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CompressibleflowFromWikipedia,thefreeencyclopedia

Compressibleflow(gasdynamics)isthebranchoffluidmechanicsthatdealswithflowshavingsignificantchangesinfluiddensity.Gases,butnotliquids,displaysuchbehavior.[1]Todistinguishbetweencompressibleandincompressibleflowingases,theMachnumber(theratioofthespeedoftheflowtothespeedofsound)mustbegreaterthanabout0.3(sincethedensitychangeisgreaterthan5%)beforesignificantcompressibilityoccurs.Thestudyofcompressibleflowisrelevanttohighspeedaircraft,jetengines,gaspipelines,commercialapplicationssuchasabrasiveblasting,andmanyotherfields.

Contents

1History2IntroductoryConcepts3MachNumberandSonicFlows4OneDimensionalFlow

4.1ConvergingDivergingLavalNozzles4.2MaximumAchievableVelocityofaGas4.3IsentropicFlowMachNumberRelationships4.4AchievingSupersonicFlow4.5NonIsentropic1DChannelFlowofaGasNormalShockWaves

5TwoDimensionalFlow5.1ObliqueShockWaves

5.1.1ShockPolarDiagram5.1.2ObliqueShockReflection

5.1.2.1SolidBoundary5.1.2.2IrregularReflection

5.2PrandtlMeyerFans5.2.1PrandtlMeyerExpansionFans5.2.2PrandtlMeyerCompressionFans

6Applications6.1SupersonicWindTunnels6.2SupersonicAircraftInlets6.3NaturalGasPipeline

7Seealso8References9Externallinks

History

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BreakdownofFluidMechanicsChart

Thestudyofgasdynamicsisoftenassociatedwiththeflightofmodernhighspeedaircraftandatmosphericreentryofspaceexplorationvehicleshowever,itsoriginsliewithasimplermachine.Atthebeginningofthe19thcentury,investigationintothebehavioroffiredbulletsledtoimprovementintheaccuracyandcapabilitiesofgunsandartillery.[2]Asthecenturyprogressed,inventorssuchasGustafdeLavaladvancedthefield,whileresearcherssuchasErnstMachsoughttounderstandthephysicalphenomenoninvolvedthroughexperimentation.

Atthebeginningofthe20thcentury,thefocusofgasdynamicsresearchshiftedtowhatwouldeventuallybecometheaerospaceindustry.LudwigPrandtlandhisstudentsproposedimportantconceptsrangingfromtheboundarylayertosupersonicshockwaves,supersonicwindtunnels,andsupersonicnozzledesign.[2]TheodorevonKrmn,astudentofPrandtl,continuedtoimprovetheunderstandingofsupersonicflow.Othernotablefigures(Meyer,Crocco,andShapiro)alsocontributedsignificantlytotheprinciplesconsideredfundamentaltothestudyofmoderngasdynamics.

Accompanyingtheimprovedconceptualunderstandingofgasdynamicswasapublicmisconceptionthatthereexistedabarriertotheattainablespeedofaircraft,commonlyreferredtoasthesoundbarrier.Intruth,theonlybarrierthatexistedforsupersonicflightwasatechnologicalbarrier.Amongstotherfactors,conventionalairfoilssawadramaticincreaseindragcoefficientwhentheflowapproachedthespeedofsound.Overcomingthelargerdragproveddifficultwithcontemporarydesigns,thustheperceptionofasoundbarrier.However,aircraftdesignprogressedsufficientlytoproducetheBellX1A.PilotedbyChuckYeager,theX1AachievedsupersonicspeedinOctober1947.[3]Thisachievementpavedthewaytothefutureofmodernaircraft,missiles,andspacecraft.

