SCRAMJETCOMBUSTION
FundamentalsandAdvances
GAUTAMCHOUBEY
DepartmentofMechanical&AerospaceEngineering, InstituteofInfrastructureTechnologyResearchand Management(IITRAM)Ahmedabad,Gujarat380026,India
MANVENDRATIWARI
DepartmentofMechanicalEngineering,NationalInstitute ofTechnologyGoa,403401,India
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1.Introduction1
GautamChoubeyandManvendraTiwari
1.1 Backgroundandmotivation1
1.2 Abriefhistoryataglance3
1.3 Thebasicstructureandlayoutofascramjetengine5
1.4 Challengesfacedbyascramjetengineduringsupersonic combustion7
1.5 Strategiesforcombatingcombustiondifficulties7
1.6 Fuelinjectorsforscramjetengines7
1.7 Workingcycle12
1.8 Governingequations14 References17
2.Scramjetcombustionmechanism23
2.1 Introduction23
2.2 Chemicalkineticsinsupersonicflow24
2.3 Basictermsrelatedtosupersoniccombustion27
2.4 Fuelsusedinsupersoniccombustion29
2.5 Strategiesforfuel–airmixing32
2.6 Turbulentcombustion40 References44
3.Factorsaffectingthescramjetperformance49
3.1 Introduction49
3.2 Potentialflameholders50
3.3 Combustionstabilizationinastrut-basedscramjetcombustor50
3.4 Flamepropagationmechanism56
3.5 Wall-mountedcavity-stabilizedcombustion58 References61
4.Advancesinscramjetfuelinjectiontechnology65
4.1 Strutinjection65
4.2 Rampinjection80 v
4.3 Cavityinjectionapproach88
4.4 Thetransverseinjectionapproach116
4.5 Emergingramjettechnologies136 References141
5.Roadmapforfutureresearch159
5.1 Challengesforstrutdesigns160
5.2 Issuesrelatedtocavity-basedinjection161
5.3 Futureresearchstrategiesinthedirectionofthetransverse injectiontechnique162
6.Pedagogyforthecomputationalapproachinsimulating supersonicflows163
6.1 Turbulencemodellinginscramjetflows163
6.2 Computationalapproachtomodelasinglestrut-based scramjetcombustor171
6.3 Numericalmodellingandsimulationdetails172
6.4 Validationstudyforasingle-strutscramjetcombustor175 References179 Index 183
Acknowledgements
Theauthorsthanktheanonymousreviewersfortheirhighlyconstructive suggestionsonthisbook.TheauthorsalsoexpressgreatgratitudetoCarrie L.Bolger,DennisMcGonagle,RafaelGuilhermeTrombaco,Manju Thirumalaivasanandtheirteamfortheircontinuousandunconditional supportincompletingthisbook.Lastbutnotleast,theauthorsthankthe Almightyforprovidingstrength,ability,knowledge,andopportunityto writethebook.
GautamChoubeya andManvendraTiwarib
aDepartmentofMechanical&AerospaceEngineering,InstituteofInfrastructureTechnologyResearchand Management(IITRAM)Ahmedabad,Gujarat,India
bDepartmentofMechanicalEngineering,NationalInstituteofTechnologyGoa,Ponda,Goa,India
1.1Backgroundandmotivation
EversincetheWrightbrothersmadetheirsuccessfulmaidenairflight, continuoustechnologicaldevelopmenthasbeenongoinginaviationand spaceapplications,and,asaresult,wehavewitnessedvariousstagesofdevelopmentsuchasgasturbineengineswithdifferentconfigurations,rocket engines,andair-breathingengines.Needlesstosay,thepower-to-weight ratioisacrucialdesignparameterandaffectstheperformanceofairtransportationvehicles.Thedesireforahypersonicflighthasmotivatedengineers fordecadestosatisfythedemandsforhigh-speedtransportandeconomical accesstospace [1–6].Inordertomeettheseneedsandambitions,significant advancementsinpropulsiontechnologyarerequired.Itisobservedthatgas turbine-drivenaircraftenginescouldachieveaspeedlimitedtoasubsonic range.Rocketpropulsiontechnologyhasemergedasasolutiontohypersonicrangeflights,butsafetyandlowspecificimpulseistheAchillesheel ofthisinnovation.Greatsuccessisachievedwhentheconceptofaramjet evolved,leadingtotheeliminationofcompressorsandturbines.These mechanicalunits’rolehasbeenreplacedbythestrategicallyconstructedgeometricalcontourofaramjet’sbodythatactsasadiffuserandanexpander. Throughthesemodifications,theramjetcouldenterintoahypersonicoperatingrangeowingtoreducedweight.Duringtheoperationoftheramjet whileithasattainedsupersonicspeed,theairflowrushingintothecombustorisretardedwiththeobjectivetoachievethedesiredthermodynamicstate andfavourablerecirculationzonesconduciveforacceleratingthecombustionprocess.However,studiesadvocatenotreducingthespeedoftheinternalairflowbelowMachnumber7inordertoreduceenergylossesthat wouldtakeplaceastheintensityofretardationincreases,and,this,inturn, willgiveimpetustoskindragfrictionandassociatedsurfaceheating problem. ScramjetCombustion
