流體力學(xué)（清華大學(xué)張兆順54講） PPT課件 134-7-1英文課件9

IX.COMPRESSIBLEFLOW符松清華大學(xué)航天航空學(xué)院2023/4/61清華大學(xué)工程力學(xué)系COMPRESSIBLEFLOWCompressibleflowisthestudyoffluidsflowingatspeedscomparabletothelocalspeedofsound.Thisoccurswhenfluidspeedsareabout30%ormoreofthelocalacousticvelocity.Then,thefluiddensitynolongerremainsconstantthroughouttheflowfield.Thistypicallydoesnotoccurwithfluidsbutcaneasilyoccurinflowinggases.Twoimportantanddistinctiveeffectsthatoccurincompressibleflowsarechokingwheretheflowislimitedbythesonicconditionthatoccurswhentheflowvelocitybecomesequaltothelocalacousticvelocityandshockwavesthatintroducediscontinuitiesinthefluidpropertiesandarehighlyirreversible.2023/4/62清華大學(xué)工程力學(xué)系COMPRESSIBLEFLOWSincethedensityofthefluidisnolongerconstantincompressibleflows,therearenowfourdependentvariablestobedeterminedthroughouttheflowfield.Thesearepressure,temperature,density,andflowvelocity.Twonewvariables,temperatureanddensity,havebeenintroducedandtwoadditionalequationsarerequiredforacompletesolution.Thesearetheenergyequationandthefluidequationofstate.Thesemustbesolvedsimultaneouslywiththecontinuityandmomentum

equationstodeterminealltheflowfieldvariables.2023/4/63清華大學(xué)工程力學(xué)系EquationsofStateandIdealGasPropertiesTwoequationsofstateareusedtoanalyzecompressibleflows:theidealgasequationofstateandtheisentropicflowequationofstate.Thefirstofthesedescribegasesatlowpressure(relativetothegascriticalpressure)andhightemperature(relativetothegascriticaltemperature).Thesecondappliestoidealgasesexperiencingisentropic(adiabaticandfrictionless)flow.2023/4/64清華大學(xué)工程力學(xué)系EquationsofStateandIdealGasPropertiesTheidealgasequationofstateisInthisequation,Risthegasconstant,andPandTaretheabsolutepressureandabsolutetemperaturerespectively.AiristhemostcommonlyincurredcompressibleflowgasanditsgasconstantisRair=1716ft2/(s2?oR)=287m2/(s2?K).2023/4/65清華大學(xué)工程力學(xué)系EquationsofStateandIdealGasPropertiesTwoadditionalusefulidealgaspropertiesaretheconstantvolumeandconstantpressurespecificheatsdefinedaswhereuisthespecificinternalenergyandhisthespecificenthalpy.Thesetwopropertiesaretreatedasconstantswhenanalyzingelementalcompressibleflows.Commonlyusedvaluesofthespecificheatsofairare:Cv=4293ft2/(s2?oR)=718m2/(s2?K)andCp=6009ft2/(s2?oR)=1005m2/(s2?K).AdditionalspecificheatrelationshipsareThespecificheatratio

kforairis1.4.2023/4/66清華大學(xué)工程力學(xué)系EquationsofStateandIdealGasPropertiesWhenundergoinganisentropicprocess(constantentropyprocess),idealgasesobeytheisentropicprocessequationofstate:CombiningthisequationofstatewiththeidealgasequationofstateandapplyingtheresulttotwodifferentlocationsinacompressibleflowfieldyieldsNote:Theaboveequationsmaybeappliedtoanyidealgasasitundergoesanisentropicprocess.

