV15FULLYDEVELOPEDLAMINARFLOWINAPIPE

VELOCITYDISTRUBUTION(IV)Integrating16FULLYDEVELOPEDLAMINARFLOWINAPIPE

VELOCITYDISTRUBUTION(V)SinceWiththeboundaryconditions

u=0atr=Ru=?atr=0

Fromphysicalconsiderationsthatthevelocitymustbedefiniteatr=0>>>Theonlywaythatthiscanbetrueisfor

c1=0henceVelocitydistribution17FULLYDEVELOPEDLAMINARFLOWINAPIPE

OTHERPARAMETERS(I)TheshearstressdistributionVolumeflowrateInfullydevelopedflow18FULLYDEVELOPEDLAMINARFLOWINAPIPE

OTHERPARAMETERS(II)AveragevelocityPointofmaximumvelocityatr=01920

FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…BothPlatesStationary(I)Consideringtheflowbetweenthetwohorizontalparallelplates.Thefluidparticlesmoveinthexdirectionparalleltotheplates,andthereisnovelocityintheyorzdirection–thatis,v=0andw=0.ThecontinuityequationreducestoForsteadyflow21FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…BothPlatesStationary(II.1)UsingtheMomentumequationsTosumtheforcesactingonthecontrolvolumeinthex-direction.METHOD122FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…BothPlatesStationary(II.1)ThepressureattheleftfaceoftheCVThepressureattherightfaceoftheCVTheshearforceatthebottomfaceoftheCVTheshearforceatthetopfaceoftheCV23FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…BothPlatesStationary(II.1)IntegratingSinceIntegrating24

FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…BothPlatesStationary(II.2)TheNavier-StokesequationsreducetoIntegratingMETHOD225

FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…BothPlatesStationary(III)

Withtheboundaryconditions

u=0aty=0u=0aty=aVelocitydistributionMETHOD1+226FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…BothPlatesStationary(IV)ShearstressdistributionVolumeflowrate

Infullydevelopedflow27FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…BothPlatesStationary(V)AveragevelocityPointofmaximumvelocityaty=a/22829FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…UpperPlatesMoving(I)Sinceonlytheboundaryconditionshavechanged,thereisnoneedtorepeattheentireanalysisofthe“bothplatesstationary”case.30FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…UpperPlatesMoving(II)Theboundaryconditionsforthemovingplatecaseareu=0aty=0u=Uaty=aVelocitydistribution31FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…UpperPlatesMoving(III)ShearstressdistributionVolumeflowrate32FULLYDEVELOPEDLAMINARFLOWBETWEENINFINITEPARALLELPLATES…UpperPlatesMoving(IV)AveragevelocityPointofmaximumvelocityat333435SHEARSTRESSDISTRIBUTION

inFullyDevelopedPipeFlowConsideringfullydevelopedlaminarorturbulentflowinahorizontalcircularpipe,theforcebalancebetweenfrictionandpressureforcesleadsto

Whichmeansthatforbothlaminarandturbulentfullydevelopedflows,theshearstressvarieslinearlyacrossthepipe,fromzeroatthecenterlinetomaximumatthepipewall.Wherec1=0(becausewecannothaveinfinitestressatthecenterline)36SHEARSTRESSDISTRIBUTION

FORLaminarFlowLaminarflowismodeledasfluidparticlesthatflowsmoothlyalonginlayers,glidingpasttheslightlyslowerorfasteronesoneitherside.ThemomentumfluxinthexdirectionacrossplaneA-Agiverisetoadragofthelowerfluidontheupperfluidandanequalbutoppositeeffectoftheupperfluidonthelowerfluid.ThesluggishmoleculesmovingupwardacrossplaneA-Amustacceleratedbythefluidabovethisplane.Therateofchangeofmomentuminthisprocessproducesashearforce.TheShearstressdistribution37SHEARSTRESSDISTRIBUTION

FORTurbulentFlow(I)Theturbulentflowisthoughtasaseriesofrandom,three-dimensionaleddytypemotions.Theflowisrepresentedby(time-meanvelocity)plusu’andv’(timerandomlyfluctuatingvelocitycomponentsinthexandydirection).TheshearstressiscalledReynoldsstressintroducedbyOsborneReynolds.Asweapproachwall,andiszeroatthewall(thewalltendstosuppressthefluctuations.)38TurbulentShearStress(Reynoldsstress)forfullydevelopedturbulentflowinapipe39SHEARSTRESSDISTRIBUTION

