全套課件：工程流體力學(xué)

2023/1/111

SchoolofJetPropulsionBeihangUniversity.FLUIDMECHANICS2023/1/112Chapter1Introduction1.1PreliminaryRemarks

Whenyouthinkaboutit,almosteverythingonthisplaneteitherisafluidormoveswithinorneara

fluid.-FrankM.WhiteWhatisafluid?2023/1/113

TheconceptofafluidAsolidcanresistashearstress(剪切應(yīng)力)byastaticdeformation,afluidcannot.Anyshearstressappliedtoafluid,nomatterhowsmall,willresultinmotionofthatfluid.Thefluidmovesanddeformscontinuouslyaslongastheshearisapplied.2023/1/114WhatisFluidMechanics

FluidMechanicsisthestudyoffluideitherinmotion(FluidDynamics流體動(dòng)力學(xué))oratrest(FluidStatics流體靜力學(xué))andsubsequenteffectsofthefluidupontheboundaries,whichmaybeeithersolidsurfacesorinterfaceswithotherfluids.2023/1/115ThefamouscollapseoftheTacomaNarrowBridgein1940Curvedshoot(Bananashoot)NospinSpinwhy2023/1/116Boeing74770.7×64.4×19.41(m)395000kgAn-22584×88.4×18.1(m)600,000kg

Howcantheairplanefly?Drag&Lift2023/1/1172023/1/118Theengineofaturbofan(渦扇)jet2023/1/119；2023/1/1110HistoryandScopeof

FluidMechanicsPre-history:Sailingshipswithoars(櫓槳)andirrigationsystemwerebothknowninprehistory2023/1/1111Archimedes(285-212BC)Parallelogramlawforadditionofvectors

Lawofbuoyancy2023/1/1112LeonardodaVinci(1452-1519)*Equationofconservationofmassinone-dimensionalsteadyflow*Experimentalist*Turbulence2023/1/1113IsaacNewton(1642-1727)LawsofmotionLawsofviscosityofNewtonianfluid2023/1/1114

18thcenturyMathematicians：Euler（歐拉）：

EulerequationBernoulli(伯努利)：BernoulliequationFrictionless(無粘)flowsolutionsD’Alembert（達(dá)朗貝爾）：

D’Alembertparadox（佯謬，疑題）Engineers：Hydraulics（水力學(xué)）relayingonexperimentChannels，Shipresistance，Pipeflows，WaveturbinePitotVenturiTorricelliPoiseuille2023/1/111519thcenturyNavier(1785-1836)

&

Stokes(1819-1905)N-Sequation

viscousflowsolutionReynolds(1842-1912)

TurbulenceFamousexperimentontransitionReynoldsNumber2023/1/111620thcenturyLudwigPrandtl

(1875-1953)Boundarytheory(1904)Tobethesinglemostimportanttoolinmodernflowanalysis.ThefatherofmodernfluidmechanicsVonkarman(1881-1963)I.taylor(1886-1975)Laidfoundationforthepresentstateoftheartinfluidmechanics2023/1/11171.2TheFluidasaContinuum(連續(xù)介質(zhì)）Density(密度)Elementalvolume（流體微團(tuán)、流體質(zhì)點(diǎn)）*Largeenoughinmicroscope（微觀）10-9mm3ofairatstandardconditionscontainsapproximately3×107molecules.Sodensityisessentiallyapointfunctionandfluidpropertiescanbethoughtofasvaryingcontinuallyinspace.*Smallenoughinmacroscope（宏觀）.Mostengineeringproblemsareconcernedwithphysicaldimensionsmuchlargerthanthislimitingvolume.2023/1/1118TheelementalvolumemustbesmallenoughinmacroscopeSuchafluidiscalledacontinuum,whichsimplymeansthatitsvariationinpropertiesissosmooththatthedifferentialcalculuscanbeusedtoanalyzethesubstance.2023/1/11191.3SomePropertiesoffluids1.viscosity(粘性)*Definition：Whenafluidissheared（剪切）,itbeginstomove.Subsequently,apairofforcesappearontheshearsurface,whichresiststheshearmotionofthefluid.Thisiscalled