Thepotentialsolutiontothisproblemhasevolvedintoscramjettechnologywhereeveryprocessresemblesaramjetandonlytheinternalflow, unliketheramjet,iskeptsupersonicwhileitentersintothecombustor.This changeofthestrategyofcombustionhaspavedthewayforachievinghypersonicspeedandisgoingtoplayavitalroleindefence,aviation,andspace applications.However,therearetechnicalchallengesaheadbeforeitcanbe successfullyimplementedindifferentdomainsofitsapplications.Theprime bottlenecksare(1)completionofthecombustionprocesswithinan extremelyshortintervalowingtothesupersonicstateofair,(2)development ofamaterialtosustainagainsthighertemperatureandtheenormousmagnitudeofthedragforce,(3)relianceonthecarriervehicleforthescramjetto operate,and(4)developmentofhi-techlaboratoriesandfacilitiesforconductingresearchandtesting.Thisrequirementisofutmostimportance becausescramjetcombustionexperimentsareextremelycomplicated,and onlylimitedruntimefacilitiesareavailablearoundtheworldtocarryout real-timeexperiments. Fig.1.1 showsthejourneystartingfromagasturbine toscramjettechnologicaldevelopment.
Fig.1.1 Journeyfromagasturbinetoscramjettechnologicaldevelopment.
1.2Abriefhistoryataglance
Theconceptofasupersonicenginecameintorealizationduringthe early90s,andthefirsttechnicalreportwaspublishedin1920.Intheeraof WorldWarII,thistechnologygainedmomentumintermsofresearchand itsapplicationsinfortifyingthenavyandairforcewithhigh-speedmissiles andaircraft.Duringthisperiodofwar,thepioneerFrenchcompany,named NordAviation,designedaircraft,whichreceivedpropulsionfrom turbo–ramjethybridtechnology.
NASAinitiatedresearchonscramjetsinthe1950s.TheBellX-1 achievedsupersonicflightin1947,and,bytheearly1960s,rapidprogress towardsfasteraircraftindicatedthatsupersonicflightwasfeasible [3].This newsencouragedthecivilaviationindustrytoresearchthepossibilityof reducingtheflighttime,but,atpresent,itisnoteconomicallyviableto switchthismode.Theprimarygoalofthecivilianairtransportistoreduce theoperatingcostratherthantoincreaseflightspeeds,andgiventhefactthat supersonicflightrequiressignificantamountsoffuel,airlinesgowithsubsonicjumbojetsratherthansupersonictransports.Theconceptofascramjet hasencouragedmanyjointventureprogrammes.In1992,theresearchwing ofRussiaandFrancecollaboratedandobservedsuccessinlaunchinga scramjetengine.Inanotherjointventurein1994,NASAandtheCentral InstituteofAviationMotors(CIAM)agreedtocarryoutaresearchprogrammeonadual-modescramjetengine,whichcontinuedfor4years [7].In1995,researchrelatedtoascramjetwasinitiatedinAustraliaat TheUniversityofQueenslandandreportedsuccessfultestingoftheHyshot scramjetunitin2002 [8].Amongtherecentdevelopments,the“National HypersonicsStrategy”hasbeenformulatedbytheUSmilitaryandNASA toinvestigatearangeofoptionsforhypersonicflight.UndertheNASA LangleyHyper-Xprogram,differentversionsoftheX-43vehicleseries havebeentested(Fig.1.2)andspeedsuptoMachnumber15havebeen achievedsofar [3,9,10]
AmongAsiancountries,Indiaisemergingasapotentialcentreof researchundertheleadershipofISROwherescramjetsaredesigned.During agroundtestingin2005,ISROreportedhavingachievedMach6forabout 7s.Furthermore,in2016,ISRO [11] reportedasuccessfultestofascramjet engine,whichwasintegratedwiththeAdvancedTechnologyVehicle (ATV)(Fig.1.3A).TheBrahMosAerospaceisstrivingtodesignand developascramjetwithatargetofMach5.Ontheotherhand,theDRDL
Fig.1.2 (A)Hyper-Xresearchvehicle(USNASALangley),(B)X-43A(USNASA),and (C)ScramjettestsatNASA-LaRC.
