工程流体力学英文版第三章pdf_.pdf

工程流体力学英文版第三章pdf_.pdf

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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（?

）EquationforIdealFlow?

3.1MethodstoStudyFluidsinMotion1.LagrangianApproach（?

）2.EulerianApproach（?

）3.SystemandControlVolume4.EulerianAccelerationAABBviewpoints:

aindividualfluidparticlebcertainpointinspaceLagrangianDescriptionofMotionisthedescriptionthateveryfluidpartideinflowfieldisobservedasafunctionoftime.Spacecoordinates?

=）,（）,（）,（tcbazztcbayytcbaxx1.LagrangianApproachAABBEulerApproachisthedescriptionthatthemotionfactorsofeveryspacepointinflowfieldareobservedasafunctionoftime.flowfielddescription.Flowfieldmotionfactorsarethecontinuousfunctionsoftimeandspace?

x,y,z?

EulerianDescriptionisutilizedwidelyinengineering.2.EulerianApproach（x,y,z）-EulerianVariables（）（）（）,xyztppxyztVVxyzt?

=?

=?

=?

3.System（?

andControlVolume?

DefinitionofaSystemAsystemreferstoaspecificmassoffluidwithintheboundariesdefinedbyaclosedsurface.ShapemaychangemassnochangeAcontrolvolumereferstoafixedregioninspace,whichdoesnotmoveorchangeshape.Thesurfacesurroundingthecontrolvolumeiscalledcontrolsurface3.System（?

andControlVolume?

ShapenochangemassmaychangeDefinitionofaControlVolume1?

1?

2?

2?

.Euleriancceleration（）,Vxyzt?

t:

position:

?

x,y,z?

tt+:

position:

velocity:

（）,xxyyzz+（）,tVxxyyzzt+?

velocity:

y?

x?

z?

0?

t?

（x,y,z）?

（）（）（）（）000,lim1lim,limxtttuxxyyzzttuxyztatuuuuuxyztxyztuxyzttxyztutuxuyuzttxtytzt+=?

=+?

=+?

000lim,lim,limtttyxzuvwttt=y?

x?

z?

0?

t?

（x,y,z）?

so?

ddxuuuuuauvwttxyz=+and?

ddddddxyzuuuuuauvwttxyzvvvvvauvwttxyzwwwwwauvwttxyz?

=+?

=+?

=+?

or:

（）VaVVt=+?

ijkxyz=+?

Similarly?

Accelerationofparticlesiscomposedoftwoparts?

1?

LocalAccelerationthechangeofvelocityateverypointwithtime.?

2?

ConvectiveAccelerationthechangeofvelocitywithpositionddpppppuvwttxyz=+dduvwttxyz=+Fordensityandpressure:

Generalform:

ddVtt=+?

.TheTotalDerivativeexample3.1velocityis:

2232Vxyiyjzk=+?

（m/s）,Whatistheaccelerationofpoint（3,1,2）.solution:

2220

（2）（3）027xuuuuauvwxyxyyxmstxyz=+=+=22200（3）（3）209yvvvvauvwxyyzmstxyz=+=+=22200（3）02464zwwwwauvwxyyzzmstxyz=+=+=So,theaccelerationofpoint（3,1,2）:

27964aijk=+?

3.2ClassificationofFluidFlowClassificationofFluidFlowBasedontheCharacteristicofFluidBasedontheStateofFlowBasedontheNumberofSpaceVariables1.BasedontheCharacteristicsofFluidIdealflowandViscousflow0or=Incompressibleflowandcompressiblefloworconst=2.BasedontheStateofFlowSteadyflowandunsteadyflow0ort=Rotational（?

）flowandirrotational（?

）flowLaminarflow（?

）andturbulentflow（?

）Subsonicflow（?

）?

Transonicflow（?

）andsupersonicflow（?

）Uniformflowandnon-uniformflow0Vors=?

3.BasedontheNumberofSpaceVariablesOnedimensionalflow（?

）Twodimensionalflow（?

）Threedimensionalflow（?