2023/4/67清華大學(xué)工程力學(xué)系A(chǔ)cousticVelocityandMachNumberTheacousticvelocity(speedofsound)isthespeedatwhichaninfinitesimallysmallpressurewave(soundwave)propagatesthroughafluid.Ingeneral,theacousticvelocityisgivenbyTheprocessexperiencedbythefluidasasoundwavepassesthroughitisanisentropicprocess.Thespeedofsoundinanidealgasisthengivenby2023/4/68清華大學(xué)工程力學(xué)系A(chǔ)cousticVelocityandMachNumberTheMachnumberistheratioofthefluidvelocityandspeedofsound,Thisnumberisthesinglemostimportantparameterinnderstandingandanalyzingcompressibleflows.2023/4/69清華大學(xué)工程力學(xué)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清華大學(xué)工程力學(xué)系IdealGasSteadyIsentropicFlowWhentheflowofanidealgasissuchthatthereisnoheattransfer(i.e.,adiabatic)orirreversibleeffects(e.g.,friction,etc.),theflowisisentropic.Thesteady-flowenergyequationappliedbetweentwopointsintheflowfieldbecomeswhereh0,calledthestagnationenthalpy,remainsconstantthroughouttheflowfield.Observethatthestagnationenthalpyistheenthalpyatanypointinanisentropicflowfieldwherethefluidvelocityiszeroorverynearlyso.2023/4/611清華大學(xué)工程力學(xué)系IdealGasSteadyIsentropicFlowTheenthalpyofanidealgasisgivenbyh=CpToverreasonablerangesoftemperature.Whenthisissubstitutedintotheadiabatic,steady-flowenergyequation,weseethatho=CpTo=constantandThus,thestagnationtemperatureToremainsconstantthroughoutanisentropicoradiabaticflowfieldandtherelationshipofthelocaltemperaturetothefieldstagnationtemperatureonlydependsuponthelocalMachnumber.2023/4/612清華大學(xué)工程力學(xué)系IdealGasSteadyIsentropicFlowIncorporationoftheacousticvelocityequationandtheidealgasequationsofstateintotheenergyequationyieldsthefollowingusefulresultsforsteadyisentropicflowofidealgases.2023/4/613清華大學(xué)工程力學(xué)系IdealGasSteadyIsentropicFlowThevaluesoftheidealgaspropertieswhentheMachnumberis1(i.e.,sonicflow)areknownasthecriticalorsonicpropertiesandaregivenby2023/4/614清華大學(xué)工程力學(xué)系IdealGasSteadyIsentropicFlowBoththecritical(sonic,Ma=1)andstagnationvaluesofpropertiesareusefulincompressibleflowanalyses.Forair(k=1.4),theseratiosbecome2023/4/615清華大學(xué)工程力學(xué)系IdealGasSteadyIsentropicFlowInallisentropicflows,allcritical(Ma=1)propertiesareconstant.Inadiabatic,butnon-isentropicflows(e.g.adiabaticflowswithfriction),a*andT*areconstant,butP*andr*mayvary.AtsonicconditionsThesevalueswillbeveryusefulinproblemsinvolvingcompressibleflowwithfrictionorheattransferconsideredlaterinthechapter.2023/4/616清華大學(xué)工程力學(xué)系IsentropicFlowExample:Airflowingthroughanadiabatic,frictionlessductissuppliedfromalargesupplytankinwhichP=500kPaandT=400K.WhataretheMachnumberMa.thetemperatureT,densityr,andfluidVatalocationinthisductwherethepressureis430kPa?2023/4/617清華大學(xué)工程力學(xué)系IsentropicFlowExample:Thepressureandtemperatureinthesupplytankarethestagnationpressureandtemperaturesincethevelocityinthistankispracticallyzero.Then,theMachnumberatthislocationis2023/4/618清華大學(xué)工程力學(xué)系IsentropicFlowExample:andthetemperatureisgivenbyTheidealgasequationofstateisusedtodeterminethedensity,2023/4/619清華大學(xué)工程力學(xué)系IsentropicFlowExample:UsingthedefinitionoftheMachnumberandtheacousticvelocity,weobtain2023/4/620清華大學(xué)工程力學(xué)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清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChangesAllflowsmustsatisfythecontinuityandmomentumrelationsaswellastheenergyandstateequations.Applicationofthecontinuityandmomentumequationstoadifferentialflow(seetextbookforderivation)yields:ThisresultrevealsthatwhenMa<1(subsonicflow),velocitychangesaretheoppositeofareachanges.Thatis,increasesinthefluidvelocityrequirethattheareadecreaseinthedirectionoftheflow.Forsupersonicflow(Ma>1),theareamustincreaseinthedirectionoftheflowtocauseanincreaseinthevelocity.ChangesinthefluidvelocitydVcanonlybefiniteinsonicflows(Ma=1)whendA=0.Theeffectofthegeometryuponvelocity,Machnumber,andpressureisillustratedinFigure1nextpage.2023/4/622清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChangesFigure12023/4/623清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChanges2023/4/624清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChangesIfthesonicconditiondoesoccurintheduct,itwilloccurattheductminimumormaximumarea.Ifthesonicconditionoccurs,theflowissaidtobechokedsincethemassflowrateandisthemaximummassflowratetheductcanaccommodatewithoutamodificationoftheductgeometry.Themaximumflowrateisalsogivenbyandforair