FORTurbulentFlow(II)Nearthewall(theviscoussublayer),thelaminarshearstressisdominant.Awayfromthewall(intheouterlayer),theturbulentshearstressisdominant.Thetransitionbetweenthesetworegionsoccursintheoverlaplayer.40TURBULENTVELOCITYPROFILESinFullyDevelopedPipeFlow(I)Considerableinformationconcerningturbulentvelocityprofileshasbeenobtainedthroughtheuseofdimensionalanalysis,andsemi-empiricaltheoreticalefforts.IntheviscoussublayerthevelocityprofilecanbewrittenindimensionlessformasWhereyisthedistancemeasuredfromthewally=R-r.iscalledthefrictionvelocity.LawofthewallIsvalidverynearthesmoothwall,for41TURBULENTVELOCITYPROFILESinFullyDevelopedPipeFlow(II)IntheoverlapregionthevelocityshouldvaryasthelogarithmofyIntransitionregionorbufferlayerforfor42TURBULENTVELOCITYPROFILESinFullyDevelopedPipeFlow(III)43TURBULENTVELOCITYPROFILESinFullyDevelopedPipeFlow(IV)Thevelocityprofileforturbulentflowthroughasmoothpipemayalsobeapproximatedbytheempiricalpower-lawequationThepower-lawprofileisnotapplicableclosetothewallTheaveragevelocityTheratiooftheaveragevelocitytothecenterlinevelocityis:Wheretheexponent,n,varieswiththeReynoldsnumber.ForRe>2x10444ENERGYCONSIDERATIONSINPIPEFLOWConsideringthesteadyflowthroughthepipingsystem,includingareducingelbow.Thebasicequationforconservationofenergy–thefirstlawofthermodynamics45ENERGYCONSIDERATIONSINPIPEFLOW—KineticEnergyCoefficientThekineticenergycoefficientInlaminarflowinapipeInturbulentflowinapipeBypower-lawvelocityprofile46ENERGYCONSIDERATIONSINPIPEFLOW—HeadLossUsingthedefinitionofkineticenergycoefficient,theenergyequationcanbewrittenEachtermhasdimensions[L]ofenergyperunitweightofflowingfluid[FL/W]=[L].HeadLoss47CALCULATIONOFHEADLOSSTotalheadloss,HL,isregardedasthesumofmajorlosses,Hl,duetofrictionaleffectsinfullydevelopedflowinconstantareatubes,andminorlosses,Hlm,resultingfromentrance,fitting,areachanges,andsoon.48MajorLosses:FrictionFactorTheenergyequationforsteadyandincompressibleflowwithzeroshaftwork

Forfullydevelopedflowthroughaconstantareapipe,Hlm=0Forhorizontalpipe,z2=z149MajorLosses:LaminarFlowInfullydevelopedlaminarflowinahorizontalpipe,thepressuredrop50MajorLosses:TurbulentFlow(I)Inturbulentflowwecannotevaluatethepressuredropanalytically;wemustresorttoexperimentalresultsandusedimensionalanalysistocorrelatetheexperimentaldata.Infullydevelopedturbulentflowthepressuredrop,△P,causedbyfrictioninahorizontalconstant-areapipeisknowntodependonpipediameter,D,pipelength,L,piperoughness,e,averageflowvelocity,V,fluiddensityρ,andfluidviscosity,μ.Applyingdimensionalanalysis,theresultwereacorrelationoftheform51MajorLosses:TurbulentFlow(II)ExperimentsshowthatthenondimensionalheadlossisdirectlyproportionaltoL/D.HencewecanwriteorTheunknownfunction,,isdefinedasthefrictionalfactor,f,52FrictionFactor?Thefrictionfactor,f,isdeterminedexperimentally.Theresults,publishedbyL.F.Moody.Todetermineheadlossforfullydevelopedflowwithknowncondition:TheReynoldsnumberisevaluated.Theroughness,e,isobtainedfromTable8.1.Thefrictionfactor,f,canbereadfromtheappropriatecurve(seeFigure8.12),attheknownvaluesofReande/D.Theheadlosscanbefoundby53RoughnessforPipes54FrictionFactorbyL.F.Moody55FormulaforFrictionFactorColebrook–Toavoidhavingtouseagraphicalmethodforobtainingfforturbulentflows.Milersuggeststhatasingleiterationwillproducearesultwithin1percentiftheinitialestimateiscalculatedfrom56MinorLossesTheflowinapipingsystemmayberequiredtopassthroughavarietyoffittings,bends,orabruptchangesinarea.Additionalheadlossesareencountered,primarilyasaresultofflowseparation.Theselosseswillbeminorifthepipingsystemincludeslonglengthsofconstantareapipe.TheminorlossesarecomputedinoneoftwowaysorWherethelosscoefficient,K,mustbedeterminedexperimentallyforeachsituation.K=Φ（Geometry）