viscosityThisresistantforceis

shearstress.(剪切應(yīng)力,內(nèi)摩擦應(yīng)力)Infact,thisshearmotionofafluidisakindofdeformation（變形）*Thenatureofviscosity：Forliquidiscohesion（結(jié)合）(movie)Forgasisthetransportofmomentum（動(dòng)量輸運(yùn)）(movie)2023/1/1120m:Coefficientofviscosity(粘性系數(shù))[FT/L2]n=m/r:Kinematicviscosity(運(yùn)動(dòng)學(xué)粘性系數(shù))[L2/T]Velocitygradient*Newtonianlawofviscosity(牛頓粘性定律，牛頓內(nèi)摩擦定律)UUu(y)xyShearstressThelinearfluid,whichfollowNewtonianresistancelaw,iscalledNewtonianflow.（牛頓流動(dòng)、牛頓流體）Thevelocitygradientisinfactakindofdeformation.Realfluid(Viscous),Idealfluid(Inviscid&Frictionless)2023/1/11212.Compressibility(壓縮性)Incompressible(不可壓):r=constMostliquidflowsaretreatedasincompressible.Only1percentincreaseifpressureincreaseby220Compressible（可壓縮）:

r=r(P.T)Gasescanalsobetreatedasincompressiblewhentheirvelocityislessthan0.3Manumbers3.StateRelationsforGasesPerfect-gasLaw（理想氣體狀態(tài)方程）2023/1/11224.ThermalConductivity(熱傳導(dǎo))

:

heatfluxinndirectionperunitareak:coefficientofthermalconductivityT:temperaturen:directionofheattransferFourier’slawofheatconduction2023/1/11231.4Twodifferentpointsofviewinanalyzingproblemsinmechanics*TheEulerianview(歐拉觀點(diǎn))andtheLagrangianview（拉格朗日觀點(diǎn)）TheEulerianviewisconcernedwiththefieldofflow,appropriatetofluidmechanics.TheLagrangian

viewfollowsanindividualparticlemovingthoughtheflow,appropriatetosolidmechanics.Thecontrastoftwoframes2023/1/1124*Flowclassification(流動(dòng)分類)AccordingtoEulerianview,anypropertyisfunctionofcoordinates（space）andtime.InCartesiansystem(直角坐標(biāo)系)，itcanbeexpressedasf(x,y,z,t)x,y,z,t:Eulerianvariablecomponent(歐拉變數(shù))f:Functionofonlyonecoordinatecomponent,one-dimensional

(一維

1-D）.Inthelikemanner,two-dimensional（二維2-D）

,three-dimensional

(三維

3-D)

:Functionoftime~~unsteady

(非定常)Otherwisesteady(定常)2023/1/1125OneTwodimensionalThreeSteadyUnsteadyCompressibleIncompressibleViscousInviscid2023/1/11261.5Streamline(流線),Pathline(跡線)&Flowfield(流場(chǎng))*Whatisastreamline

Astreamlineisthelineeverywheretangenttothevelocityvectoratagiveninstant.2023/1/1127

A

pathlineistheactualpathtraversedbyagivenfluidparticles.Forsteadyflow:Streamline=Pathline*WhatisapathlinePathlinesinsteadyflowPathlinesinunsteadyflow2023/1/1128FlowPattern(流型、流普、流線族）Streamsurface(流面)&Streamtube(流管）Flowpattern:asetofstreamlinesStreamsurface:acollectionofallthestreamlinespassingthroughalinewhichisnotastreamline.Streamlinecannotintersect（相交），exceptforsingularitypoint(奇點(diǎn))Streamtube:aclosedcollectionofstreamlines.Noflowacrossstreamtubewalls2023/1/1129Flowfield(流場(chǎng))

：Inagivenflowsituation,thepropertiesofthefluidarefunctionsofpositionandtime,namelyspace-timedistributionsofthefluidproperties.2023/1/1130Streamlineequation(流線方程)ds->Infinitesimal(無窮小)dydxds2023/1/1131Example:Giventhesteadytwo-dimensionalvelocitydistributionu=kx,v=-ky,w=0,wherekisapositiveconstant.Computeandplotthestreamlinesoftheflow,includingdirection.Solution:

Sincetime(t)doesnotappearexplicitly,themotionissteady,sothatstreamlines,pathlineswillcoincide.Sincew=0,themotionistwo-dimensional.Integrating:Hyperbolas(雙曲線)2023/1/1132Direction:

u=kx,v=-ky

QuadrantI（第一象限)(x>0,y>0)u>0,v<0

Atthepointo:u=v=0Singularitypoint,(匯)xyo2023/1/11331.5Surfaceforce(表面力)andbodyforce（質(zhì)量力，體積力）Surfaceforceactscontinuouslyonthesidesurfacesoffluidelements.Pressure,friction.Contactsurfaceforceperunitarea(單位面積)（應(yīng)力）Bodyforceactsontheentiremassoftheelement.Gravity,electromagnetic.NocotactPerunitmass(單位質(zhì)量)g2023/1/1134Homework1.Giventhevelocitydistribution:u=-cy,v=cx,w=0Wherecisapositiveconstant.Computeandplotthestreamlinesoftheflow.2.Givenvelocitydistribution:u=x+t,v=-y+t,w=0(tistime)Findthestreamlinepassingthroughpoint(-1,-1)attheinstantt=0.35White: Chapter2潘錦珊： 第一章Chapter2

PressureDistribution

inafluid

(FluidStatics

Basic)36Definition:Unit:(SI)(PoundperSquireInch)Verticaltothesurfaceandpointintoit.Atanypoint,pressureisindependentoforientation.PropertiesofPressurePressure37Atanypointinastaticfluid,pressureisindependentoforientation.Verification:When(up)Forcesonleftandupsurface:38FluidMechanicsAerodynamicsFluidatrestFluidStaticsFluidDynamicsFluidinmotion39Pressureistheonlysurfaceforce.Pressuredistributionrelatestobodyforceonly.Dams（水壩）Buoyancyrelatedinstrument（利用浮力的裝置）

Fluidpowersystem（液壓驅(qū)動(dòng)系統(tǒng)）Connectedvessel（連通器）……Applications:§2.1Fluid@rest40Consideracubeinastaticfluid Pressureatthecenterisp; Bodyforcesaredxdydz§2.2EquilibriumofaFluidElement41dxdydzdxdydzPressure:Bodyforce:42Inxdirection:ForceonleftsurfaceForceonrightsurfaceBodyforceinxdirectionEulerEquilibriumEquations(Euler1775)43Pressureincreaseinthedirectionofbodyforce.Surfacesinfluidwithsamepressure,verticaltobodyforceeverywhere,ingravityfielditisahorizontalplane.Equipressuresurface（等壓面）44xzzhz0p01Basicrule:Generalsolution2Boundarycondition:§2.3

PressureDistributionunderGravity45PressureatfreesurfacePressureduetoweightontopAnypointwiththesamedepthhunderfreesurfacehasthesamepressure.equipressuresurface（等壓面）Freesurfaceisanequipressuresurface46p0Watermanometer（水柱壓力計(jì)）phPressuresourceConnectedwatertubeApplication?Absolutepressure（絕對(duì)壓力）Relativepressure（相對(duì)壓力）Gaugepressure（表壓力）Vacuumdegree（真空度）Pressuremeasurementh(mmH2O)h+p0

(mmH2O)pA絕壓pG表壓47P2.7Homework:48Centroid（形心，重心）hcycChyxAαyxp0FindtotalforceP§2.4HydrostaticForceonPlaneSurface49hc:

depthofcentroidTheforceonasubmergedplaneequalsthepressureattheplatecentertimestheplatearea,independentoftheshapeoftheplateortheangle.CenterofpressureIsthecenterofpressureatcentroid?hcycChdydDhyxAαyxp050hcycChdydDhyxAαyxp0Momenttoxaxis51ExampleThegateis5mwide,ishingedatpointB,andrestagainstasmoothwallatpointA.FindTheforceonthegateexertedbyseawaterpressure,ThehorizontalforcePxexertedbythewallatpointA52(a)Centroid:3maboveBSolution:(b)LcL53HooverDamChannel

SelectadAandfindthethreeforcesonitIntegration§2.5HydrostaticForcesonCurvedSurfaces54Conclusion:x1.Horizontalforces:xOzAAx2.Verticalforces:OzV55Example:Findtheforcesactingonthehemi-sphericalcovers.RFOxyH45oSolution:56§2.6.1UniformLinearAcceleration（恒加速度直線運(yùn)動(dòng)）aX=-agxGravityBodyforceInertiaForce§2.6Fluidinrigidbodymotion57Boundarycondition:Atfreesurface（自由液面）Equipressuresurface（等壓面）EulerEquilibriumEquations58Acupofcoffeeis7cmdeepatrest.1.Whetheritwillspilloutwhileax=7m/s2?2.GagepressureofpointA?Example:Solution:Itwillnotspillout!59§2.6.2Rigidbodyrotation（整體旋轉(zhuǎn)）gω2rzfOω2yωxyROfω2xryxθBodyforce:Equipressuresurface:dp=060Paraboladish（拋物面）Freesurface:gω2rzfOz0Howtofindz0?旋轉(zhuǎn)拋物面體的體積是同底面積和高的圓柱體積的一半。61P2.64P2.97(selective) P2.147P2.152Homework:3.1Systems(體系)versusControlVolumes

(控制體)

System：anarbitraryquantityofmassoffixedidentity.

