全套课件：工程流体力学

2023/1/111

SchoolofJetPropulsionBeihangUniversity.FLUIDMECHANICS2023/1/112Chapter1Introduction1.1PreliminaryRemarks

Whenyouthinkaboutit,almosteverythingonthisplaneteitherisafluidormoveswithinorneara

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

TheconceptofafluidAsolidcanresistashearstress(剪切应力)byastaticdeformation,afluidcannot.Anyshearstressappliedtoafluid,nomatterhowsmall,willresultinmotionofthatfluid.Thefluidmovesanddeformscontinuouslyaslongastheshearisapplied.2023/1/114WhatisFluidMechanics

FluidMechanicsisthestudyoffluideitherinmotion(FluidDynamics流体动力学)oratrest(FluidStatics流体静力学)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(涡扇)jet2023/1/119；2023/1/1110HistoryandScopeof

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

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

18thcenturyMathematicians：Euler（欧拉）：

EulerequationBernoulli(伯努利)：BernoulliequationFrictionless(无粘)flowsolutionsD’Alembert（达朗贝尔）：

D’Alembertparadox（佯谬，疑题）Engineers：Hydraulics（水力学）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(连续介质）Density(密度)Elementalvolume（流体微团、流体质点）*Largeenoughinmicroscope（微观）10-9mm3ofairatstandardconditionscontainsapproximately3×107molecules.Sodensityisessentiallyapointfunctionandfluidpropertiescanbethoughtofasvaryingcontinuallyinspace.*Smallenoughinmacroscope（宏观）.Mostengineeringproblemsareconcernedwithphysicaldimensionsmuchlargerthanthislimitingvolume.2023/1/1118TheelementalvolumemustbesmallenoughinmacroscopeSuchafluidiscalledacontinuum,whichsimplymeansthatitsvariationinpropertiesissosmooththatthedifferentialcalculuscanbeusedtoanalyzethesubstance.2023/1/11191.3SomePropertiesoffluids1.viscosity(粘性)*Definition：Whenafluidissheared（剪切）,itbeginstomove.Subsequently,apairofforcesappearontheshearsurface,whichresiststheshearmotionofthefluid.Thisiscalled

viscosityThisresistantforceis

shearstress.(剪切应力,内摩擦应力)Infact,thisshearmotionofafluidisakindofdeformation（变形）*Thenatureofviscosity：Forliquidiscohesion（结合）(movie)Forgasisthetransportofmomentum（动量输运）(movie)2023/1/1120m:Coefficientofviscosity(粘性系数)[FT/L2]n=m/r:Kinematicviscosity(运动学粘性系数)[L2/T]Velocitygradient*Newtonianlawofviscosity(牛顿粘性定律，牛顿内摩擦定律)UUu(y)xyShearstressThelinearfluid,whichfollowNewtonianresistancelaw,iscalledNewtonianflow.（牛顿流动、牛顿流体）Thevelocitygradientisinfactakindofdeformation.Realfluid(Viscous),Idealfluid(Inviscid&Frictionless)2023/1/11212.Compressibility(压缩性)Incompressible(不可压):r=constMostliquidflowsaretreatedasincompressible.Only1percentincreaseifpressureincreaseby220Compressible（可压缩）:

r=r(P.T)Gasescanalsobetreatedasincompressiblewhentheirvelocityislessthan0.3Manumbers3.StateRelationsforGasesPerfect-gasLaw（理想气体状态方程）2023/1/11224.ThermalConductivity(热传导)

:

heatfluxinndirectionperunitareak:coefficientofthermalconductivityT:temperaturen:directionofheattransferFourier’slawofheatconduction2023/1/11231.4Twodifferentpointsofviewinanalyzingproblemsinmechanics*TheEulerianview(欧拉观点)andtheLagrangianview（拉格朗日观点）TheEulerianviewisconcernedwiththefieldofflow,appropriatetofluidmechanics.TheLagrangian

viewfollowsanindividualparticlemovingthoughtheflow,appropriatetosolidmechanics.Thecontrastoftwoframes2023/1/1124*Flowclassification(流动分类)AccordingtoEulerianview,anypropertyisfunctionofcoordinates（space）andtime.InCartesiansystem(直角坐标系)，itcanbeexpressedasf(x,y,z,t)x,y,z,t:Eulerianvariablecomponent(欧拉变数)f:Functionofonlyonecoordinatecomponent,one-dimensional