）4.Steadyflowandunsteadyflowistheflowwhosemotionfactorsdontchangewithtime.Thatis:

SteadyFlow（）,VVxyz=?

0Vt=?

H=C?

Unsteadyflowistheflowthatatleastoneofitsmotionfactorschangeswithtime.ThatisunsteadyFlow（）,VVxyzt=?

0Vt?

H?

H?

H?

51-D,2-Dand3-DFlowOne-dimensionalFlow:

（2）cross-sectionalaveragevaluesSfluidmotionfactorsarefunctionofaspacecoordinate.

（1）Idealflow.（3）motionfactorsarefunctionsofcurvedcoordinatess.（,）xt=（,）xt=（,）st=（,）st=Two-dimensionalFlow:

fluidmotionfactorsarefunctionoftwospacecoordinates.（Notonlylimitedtorectangularcoordinates）.Fluidflowsmotionfactorsarefunctionsofthreespacecoordinates.Forexample:

Waterflowinanaturalriverwhosecrosssectionshapeandmagnitudechangealongthedirectionofflow;waterflowsaroundtheship.Three-dimensionalFlow:

Apathlineisthetraceafterasingleparticletravelsinafieldofflowoveraperiodoftime.

（1）.DefinitionKinescope1Kinescope2?

3.3Pathline（?

）andStreamline（?

）1.Pathline

（2）?

EquationofPathlineu?

v?

warefunctionsofbothtimetandspace（x?

y?

z）.Heretisanindependentvariabledydxdzuvwdt=AStreamlineisacurvethatshowthedirectionofanumberofparticlesattheatthesameinstantoftime.Thecurveindicatesthevelocityvectorsofanypointsoccupyingonthestreamline.Kinescope2.Streamline1.DefinitionaV?

bV?

cV?

dV?

eV?

2?

EquationofStreamlineSelectpointAinstreamline,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=3?

CharacterofStreamlineb?

Atthesameinstantoftime,streamlinescannotintersect.c?

Streamlinecantbeafoldingline,butasmoothcurve.U2L1L2U1d.Insteadyflow,streamlinesandpathlinescoincide.mt2StreamlinePathLinet1mt1mt3t3t2（b）UnsteadyFlowa?

streamline:

manyfluidparticles,oneinstantoftime（pathline:

afluidparticle,aperiodoftime）1ux=+vy=Example?

Findthestreamlineequationofpoint?

1?

2?

.Solution?

ddxyuv=so?

dd1xyxy=+integrateit:

（）ln1lnlnxyC+=+or（）1xyC+=thestreamlineequationofpoint?

1?

2?

:

（）14xy+=Thevelocityofa2-dflowfieldis:

?

3.4Streamtube（?

）andDischarge（?

）Consideraclosedcurve（notstreamline）intheflowfield,thendrawstreamlinesthrougheverypointonit,soastoformatube-shapingspacewhosewallsarestreamlines.Thistubeiscalledthestreamtube.Fluidfullingthestreamtubeiscalledthetubeflowandthelimitofatubeflowisastreamline.Streamtubeandtubeflow1?

Streamtube?

（?

）2?

TubeFlow:

（?

）Thesectionisperpendiculartothedirectionoffluidflow（suchaspipeflowandchannelflow）.1122CrossSection4.1TotalFlowAnalysisMethodIfthewallsofthestreamtubeareextendedtotheflowfieldboundary,thefluidflowwithintheboundaryistheTotalFlow.3?

TotalFlow?

（?

）4?

CrossSection?

（?

）Amountoffluidpassthroughacrosssection（suchasthesectioninthechannelorpipe）perunittime.Volumedischarge?

m3/s?

Massdischarge?

kg/s?

Weightdischarge?

N/s?

5.Discharge?

（?

）AQVdA=?

mAQVdA=?

gAQgVdA=?

6.MeanVelocity（?

）Thevelocitiesofpointsonthesamecrosssectioninthetotalflowaredifferent,sousuallyanaveragevelocityisusedinsteadoftherealvelocityoverthecrosssection,thisaveragevelocityiscalledthemeanvelocity.QVA=1?

