PARTIFUNDAMENTALPRINCIPLES（基本原理）InpartI,wecoversomeofthebasicprinciplesthatapplytoaerodynamicsingeneral.ThesearethepillarsonwhichallofaerodynamicsisbasedChapter2Aerodynamics:SomeFundamentalPrinciplesandEquationsThereissogreatadifferencebetweenafluidandacollectionofsolidparticlesthatthelawsofpressureandofequilibriumoffluidsareverydifferentfromthelawsofthepressureandequilibriumofsolids.JeanLeRondd’Alembert,17682.1IntroductionandRoadMapPreparationoftoolsfortheanalysisofaerodynamicsEveryaerodynamictoolwedevelopedinthisandsubsequentchaptersisimportantfortheanalysisandunderstandingofpracticalproblemsOrientationofferedbytheroadmap2.2ReviewofVectorrelations2.2.1to2.2.10Skippedover2.2.11Relationsbetweenline,surface,andvolumeintegralsThelineintegralofAoverCisrelatedtothesurfaceintegralofA(curlofA)overSbyStokes’theorem:WhereaeraSisboundedbytheclosedcurveC:ThesurfaceintegralofAoverSisrelatedtothevolumeintegralofA(divergenceofA)overVbydivergence’theorem:WherevolumeVisboundedbytheclosedsurfaceS:Ifprepresentsascalarfield,avectorrelationshipanalogoustodivergencetheoremisgivenbygradienttheorem:2.3Modelsofthefluid:controlvolumesandfluidparticlesImportancetocreatephysicalfeelingfromphysicalobservation.Howtomakereasonablejudgmentsondifficultproblems.

Inthischapter,basicequationsofaerodynamicswillbederived.PhilosophicalprocedureinvolvedwiththedevelopmentoftheseequationsInvokethreefundamentalphysicalprincipleswhicharedeeplyentrenchedinourmacroscopicobservationsofnature,namely,a.Massisconserved,that’stosay,masscanbeneithercreatednordestroyed.b.Newton’ssecondlaw:force=mass?accelerationc.Energyisconserved;itcanonlychangefromoneformtoanother2.Determineasuitablemodelofthefluid.3.Applythefundamentalphysicalprincipleslistedinitem1tothemodelofthefluiddeterminedinitem2inordertoobtainmathematicalequationswhichproperlydescribethephysicsoftheflow.Emphasisofthissection:Whatisasuitablemodelofthefluid?Howdowevisualizethissquishysubstanceinordertoapplythethreefundamentalprinciples?Threedifferentmodelsmostlyusedtodealwithaerodynamics.

finitecontrolvolume（有限控制體）infinitesimalfluidelement（無限小流體微團）molecular（自由分子）

2.3.1FinitecontrolvolumeapproachDefinitionoffinitecontrolvolume:

aclosedvolumesculpturedwithinafiniteregionoftheflow.ThevolumeiscalledcontrolvolumeV,andthecurvedsurfacewhichenvelopsthisregionisdefinedascontrolsurfaceS.Fixedcontrolvolumeandmovingcontrolvolume.Focusofourinvestigationforfluidflow.2.3.2InfinitesimalfluidelementapproachDefinitionofinfinitesimalfluidelement:

aninfinitesimallysmallfluidelementintheflow,withadifferentialvolume.ItcontainshugelargeamountofmoleculesFixedandmovinginfinitesimalfluidelement.Focusofourinvestigationforfluidflow.Thefluidelementmaybefixedinspacewithfluidmovingthroughit,oritmaybemovingalongastreamlinewithvelocityVequaltotheflowvelocityateachpointaswell.2.3.3MoleculeapproachDefinitionofmoleculeapproach:

Thefluidpropertiesaredefinedwiththeuseofsuitablestatisticalaveraginginthemicroscopewhereinthefundamentallawsofnatureareapplieddirectlytoatomsandmolecules.Insummary,althoughmanyvariationsonthethemecanbefoundindifferenttextsforthederivationofthegeneralequationsofthefluidflow,theflowmodelcanbeusuallybecategorizedunderoneoftheapproachdescribedabove.2.3.4PhysicalmeaningofthedivergenceofvelocityDefinitionof:

isphysicallythetimerateofchangeofthevolumeofamovingfluidelementoffixedmassperunitvolumeofthatelement.Analysisoftheabovedefinition:Step1.Selectasuitablemodeltogiveaframeunderwhichtheflowfieldisbeingdescribed.

amovingcontrolvolumeisselected.Step2.Selectasuitablemodeltogiveaframeunderwhichtheflowfieldisbeingdescribed.

amovingcontrolvolumeisselected.Step3.Howaboutthecharacteristicsforthismovingcontrolvolume?volume,controlsurfaceanddensitywillbechangingasitmovestodifferentregionoftheflow.Step4.ChanginvolumeduetothemovementofaninfinitesimalelementofthesurfacedS

over

.

