流体力学（清华大学张兆顺54讲） PPT课件 133-7-1英文课件8

IX.COMPRESSIBLEFLOW符松清华大学航天航空学院2023/4/61清华大学工程力学系Compressibleflowisthestudyoffluidsflowingatspeedscomparabletothelocalspeedofsound.Thisoccurswhenfluidspeedsareabout30%ormoreofthelocalacousticvelocity.Then,thefluiddensitynolongerremainsconstantthroughouttheflowfield.Thistypicallydoesnotoccurwithfluidsbutcaneasilyoccurinflowinggases.Twoimportantanddistinctiveeffectsthatoccurincompressibleflowsarechokingwheretheflowislimitedbythesonicconditionthatoccurswhentheflowvelocitybecomesequaltothelocalacousticvelocityandshockwavesthatintroducediscontinuitiesinthefluidpropertiesandarehighlyirreversible.2023/4/62清华大学工程力学系Sincethedensityofthefluidisnolongerconstantincompressibleflows,therearenowfourdependentvariablestobedeterminedthroughouttheflowfield.Thesearepressure,temperature,density,andflowvelocity.Twonewvariables,temperatureanddensity,havebeenintroducedandtwoadditionalequationsarerequiredforacompletesolution.Thesearetheenergyequationandthefluidequationofstate.Thesemustbesolvedsimultaneouslywiththecontinuityandmomentum

equationstodeterminealltheflowfieldvariables.2023/4/63清华大学工程力学系EquationsofStateandIdealGasPropertiesTwoequationsofstateareusedtoanalyzecompressibleflows:theidealgasequationofstateandtheisentropicflowequationofstate.Thefirstofthesedescribegasesatlowpressure(relativetothegascriticalpressure)andhightemperature(relativetothegascriticaltemperature).Thesecondappliestoidealgasesexperiencingisentropic(adiabaticandfrictionless)flow.2023/4/64清华大学工程力学系TheidealgasequationofstateisInthisequation,Risthegasconstant,andPandTaretheabsolutepressureandabsolutetemperaturerespectively.AiristhemostcommonlyincurredcompressibleflowgasanditsgasconstantisRair=1716ft2/(s2∙oR)=287m2/(s2∙K).2023/4/65清华大学工程力学系Twoadditionalusefulidealgaspropertiesaretheconstantvolumeandconstantpressurespecificheatsdefinedaswhereuisthespecificinternalenergyandhisthespecificenthalpy.Thesetwopropertiesaretreatedasconstantswhenanalyzingelementalcompressibleflows.Commonlyusedvaluesofthespecificheatsofairare:Cv=4293ft2/(s2∙oR)=718m2/(s2∙K)andCp=6009ft2/(s2∙oR)=1005m2/(s2-K).AdditionalspecificheatrelationshipsareThespecificheatratio

kforairis1.4.2023/4/66清华大学工程力学系Whenundergoinganisentropicprocess(constantentropyprocess),idealgasesobeytheisentropicprocessequationofstate:CombiningthisequationofstatewiththeidealgasequationofstateandapplyingtheresulttotwodifferentlocationsinacompressibleflowfieldyieldsNote:Theaboveequationsmaybeappliedtoanyidealgasasitundergoesanisentropicprocess.

2023/4/67清华大学工程力学系AcousticVelocityandMachNumberTheacousticvelocity(speedofsound)isthespeedatwhichaninfinitesimallysmallpressurewave(soundwave)propagatesthroughafluid.Ingeneral,theacousticvelocityisgivenbyTheprocessexperiencedbythefluidasasoundwavepassesthroughitisanisentropicprocess.Thespeedofsoundinanidealgasisthengivenby2023/4/68清华大学工程力学系TheMachnumberistheratioofthefluidvelocityandspeedofsound,Thisnumberisthesinglemostimportantparameterinnderstandingandanalyzingcompressibleflows.2023/4/69清华大学工程力学系MachNumberExample:Anaircraftfliesataspeedof400m/s.Whatisthisaircraft’sMachnumberwhenflyingatstandardsea-levelconditions(T=289K)andatstandard15,200m(T=217K)atmosphereconditions?Atstandardsea-levelconditions,