Fig.1.3 (A)AdvancedTechnologyVehicle(ATV,ISRO) [10] and(B) ‘HypersonicTechnologyDemonstratorVehicle’ (HSTDV,DRDO) [11]
Fig.1.4 Characteristicflighttrajectoriesfordifferentaerospacesystemsincluding hypersonicscramjet-poweredair-breathingvehicles.
hasinitiatedaprogramme,namely,‘HypersonicTechnologyDemonstrator Vehicle’(HSTDV)(Fig.1.3B)withthebeliefinproposingavehiclethat wouldbeabletoattainMach6.5 [12].
Hypersonicair-breathingspacecrafthavetheabilitytocarrypayloadstoa lowEarthorbitatahighrate.However,duetooxidizershortage,escaping theatmosphereisnoteasy,sothepayloadmustshifttoasecond,rocketbasedstageatMach10–12toachievehypervelocitiesintheorderofMach 20–25. Fig.1.4 illustratestheschematicdiagramofflighttrajectoriesfordifferentaerospacesystemsincludinghypersonicscramjet-poweredairbreathingvehicles [3]
1.3Thebasicstructureandlayoutofascramjetengine
Scramjetisanext-generationtechnologyandplausiblypromisingin reducingflighttimes,whichisaparameterofgreatinterestandimportance inthedomainofaviation,defence,andspacetechnologicaldevelopments. Scramjetisanimprovedversionoframjettechnologyintermsofgaining higherflightvelocity.Thecreditforthisachievementgoestotheideaof keepingthefluidflowsupersonicwhileitisrushingintothecombustor,
andthisdemonstratedattainmentofahigherMachnumberthantheramjet enginesduetosupersoniccombustion.Incontrasttothis,inramjet,theflow isretardedtothesubsoniclevelbeforeitentersthecombustor.Theidea behindreducingtheflowleveltothesubsoniclevelwastogivetimeforfluid mixingforachievingcompletecombustion.
Atypicalconceptualdiagramofascramjetisshownin Fig.1.5[10,13].A scramjetpropulsionsystemisanintegrationofascramjetengineandapropulsionvehicle.Thescramjetenginerequiresthesupportofthepropulsion vehicletomeettheminimumrequirementofmaintainingspeedwithan equivalentvalueofMachnumber3,foritscontinuousworkingandattendinghypersonicspeed.So,thescramjetengineisequippedwithfourkey components,namely,aninternalinlet,anisolator,acombustor,andan internalnozzle,whereasthefore-bodyandaft-bodyrepresentthepropulsionvehiclecomponents.Ascramjetengineismountedinsuchawaythat itacquirestheentirelowersurfaceofthevehiclebody.Thevehicleforebodyistheheartoftheairinductionsystem,whereasthevehicleaft-body isthebackboneofthenozzlecomponent.