Whatarestreamlineandpathline?

Whataretheirdifferences?

2?

Whatarestreamlineandpathlinescharacters?

3?

Isthereanystreamlinesinrealflow?

Whatispurposeofintroducingtheconceptofstreamline?

QuestionsStreamlinesarecurvesthatshowthemeandirectionofanumberofparticlesatthesameinstantoftime;curvesaretangenttothevelocityvectorsofanypointsoccupyingonthestreamline.Pathlineisthetraceafterasingleparticletravelsinafieldofflowoveraperiodoftime.a?

Atthesameinstantoftime,streamlinescannotintersect.b?

Streamlinecantbeafoldingline,itmustbeasmoothcurve.c?

Streamlineclusterdensityreflectstheamountofvelocity.（Densestreamlinesmeanlargevelocity;whilesparsestreamlinesmeansmallvelocity.）No.Itisconvenienttoanalyzethefluidflowanddefinethetrendsoffluidflow.5?

Whatarethecharactersofsteadyflow?

6?

WhataretheresearchobjectsofEulerianDescriptionandLagrangianDescriptionofMotion?

Inengineering,whichdescriptionofmotionisutilized?

Steadyflowistheflowwhosehydraulicmotionfactorsdontchangewithtime;Insteadyflow,streamlineandpathlinecoincide,andlocalaccelerationisequaltozero.0tA=?

TheresearchobjectofEulerianDescriptionisflowfieldandtheresearchobjectofLagrangianDescriptionofMotionisfluidparticles;Inengineering,EulerianDescriptionisusedwidely.?

3.5ContinuityEquationforSteadyFlowinaConduit（1-D）?

）1.Alltheparametersdonotvaryacrossthecross-sections1-Dflow:

1?

1?

2?

2?

s?

V?

A?

V?

A?

2.Onlythemeanparametersofthecross-sectionsareconsidered.or:

?

3.5ContinuityEquationforSteadyFlowinaConduit（1-D）?

）Consideracontrolvolumeandtheconditionsbelow:

?

1?

Nofluidcanleaveorenterthecontrolvolumethroughthetubewall?

2?

Fluidisacontinuum,andthereisnogapinthetubeflow?

3?

Ignorethepossibilitythatthemassturnstoenergy.1?

1?

2?

2?

s?

V?

A?

V?

A?

1.Selectacontrolvolume1122inaflowfieldvolume?

leftsurfaceA1（crosssection）:

A1,V1,1rightsurfaceA2（crosssection）:

A2,V2,1V2V1?

1?

2?

2?

s?

V?

A?

V?

A?

2?

3.5ContinuityEquationforSteadyFlowinaConduit（1-D）（?

）Basedonthelawofconservationofmass,thereis1?

1?

2?

2?

s?

V?

A?

V?

A?

12111222AA?

VdAVdAd?

t=?

leftsurfaceflowin（dt）:

1111AVdAdt?

rightsurfaceflowout（dt）:

2222AVdAdt?

Masschange（dt）:

?

dtd?

t?

Physicalmeaning?

Thenetmassdischargeenteringthecontrolvolumeisequaltothemassincreasedinunittimeduetothechangeindensity.Fitfor?

Steadyflow,unsteadyflow,compressibleandincompressiblefluid,idealfluidandrealfluid.12111222AA?

VdAVdAd?

t=?

Continuityequationintotalflow（generalform）12111222AA?

VdAVdAd?

t=?

Ifdensitydonotvaryacrosstheinletandoutletareas:

Fitfor?

Allsteadyflowswithinsolidboundary,includingcompressibleandincompressiblefluid,idealfluidandrealfluid.0t=ContinuityEquationinSteadyTotalFlowForsteadyflow:

Thus:

AVdAVQAA?

=121212VAVA=121112220AAVdAVdA=?

112212VAVA=12111222AA?

VdAVdAd?

t=?

ContinuityEquationinIncompressibleTotalFlowPhysicalmeaning?

Fortheincompressible