Everythingexternaltothissystemisdenotedbythetermsurroundings,andthesystemisseparatedfromitssurroundingsbyit‘sboundariesthroughwhichnomassacross.(Lagrange拉格朗日)Chapter3IntegralRelations（積分關(guān)系式）

foraControlVolumeinOne-dimensionalSteadyFlows

ControlVolume(CV):

In

theneighborhoodofourproductthefluidformstheenvironmentwhoseeffectonourproductwewishtoknow.Thisspecificregioniscalledcontrolvolume,withopenboundariesthroughwhichmass,momentumandenergyareallowedtoacross.(Euler歐拉)FixedCV,movingCV,deformingCV3.2BasicPhysicalLawsofFluidMechanicsAllthelawsofmechanicsarewrittenforasystem,whichstatewhathappenswhenthereisaninteractionbetweenthesystemandit’ssurroundings.IfmisthemassofthesystemConservationofmass(質(zhì)量守恒)Newton’ssecondlawAngularmomentumFirstlawofthermodynamic

Itisrarethatwewishtofollowtheultimatepathofaspecificparticleoffluid.Insteaditislikelythatthefluidformstheenvironmentwhoseeffectonourproductwewishtoknow,suchashowanairplaneisaffectedbythesurroundingair,howashipisaffectedbythesurroundingwater.Thisrequiresthatthebasiclawsberewrittentoapplytoaspecificregionintheneighboredofourproductnamelyacontrolvolume(CV).TheboundaryoftheCViscalledcontrolsurface(CS)BasicLawsforsystemforCV3.3TheReynoldsTransportTheorem(RTT)雷諾輸運(yùn)定理1122isCV.1*1*2*2*issystemwhichoccupiestheCVatinstantt.:Theamountofperunitmass

ThetotalamountofintheCVis:t+dtt+dttts:anypropertyoffluidt+dtt+dtttsInthelikemanner

s1-Dflow

:

isonlythefunctionofs.Forsteadyflow:t+dtt+dtttdsRTTIfthereareseveralone-Dinletsandoutlets:Steady,1-Donlyininletsandoutlets,nomatterhowtheflowiswithintheCV.3.3Conservationofmass(質(zhì)量守恒)(ContinuityEquation)f=mb=dm/dm=1Massflux(質(zhì)量流量)Forincompressibleflow:體積流量LeonardodaVinciin1500Ifonlyoneinletandoneoutlet

壺口瀑布是我國著名的第二大瀑布。兩百多米寬的黃河河面，突然緊縮為50米左右，跌入30多米的壺形峽谷。入壺之水，奔騰咆哮，勢(shì)如奔馬，浪聲震天，聲聞十里?！包S河之水天上來”之驚心動(dòng)魄的景觀。

Example:Ajetengineworkingatdesigncondition.AttheinletofthenozzleAttheoutletPleasefindthemassfluxandvelocityattheoutlet.GivengasconstantT1=865K，V1=288m/s，A1=0.19㎡；

T2=766K，A2=0.1538㎡

R=287.4J/kg.K。

SolutionAccordingtotheconservationofmassHomework:P185P3.12,P189P3.36

3.4TheLinearMomentumEquation(動(dòng)量方程)