Historicallytwopathsofresearchhavebeenused,inordertofurthergasdynamicsknowledge.Experimentalgasdynamicscomesintheformofwindtunnelmodelexperimentsandshocktubeswiththeuseofopticaltechniquestodocumentthefindings.Computationalfluiddynamicsappliessupercomputingpowertoanalyzeavarietyofgeometriesandflowcharacteristics.Bothinternalandexternalflowscanbeevaluated.Althoughnotacompletesubstituteforexperimentalconfirmation,computationalgasdynamicsisaninexpensivealternativethatcontinuestoincreaseincapability.

IntroductoryConcepts

Thereareseveralassumptionsusedwhendevelopingcalculationsforcompressibleflow.Fluidsarecomposedofmolecules,whichmeansthatdifferentiatingbetweenallmoleculesinasystemmakescalculationsnearlyimpossible.However,thecontinuumassumptionstatesthatthedifferencesbetweenmoleculesisnegligibleandflowcanbeconsideredacontinuoussubstance.Thisassumptionspansoverabroadreachofmostofgasdynamicsonlywhenlookingatrarefiedgasdynamicsdoesthemotionofindependentmoleculesbecomeimportant.

Thenextassumptionmadeisnoslipconditionwheretheflowvelocityatasolidsurfaceisequaltothevelocityofthesurface.Manytimesthetheflowatthesurfaceorwalliszero.Thenoslipconditionestablishesthattheflowisviscousandasaresultdevelopstheneedforaboundarylayer.

Mostproblemsinincompressibleflowhavetwounknowns:pressureandvelocity.Theseunknownsweresolvedbyusingtheunderlyingprinciplesfromthecontinuityandlinearmomentumconservationequations.Incompressibleflowpressureandvelocityremainunknownbutdensityandtemperaturealsobecomeafactor.Thishintsattheneedfortwoadditionalequationsinordertosolve:equationofstateforgasandtheconservationofenergyequation.

Thesetypesoffluiddynamicsquestionshavetwotypesofreferencesframes,thelagrangianandeulerian.Thelagrangianapproachfollowsaparticularparticleoragroupofparticlesoffixedidentity.Theeulerianreferenceframeisdifferentinthatitdoesnotmovewiththeparticles,ratheritisafixedframeorcontrolvolumethatfluidcanflowthrough.Sincecompressibleflowhasawiderangeoffieldsandpotentialproblemsbothframesareneededformoreindepthproblemanalysis.

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MachNumberandSonicFlows

Machnumber(M)isdefinedastheratioofthespeedofanobjecttothespeedofsound.Mcanrangefrom0to,butthisbroadrangeisbrokenupintoseveralflowregimes.Theseregimesaresubsonic,transonic,supersonic,hypersonic,andhypervelocityflow.Forinstance,inairatroomtemperature,thespeedofsoundisabout340m/s(760mph).ThefigurebelowillustratesthespectrumofMachnumberflowregimes.

MachNumberFlowRegimes

Asanobjectacceleratesfromsubsonictowardsupersonicspeed,certainregimesofwavephenomenaoccur.Toillustratethesechanges,thefigurebelowshowsastationarypoint(M=0)thatemitssymmetricsoundwaves.Onecanthinkofthispointasaboomboxfloatingintheairandprojectingsoundwavesinalldirections.Fromthisstationarypoint,theboomboxbeginstoacceleratetoasubsonicspeed.Astheboomboxaccelerates,thesoundwavesitcreatespileupinthedirectionofmotionandstretchoutintheoppositedirection.Whentheboomboxreachessonicspeed(M=1),itistravellingatthesamespeedasthesoundwavesitcreates.Therefore,aninfinitenumberofthesewavesstackupinthedirectionofmotiontoformashockwave.Uponachievingsupersonicflow,theboomboxleavesitspressurewavesbehind.Whenthisoccurs,thepressurewavescreateanangleknownastheMachwaveangle(orDopplerangle),:

wherearepresentsthespeedofsoundinairandVrepresentsthevelocityoftheobject.AlthoughnamedforAustrianphysicistErnstMach,theseobliquewaveswereactuallyfirstdiscoveredbyChristianDoppler.