andforair

2023/4/625清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChanges2023/4/626清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChanges2023/4/627清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChangesThelocalstagnationpressureisThecritical,sonic-throatareaisdeterminedfrom2023/4/628清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChangesNotethatthisistheminimumthroatareathatmustactuallyoccurintheductinorderfortheflowtobecomesupersonic.ThemassflowisgivenbyForparts(e)and(f),weknowA2/A*asgivenbelowandmustthereforesolveEqn.9.45forthevaluesofMa2thatwillyield(e)thesubsonicsolutionor(f)thesupersonicsolution.Use9.28atoobtainthepressure.and2023/4/629清華大學(xué)工程力學(xué)系IsentropicFlowwithAreaChangesThisiseasilyaccomplishedwiththeEESorsomeothercomputerbasediterativesoftwaretoyieldthefollowing:(e)subsonicsolution-Ma2=0.6758P2=415kPaor(f)supersonicsolution-Ma2=1.4001P2=177kPaNotethatforthesupersonicsolution,thepressurehasdecreasedtoalowervalueandsonicconditionsmusthaveoccurredatthethroatbetween1and2.2023/4/630清華大學(xué)工程力學(xué)系NormalShockWavesUndertheappropriateconditions,verythin,highlyirreversiblediscontinuitiescanoccurinotherwiseisentropiccompressibleflows.Thesediscontinuitiesareknownasshockwaveswhichwhentheyareperpendiculartotheflowvelocityvectorarecallednormalshockwaves.Anormalshockwaveinaone-dimensionalflowchannelisillustratedinFigure2.Figure22023/4/631清華大學(xué)工程力學(xué)系NormalShockWavesApplicationofthesecondlawofthermodynamicstothethin,adiabaticnormalshockwaverevealsthatnormalshockwavescanonlycauseasharpriseinthegaspressureandmustbesupersonicupstreamandsubsonicdownstreamofthenormalshock.RarefactionwavesthatresultinadecreaseinpressureandincreaseinMachnumberareimpossibleaccordingtothesecondlaw.Applicationoftheconservationofmass,momentum,andenergyequationsalongwiththeidealgasequationofstatetoathin,adiabaticcontrolvolumesurroundinganormalshockwaveyieldstheresultsshowninthefollowingtable.2023/4/632清華大學(xué)工程力學(xué)系NormalShockWavesItisnotedthatinmanycompressibleflowproblemswithnormalshocks,thelocationoftheshockisunknown.Fromtheequationsshownnextpage,thisismostreadilyspecifiedbyfindingthemachnoupstreamoftheshock,Ma1.However,formostproblemsthisrequiresaniterativesolutionofoneofthefollowingequations,dependingonthegiveninformation.2023/4/633清華大學(xué)工程力學(xué)系NormalShockRelations2023/4/634清華大學(xué)工程力學(xué)系NormalShockRelationsWhenusingtheseequationstorelateconditionsupstreamanddownstreamofanormalshockwave,keepthefollowingpointsinmind:UpstreamMachnumbersarealwayssupersonicwhiledownstreamMachnumbersaresubsonic.Stagnationpressuresanddensitiesdecreaseasonemovesdownstreamacrossanormalshockwavewhilethestagnationtemperatureremainsconstant(aconsequenceoftheadiabaticflowcondition).Pressuresincreasegreatlywhiletemperatureanddensityincreasemoderatelyacrossashockwaveinthedownstreamdirection.Thecritical/sonicthroatareachangesacrossanormalshockwaveinthedownstreamdirectionand.Shockwavesareveryirreversiblecausingthespecificentropydownstreamoftheshockwavetobegreaterthanthespecificentropyupstreamoftheshockwave.2023/4/635清華大學(xué)工程力學(xué)系NormalShockRelationsMovingnormalshockwavessuchasthosecausedbyexplosions,spacecraftreenteringtheatmosphere,andotherscanbeanalyzedasstationarynormalshockwavesbyusingaframeofreferencethatmovesatthespeedoftheshockwaveinthedirectionoftheshockwave.2023/4/636清華大學(xué)工程力學(xué)系Example:NormalShockinaConverging-DivergingNozzle2023/4/637清華大學(xué)工程力學(xué)系Example:NormalShockinaConverging-DivergingNozzle2023/4/638清華大學(xué)工程力學(xué)系2023/4/639清華大學(xué)工程力學(xué)系Example:NormalShockinaConverging-DivergingNozzle2023/4/640清華大學(xué)工程力學(xué)系Example:NormalShockinaConverging-DivergingNozzleThesolutionofthisequationgivesanswer(b)Ma4=0.483.NowthattheMachnumberat4isknown,wecanproceedtoapplytheisentropicrelationstoobtainanswers(c)and(d).Note:Observehowthesonicareadownstreamfromtheshockisnotthesameasupstreamoftheshock.Also,observetheuseofthearearatiostodeterminetheMachnumberatthenozzleexit.2023/4/641清華大學(xué)工程力學(xué)系Example:NormalShockinaConverging-DivergingNozzleThefollowingstepscanbeusedtosolvemostone-dimensionalcompressibleflowproblems.Clearlyidentifytheflowconditions:e.g.,isentropicflow,constantstagnationtemperature,constantstagnationpressure,etc.Usetheflowconditionrelationships,tables,orsoftwaretodeterminetheMachnumberatlocationsofinterestintheflowfield.OncetheMachnumberisknownatthelocationsofinterest,onecanproceedtousetheflowrelations,tables,orsoftwaretodetermineotherflowpropertiessuchasfluidvelocity,pressure,andtemperature.Thismayrequirethereductionofpropertyratiostotheproductofseveralratios,aswasdonewiththearearatiointheaboveexampletoobtaintheanswer.2023/4/642清華大學(xué)工程力學(xué)系OperationofConverging-DivergingNozzlesAconverging-divergingnozzlelikethatshowninFigure3canoperateinseveraldifferentmodesdependingupontheratioofthedischargeandsupplypressurePd/Ps.Thesemodesofoperationareillustratedonthepressureratio–axialpositiondiagramofFigure3.Figure32023/4/643清華大學(xué)工程力學(xué)系OperationofConverging-DivergingNozzlesMode(a) Theflowissubsonicthroughoutthenozzle,supply,anddischargechambers.Withoutfriction,thisflowisalsoisentropicandtheisentropicflowequationsmaybeusedthroughoutthenozzle.Sonicconditionsarenotreachedatthethroat.Mode(b) Theflowisstillsubsonicandisentropicthroughoutthenozzleandchambers.TheMachnumberatthenozzlethroatisnowunity.Atthethroat,theflowissonic,thethroatischoked,andthemassflowratethroughthenozzlehasreacheditsupperlimitforthegivengeometryandPo,To.Furtherreductionsinthedischargetankpressurewillnotincreasethemassflowrateanyfurther.