ThetotalchangeinvolumeofthewholecontrolvolumeoverthetimeincrementisobviouslygivenasbellowStep5.Iftheintegralaboveisdividedby

.theresultisphysicallythetimeratechangeofthecontrolvolume

Step6.ApplyingGausstheorem,wehave

Step7.Asthemovingcontrolvolumeapproachestoainfinitesimalvolume,.Thentheaboveequationcanberewrittenas

Assumethatissmallenoughsuchthatisthesamethroughout.Then,theintegralcanbeapproximatedas,wehaveorDefinitionof:

isphysicallythetimerateofchangeofthevolumeofamovingfluidelementoffixedmassperunitvolumeofthatelement.Anotherdescriptionofand:Assumeisacontrolsurfacecorrespondingtocontrolvolume,whichisselectedinthespaceattime.Attimethefluidparticlesenclosedbyattimewillhavemovedtotheregionenclosedbythesurface.ThevolumeofthegroupofparticleswithfixedidentityenclosedbyattimeisthesumofthevolumeinregionAandB.Andattime,thisvolumewillbethesumofthevolumeinregionBandC.Astimeintervalapproachestozero,coincideswith.Ifisconsideredasafixedcontrolvolume,then,theregioninAcanbeimaginedasthevolumeenterintothecontrolsurface,Cleaveout.Basedontheargumentabove,theintegralofcanbeexpressedasvolumefluxthroughfixedcontrolsurface.Further,canbeexpressedastherateatwhichfluidvolumeisleavingapointperunitvolume.Theaveragevalueofthevelocitycomponentontheright-handxfaceisTherateofvolumeflowoutoftheright-handxfaceisThatintotheleft-handxfaceisThenetoutflowfromthexfacesisperunittimeThenetoutflowfromallthefacesinx,y,zdirectionsperunittimeisThefluxofvolumefromapointis2.4ContinuityequationInthissection,wewillapplyfundamentalphysicalprinciplestothefluidmodel.Moreattentionshouldbegivenforthewayweareprogressinginthederivationofbasicflowequations.DerivationofcontinuityequationStep1.Selectionoffluidmodel.Afixedfinitecontrolvolumeisemployedastheframefortheanalysisoftheflow.Herein,thecontrolsurfaceandcontrolvolumeisfixedinspace.Step2.Introductionoftheconceptofmassflow.LetagivenareaAisarbitrarilyorientedinaflow,thefiguregivenbellowisanedgeview.IfAissmallenough,thenthevelocityVovertheareaisuniformacrossA.ThevolumeacrosstheareaAintimeintervaldt

canbegivenasThemassinsidetheshadedvolumeisThemassflowthroughisdefinedasthemasscrossingAperunitsecond,anddenotedasorTheequationabovestatesthatmassflowthroughAisgivenbytheproductAreaXdensityXcomponentofflowvelocitynormaltotheareamassfluxisdefinedasthemassflowperunitareaStep3.

Physicalprinciple

Masscanbeneithercreatednordestroyed.Step4.Descriptionoftheflowfield,controlvolumeandcontrolsurface.DirectionalelementarysurfaceareaonthecontrolsurfaceElementaryvolumeinsidethefinitecontrolvolumeStep5.Applythemassconservationlawtothiscontrolvolume.NetmassflowoutofcontrolvolumethroughsurfaceSTimeratedecreaseofmassinsidecontrolvolumeVorStep6.MathematicalexpressionofBTheelementalmassflowacrosstheareaisThephysicalmeaningofpositiveandnegativeofThenetmassflowoutofthewholecontrolsurfaceS