(一维

1-D）.Inthelikemanner,two-dimensional（二维2-D）

,three-dimensional

(三维

3-D)

:Functionoftime~~unsteady

(非定常)Otherwisesteady(定常)2023/1/1125OneTwodimensionalThreeSteadyUnsteadyCompressibleIncompressibleViscousInviscid2023/1/11261.5Streamline(流线),Pathline(迹线)&Flowfield(流场)*Whatisastreamline

Astreamlineisthelineeverywheretangenttothevelocityvectoratagiveninstant.2023/1/1127

A

pathlineistheactualpathtraversedbyagivenfluidparticles.Forsteadyflow:Streamline=Pathline*WhatisapathlinePathlinesinsteadyflowPathlinesinunsteadyflow2023/1/1128FlowPattern(流型、流普、流线族）Streamsurface(流面)&Streamtube(流管）Flowpattern:asetofstreamlinesStreamsurface:acollectionofallthestreamlinespassingthroughalinewhichisnotastreamline.Streamlinecannotintersect（相交），exceptforsingularitypoint(奇点)Streamtube:aclosedcollectionofstreamlines.Noflowacrossstreamtubewalls2023/1/1129Flowfield(流场)

：Inagivenflowsituation,thepropertiesofthefluidarefunctionsofpositionandtime,namelyspace-timedistributionsofthefluidproperties.2023/1/1130Streamlineequation(流线方程)ds->Infinitesimal(无穷小)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(双曲线)2023/1/1132Direction:

u=kx,v=-ky

QuadrantI（第一象限)(x>0,y>0)u>0,v<0

Atthepointo:u=v=0Singularitypoint,(汇)xyo2023/1/11331.5Surfaceforce(表面力)andbodyforce（质量力，体积力）Surfaceforceactscontinuouslyonthesidesurfacesoffluidelements.Pressure,friction.Contactsurfaceforceperunitarea(单位面积)（应力）Bodyforceactsontheentiremassoftheelement.Gravity,electromagnetic.NocotactPerunitmass(单位质量)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潘锦珊： 第一章Chapter2

PressureDistribution

inafluid

(FluidStatics

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

Fluidpowersystem（液压驱动系统）Connectedvessel（连通器）……Applications:§2.1Fluid@rest40Consideracubeinastaticfluid Pressureatthecenterisp; Bodyforcesaredxdydz§2.2EquilibriumofaFluidElement41dxdydzdxdydzPressure:Bodyforce:42Inxdirection:ForceonleftsurfaceForceonrightsurfaceBodyforceinxdirectionEulerEquilibriumEquations(Euler1775)43Pressureincreaseinthedirectionofbodyforce.Surfacesinfluidwithsamepressure,verticaltobodyforceeverywhere,ingravityfielditisahorizontalplane.Equipressuresurface（等压面）44xzzhz0p01Basicrule:Generalsolution2Boundarycondition:§2.3

PressureDistributionunderGravity45PressureatfreesurfacePressureduetoweightontopAnypointwiththesamedepthhunderfreesurfacehasthesamepressure.equipressuresurface（等压面）Freesurfaceisanequipressuresurface46p0Watermanometer（水柱压力计）phPressuresourceConnectedwatertubeApplication?Absolutepressure（绝对压力）Relativepressure（相对压力）Gaugepressure（表压力）Vacuumdegree（真空度）Pressuremeasurementh(mmH2O)h+p0

(mmH2O)pA绝压pG表压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（恒加速度直线运动）aX=-agxGravityBodyforceInertiaForce§2.6Fluidinrigidbodymotion57Boundarycondition:Atfreesurface（自由液面）Equipressuresurface（等压面）EulerEquilibriumEquations58Acupofcoffeeis7cmdeepatrest.1.Whetheritwillspilloutwhileax=7m/s2?2.GagepressureofpointA?Example:Solution:Itwillnotspillout!59§2.6.2Rigidbodyrotation（整体旋转）gω2rzfOω2yωxyROfω2xryxθBodyforce:Equipressuresurface:dp=060Paraboladish（抛物面）Freesurface:gω2rzfOz0Howtofindz0?旋转抛物面体的体积是同底面积和高的圆柱体积的一半。61P2.64P2.97(selective) P2.147P2.152Homework:3.1Systems(体系)versusControlVolumes

(控制体)

System：anarbitraryquantityofmassoffixedidentity.