（Newton’sSecondLaw

）Newton’ssecondlaw:NetforceonthesystemorCV(體系或控制體受到的合外力)：Momentumflux(動(dòng)量流量)1-Din&outsteadyRTTFfluxForonlyoneinletandoneoutletAccordingtocontinuity2－out,1－inExample:Afixedcontrolvolumeofastreamtubeinsteadyflowhasauniforminlet(r1,A1,V1)andauniformexit(r2,A2,V2).Findthenetforceonthecontrolvolume.Solution:Neglecttheweightofthefluid.Findtheforceonthewaterbytheelbowpipe.Example:1212Solution:selectcoordinate,controlvolumeInthelikemannerFindtheforcetofixtheelbow.Solution:coordinate,CVNetforceonthecontrolvolume:WhereFexistheforceontheCVbypipe,(onelbow)12FexSurfaceforce:(1)Forcesexposedbycuttingthoughsolidbodieswhichprotrudeintothesurface.(2)Pressure,viscousstress.AfixedvaneturnsawaterjetofareaAthroughanangleqwithoutchangingitsvelocitymagnitude.Theflowissteady,pressurepaiseverywhere,andfrictiononthevaneisnegligible.FindtheforceFappliedtovane.AwaterjetofvelocityVjimpingesnormaltoaflatplatewhichmovestotherightatvelocityVc.Findtheforcerequiredtokeeptheplatemovingatconstantvelocityandthepowerdeliveredtothecartifthejetdensityis1000kg/m3thejetareais3cm2,andVj=20m/s,Vc=15m/sNeglecttheweightofthejetandplate,andassumesteadyflowwithrespecttothemovingplatewiththejetsplittingintoanequalupwardanddownwardhalf-jet.Homework:P190-p3.46P191-p3.50P192-p3.54P192-p3.58Derivethethrust(推力)equationforthejetengine.airdragisneglectSolution::massfluxoffuelxBalancewiththrustCoordinate,CV

Example:Inagroundtestofajetengine,pa=1.0133×105N/m2,Ae=0.1543m2,Pe=1.141×105N/m2,Ve=542m/s,.Findthethrustforce.Solution:F16R=65.38KNxcoordinateArocketmovingstraightup.LettheinitialmassbeM0,andassumeasteadyexhaustmassflowandexhaustvelocityverelativetotherocket.Iftheflowpatternwithintherocketmotorissteadyandairdragisneglect.Derivethedifferentialequationofverticalrocketmotionv(t)andintegrateusingtheinitialconditionv=0att=0.Example:Solution:TheCVenclosetherocket,cutsthroughtheexitjet,andacceleratesupwardatrocketspeedv(t).coordinatezv(t)Z-momentumequation:v(t)z3.5TheAngular-MomentumEquation(Angular-Momentum)：Netmoment(合力矩)Example:Centrifugal(離心)pumpThevelocityofthefluidischangedfromv1tov2anditspressurefromp1top2.Find(a).anexpressionforthetorqueT0whichmustbeappliedthosebladestomaintainthisflow.(b).thepowersuppliedtothepump.

blade

wForincompressibleflow1-DContinuity:Solution:TheCVischosen.blade

w

PressurehasnocontributiontothetorquearebladerotationalspeedsWorkonperunitmassHomework:P192-p3.55;P194-p3.68,p3.78;P200-p3.114,p3.116

BriefReviewBasicPhysicalLawsofFluidMechanics:TheReynoldsTransportTheorem:TheLinearMomentumEquation:TheAngular-MomentumTheorem:ConservationofMass:ReviewofFluidStaticsEspecially:

Question

Whenfluidflowing…

Bernoulli(1700~1782)Whatrelationsarethereinvelocity,heightandpressure?SeveralTragediesinHistory:

Alittlerailwaystationin19thRussia.The‘Olimpic’shipwreckinthePacificThebumpingaccidentofB-52bomberoftheU.S.airforcein1960s.3.6FrictionlessFlow:

TheBernoulliEquation1.DifferentialFormofLinearMomentumEquationElementalfixedstreamtubeCVofvariableareaA(s),andlengthds.Linearmomentumrelationinthestreamwisedirection:one-D,steady,frictionlessflowForincompressibleflow,r=const.Integralbetweenanypoints1and2onthestreamline:AQuestion:

IstheBernoulliequationamomentumorenergyequation?Hydraulicandenergygradelinesforfrictionlessflowinaduct.Example1:Findarelationbetweennozzledischargevelocityandtankfree-surfaceheighth.Assumesteadyfrictionlessflow.1,2maximuminformationisknownordesired.h12V2Solution:h12V2Continuity:Bernoulli:Torricelli1644AccordingtotheBernoulliequation,thevelocityofafluidflowingthroughaholeinthesideofanopentankorreservoirisproportionaltothesquarerootofthedepthoffluidabovethehole.Thevelocityofajetofwaterfromanopenpopbottlecontainingfourholesisclearlyrelatedtothedepthofwaterabovethehole.Thegreaterthedepth,thehigherthevelocity.ReviewofBernoulliequationThedimensionsofabovethreeitemsarethesameoflength!Example1:Findarelationbetweennozzledischargevelocityandtankfree-surfaceheighth.Assumesteadyfrictionlessflow.V2h12