Theresponsibilityofprovidingcompressedairtothecombustorsis sharedamongthefore-body,internalinlet,andisolatorunit.Asthefree streamofairistakeninviathefore-bodyandinternalinlet,thecompression processbegins,andthermodynamicstatesofairstartchangingalongwiththe reductioninitsMachnumber.Theprocessofcompressioncontinuesinthe isolator,whichalsohastheresponsibilityofprotectingtheinletsection fromthecatastrophiceffectofbackfirefromthecombustor.Completecombustionisanidealexpectationfromaproperlydesignedcombustor.The expansionsystem,whichisanintegrationoftheinternalnozzleandvehicle
Fig.1.5 Schematicdiagramofascramjetengine [13].
aft-body,allowshigh-pressure,high-temperaturegasestoexpandtothe lowestpossiblepressuresothatthemaximumpossiblethrustmaybe generated.
1.4Challengesfacedbyascramjetengineduring supersoniccombustion
Thescramjetengineisamechanicallysimplesystemduetothe absenceofmovingparts.Nevertheless,thecombustioninascramjetengine isahighlycomplexphenomenon.Astheairentersintothecombustion chamberatasupersonicspeed,theHerculeantaskofmixingfuelwithair andsubsequentlycompletingthecombustionposesmajorchallengesto theresearchcommunity.Besides,thesupersonicstateofairinthecombustoradverselyaffectstheperformanceofthefuelinjectorbyreducingthefuel injector’spenetrationdepthowingtohigherairpressure.Thisnecessitates theneedtorevisitanddevelopafuelinjectororstrategiesforinjectingthe fuelsuccessfullyagainsttheprevailinghigh-pressureairinthecombustor. Furthermore,ashortresidencetimeofairinthecombustionchamber, owingtoitssupersonicstate,isahighlydiscouragingsituationforthecompletionofchemicalreactions.Ascombustioninitiationconsumesan extremelysmallbutdefinitetimeandiscomparablewiththeairresidence timeinthecombustor,andthisdeprivesthefuelmoleculesofgettingoxygenmolecules,thisresultsincombustionsuppressionornoinitiationatall.
1.5Strategiesforcombatingcombustiondifficulties
Theabove-mentionedtechnicaldifficultiescanbehandled(a)by injectingcombustion-enhancingradicalsusingaplasmatorchand(b)bycreatingrecirculationzonesusingaerodynamicbodiessuchaswedges,ramps, orcavitiestoslowdowntheflowandprovideanenvironmentwhereastable flamecanexistandhelpincombustingtheavailablefuel.Inanutshell,the solutiontotheexistingchallengesinscramjettechnologiesliesinthedevelopmentofcombustors,fuelinjectors,fuelswithashorterdurationofchemicalreactions,andcombustion.
1.6Fuelinjectorsforscramjetengines
Combustion,beingahighlycomplexchemicalphenomenon,essentiallyinvolvesthepreparationofacombustiblemixtureandsubsequentlya
sustainableflamepropagationtoensurecompletecombustion.Amixtureis saidtobecombustibleifitissurroundedbytherequisiteamountofoxygen andasourceofignition [14,15].Fuelinjectorssupplyfuelsconsistingof finelydividedparticlesthatseekoxygenwhenencounteringincomingpressurizedair.So,theroleofafuelinjectorishighlycriticalinthecontextofthe preparationofthecombustiblemixtureandstableflamepropagation.However,forthiswholeevent,adefinitedurationoftimeisrequired,whichis hardlyavailablewhenair,inthecaseofscramjetengines,entersintothe combustoratasupersonicspeedandremainsinthesamestateofmotion whilepassingthroughit.Thepreparationofthecombustiblemixtureduring anextremelyshortstayofairisamajorhurdleforthesuccessofscramjet technology.Thischallengehasbeenacceptedbyresearchers,andcontinuouseffortsarepouringinforaugmentingair–fuelmixing;thevariousproposedstrategiesare(1)wallinjection,(2)wallinjectionwithcavity,(3)ramp injection,and(4)strutinjection.Besides,researchisalsoongoingwith respecttoscramjetfuels,rangingfromhydrocarbonstogreenfuels(e.g. hydrogen)andassociatedprocesseswiththeobjectivetoreduceignition delayandachievecompletecombustioninthatconciseduration.