WhereLeisanequivalentlengthofstraightpipe.57MinorLoss:Inlets58MinorLoss:ExitslosscoefficientK=159MinorLoss:SuddenEnlargementsandContractions60MinorLoss:

Gradualcontractions-Nozzles61MinorLoss:

GradualEnlargements–Diffusers(I)PressurerecoverycoefficientCp

vsheadloss??62MinorLoss:

GradualEnlargements–Diffusers(II)Cp

vsheadloss??Ifgravityisneglected,theenergyequationforsteadyandincompressibleflowwithzeroshaftwork,andα1=α2=1.0

63MinorLoss:PipeBends64MinorLoss:ValvesandFitting65NoncircularDucts(I)Theempiricalcorrelationsforpipeflowmaybeusedforcomputationsinvolvingnoncircularducts,providedtheircrosssectionsarenottooexaggerated.Thecorrelationforturbulentpipeflowareextendedforusewithnoncirculargeometriesbyintroducingthehydraulicdiameter,definedasForacircularductForarectangularductofwidthbandheighthWhereAiscross-sectionalarea,andPiswettedperimeter66NoncircularDucts(II)Thehydraulicdiameterconceptcanbeappliedintheapproximaterange¼<ar<4.Sothecorrelationsforpipeflowgiveacceptablyaccurateresultsforrectangularducts.Lossescausedbysecondaryflowsincreaserapidlyformoreextremegeometry,sothecorrelationsarenotapplicabletowide,flatducts,ortoductsoftriangularorotherirregularshapes.67

LossesCausedbySecondaryFlowsCarefullydesignedguidevaneshelpdirecttheflowwithlessunwantedswirlanddisturbances.68SOLUTIONOFPIPEFLOWPROBLEMSTheenergyequation,relatingtheconditionsatanytwopoints1and2forasingle-pathpipesystem

byjudiciouschoiceofpoints1and2wecananalyzenotonlytheentirepipesystem,butalsojustacertainsectionofitthatwemaybeinterestedin.MajorlossMinorlossor69Single-PathSystems(I)FindpressuredropΔp,foragivenpipe(LandD),andflowrate,andQFindLforagivenΔp,D,andQFindQforagivenΔp,L,andDFindDforagivenΔp,L,andQ70FindΔpforagivenL,D,andQTheenergyequationTheflowrateleadstotheReynoldsnumberandhencethefrictionfactorfortheflow.Tabulateddatacanbeusedforminorlosscoefficientsandequivalentlengths.Theenergyequationcanthenbeusedtodirectlytoobtainthepressuredrop.71

FindLforagivenΔp,D,andQTheenergyequationTheflowrateleadstotheReynoldsnumberandhencethefrictionfactorfortheflow.Tabulateddatacanbeusedforminorlosscoefficientsandequivalentlengths.Theenergyequationcanthenberearrangedandsolveddirectlyforthepipelength.72

FindQforagivenΔp,L,andDThesetypesofproblemsrequiredeithermanualiterationoruseofacomputerapplication.TheunknownflowrateorvelocityisneededbeforetheReynoldsnumberandhencethefrictionfactorcanbefound.First,wemakeaguessforfandsolvetheenergyequationforVintermsofknownquantitiesandtheguessedfrictionfactorf.ThenwecancomputeaReynoldsnumberandhenceobtainanewvaluef