Example2:Findvelocityintherighttube.hABInlikemanner:VExample3:FindvelocityintheVenturitube.12AsafluidflowsthroughaVenturitube,thepressureisreducedinaccordancewiththecontinuityandBernoulliequations.Example4:Estimaterequiredtokeeptheplateinabalancestate.(Assumetheflowissteadyandfrictionless.)Solution:Forplate,bylinealmomentumequation,byBernoulliequation,Example5:Firehose,Q=1.5m3/minFindtheforceonthebolts.1122Solution:Bycontinuity:ByBernoulli:1122Bymomentum:Example6:Findtheaero-forceontheblade (cascade).ABDCSSSolution:ABDCSSBycontinuity,葉片越彎，做功量越大。ABDCSSByBernoulli,BernoulliEquationforcompressibleflowSpecific-heatratioForisentropicflow:GasWeightneglectFornozzle:Fordiffuser:ExtendedBernoulliEquationForcompressor

多變壓縮功Forturbine

多變膨脹功Homework!Page206:P3.158,P3.161Page207:P3.164,P3.165《氣體動(dòng)力學(xué)》第二章習(xí)題第一部分：Page2033題Reviewofexamples:V12AnalysisChooseyourcontrolvolumnBodyforceandSurfaceforceSolution1122xFindtheaero-forceontheblade (cascade).葉片越彎，做功量越大。ABDCSSByBernoulli,3.7TheEnergyEquation

ConservationofEnergyVarioustypesofenergyoccurinflowingfluids.Workmustbedoneonthedeviceshowntoturnitoverbecausethesystemgainspotentialenergyastheheavy(dark)liquidisraisedabovethelight(clear)liquid.Thispotentialenergyisconvertedintokineticenergywhichiseitherdissipatedduetofrictionasthefluidflowsdownramporisconvertedintopowerbytheturbineanddissipatedbyfriction.Thefluidfinallybecomesstationaryagain.Theinitialworkdoneinturningitovereventuallyresultsinaveryslightincreaseinthesystemtemperature.

EnergyPerUnitMass1122eFirstLawsofThermodynamicsConservationofEnergy1122Theenergyequation!Example:Asteadyflowmachinetakesinairatsection1anddischargeditatsection2and3.Thepropertiesateachsectionareasfollows:sectionA,Q,T,P,PaZ,m10.042.82110000.320.091.13814401.230.021.4100?0.4CV(1)(2)(3)110KWWorkisprovidedtothemachineattherateof110kw.Findthepressure(abs)andtheheattransfer.AssumethatairisaperfectgaswithR=287,Cp=1005.Solution:Massconservation:Byenergyequation:CV(1)(2)(3)110KW124Chapter4DifferentialRelationsForViscousFlow4.1PreliminaryRemarks*

TwowaysinanalyzingfluidmotionSeekinganestimateofgrosseffectsoverafiniteregionorcontrolvolume.

Integral

(2)Seekingthepoint-by-pointdetailsofaflowpatternbyanalyzinganinfinitesimalregionoftheflow.

Differential125TurbulentFlow

VS.

LaminarFlow*

TwoformsofflowTurbulent（湍，紊）flow,laminar（層）flow*

ViscousflowViscosityisinherentnatureofrealfluid.Strain(剪切)isverystrongininternalflow.TransitionReynolds

numberOsbroneReynoldsReynoldstank慣性力/粘性力1264.2TheAccelerationFieldofaFluidLocalaccelerationunsteadyConvectiveaccelerationnonuniformNonlinearterms127InthelikemannerAnypropertyΦSubstantial(Material)derivative隨體（物質(zhì)、全）導(dǎo)數(shù)128ExampleGiven.Findtheaccelerationofaparticle.129Xinlet(massflow)XoutletdxyzxdzdyInfinitesimalfixedCVXflowout4.3DifferentialEquationofMassConservationInthelikemanner