1.6.1Wallinjection
Inthisstrategyoffuelinjection,slotsareincorporatedintothewallandthe fuelsaredirectedeitherorthogonaloratsomeangletothecoreflow [15–19].Thisplanoffuelinjectioninvolveslittledesigncomplexity,but, duetonormalimpingementoffuelstreams,variousshockandrecirculation regionsaredeveloped [20,21] (Fig.1.6).Thismethodoffuelinjectionhas notreceivedmuchencouragementasthisapproachhasdemonstratedshallowfuelpenetrationintotheairstream.
However,researchhasbeencarriedoutinviewtostudytheeffectof holeshapeandarrangementonthecombustionprocess.Itisobservedthat whentheholesarelocatedtoofar,fuelmixingisnotadequate,whereasplacingholesatasmallerpitchresultsinfluidexpansion-relatedissuesowingto spacestarvation [24–30].Inanotherstrategy,fuelinjectionbehindthewallis reported,butthisapproachdidnotmeettheexpectationofair–fuelmixing owingtotheweekrecirculationzonedevelopment [31–37]
1.6.2Wallinjectionwithcavity
Theideaofinjectingfuelthroughacavityimpressioninthewallresultedin betterair–fuelmixing.Thecreditgoestoinducedacousticvibrations,which
Fig.1.6 (A)Schematicdiagramof2Dand(B)3Dflowfieldsinatransverseinjectionbasedsupersoniccombustor [21–23].
originateincavitiesthroughprevailingunstableshearlayers.Thewall-based cavityinjection [38–49] offersimprovedthermohydraulicperformancein thesensethatbettermixingofair–fuelisachievedatrelativelylowerfrictionallossesowingtotheaerodynamiccontourofthecavity.Besides,itacts asapotentialsourceofignitionandassistsinachievingtheobjectiveofcompletecombustioninanextremelyshortspanofavailabletime.Furthermore, itisobservedthatthedimensionsofthecavityhaveastrongbearingon ignitiontimeandvortexgenerationforpromotingair–fuelmixing.Studies havebeenconductedtobringforwardtheeffectofvaryingtheratioofthe
cavity’sbasesurfacelength(L)toitsdepth(D).Itisreportedthatthedepthof thecavitygovernstheignitiontimebasedonthefreestreamconditions.In contrast,thelengthofthecavityinfluencestheintensityofvortexgeneration andconsequentlyimpactstheair–fuelmixinginsidethecavity.Depending onthe L/D ratio,twotypesofcavityflow,asshownin Fig.1.7A,areidentified:(1)opencavityflowand(2)closedcavityflow [46].
Opencavityflownormallyoccursforcavitieswith L/D < 10,whereas closedcavityflowisobservedfor L/D > 10 [46].Studiesrevealthattheopen cavityflowoptionismoreconduciveforscramjettechnologybecauseofthe
Fig.1.7 (A)Schematicdiagramofdifferentflowfieldsoveracavityfordifferentlengthto-depthratios, L/D [31].(B)Smalldisturbancescreatedbyafueljetincavity-based injection [47].
prevailinglargerdragforcewithclosedcavityflow.Thisconclusionisbased onthefactthatinopencavityflow,thereattachmentphenomenontakes placesomewhereontherearsideofthecavitywhilethesameeventoccurs onthecavityfloor,resultinginlargershearlosses(Fig.1.7B).
1.6.3Rampinjection
ThecredittothisapproachgoestoBurtonetal. [50] whoproposedawallmountedrampwithanintegratedsetofnozzlesforalmostaxialfuelinjection [51].Itwaspossibletostudytheeffectoframpinclinationwithrespect totheaxis,whichresultedinthedevelopmentofthefollowingtwoconfigurations,namely,asweptcompressionrampandasweptexpansionramp,as shownin Fig.1.8.Thestudyofasweptcompressionrampdemonstrated improvedair–fuelmixingduetothepassagebetweentherampandwallactingasadiffuser,which,inturn,increasesthedensityofthemixture,and, also,recirculationzonesaresetupowingtotheflowseparationfrom thewall.
Fig.1.8 Schematicofa(A)compressionramp [52,53],(B)asweptcompressionramp injector [21,50,53],and(C)aphysicalrampinjector [54].
