工程流体力学(英文版)第二章.pdf.pdf

工程流体力学(英文版)第二章.pdf.pdf

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ChapterTwoFluidStatics（?

WhatisFluidStaticsGeneralRulesoffluidatrest,andtheirengineeringapplication.fluidatrestfluidinequilibriumEquilibrium（a=0）relativeequilibrium（a=0）CharacteristicofFluidatrestu=0du=00dudy=Contents2.1Staticalpressureintensityanditscharacteristic2.2DifferentialEquationofFluidEquilibrium2.3PressureDistributionintheStaticFluid2.4PressureMearurements2.5FluidinRelativeEquilibriumFluid.2.6FluidStaticForceonPlaneandCurvedArea2.2.1DefinitionofstaticalpressureintensityNormalforceactingoverperunitareaofastaticfluid2.1Staticalpressureintensityanditscharacteristic?

0lim?

nAFpA=Unit:

PaorN/m22.2.1Characteristic1?

direction2?

magnitudeThepressureatapointinafluidatrestisthesameinalldirections.Ithasnothingtodowiththenormaldirectionoftheactingsurface.NegativeNormalForcePositive-PullingShearing?

Thereisonlycompressivestress（orpressure）inafluidatrest,andthedirectionofpressureisthesameasthedirectionofinwardnormallineofactingpoint.Fluidatrestcannotbearpullingforcebecauseofthetrendstoflow.xyznpppp=

（1）SelectatriangularprismelementOABC,dx,dy,dzpx,py,pz,pnarepressureintensityactedontherespectivesurface.Thesurfacepressure:

1dd2xpyz1dd2ypxz1dd2zpxydnpA,nisnormaldirectionofinclinedsurfaceABC.y?

x?

z?

0?

pz?

py?

px?

pn?

dy?

dx?

dz?

C?

A?

B?

Considerforcecomponentsinxdirection:

Surfaceforces:

Massforces?

（2）Forceanalysis:

?

OAC:

?

OAB:

?

OBC:

?

ABC:

1dd2xpyz1dd2ypxz1dd2zpxydnpA1ddd6xfxyz1ddd6yfxyz1ddd6zfxyz,（3）Equationoffluidinequilibrium0F=?

11cos（,）026xnxpdydzpdAnxfdxdydz+=y?

x?

z?

0?

pz?

py?

px?

pn?

dy?

dx?

dz?

C?

A?

B?

Similarly?

Whentheelementshrinkstoapointo,dx0?

thusTheresultsshowthatthepressuresareindependentofdirectionbecauseisarbitrary.Hencethepressureatapointonastaticfluidisthesameinalldirections.n?

n?

n?

And:

1dcos（,）dd2Anxyz=So:

111ddddddd0226xnxpyzpyzfxyz+=1d03xnxppfx+=y?

x?

z?

0?

pz?

py?

px?

pn?

dy?

dx?

dz?

C?

A?

B?

xnpp=xyznpppp=（）,ppxyz=11cos（,）026xnxpdydzpdAnxfdxdydz+=1DifferentialEquationsofaFluidinEquilibrium-EulerEquilibriumEquations2PressureDifferenceEquation3ForcePotentialFunction2SurfaceofEqualPressure2.2DifferentialEquationofFluidEquilibrium2.2.1DifferentialEquationsofaFluidinEquilibrium-EulerEquilibriumEquations2.2DifferentialEquationsofaFluidinEquilibriumConsiderthesixsurfacesofinfinitesimalelementinequilibriumfluid.Itssidesaredx,dy,dz.Assumethepressureatthecenteroftheelementisp（x,y,z）=p.2.2DifferentialEquationofFluidEquilibriumy?

x?

z?

dy?

dx?

dz?

2.2DifferentialEquationsofaFluidinEquilibriumConsiderforcecomponentsinydirection?

Massforces?

y?

x?

z?

dy?

dx?

dz?

Surfaceforces:

d2Bpyppy=d2Cpyppy=+d（）dd2pypxzyd（）dd2pypxzy+left:

right:

dddyfxyzThereis?

Fy=0inydirectionbecausetheelementisinequilibrium:

dd（）dd（）ddddd022ypypypxzpxzfxyzyy+=（）200（）（）（）（）2fxfxxfxfxxx+=+DifferentialEquationsofaFluidinEquilibrium?

EulerEquilibriumEquations?

condition:

?

or?

2.2DifferentialEquationsofaFluidinEquilibriumdd（）dd（）ddddd022ypypypxzpxzfxyzyy+=10ypfy=101010xyzpfxpfypfz?