FlowoutofftheCV130LossofmassintheCV131ForsteadyflowForincompressibleflowExample1Underwhatconditionsdoesthevelocityfieldrepresentsanincompressibleflowwhichconservesmass?(where)132SolutionContinuityforincompressibleflowExample2Anincompressiblevelocityfield:u=a(x2-y2),w=b,a,bareconst,whatv=?SolutionAnarbitraryfunctionofx,z,t133Assignment:P264:P4.1(a),P4.2,P4.4,P4.9(a)134Newton’ssecondlaw4.4DifferentialEquationofLinearMomentumdxdzdyElementalvolumeWhatarethesurfaceforcesFs

ontheelementalvolume?135Surfaceforceonanelementalvolume:dxdzdyVectorSumSurfacestressNetSurfaceForce:136MomentumequationInthelikemannerItisnotthesestressesbuttheirgradient,whichcauseanetforceonthedifferentialvolume.137Tensor

張量138ConstitutiveRelation

本構(gòu)Newton’sLaw(廣義牛頓內(nèi)摩擦定律)

Momentumequation(角標(biāo)表示法)139SubstituteNewton’sConstitutiveRelationintoMENewtonfluid,linearfluid(牛頓流體，線性流體）140N-SEquation141ForincompressibleflowForinviscidflowFor2-D,steady,incompressibleflow1421434.5TheDifferentialEquationofEnergyInfinitesimalfluidelementdxdzdyThefirstthermodynamiclaw144(1)Thermalconductivity(2)othersX:HeatflowdxdydzAccordingFourier’sLaw:145質(zhì)量力做功和表面力做功Bodyforce146SurfaceforcedxdydzX:X:NetpowerY,Z147NetpowerbyFsleft148單位體積內(nèi)能變化率熱傳導(dǎo)等傳熱變型時(shí)表面應(yīng)力做功149變型時(shí)表面應(yīng)力做功壓力做膨脹功粘性耗散Φ>0由連續(xù)方程：根據(jù)熱力學(xué)公式，熵s、焓h和壓強(qiáng)p、密度ρ的關(guān)系為：150已知D?=CvDT,Dh=CpDT151SummaryoftheEquations152153154equationunknownvariablescontinuity1r,u,v,wmomentum3p,ru,v,wenergy1p,ru,v,w,Tperfectgas1p,r,TToSolveAFlow…1554.6Initial(初始)andBoundary（邊界）ConditionsfortheBasicEquationsInitialConditions：t=t0

:BoundaryConditions：(Noslip)VelocityWallIfthewallisstationaryTemperature156DuetothehighlycomplexoftheN-Sequations,onlyafewparticularsolutionswerefounduptonow.Formostproblems,theequationsmustbesolvednumerically,whichisabrandnewcoursecalledCFD(ComputationalFluidDynamics)

Flowpassacylinder

Anexperimentresult

AcomputationresultSolvingtheN-Sequationsnumerically157xyohhUFlowbetweentwoparallelwalls,Steady,incompressible,neglectbodyforce,2-DContinuity:Momentum:4.7

ExactsolutionsofN-SEquations158=constantIntegraterelativetoyBoundarycondition:xyohhU159Applytheboundaryconditionxyoh-hu(y)Uxyoh-hWhenU=0PoiseuilleflowWhenSimpleCouetteflow160WhenWhenConsideraspecialcaseGeneralcase:yxoUxyoUxoyU161Q=0:yxoVolumeflowrateQ=u*dy162Thewallshearstresses1634.8DynamicalSimilarity&NondimensionalizationFlowpassacylinder

D=5cm

D=10cmMeasurementforWingtipvortex164N-Sequation,2-D,steady,nobodyforce,incompressibleUseU,LasreferencevelocityandlengthDimensionlessquantitiesxdirection:4.8.1NondimensionalizationofN-SEquation165Boundaryconditionsneedtobenormalizedtoo…166Forsteady,incompressible,nobodyforceflow,iftwogeometricallysimilarflowfieldshassameReynoldsnumber,thentheyhavesimilarflowstructurewhensameboundaryconditionsareprovided.WhyReynoldsnumber?167Inertiaforce/viscousforce兩個(gè)鐵球同時(shí)落地？168Flowpassacylinder

D=5cm

D=10cmFlowpassasquare

Re=50

Re=10000Theflowfieldsfortwoobjectsofthesameshapebutdifferentsizearesaidtobegeometricallysimilar.If,inaddition,theReynoldsnumberarethesame,thetwoflowsaresaidtobedynamicallysimilar,sincetheratioofrelevantforcesarethesameinthetwocases.4.8.2D