工程流体力学(英文版)第三章.pdf

Chapter3ConceptsandBasicEquationofFluidsinmotion(onedimension,idealfluids)Contents1.MethodstoStudyFluidsinMotion2.FlowClassification3.Pathline()andStreamline()4.Streamtube()andDischarge()5.ContinuityEquationforSteadyFlowinaConduit6.MotionDifferentialEquationforIdeal1-DFlow&TheBernoulliEquationAlongaStreamlineContents7.DifferentialEquationforIdealFlowalongNormalLine8.TheBernoulliEquationfor1-Dpipeflow9.ApplicationofTheBernoulliEquation10.TheLinear-Momentum()EquationandMoment-of-Momentum()EquationforIdealFlow3.1MethodstoStudyFluidsinMotion1.LagrangianApproach()2.EulerianApproach()3.SystemandControlVolume4.EulerianAccelerationA′•A•••B′Bviewpoints:aindividualfluidparticlebcertainpointinspaceLagrangianDescriptionofMotionisthedescriptionthateveryfluidpartideinflowfieldisobservedasafunctionoftime.Spacecoordinates

===),,,(),,,(),,,(tcbazztcbayytcbaxx1.LagrangianApproachA′•A•••B′BEulerApproachisthedescriptionthatthemotionfactorsofeveryspacepointinflowfieldareobservedasafunctionoftime.——flowfielddescription.Flowfieldmotionfactorsarethecontinuousfunctionsoftimeandspacex,y,zEulerianDescriptionisutilizedwidelyinengineering.2.EulerianApproach(x,y,z)-----EulerianVariables()()(),,,,,,,,,xyztppxyztVVxyzt=
=
=

ρρ3.System(andControlVolumeDefinitionofaSystemAsystemreferstoaspecificmassoffluidwithintheboundariesdefinedbyaclosedsurface.ShapemaychangemassnochangeAcontrolvolumereferstoafixedregioninspace,whichdoesnotmoveorchangeshape.Thesurfacesurroundingthecontrolvolumeiscalledcontrolsurface3.System(andControlVolumeShapenochangemassmaychangeDefinitionofaControlVolume1122.Euleriancceleration(),,,Vxyzt
t:position:x,y,ztt+δ:position:velocity:(),,xxyyzz+++δδδ(),,,tVxxyyzzt++++
δδδδvelocity:yxz0••t(x,y,z)()()()()000,,,,,,lim1lim,,,,,,limxtttuxxyyzzttuxyztatuuuuuxyztxyztuxyzttxyztutuxuyuzttxtytzt→→→++++−=∂∂∂∂=++++−∂∂∂∂ ∂∂∂∂=+++∂∂∂∂ δδδδδδδδδδδδδδδδδδδδδ000lim,lim,limtttyxzuvwttt→→→===δδδδδδδδδyxz0••t(x,y,z)soddxuuuuuauvwttxyz∂∂∂∂==+++∂∂∂∂andddddddxyzuuuuuauvwttxyzvvvvvauvwttxyz∂∂∂∂==+++
∂∂∂∂

∂∂∂∂==+++∂∂∂∂

∂∂∂∂==+++
∂∂∂∂or:()VaVVt∂=+⋅∇∂



ijkxyz∂∂∂∇=++∂∂∂


SimilarlyAccelerationofparticlesiscomposedoftwoparts1LocalAccelerationthechangeofvelocityateverypointwithtime.2ConvectiveAccelerationthechangeofvelocitywithpositionddpppppuvwttxyz∂∂∂∂=+++∂∂∂∂dduvwttxyzρρρρρ∂∂∂∂=+++∂∂∂∂Fordensityandpressure:Generalform:ddVtt∂=+⋅∇∂
.TheTotalDerivativeexample3.1velocityis:2232Vxyiyjzk=−+



(m/s),Whatistheaccelerationofpoint(3,1,2).solution:2220(2)(3)027xuuuuauvwxyxyyxmstxyz∂∂∂∂=+++=+⋅+−⋅+=∂∂∂∂22200(3)(3)209yvvvvauvwxyyzmstxyz∂∂∂∂=+++=+⋅+−⋅−+⋅=∂∂∂∂22200(3)02464z∂∂∂∂=+++=+⋅+−⋅+⋅=∂∂∂∂So,theaccelerationofpoint(3,1,2):27964aijk=++



3.2ClassificationofFluidFlowClassificationofFluidFlowBasedontheCharacteristicofFluidBasedontheStateofFlowBasedontheNumberofSpaceVariables1.BasedontheCharacteristicsofFluidIdealflowandViscousflow0orμ=≠Incompressibleflowandcompressiblefloworconstρ=≠2.BasedontheStateofFlowSteadyflowandunsteadyflow0ort∂=≠∂Rotational()flowandirrotational()flowLaminarflow()andturbulentflow()Subsonicflow()Transonicflow()andsupersonicflow()Uniformflowandnon-uniformflow0Vors∂=≠∂
3.BasedontheNumberofSpaceVariablesOnedimensionalflow()Twodimensionalflow()Threedimensionalflow()4.Steadyflowandunsteadyflowistheflowwhosemotionfactorsdon′tchangewithtime.Thatis:SteadyFlow(),,,VVxyz=

0Vt∂=∂
H=CUnsteadyflowistheflowthatatleastoneofitsmotionfactorschangeswithtime.ThatisunsteadyFlow(),,,VVxyzt=

0Vt∂≠∂
HHH51-D,2-Dand3-DFlowOne-dimensionalFlow:(2)cross-sectionalaveragevaluesSfluidmotionfactorsarefunctionofaspacecoordinate.(1)Idealflow.(3)motionfactorsarefunctionsofcurvedcoordinatess.(,)xtϕϕ=(,)xtϕϕ=(,)stϕϕ=(,)stϕϕ=Two-dimensionalFlow:fluidmotionfactorsarefunctionoftwospacecoordinates.(Notonlylimitedtorectangularcoordinates).Fluidflow′smotionfactorsarefunctionsofthreespacecoordinates.Forexample:Waterflowinanaturalriverwhosecrosssectionshapeandmagnitudechangealongthedirectionofflow;waterflowsaroundtheship.Three-dimensionalFlow:Apathlineisthetraceafterasingleparticletravelsinafieldofflowoveraperiodoftime.(1).DefinitionKinescope1Kinescope23.3Pathline()andStreamline()1.Pathline(2)EquationofPathlineuvwarefunctionsofbothtimetandspace(xyz).Heretisanindependentvariabledydxdzuvwdt===AStreamlineisacurvethatshowthedirectionofanumberofparticlesattheatthesameinstantoftime.Thecurveindicatesthevelocityvectorsofanypointsoccupyingonthestreamline.Kinescope2.Streamline1.DefinitionaV
bV
cV
dV
eV
2EquationofStreamlineSelectpointAinstreamline,dsisadifferentialarclength,uisthevelocityatpointAdsdxidyjdzk=++



dsuAVuivjwk=++



Directionalcosinebetweenvelocityvectorandcoordinatescos(,)vVyV=
cos(,)uVxV=
cos(,)wVzV=
Directionalcosinebetweendsandcoordinatescos(,)dxdsxds=
cos(,)dydsyds=
cos(,)dzdszds=
//dsV

dydxdzuvw==——StreamlineequationSo,velocityvectoristangenttostreamlinecos(,)cos(,)dsxVx=

cos(,)cos(,)dsyVy=

cos(,)cos(,)dszVz=

udxVds=vdyVds=wdzVds=uvwVdxdydzds===3CharacterofStreamlinebAtthesameinstantoftime,streamlinesca