Step7.MathematicalexpressionofCThemasscontainedinsidetheelementalvolumeVisThemassinsidetheentirecontrolvolumeisThetimerateofincreaseofthemassinsideVisThetimerateofdecreaseofthemassinsideVisStep8.FinalresultofthederivationLetB=C,thenwegetorDerivationwithmovingcontrolvolumeMassattimeMassattimeBasedonmassconservationlawConsiderthelimitsasThenwegetthemathematicaldescriptionofthemassconservationlawwiththeuseofmovingcontrolvolumeWhythefinalresultsderivedwithdifferentfluidmodelarethesame??Step9.NotesfortheContinuityEquationaboveThecontinuityequationaboveisinintegralform,itgivesthephysicalbehaviouroverafiniteregionofspacewithoutdetailedconcernsforeverydistinctpoint.Thisfeaturegivesusnumerousopportunitiestoapplytheintegralformofcontinuityequationforpracticalfluiddynamicoraerodynamicproblems.Ifwewanttogetthedetailedperformanceatagivenpoint,then,whatshallwedealwiththeintegralformabovetogetapropermathematicdescriptionformassconservationlaw?Step10.

ContinuityEquationinDifferentialformControlvolumeisfixedinspaceTheintegrallimitisnotthesameTheintegrallimitisthesameorApossiblecasefortheintegraloverthecontrolvolumeIfthefinitecontrolvolumeisarbitrarilychoseninthespace,theonlywaytomaketheequationbeingsatisfiedisthat,theintegrandoftheequationmustbezeroatallpointswithinthecontrolvolume.Thatis,Thatisthecontinuityequationinapartialdifferentialform.ItconcernstheflowfieldvariablesatapointintheflowwithrespecttothemassconservationlawItisimportanttokeepinmindthatthecontinuityequationsinintegralformanddifferentialformareequallyvalidstatementsofthephysicalprinciplesofconservationofmass.theyaremathematicalrepresentations,butalwaysrememberthattheyspeakwords.Step11.

LimitationsoftheequationsderivedContinuumflowormolecularflowAsthenatureofthefluidisassumedasContinuumflowinthederivationsoItsatisfiesonlyforContinuumflowSteadyfloworunsteadyflowItsatisfiesbothsteadyandunsteadyflowsviscousfloworinviscidflowItsatisfiesbothviscousandinviscidflowsCompressiblefloworincompressiblwflowItsatisfiesbothCompressibleandincompressiblwflowsDifferencebetweensteadyandunsteadyflowUnsteadyflow:Theflow-fieldvariablesareafunctionofbothspatiallocationandtime,thatisSteadyflow:Theflow-fieldvariablesareafunctionofspatiallocationonly,thatisForsteadyflow:Forsteadyincompressibleflow:2.5MomentumequationNewton’ssecondlawwhereForceexertedonabodyofmassMassofthebodyAccelerationConsiderafinitemovingcontrolvolume,themassinsidethiscontrolvolumeshouldbeconstantasthecontrolvolumemovingthroughtheflowfield.Sothat,Newton’ssecondlawcanberewrittenasDerivationofmomentumequationStep1.Selectionoffluidmodel.Afixedfinitecontrolvolumeisemployedastheframefortheanalysisoftheflow.Step2.

Physicalprinciple

Force=timeratechangeofmomentumStep3.ExpressionoftheleftsideoftheequationofNewton’ssecondlaw,i.e.,theforceexertedonthefluidasitflowsthroughthecontrolvolume.Twosourcesforthisforce:Bodyforces:overeverypartofV2.Surfaceforces:overeveryelementalsurfaceofSBodyforceonaelementalvolumeBodyforceoverthecontrolvolumeSurfaceforcesoverthecontrolsurfacecanbedividedintotwoparts,oneisduetothepressuredistribution,andtheotherisduetotheviscousdistribution.PressureforceactingontheelementalsurfaceNote:indicationofthenegativesignCompletepressureforceovertheentirecontrolsurfaceThesurfaceforceduetotheviscouseffectissimplyexpressedbyTotalforceactingonthefluidinsidethecontrolvolumeasitissweepingthroughthefixedcontrolvolumeisgivenasthesumofalltheforceswehaveanalyzedStep4.ExpressionoftherightsideoftheequationofNewton’ssecondlaw,i.e.,thetimeratechangeofmomentumofthefluidasitsweepsthroughthefixedcontrolvolume.MovingcontrolvolumeLetbethemomentumofthefluidwithinregionA,