=?

=?

=?

1grad0fp=?

EquilibriumandrelativeequilibriumCompressibleandincompressibleflowPhysicalMeaning:

Forthefluidinequilibrium,surfaceforcecomponentspermassfluidareequaltomassforcecomponentspermassfluid.Pressurevariationrateinaxesdirections?

）areequaltomassforcecomponentsperunitvolumeinaxesdirections?

fx,fy,fz）respectively.zpypxp,?

or?

2.2DifferentialEquationsofaFluidinEquilibrium101010xyzpfxpfypfz?

=?

=?

=?

1grad0fp=?

2.2.2PressureDifferenceEquation（GeneralDifferentialEquationsofaFluidinEquilibrium）101010xyzpfxpfypfz?

=?

=?

=?

ddddppppxyzxyz=+1ddd（ddd）xyzpppfxfyfzxyzxyz+=+?

p=p（x,y,z）Multiplyeveryequationinequationgroup

（1）withdx,dy,dzrespectively,thenaddthem:

?

thetotaldifferentialofpressureis:

2.2DifferentialEquationsofaFluidinEquilibriumd（ddd）xyzpfxfyfz=+2.2.3ForcePotentialFunction（）xyzdpfdxfdyfdz=+Ifthedensityisaconstant:

Defineaforcepotentialfunction:

p=（）xyzdpfdxfdyfdz=+（）pdddxdydzxyy?

=?

xyzfxfyfz?

=?

=?

=?

2.2.4EquipressureSurfaceEquipressureSurfaceisasurfacethatthepressureofeverypointinliquidisequal.Commonequipressuresurfacesarefreeliquidsurfaceandinterfaceoftwounmixedfluidsinequilibrium.massforceofanypointontheequipressuresurfaceinequilibriumfluidisperpendiculartotheequipressuresurface.KinescopeCartoon2.2DifferentialEquationsofaFluidinEquilibriumd0p=ddd0xyzfxfyfz+=0fdr=?

Importantcharacterofequipressuresurface:

xyxffifjfkdrdxidyjdzk?

=+?

=+?

（）xyzdpfdxfdyfdz=+Proving?

ConsiderafluidparticleMontheequipressuresurface,drisadifferentialdistanceontheequipressuresurface.Assumetheunitmassforceoftheparticleis?

Attheequipressuresurfaceinequilibriumfluid:

drdxidyjdzk=+?

（）0xyzdpfdxfdyfdz=+=Massforceisperpendiculartodrdrislinevectorontheequipressuresurface0fdr=?

Massforceisperpendiculartoequipressuresurfacexyzdris?

xyxffifjfk=+?

0fdr=?

2.3.1BasicEquationofStaticFluidunderGravity1.BasicEquationGeneraldifferentialequationofafluidinequilibrium?

（）xyzdpfdxfdyfdz=+0xyzfffg=?

0dpgdzdzdpg=+=c=pzCg+=ABABppzzgg+=+（uniform,impressiblefluid,undertheactionofgravity）2.3PressureDistributionintheStaticFluidorBasicequationoffluidstaticsz（m）?

theelevationheightabovedatumsurfaceo-o.ElevationHeadp/g（m）?

risingheightoffluidwithunitweightundertheactionofpressureP.PressureHead（m）:

Totalheightisaconstant.PiezometricHead2.GeometricalmeaningABABppzzgg+=+pzCg+=1212ppzzgg+=+pzg+Foranypointsinastaticfluidundergravity,theirunitpotentialenergyarethesame.3.PhysicalmeaningZ（Nm/N）?

elevationpotentialenergyperunitweightoffluidp/g（Nm/N）?

pressurepotentialenergyperunitweightoffluid（Nm/N）:

Totalpotentialenergyperunitweightoffluid.Cartoonpzg+Hhzzoop0ApzCg+=So,pressureoffluidstatics?

2.3.2PressureDistributioninaStaticFluidunderGravityEqui-pressureSurface?

hcorzc=Atthefreesurface?

dz=-dhdpgdz=let?

ddpgh=h=0?

p=p01pghC=+0ppgh=+Conclusions?

1?

Pressureatapointinastaticfluidundergravityincreaseslinearlywithdepth.2?

Pressureatapointinastaticfluidundergravityisequaltothesumofthepressureatthefreesurfaceandthefluidspecificweighttimingdepth.3?

Equipressuresurfaceinastaticfluidundergravityisahorizontalplane.4?