ExplanationofSonicMotion

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OneDimensionalFlow

Onedimensional(1D)flowreferstotheflowofgasthroughaductorchannelinwhichtheflowparametersareassumedtochangesignificantlyalongonlyonespatialdimension,namely,theductlength.Inanalyzingthe1Dchannelflow,anumberofassumptionsaremade:

Ratioofductlengthtowidth(L/D)isabout5(inordertoneglectfrictionandheattransfer),Steadyvs.UnsteadyFlow,Flowisisentropic(i.e.areversibleadiabaticprocess),Idealgaslaw(i.e.P=RT)

ConvergingDivergingLavalNozzles

Asthespeedofaflowacceleratesfromthesubsonictothesupersonicregime,thephysicsofnozzleanddiffuserflowsisaltered.Usingtheconservationlawsoffluiddynamicsandthermodynamics,thefollowingrelationshipforchannelflowisdeveloped(combinedmassandmomentumconservation):

,

wheredPisthedifferentialchangeinpressure,MistheMachnumber,isthedensityofthegas,Visthevelocityoftheflow,Aistheareaoftheduct,anddAisthechangeinareaoftheduct.Thisequationstatesthat,forsubsonicflow,aconvergingduct(dA0)decreasesvelocityoftheflow.Forsupersonicflow,theoppositeoccursduetothechangeofsignof(1M2).Aconvergingduct(dA0)increasesthevelocityoftheflow.AtMach=1,aspecialcaseoccursinwhichtheductareamustbeeitheramaximumorminimum.Forpracticalpurposes,onlyaminimumareacanaccelerateflowstoMach1andbeyond.SeeTableofSubSupersonicDiffusersandNozzles.

TableshowingthereversalinthephysicsofnozzlesanddiffuserswithchangingMachNumbers

Therefore,toaccelerateaflowtoMach1,anozzlemustbedesignedtoconvergetoaminimumcrosssectionalareaandthenexpand.ThistypeofnozzletheconvergingdivergingnozzleiscalledadeLavalnozzleafterGustafdeLaval,whoinventedit.Assubsonicflowenterstheconvergingductandtheareadecreases,theflowaccelerates.Uponreachingtheminimumareaoftheduct,alsoknownasthethroatofthenozzle,theflowcanreachMach1.Ifthespeedoftheflowistocontinuetoincrease,itsdensitymustdecreaseinordertoobeyconservationofmass.Toachievethisdecreaseindensity,theflowmustexpand,andtodoso,theflowmustpassthroughadivergingduct.SeeimageofdeLavalNozzle.

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NozzledeLavaldiagram

MaximumAchievableVelocityofaGas

Ultimately,becauseoftheenergyconservationlaw,agasislimitedtoacertainmaximumvelocitybasedonitsenergycontent.Themaximumvelocity,Vmax,thatagascanattainis:

wherecpisthespecificheatofthegasandTtisthestagnationtemperatureoftheflow.

IsentropicFlowMachNumberRelationships

Usingconservationslawsandthermodynamics,anumberofrelationshipsoftheform

canbeobtained,whereMistheMachnumberandistheratioofspecificheats(1.4forair).SeetableofIsentropicFlowMachNumberRelationships.

IsentropicFlowRelationshipTable.Equationstorelatethefieldpropertiesinisentropicflow.

AchievingSupersonicFlow

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Aspreviouslymentioned,inorderforaflowtobecomesupersonic,itmustpassthroughaductwithaminimumarea,orsonicthroat.Additionally,anoverallpressureratio,Pb/Pt,ofapproximately2isneededtoattainMach1.OnceithasreachedMach1,theflowatthethroatissaidtobechoked.Becausechangesdownstreamcanonlymoveupstreamatsonicspeed,themassflowthroughthenozzlecannotbeaffectedbychangesindownstreamconditionsaftertheflowischoked.