B,andC.forinstance,Attime,themomentuminsideisAttime,themomentuminsideisThemomentumchangeduringthetimeintervalorAsthetimeintervalapproachestozero,theregionBwillcoincidewithSinthespace,andthetwolimitsNetmomentumflowoutofcontrolvolumeacrosssurfaceSTimeratechangeofmomentumduetounsteadyfluctuationsofflowpropertiesinsideVTheexplanationsabovehelpsustomakeabetterunderstandingoftheargumentsgiveninthetextbookbellowNetmomentumflowoutofcontrolvolumeacrosssurfaceSTimerateofchangeofmomentumduetounsteadyfluctuationsofflowpropertiesinsidecontrolvolumeVStep5.MathematicaldescriptionofmassflowacrosstheelementalareadSismomentumflowacrosstheelementalareadSisThenetflowofmomentumoutofthecontrolvolumethroughSisStep6.MathematicaldescriptionofThemomentumintheelementalvolumedV

isThemomentumcontainedatanyinstantinsidethecontrolvolumeV

isItstimeratechangeduetounsteadyflowfluctuationisBeawareofthedifferencebetweenandStep7.FinalresultofthederivationCombinetheexpressionsoftheforcesactingonthefluidandthetimeratechangeduetotermand,respectively,accordingtoNewton’ssecondlowIt’sthemomentumequationinintegralformIt’savectorequationAdvantagesformomentumequationinintegralformStep8.

MomentumEquationinDifferentialformTrytorearrangetheeveryintegralstosharethesamelimitgradienttheoremcontrolvolumeisfixedinspaceThenwegetSplitthisvectorequationasthreescalarequationswithMomentumequationinxdirectionisdivergencetheoremAsthecontrolvolumeisarbitrarychosen,thentheintegrandshouldbeequaltozeroatanypoint,thatisxdirectionydirectionzdirectionTheseequationscanappliedforunsteady,3Dflowofanyfluid,compressibleorincompressible,viscousorinviscid.SteadyandinviscidflowwithoutbodyforcesEuler’sEquationsandNavier-StokesequationsWhetherthe

viscouseffectsarebeingconsideredornotEulersEquations:inviscidflowNavier-Stokesequations:viscousflowDeepunderstandingofdifferenttermsincontinuityandmomentumequationsTimeratechangeofmassinsidecontrolvolumeTimeratechangeofmomentuminsidecontrolvolumeNetflowofmassoutofthecontrolvolumethroughcontrolsurfaceSNetflowofvolumeoutofthecontrolvolumethroughcontrolsurfaceSNetflowofmomentumoutofthecontrolvolumethroughcontrolsurfaceSBodyforcethroughoutthecontrolvolumeVSurfaceforceoverthecontrolsurfaceSWhatwecanforeseetheapplicationsforaerodynamicproblemswithbasicflowequationsonhand?IfthesteadyincompressibleinviscidflowsareconcernedPartialdifferentialequationforvelocityPartialdifferentialequationforvelocityandpressure2.6Anapplicationofthemomentumequation:dragofa2DbodyHowtodesigna2Dwindtunneltest?Howtomeasuretheliftanddragexertedontheairfoilbythefluid?AselectedcontrolvolumearoundanairfoilDescriptionsofthecontrolvolume1.Theupperandlowerstreamlinesfaraboveandbelowthebody(abandhi).2.Linesperpendiculartotheflowvelocityfaraheadandbehindthebody(ai

andbh)3.Acutthatsurroundsandwrapsthesurfaceofthebody(cdefg)1.Pressureatabandhi.2.Pressureataiandbh