andat15,200m,.Theaircraft’sMachnumbersarethenNote: Althoughtheaircraftspeeddidnotchange,theMachnumberdidchangebecauseofthechangeinthelocalspeedofsound.2023/4/610清华大学工程力学系IdealGasSteadyIsentropicFlowWhentheflowofanidealgasissuchthatthereisnoheattransfer(i.e.,adiabatic)orirreversibleeffects(e.g.,friction,etc.),theflowisisentropic.Thesteady-flowenergyequationappliedbetweentwopointsintheflowfieldbecomeswhereh0,calledthestagnationenthalpy,remainsconstantthroughouttheflowfield.Observethatthestagnationenthalpyistheenthalpyatanypointinanisentropicflowfieldwherethefluidvelocityiszeroorverynearlyso.2023/4/611清华大学工程力学系Theenthalpyofanidealgasisgivenbyh=CpToverreasonablerangesoftemperature.Whenthisissubstitutedintotheadiabatic,steady-flowenergyequation,weseethatho=CpTo=constantandThus,thestagnationtemperatureToremainsconstantthroughoutanisentropicoradiabaticflowfieldandtherelationshipofthelocaltemperaturetothefieldstagnationtemperatureonlydependsuponthelocalMachnumber.2023/4/612清华大学工程力学系Incorporationoftheacousticvelocityequationandtheidealgasequationsofstateintotheenergyequationyieldsthefollowingusefulresultsforsteadyisentropicflowofidealgases.2023/4/613清华大学工程力学系ThevaluesoftheidealgaspropertieswhentheMachnumberis1(i.e.,sonicflow)areknownasthecriticalorsonicpropertiesandaregivenby2023/4/614清华大学工程力学系Boththecritical(sonic,Ma=1)andstagnationvaluesofpropertiesareusefulincompressibleflowanalyses.Forair(k=1.4),theseratiosbecome2023/4/615清华大学工程力学系Inallisentropicflows,allcritical(Ma=1)propertiesareconstant.Inadiabatic,butnon-isentropicflows(e.g.adiabaticflowswithfriction),a*andT*areconstant,butP*andr*mayvary.AtsonicconditionsThesevalueswillbeveryusefulinproblemsinvolvingcompressibleflowwithfrictionorheattransferconsideredlaterinthechapter.2023/4/616清华大学工程力学系IsentropicFlowExample:Airflowingthroughanadiabatic,frictionlessductissuppliedfromalargesupplytankinwhichP=500kPaandT=400K.WhataretheMachnumberMa.thetemperatureT,densityr,andfluidVatalocationinthisductwherethepressureis430kPa?2023/4/617清华大学工程力学系Thepressureandtemperatureinthesupplytankarethestagnationpressureandtemperaturesincethevelocityinthistankispracticallyzero.Then,theMachnumberatthislocationis2023/4/618清华大学工程力学系andthetemperatureisgivenbyTheidealgasequationofstateisusedtodeterminethedensity,2023/4/619清华大学工程力学系UsingthedefinitionoftheMachnumberandtheacousticvelocity,weobtain2023/4/620清华大学工程力学系SolvingCompressibleFlowProblemsCompressibleflowproblemscomeinavarietyofforms,butthemajorityofthemcanbesolvedasfollows:Usetheappropriateequationsandreferencestates(i.e.,stagnationandsonicstates)todeterminetheMachnumberatallflowfieldlocationsinvolvedintheproblem.Determinewhichconditionsarethesamethroughouttheflowfield(e.g.thestagnationpropertiesarethesamethroughoutanisentropicflowfield).Applytheappropriateequationsandconstantconditionstodeterminethenecessaryremainingpropertiesintheflowfield.Applyadditionalrelations(i.e.equationofstate,acousticvelocity,etc.)tocompletethesolutionoftheproblem.MostcompressibleflowequationsareexpressedintermsoftheMachnumber.Youcansolvetheseequationsexplicitlybyrearrangingtheequation,byusingtables,orbyprogrammingthemwithspreadsheetorEESsoftware.2023/4/621清华大学工程力学系IsentropicFlowwithAreaChangesAllflowsmustsatisfythecontinuityandmomentumrelationsaswellastheenergyandstateequations.Applicationofthecontinuityandmomentumequationstoadifferentialflow(seetextbookforderivation)yields:ThisresultrevealsthatwhenMa<1(subsonicflow),Ma-1<0andvelocitychangesaretheoppositeofareachanges.Thatis,increasesinthefluidvelocityrequirethattheareadecreaseinthedirectionoftheflow.Forsupersonicflow(Ma>1),Ma-1>0andtheareamustincreaseinthedirectionoftheflowtocauseanincreaseinthevelocity.ChangesinthefluidvelocitydVcanonlybefiniteinsonicflows(Ma=1)whendA=0.Theeffectofthegeometryuponvelocity,Machnumber,andpressureisillustratedinFigure1nextpage.2023/4/622清华大学工程力学系Figure12023/4/623清华大学工程力学系2023/4/624清华大学工程力学系Ifthesonicconditiondoesoccurintheduct,itwilloccurattheductminimumormaximumarea.Ifthesonicconditionoccurs,theflowissaidtobechokedsincethemassflowrateandisthemaximummassflowratetheductcanaccommodatewithoutamodificationoftheductgeometry.Themaximumflowrateisalsogivenbyandforair