Extended:

whilethepressureatapointandthedepthdifferencebetweentwopointsareknown,thepressureatanotherpointcanbecalculated.0ppgh=+Exercise?

whichstaticequationistruebasedonFigure（?

2）?

1212ppzzgg+=+3232ppzzgg+=+Question?

Totalpotentialenergyatanypointtothesamedatumplaneperunitweightfluidundergravityatrest_?

A.increasesduetoincreasesindepth?

B.isconstant?

C.decreasesduetoincreasesindepth?

D.isuncertain.Returnb.RelativePressure?

Knownas“gagepressure”,pressurethatismeasuredrelativetolocalatmosphericpressure.p=pabspa.pcanbepositive,negativeorzero.c.Vacuum?

Itisthenegativerelativepressure.Thestatethattheabsolutepressureislessthanatmosphericpressure.a.AbsolutePressure?

Pressurethatismeasuredaboveabsolutevacuum（absolutezero）.weexpressitwithpabs?

pabs0.2.4.PressureExpressionandMearurements2.4.1?

ExpressionofPressureCartoondifferentreferenceAttention?

Relativepressureisusedincalculationwithoutextraexplanation.Vacuumvaluepv）（aabsabsappppp1atm:

111Appgh=+2appgh=+12pp=11Aappghgh=+1?

2isEquipressureSurface,so?

1MApgh=if:

Aappgh=+gagepressure:

1atm:

vacuum:

111Appghgh=+2app=12pp=11Aappghgh=1?

2isequipressuresurface,so?

11vApghgh=+vApgh=1p2p3?

C.p1p2p3?

D.p2p11atm:

Aappgh=+gagepressure:

1atm:

aAppgh=+gagepressure:

MApgh=MApgh=IfthepressureofmeasuredpointAisverysmall,thefollowingtwomeanscanbeusedtoliftmeasureaccuracy,throughenlargingpiezometrictubereadonthescale,?

1?

Piezometrictubemaybeinclined.Ifthereadonthescaleisl,thepressurehead（verticalheight）his:

pA=?

gh=?

glsin?

2?

Putinaliquidwithlowdensity.Itsdensity?

1sinhl=rising:

falling:

212hlAA=12hhh=+（）（）21212sinaappghhpglAA=+=+12AApa?

C.p0pa?

D.Uncertain.A.1.5m?

B.1.125m?

C.2m?

D.11.5m?

Choice2?

Thefreesurfaceinpiezometrictubeis1.5mhigherthanliquidsurfaceinaclosedvesselasshowninfigure,.Whatistherelativepressureintheheightofwatercolumn?

Liquidinthevesselisgasoline.?

=7.35KN/m3?

1.Intraditionalexperiment?

whyisthemercuryusedasworkingfluidofU-tubepiezometer?

1?

Lowcompressibility?

2?

Lowevaporationpressure?

3?

Highdensity.2.Twofluidsareinthesamevesselshowninfigure,12,andtwopiezometrictubesarefixedonthevesselwall.Istheliquidelevationofpiezometrictubeshowninfiguretrue?

12YesExample2.2asshowninFig2.2,formercury=13600kg/m3?

andwater1=1000kg/m3?

h=15cm?

whatisthepressuredifferencebetweenpointAandC.Solution?

so?

411211Appghpgh=+=+312112Cppghpgh=+=+（）21112Acppppghh=+（）2121ppghhgh=（）1121Acppghghhghgh=+=（）21360010009.80.1518522NmAcpp=so?

andExample2.3pMA=0.25atm?

=13600kg/m3?

=800kg/m3?

h=0.5m?

h1=200mm?

h2=250mm?

h3=220mm?

whatisthepressureinvesselB.Solution?

for1-1?

2-2?

3-3?

pM=p4?

and?

Fromtheseequations?

and?

（）11MAppghh=+?

211ppgh=?

323ppgh=+?

432Mpppgh=?

（）132MMAppghghghgh=+?

226100NmMp=ChineseText:

P36:

2-1,2-3,2-4,2-8,2-9ExerciseInarelativeequilibriumfluid,thereisinertiaforcesbesidesgravity.2.5FluidinRelativeEquilibrium0aa.UniformLinearAccelerationb.UniformRotationaboutaVerticalAxisRelativeEquilibrium?

Relativerestorequilibriumstatethatthereisnorelativemotionbetweenfluidparticlesorbetweenfluidandcontainer.Thereisnorelativemotion,sothereisnoshearingstressexistinginfluidorbetweenfluidandwall.a.UniformLinearAccelerationConsidertheunitmassforcesincludinginertiaforceandgravity:

So?

C