.,velocity,3.Thepressureforceoverthesurfaceabhi4.Thesurfaceforceondefbythepresenceofthebody,thisforceincludestheskinfrictiondrag,anddenotedasperunitspan.5.Thesurfaceforcesoncdandfgcanceleachother.6.Thetotalsurfaceforceontheentirecontrolvolumeis7.ThebodyforceisnegligibleApplytomomentumequation,wehaveforsteadyflowNote:it’savectorequation.Ifweonlyconcernthexcomponentoftheequation,withrepresentsthexcomponentof.Asboundariesofthecontrolvolumeabhiarechosenfarawayfromthebody,thepressureperturbationduetothepresenceofthebodycanbeneglected,thatmeans,thepressurethereequaltothefreestreampressure.Ifthepressuredistributionoverabhiisconstant,thenSothatAsab,hi,defarestreamlines,thenAscd,fg

areareadjacenttoeachother,thenTheonlycontributiontomomentumflowthroughthecontrolsurfacecomefromtheboundariesaiandbh.FordS=dy(1),themomentumflowthroughthecontrolsurfaceisNote:Thesigninfrontofeachintegralsontherighthandsideoftheequation2.TheintegrallimitsforeachintegralsontherighthandsideoftheequationConsidertheintegralformofthecontinuityequationforsteadyflow,orAsisaconstantThefinalresultgivesthedragperunitspanThedragperunitspancanbeexpressedintermsoftheknownfreestreamvelocityandflow-fieldproperties,acrossaverticalstationdownstreamofthebody.PhysicalmeaningbehindtheequationMassflowoutofthecontrolvolumeVelocitydecrementMomentumdecrementpersecondForincompressibleflow,thatis,thedensityisconstant2.6.1CommentsWiththeapplicationofmomentumprincipletoalarge,fixedcontrolvolume,anaccurateresultforoverallquantitysuchasdragonabodycanbepredictedwithknowingthedetailedflowpropertiesalongthecontrolsurface.Thattosay,itisunnecessarytoknowthethedetailsalongthesurfaceofthebody.2.7EnergyequationContinuityequationMomentumequationUnknowns:ForsteadyincompressibleinvicidflowsForcompressibleflowsisanadditionalvariable,andthereforeweneedanadditionalfundamentalequationtocompletethesystem.Thisfundamentalequationistheenergyequation,whichwearegoingtodevelop.Twoadditionalflow-fieldvariableswillappeartotheenergyequation,thatisinternalenergyandtemperature.Energyequationisonlynecessaryforcompressibleflows.Physicalprinciple(firstlawofthermodynamics)Energycanbeneithercreatednordestroyed;itcanonlychangeinform

DefinitionsofsystemandinternalenergyperunitmasseDefinitionofsurroundingsHeattransferredfromthesurroundingstothesystemWorkdoneonthesurroundingsbythesystemChangeofinternalenergyinsystemduetotheheattransferredandtheworkdoneAsenergyisconserved,soApplythefirstlawtothefluidflowingtroughthefixedcontrolvolume,andletB1=rateofheataddedtofluidinsidecontrolvolumefromsurroundings.B2=rateofworkdoneonfluidinsidecontrolvolume.B3=rateofchangeofenergyoffluidasitflowsthroughcontrolvolume.Asfirstlawshouldbesatisfied,thenB1+B2=B3Actuallyspeaking,theequationaboveisapowerequation.RateofvolumetricheatingIftheflowisviscousB1=Rateofvolumetricheating=TheforceincludesthreepartsPressureforce,bodyforceandskinfrictionforce

RateofworkdoneonfluidinsideVduetopressureforceonSRateofworkdoneonfluidinsideVduetobodyforceB2=Sincethefluidinsidethecontrolvolumeisnotstationary,itismovingatthelocalvelocitywithaconsequentkineticenergyperunitmass,so,thetotalenergyperunitmassisNetrateofflowoftotalenergyacrosscontrolsurfaceSTimeratechangeoftotalenergyinsideVduetotransientvariationsofflow-fieldpropertiesB3=B1+B2=B3EnergyequationinintegralformNotesinthetextbookEnergyequationinpartialdifferentialformIftheflowissteady,inviscid,adiabatic,withoutbodyforceAfterapplythreefundamentalphysicalprinciples,wehavederivedthreebasicequationsforfluidflow.Andtherearethreevariables,suchasForcaloricallyperfectgasesThen,onemorepropertyisadded,butwithperfectgasequationContinuity,momentumandenergyequationwithtwoadditionalequationsarefiveindependentequations,andtherefiveunknowns.Sothatwehavegotaclosedsystemfortheflowproblems.2.8Interimsummary2.9SubstantialderivativesFocusoureyeonainfinitesimalfluidelementmovingthroughaflowfield.ThevelocityfieldcanbegivenasThedensitycanbegivenasWiththeuseofTaylorseriesexpansionaboutpoint1Dividingbyincartesiancoordinates,thenSubstantialderivativeLocalderivativeconvectivederivative2.10FundamentalequationsintermofsubstantialderivativeInthissection,thecontinuity,momentumandenergyequationswillbegivenintermsofsubstantialderivativeThecontinuityequationindifferentialformisorSinceSoThisisthecontinuityequationintermsofsubstantialderivativeThexcomponentofthemomentumequationindifferentialformisorContinuityEquationhenceInthesamewaywecangetthesearethemomentumequationsintermsofsubstantialderivativeinx,y,zdirectionsrespectivelyEnergyequationintermsofsubstantialderivativeDetaileddescriptionsforthecomparisonbetweenthebasicflowequationsindifferentforms,refertothetextbook2.11Pathlines

andstreamlinesofaflowSkippedover2.12Angularvelocity,vorticityandstrainInthissection,moreattentionwillbepaidtoexaminetheorientationofthefluidelementanditsshapeasitmovesthroughastreamlineintheflowfield.Animportantquantity,vorticity,willbeintroduced.MotionofafluidelementalongastreamlineTrytosetuptherelationshipsbetweenwithandDistanceinydirectionthatAmovesduringtimeincrementDistanceinydirectionthatCmovesduringtimeincrementNetdisplacementinydirectionofCrelativetoASinceisasmallangleSimilarlyDefinition:angularvelocityofthefluidelementistheaverageangularvelocityoflinesABandAC,theyareperpendiculartoeachotheratthetimetvorticityInavelocityfield,thecurlofthevelocityisequaltothevorticityIfateverypointinaflowfield,theflowiscalledrotational.ThisimpliesthatthefluidelementshaveafiniteangularvelocityIfateverypointinaflow,theflowiscalledirrotational.Thisimpliesthatthefluidelementhavenoangularvelocity;theirmotionthroughspaceisapuretranslationDefinitionofstrain:thestrainofthefluidelementinxyplaneisthechangeofink,wherethepositivestraincorrespondstoadecreasingk.andkistheanglebetweensidesABandAC,theyareperpendiculartoeachotheratthetimetStrain=ThetimerateofstraininxyplaneisInthematrixabovewhichiscomposedofvelocityderivatives,thediagonaltermsrepresentthedilatation(擴張)

ofafluidelement.Theoffdiagonaltermsareassociatedwithrotationandstrainoffluidelement.Relationsbetweenviscouseffectandrotationofafluidelement.Irrotationalandrotationalflowsinpracticalaerodynamicproblems2.13CirculationImportanttoolforwetoobtainsolutionsforsomeverypracticalandexcitingaerodynamicproblems.Circulationcanbeusedtocalculateliftexertedonanairfoilwithunitspan.Definitionofcirculation

Note:thenegativesigninfrontofthelineintegralStokes’theoremReferringtothevectoranalysis,whatisthephysicalmeaningthattheequationbellowspeak?2.14StreamfunctionInasteady2Dsteadyflow,thedifferentialformofstreamlinescanbeexpressedasIfareknownfunctionsof,then,aftertheequationabovebeingintegrated,wecangetthealgebraicequationofthestreamlineForeachstreamline,isaconstant.Itsvaluevarieswithdifferentstreamlines.Replacingthesymbolwith,thenwehaveThefunctioniscalledstreamfunction.Differentvalueofthe,i.e,,representsdifferentstreamlinesintheflowfield.TwostreamlinesrespectingwithdifferentvaluesofPhysicalmeaningofthestreamfunctionArbitrarinessoftheintegrandconstantDifferenceinstreamfunctionbetweentwoindividualstreamlinesMassflowbetweenthetwostreamlinesab

andcd.(perunitdepthperpendiculartothepage)Howtoremovethearbitrarinessoftheconstantofintegration?Whatwillbethemassflowthroughanarbitrarycurveconnectingtwopointsonastreamline?Forasteadyflow,themassflowinsideagivenstreamtubeisconstant.Forasteadyflow,thecontinuityequationshouldbesatisfied,then,themassflowthroughaclosedcurveCiszero.ThatmeansthemassflowthroughL1isthesametothatofL2.Forasteadyflowandphysicalpossibleflows,the