Everythingexternaltothissystemisdenotedbythetermsurroundings,andthesystemisseparatedfromitssurroundingsbyit‘sboundariesthroughwhichnomassacross.(Lagrange拉格朗日)Chapter3IntegralRelations（积分关系式）

foraControlVolumeinOne-dimensionalSteadyFlows

ControlVolume(CV):

In

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

Itisrarethatwewishtofollowtheultimatepathofaspecificparticleoffluid.Insteaditislikelythatthefluidformstheenvironmentwhoseeffectonourproductwewishtoknow,suchashowanairplaneisaffectedbythesurroundingair,howashipisaffectedbythesurroundingwater.Thisrequiresthatthebasiclawsberewrittentoapplytoaspecificregionintheneighboredofourproductnamelyacontrolvolume(CV).TheboundaryoftheCViscalledcontrolsurface(CS)BasicLawsforsystemforCV3.3TheReynoldsTransportTheorem(RTT)雷诺输运定理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(质量守恒)(ContinuityEquation)f=mb=dm/dm=1Massflux(质量流量)Forincompressibleflow:体积流量LeonardodaVinciin1500Ifonlyoneinletandoneoutlet

壶口瀑布是我国著名的第二大瀑布。两百多米宽的黄河河面，突然紧缩为50米左右，跌入30多米的壶形峡谷。入壶之水，奔腾咆哮，势如奔马，浪声震天，声闻十里。“黄河之水天上来”之惊心动魄的景观。

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(动量方程)

（Newton’sSecondLaw

）Newton’ssecondlaw:NetforceonthesystemorCV(体系或控制体受到的合外力)：Momentumflux(动量流量)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(离心)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,叶片越弯，做功量越大。ABDCSSByBernoulli,BernoulliEquationforcompressibleflowSpecific-heatratioForisentropicflow:GasWeightneglectFornozzle:Fordiffuser:ExtendedBernoulliEquationForcompressor

多变压缩功Forturbine

多变膨胀功Homework!Page206:P3.158,P3.161Page207:P3.164,P3.165《气体动力学》第二章习题第一部分：Page2033题Reviewofexamples:V12AnalysisChooseyourcontrolvolumnBodyforceandSurfaceforceSolution1122xFindtheaero-forceontheblade (cascade).叶片越弯，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（层）flow*

ViscousflowViscosityisinherentnatureofrealfluid.Strain(剪切)isverystrongininternalflow.TransitionReynolds

numberOsbroneReynoldsReynoldstank惯性力/粘性力1264.2TheAccelerationFieldofaFluidLocalaccelerationunsteadyConvectiveaccelerationnonuniformNonlinearterms127InthelikemannerAnypropertyΦSubstantial(Material)derivative随体（物质、全）导数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

张量138ConstitutiveRelation

本构Newton’sLaw(广义牛顿内摩擦定律)

Momentumequation(角标表示法)139SubstituteNewton’sConstitutiveRelationintoMENewtonfluid,linearfluid(牛顿流体，线性流体）140N-SEquation141ForincompressibleflowForinviscidflowFor2-D,steady,incompressibleflow1421434.5TheDifferentialEquationofEnergyInfinitesimalfluidelementdxdzdyThefirstthermodynamiclaw144(1)Thermalconductivity(2)othersX:HeatflowdxdydzAccordingFourier’sLaw:145质量力做功和表面力做功Bodyforce146SurfaceforcedxdydzX:X:NetpowerY,Z147NetpowerbyFsleft148单位体积内能变化率热传导等传热变型时表面应力做功149变型时表面应力做功压力做膨胀功粘性耗散Φ>0由连续方程：根据热力学公式，熵s、焓h和压强p、密度ρ的关系为：150已知Dû=CvDT,Dh=CpDT151SummaryoftheEquations152153154equationunknownvariablescontinuity1r,u,v,wmomentum3p,ru,v,wenergy1p,ru,v,w,Tperfectgas1p,r,TToSolveAFlow…1554.6Initial(初始)andBoundary（边界）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两个铁球同时落地？168Flowpassacylinder

D=5cm

D=10cmFlowpassasquare

Re=50

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