2023/4/644清華大學(xué)工程力學(xué)系OperationofConverging-DivergingNozzlesMode(c)Ashockwavehasnowformedinthedivergingsectionofthenozzle.Theflowissubsonicbeforethethroat,sameasmode(b),thethroatischoked,sameasmode(b),andtheflowissupersonicandacceleratingbetweenthethroatandjustupstreamoftheshock.Theflowisisentropicbetweenthesupplytankandjustupstreamoftheshock.Theflowdownstreamoftheshockissubsonicanddecelerating.Theflowisalsoisentropicdownstreamoftheshocktothedischargetank.Theflowisnotisentropicacrosstheshock.Isentropicflowmethodscanbeappliedupstreamanddownstreamoftheshockwhilenormalshockmethodsareusedtorelateconditionsupstreamtothosedownstreamoftheshock.

Mode(d)Thenormalshockisnowlocatedattheplaneofthenozzleexit.Isentropicflownowexiststhroughoutthenozzleuptotheshock.Theflowatthenozzleexitissupersonicupstreamoftheshockandsubsonicdownstreamoftheshock.Theflowadjuststoflowconditionsinthedischargetank,notthenozzle.Isentropicflowmethodscanbeappliedthroughoutthenozzle.

2023/4/645清華大學(xué)工程力學(xué)系OperationofConverging-DivergingNozzlesMode(e) Aseriesoftwo-dimensionalshocksareestablishedinthedischargetankdownstreamofthenozzle.Theseshocksservetodeceleratetheflow.Theflowisisentropicthroughoutthenozzle,sameasmode(d).Mode(f) Thepressureinthedischargetankequalsthepressurepredictedbythesupersonicsolutionofthenozzleisentropicflowequations.Thepressureratioisknownasthesupersonicdesignpressureratio.Flowisisentropiceverywhereinthenozzle,sameasmode(d)and(e),andinthedischargetank.

Mode(g) Aseriesoftwo-dimensionalshocksareestablishedinthedischargetankdownstreamofthenozzle.Theseshocksservetodeceleratetheflow.Theflowisisentropicthroughoutthenozzle,sameasmodes(d),(e),and(f).2023/4/646清華大學(xué)工程力學(xué)系Example9.9Aconverging-divergingnozzlehasthefollowingvalues:At=.002m2,Ae=.008m2,

Po=1000kPa,To=500?K.Find:Peandmassflowratefor(a)supersonicdesignconditions(b)Pb=300kPa,and(c)Pb=900kPa.k=1.4

2023/4/647清華大學(xué)工程力學(xué)系Example9.92023/4/648清華大學(xué)工程力學(xué)系Example9.9Eqn.9.34isusedtoobtainthedesignexitpressure.TheflowrateatdesignconditionsisobtainedfromEqn.9.46b.2023/4/649清華大學(xué)工程力學(xué)系Example9.9(b)NozzlebackpressureisPb=300kPa.SincePb=300kPa>29.3kPa,referringtoFig.3,wemustdeterminewhetherthiscorrespondstoconditiona,b,c,d,ore.Firstdeterminetheconditionforchokedflow,butsubsonicthroughoutthenozzle(casebinFig.3).AgainusingEqn’s9.45and9.34,solveforthesubsonicvalueofMeandPethatyieldsanarearatioof4.

Me=0.1465andPe

=985kPa

2023/4/650清華大學(xué)工程力學(xué)系Example9.9Since985kPa>Pb>29.3kPa,wehaveanormalshocksomewhereinthenozzle.Sincetheshockisupstreamofthenozzleexit,theexitmustbesubsonic,thethroatmustbesonicandchokedandthefollowingconditionsexit:and ReferringtoFig.3,oncethebackpressurehasdecreasedtoavaluewherethethroatischoked(conditionB),allflowconditionsforbackpressureslessthanconditionBarealsochokedandtheflowrateremainsconstant.2023/4/651清華大學(xué)工程力學(xué)系Example9.9(c) NozzlebackpressureisPb