andforair

2023/4/625清华大学工程力学系2023/4/626清华大学工程力学系2023/4/627清华大学工程力学系ThelocalstagnationpressureisThecritical,sonic-throatareaisdeterminedfrom2023/4/628清华大学工程力学系Notethatthisistheminimumthroatareathatmustactuallyoccurintheductinorderfortheflowtobecomesupersonic.ThemassflowisgivenbyForparts(e)and(f),weknowA2/A*asgivenbelowandmustthereforesolveEqn.9.45forthevaluesofMa2thatwillyield(e)thesubsonicsolutionor(f)thesupersonicsolution.Use9.28atoobtainthepressure.and2023/4/629清华大学工程力学系ThisiseasilyaccomplishedwiththeEESorsomeothercomputerbasediterativesoftwaretoyieldthefollowing:(e)subsonicsolution-Ma2=0.6758P2=415kPaor(f)supersonicsolution-Ma2=1.4001P2=177kPaNotethatforthesupersonicsolution,thepressurehasdecreasedtoalowervalueandsonicconditionsmusthaveoccurredatthethroatbetween1and2.2023/4/630清华大学工程力学系NormalShockWavesUndertheappropriateconditions,verythin,highlyirreversiblediscontinuitiescanoccurinotherwiseisentropiccompressibleflows.Thesediscontinuitiesareknownasshockwaveswhichwhentheyareperpendiculartotheflowvelocityvectorarecallednormalshockwaves.Anormalshockwaveinaone-dimensionalflowchannelisillustratedinFigure2.Figure22023/4/631清华大学工程力学系Applicationofthesecondlawofthermodynamicstothethin,adiabaticnormalshockwaverevealsthatnormalshockwavescanonlycauseasharpriseinthegaspressureandmustbesupersonicupstreamandsubsonicdownstreamofthenormalshock.RarefactionwavesthatresultinadecreaseinpressureandincreaseinMachnumberareimpossibleaccordingtothesecondlaw.Applicationoftheconservationofmass,momentum,andenergyequationsalongwiththeidealgasequationofstatetoathin,adiabaticcontrolvolumesurroundinganormalshockwaveyieldstheresultsshowninthefollowingtable.2023/4/632清华大学工程力学系Itisnotedthatinmanycompressibleflowproblemswithnormalshocks,thelocationoftheshockisunknown.Fromtheequationsshownnextpage,thisismostreadilyspecifiedbyfindingthemachnoupstreamoftheshock,Ma1.However,formostproblemsthisrequiresaniterativesolutionofoneofthefollowingequations,dependingonthegiveninformation.2023/4/633清华大学工程力学系NormalShockRelations2023/4/634清华大学工程力学系Whenusingtheseequationstorelateconditionsupstreamanddownstreamofanormalshockwave,keepthefollowingpointsinmind:UpstreamMachnumbersarealwayssupersonicwhiledownstreamMachnumbersaresubsonic.Stagnationpressuresanddensitiesdecreaseasonemovesdownstreamacrossanormalshockwavewhilethestagnationtemperatureremainsconstant(aconsequenceoftheadiabaticflowcondition).Pressuresincreasegreatlywhiletemperatureanddensityincreasemoderatelyacrossashockwaveinthedownstreamdirection.Thecritical/sonicthroatareachangesacrossanormalshockwaveinthedownstreamdirectionand.Shockwavesareveryirreversiblecausingthespecificentropydownstreamoftheshockwavetobegreaterthanthespecificentropyupstreamoftheshockwave.2023/4/635清华大学工程力学系Movingnormalshockwavessuchasthosecausedbyexplosions,spacecraftreenteringtheatmosphere,andotherscanbeanalyzedasstationarynormalshockwavesbyusingaframeofreferencethatmovesatthespeedoftheshockwaveinthedirectionoftheshockwave.2023/4/636清华大学工程力学系Example:NormalShockinaConverging-DivergingNozzle2023/4/637清华大学工程力学系2023/4/638清华大学工程力学系2023/4/639清华大学工程力学系ReviewofVelocity-PotentialConceptsThischapterpresentsexamplesofproblemsandtheirsolutionsforwhichthe