Ontheotherhand,inthecaseofasweptexpansionramp,abettercombustionefficiencyisrecordedbecausetheexpansionofgasesinduces favourableconditionsforthegenerationofvortices,which,inturn,augmentsthemixinglargelybesidesprovidingfilmcoolingtothecombustion chamberwalls.Thisstudypraisedaxialflowinjectionssincethisstrategy exhibitedadeeperfuelpenetrationcomparedtothenormalfuelinjection besidesgenerationoffavourablefluidmixingmotions [53,54].
1.6.4Strutinjection
Thisisthemostrecentfuelinjectiontechnology,whichhasattributesof multi-directionalandmulti-locationfuelinjectionfacilities [55–58].Various researchplanshavebeeninitiatedforstudyingmultiplestrategiesrelatedtoits mountingsandshapesofthenozzleopeningandtheirdistributions.Whileitis possibletolocatethestrutinjectionunitcentrally,studiesarealsoreported whentheyareinstalledonthefloorandtheroofofthecombustionchamber.
Thestrutinjection [59–65] planappearstobehighlypromisingasthe challengeofcompletecombustionwithinanextremelyshortprevailing durationispotentiallytangibleduetothefollowinginherentdistinctadvantages:(1)itgivesscopeforinitiatingcombustionatvariouslocationsinthe combustionchamber,therebyacceleratingthecombustionphenomenon and(2)forimprovingair–fuelmixing,strutinjectorscanstrategicallyinject thefuel,bothinaxialandnormaldirections(3).Asustainableflame,which lateractsasasourceofignition,ismaintainedduetotherecirculationzone, andahightemperatureprevailsbehindthebaseofthestrutowingtothe generationofshockwavesinthewakeportion(Fig.1.9).
1.7Workingcycle
TheworkingcycleofascramjetenginehasmappingwiththeBrayton cycleinthesensethatitalsohastwoadiabaticandtwoisobaricprocesses [2]. Unliketheconventionalgasturbinecycle,airundergoesvariousstagesof compression,whichstartsastheairencountersthefore-body,anditcontinueswithintheisolatoradiabatically.Then,combustiontakesplacein thecombustor,and,finally,expansiontakesplaceintheaft-body,which impartsthrusttothescramjet.Technically,thescramjetengineoperation isanopencyclewhereproductsofcombustionaredischargedintotheatmosphere,andthecyclecontinueswiththeintakeoffreshairandfuel.This cycleispresentedontheh–splane,asshownin Fig.1.10.Furthermore,
Fig.1.9 Strutinjection [55]
Fig.1.10 Actualscramjetcycle.
forcomparisonandstudiesofassociatedirreversibilityinvariousprocesses, theidealcycleisalsopresented.Intheactualcycle, •State0indicatesthestateofthefreeairstream.
•Process“0–a”depictsthepressureriseduetorammingaction.
•Process“a–b”reflectsthecontinuationofthecompressionprocessinthe fore-bodysectionwhosegeometricalcontourofgraduallyreducingarea offersalongersectiontoactasadiffuser.
•Process“b–c”isexecutedinanisolatorunit.Thepresenceofanisolator servesadualpurpose.Itprotectsthefore-bodyinletsectionfrom receivingthereverseblowofhighpressure-heatedgasesduringthe vehementcombustionprocessthattakesplaceinthecombustorby splittingtheirstrength.Besides,isolatorsalsoactasadiffuserwhileair ispassingthroughitandrisestheairpressurepriortoitsentryinto thecombustor.Althoughtheincorporationoftheisolatorhasundesirableeffectssuchasincreasedpressuredrops,increaseinweightthatthe scramjetengineneedstobear,andanadditionalloadonthermal management,duetoitsfavourabledualactionsmentionedabove,its inclusioninthesystembecomesimperative.
•Process“c–d”isthecombustionprocessthattakesplaceinthecombustorwhenhotcompressedaircombineswiththefuelsprayedbythe injectors.Thisresultsinthegenerationofextremelyhotandhighpressuregases.
•Duringtheprocess‘d–e’,thehigh-pressure,high-temperaturegases expandviathenozzlesectionwheretheyacquireextremelyhighkinetic energyoftheorderofasupersonicvalueandhenceimpartthrusttothe scramjetvehicle.