=985kPa.SincethispressureisveryclosetoconditionB(P=985kPa),wemusthaveanembeddednormalshockrepresentedbyconditionCinFig.3.AsinPartb,sinceweknowwehaveanembeddedshockveryclosetoconditionC,weagainmusthavesonic,chokedconditionsatthethroatandsubsonicconditionsfromtheshocktotheexit.Thus,weagainhaveand Wehavenot,however,determinedthelocationoftheembeddedshock.2023/4/652清華大學(xué)工程力學(xué)系Example9.9Whiletheprocedureissomewhatcumbersome,itwillbepresentedherefortheconditionsofpartc.Thebasicprocessinvolvesassumingthenozzlearea,Ax,justupstreamoftheembeddedshock,andthenproceedingbasedonthisassumedvalueacrosstheembeddedshocktotheendofthenozzle,inordertomatchthegivenbackpressureandexitarea.Whilethesolutioninvolvesaniterativetrialanderrorprocess,itiseasilydevelopedusingacomputer.2023/4/653清華大學(xué)工程力學(xué)系Example9.92023/4/654清華大學(xué)工程力學(xué)系Example9.92023/4/655清華大學(xué)工程力學(xué)系Example9.92023/4/656清華大學(xué)工程力學(xué)系Example9.92023/4/657清華大學(xué)工程力學(xué)系Example9.9Ifthisvalueofexitareadoesnotmatchthegivenexitarea,repeattheprocesswithanewassumesvalueofMax.Severalkeypointsimportanttothisanalysisaresummarizedasfollows:Theflowbetweenthenozzlethroatandjustupstreamofthenormalshockisisentropicwiththefollowingconditions:A*=const.,To=const.,Po=const.,andthusisentropic,compressibleflowequationscanbeusedinthisarea.Theflowfromjustdownstreamofthenormalshocktothenozzleexitisalsoisentropicwiththefollowingconditions:A*=const.,To=const.,Po=const.,andthusisentropic,compressibleflowequationscanbeusedinthisarea.WhileTo=constantacrossanormalshock,A*andPochange.

2023/4/658清華大學(xué)工程力學(xué)系Example9.9Note:Duetoconservationofmass,itisalsotruethatacrossanormalshockThiscanalsobeusedtodetermineconditionsacrossanormalshock.2023/4/659清華大學(xué)工程力學(xué)系A(chǔ)diabatic,ConstantDuctAreaCompressibleFlowwithFrictionWhencompressiblefluidsflowthroughinsulated,constant-areaducts,theyaresubjecttoMoody-likepipe-frictionwhichcanbedescribedbyanaverageDarcy-Weisbachfrictionfactor.Applicationoftheconservationofmass,momentum,andenergyprinciplesaswellastheidealgasequationofstateyieldsthefollowingsetofworkingequations.2023/4/660清華大學(xué)工程力學(xué)系A(chǔ)diabatic,ConstantDuctAreaCompressibleFlowwithFrictionwheretheasteriskstateisthesonicstateatwhichtheflowMachnumberisone.L*isthelengthofductrequiredtodevelopfromMatosonicconditions.Thissonicstateisconstantthroughouttheductandmaybeusedtorelateconditionsatonelocationintheducttothoseatanotherlocation.ThelengthoftheductbetweentwogivenvaluesofMaisgivenby2023/4/661清華大學(xué)工程力學(xué)系CompressibleFlowwithFrictionExample:Airentersa0.01-m-diameterduct()withMa=0.05.Thepressureandtemperatureattheductinletare1.5MPaand400K.Whatarethe(a)Machnumber,(b)pressure,and(c)temperatureintheduct50mfromtheentrance?2023/4/662清華大學(xué)工程力學(xué)系CompressibleFlowwithFrictionExample:2023/4/663清華大學(xué)工程力學(xué)系CompressibleFlowwithFrictionExample:WecannowwritefortheductexitthatorThesolutionofthesecondoftheseequationsgivesanswer(a)Ma2=0.145.2023/4/664清華大學(xué)工程力學(xué)系CompressibleFlowwithFrictionExample:Writingthefollowingexpressionforpressureratiosyieldsfor(b),2023/4/665清華大學(xué)工程力學(xué)系CompressibleFlowwithFrictionExample:Applicationofthetemperatureratiosyieldsanswer(c),Itisnotedthatinbothofthepreviousexpressions,equal1asthesonicreferenceconditionsareconstantbetweentwopoints.2023/4/666清華大學(xué)工程力學(xué)系CompressibleFlowwithFrictionExample:ThisexampledemonstrateshowMachnumberchangesinadiabaticfrictionalflowinaduct.Whentheflowattheinlettotheductissubsonic,theMachnumberincreasesastheductgetslonger.Whentheinletflowissupersonic,theMachnumberdecreasesastheductgetslonger.AplotofthespecificentropyofthefluidasafunctionoftheductMachnumber(andthereforelength)ispresentedinFigure4forbothsubsonicandsupersonicflow.Figure42023/4/667清華大學(xué)工程力學(xué)系CompressibleFlowwithFrictionExample:TheseresultsclearlyillustratethattheMachnumberintheductapproachesunityasthelengthoftheductisincreased.Oncethesonicconditionexistsattheductexit,theflowbecomeschoked.Thisfigurealsodemonstratesthattheflowcanneverproceedfromsubsonictosupersonic(orsupersonictosubsonic)flow,asthiswouldresultinaviolationofthesecondlawofthermodynamics.Othercompressibleflowsinconstantareaductssuchasisothermalflowwithfrictionandfrictionlessflowwithheatadditionmaybeanalyzedinasimilarmannerusingtheequationsappropriatetoeachflow.Manyoftheseflowsalsodemonstratechokingbehavior.

2023/4/668清華大學(xué)工程力學(xué)系FrictionlessDuctFlowwithHeatTransferWenowaddconsiderationoftheeffectofheattransfertoourcompressibleflowdiscussion.WewillconsiderthecaseshowninFig.9.16.Fig.9.16Elementalcontrolvolumeforfrictionlessflowinaconstantareaductwithheattransfer.Thelengthoftheductwouldonlybedeterminediftheheattransferperunitareaorperunitlengthwereknownfortheproblem.2023/4/669清華大學(xué)工程力學(xué)系FrictionlessDuctFlowwithHeatTransfer2023/4/670清華大學(xué)工程力學(xué)系FrictionlessDuctFlowwithHeatTransferApplicationsoftheidealgasequationanddefinitionofMachno.tothepreviousequationsyieldthefollowingexpressionsforflowpropertiesintermsofMachnumber.2023/4/671清華大學(xué)工程力學(xué)系Example9.14Afuel-airmixture,approximatedasairwithk=1.4,entersaductcombustionchamberatV1=75m/s,P1=150kPa,andT1=300?K.Theheatadditionfromthecombustionis900kJ/kgofmixture.Compute(a)theexitpropertiesV2,P2,andT2and(b)thetotalheatadditionwhichwouldhavecausedasonicexitflow.2023/4/672清華大學(xué)工程力學(xué)系Example9.142023/4/673清華大學(xué)工程力學(xué)系Example9.142023/4/674清華大學(xué)工程力學(xué)系Example9.14WiththeMachnumbersatpoints1and2andTableB4orthepreviousequation,wecantabulatethedesiredpropertyratios.2023/4/675清華大學(xué)工程力學(xué)系Example9.14Theexitpropertiesarenowobtainedfrom2023/4/676清華大學(xué)工程力學(xué)系Example9.14Theheatadditionnecessarytodrivetheflowtosonicconditionsisdeterminedfromthedifferenceinthestagnationtemperaturesattheinletandatsonicconditions.Notethatsinceitisnotpossiblefromtheflowtoproceedpastsonicconditionsthisisalsothemaximumpossibleheattransfer.2023/4/677清華大學(xué)工程力學(xué)系ObliqueShockWavesBodiesmovingthroughacompressiblefluidatspeedsexceedingthespeedofsoundcreateashocksystemshapedlikeacone.Thehalf-angleofthisshockconeisgivenby

ThisangleisknownastheMachangle.Theinterioroftheshockconeiscall