1.8Governingequations
Ascramjetengineinvolvesintricateflowprocessesowingtothe highMachnumberwhilestrivingt oachievecompletecombustionwith theaidofstrategiesforpromotingfuel –airmixing.Hence,thisnecessitates asimultaneousconsiderationofconse rvationlawsassociatedwithfluids andchemicalaspects [23,66 –73].Thederivationofpertinentequations assumescontinuumideologysincethe densityoffluidatitsvariousstages ofoperationsissufficientlyhightosupportthisassumption.Thefollowing sectionsdealexclusivelywiththecontinuity,momentum,andenergy equations [2]
1.8.1Conservationofmass
Essentially,themassofaworkingfluidcrossinganysectionofaclosedchannelremainsconstant,asisalsothecaseforascramjet.Thisprincipleisdemonstratedbytheequation
wheretheoperator D Dt denotesthematerialderivativeexpressedas
UsingEq. (1.2),thecontinuityequationcanbewrittenas
1.8.2Conservationofmomentum
Ahugemomentumexchangeisinvolvedwhilethefluidissloweddownin thediffusersectionorisgettingacceleratedduringthecombustionprocessin ascramjetunit.Thisinducesalargeamountofthrust,whichisafavourable effect,whilebeingaccompaniedbyanundesirablegrowthinwallshear stress.Theestimateoftheforcesinducedduetomomentumexchange becomesparamountandcanbeworkedoutusingtheconservationprinciple whoseexpressionisgivenas
where rP signifiesthehydrostaticpressureactingonthevolume,rτ stands forviscousstresstensorthatactsatthefluidelementboundary,and F b isthe bodyforces.Thetensorformofthemomentumconservationequationcan berepresentedas
1.8.3Conservationofenergy
Atotalenergyconservationconceptincludesmechanicalaswellasheat energy.However,inthecontextofscramjettechnologies,mechanical
energyisnotinvolvedandhencecanbeignored.Theequationcanbewrittenas
where rq signifiestheconductionheattransferrateand Q isthebulkheat source.Thetensorformoftheenergyconservationequationispresentedas
Whenenthalpy h ¼ e + pv isusedintheenergyconservationequation,then theequationiswrittenas
Forinviscid,steadyflowassumptions,bodyforceandheattransferaretotally neglected.Thus,inthatparticularcase,theenergyequationreducesto
1.8.4Conservationofspecies
Thecombustionphenomenoninascramjetinvolvesalargenumberof chemicalactivities.Hence,theconceptofspeciesconservationandassociatedchemicallawsneedtobetakenintoaccount.Theequationsofspecies areasfollows:
wherethesuperscript i indicatesthespecies i andthediffusionvelocitieshave notbeenconsideredinthisequation.Theproductionterm ω i isderived fromtheanalysesofchemicalkinetics.
1.8.5Equationsofstate
Thethermodynamicstateoftheworkingfluidwhileitisundergoingcompressionandexpansionprocessesinascramjetcanbepredictedusingequationsofstateasgivenbelow.
Generally,thesizeofthemoleculesaswellasintermolecularforcesare neglectedinperfectgasassumptions,and,hence,thesimplifiedformof theequationofstateis
1.8.6Fourier’slawofheattransfer
Ascramjet,duringitsoperation,encountershugeairdragowingtothe developmentoftheboundarylayerinthevicinityofthecombustorwall, andtheboundarylayeressentiallyhousesastagnantfluidlayerduetothe prevailingfrictionbetweenthefluidandthewall.Hence,duringcombustion,heatexchangetakesplacethroughtheconductionmode,whichcanbe calculatedusingFourier’slaw.
Here,thenegativesignspecifiesthatthedirectionofheattransferisfromthe high-temperaturetothelow-temperatureregion [74]
1.8.7Theshearstress
Ahighairdragisamajorconcerninascramjetengine.Intenseairdrag inducesextremelyhighshearstressesandresultsinthermalfailureofthesurfacematerial.Hence,theestimateofshearstressesdevelopedduringthe scramjetoperationisparamountfromasafetypointofviewaswellasfor testingbetterthermalresistingmaterials.ForNewtonianfluids,itisa well-establishedfactthatshearstressisproportionaltothestrainrate.
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