NonIsentropic1DChannelFlowofaGasNormalShockWaves

Normalshockwavesareshockwavesthatareperpendiculartothelocalflowdirection.Theseshockwavesoccurwhenpressurewavesbuildupandcoalesceintoanextremelythinshockwavethatconvertsusefulenergyintoheat.Becausealossofenergyoccursoverthethinshockwave,theshockisconsiderednonisentropicandenthalpyincreasesacrosstheshock.Whenanalyzinganormalshockwave,onedimensional,steady,andadiabatic(stagnationtemperaturedoesnotchangeacrosstheshockwave)flowofaperfectgasisassumed.

TheRankineHugoniotEquationsrelateconditionsbeforeandafteranormalshockwave.

Normalshockwavescanoccurintworeferenceframes:thestandingnormalshockandthemovingshock.Theflowbeforeanormalshockwavemustbesupersonic,andtheflowafteranormalshockmustbesubsonic.TheRankineHugoniotequationsareusedtosolvefortheflowconditions.

TwoDimensionalFlow

Althoughonedimensionalflowcanbedirectlyanalyzed,itismerelyaspecializedcaseoftwodimensionalflow.Itfollowsthatoneofthedefiningphenomenaofonedimensionalflow,anormalshock,islikewiseonlyaspecialcaseofalargerclassofobliqueshocks.Further,thenamenormaliswithrespecttogeometryratherthanfrequencyofoccurrence.Obliqueshocksaremuchmorecommoninapplicationssuchas:aircraftinletdesign,objectsinsupersonicflight,and(atamorefundamentallevel)supersonicnozzlesanddiffusers.Dependingontheflowconditions,anobliqueshockcaneitherbeattachedtotheflowordetachedfromtheflowintheformofabowshock.

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AttachedshockwaveshownonaX15Modelinasupersonicwindtunnel

Bowshockexampleforabluntbody

Diagramofobstruction

Shockpolardiagram

ObliqueShockWaves

Obliqueshockwavesaresimilartonormalshockwaves,buttheyoccuratangleslessthan90withthedirectionofflow.Whenadisturbanceisintroducedtotheflowatanonzeroangle(),theflowmustrespondtothechangingboundaryconditions.Thusanobliqueshockisformed,resultinginachangeinthedirectionoftheflow.

ShockPolarDiagram

Basedonthelevelofflowdeflection(),obliqueshocksarecharacterizedaseitherstrongorweak.Strongshocksarecharacterizedbylargerdeflectionandmoreentropylossacrosstheshock,withweakshocksastheopposite.Inordertogaincursoryinsightintothedifferencesintheseshocks,ashockpolardiagramcanbeused.Withthestatictemperatureaftertheshock,T*,knownthespeedofsoundaftertheshockisdefinedas,

withRasthegasconstantandasthespecificheatratio.TheMachnumbercanbebrokenintoCartesiancoordinates

withVxandVyasthexandycomponentsofthefluidvelocityV.WiththeMachnumberbeforetheshockgiven,alocusofconditionscanbespecified.Atsomemaxtheflowtransitionsfromastrongtoweakobliqueshock.With=0,anormalshockisproducedatthelimitofthestrongobliqueshockandtheMachwaveisproducedatthelimitoftheweakshockwave.

ObliqueShockReflection

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PrandtlMeyerExpansionFanDiagram

BasicPMCompressiondiagram

Duetotheinclinationoftheshock,afteranobliqueshockiscreated,itcaninteractwithaboundaryinthreedifferentmanners,twowhichareexplainedbelow.

SolidBoundary

Incomingflowisfirstturnedbyanglewithrespecttotheflow.Thisshockwaveisreflectedoffthesolidboundary,andtheflowisturnedbytoagainbeparallelwiththeboundary.Itisimportanttonotethateachprogressiveshockwaveisweakerandthewaveangleisincreased.

IrregularReflection

Anirregularreflectionismuchlikethecasedescribedabove,withthecaveatthatislargerthanthemaximumallowableturningangle.Thusadetachedshockisformedandamorecomplicatedreflectionoccurs.

PrandtlMeyerFans

PrandtlMeyerfanscanbeexpressedasbothcompressionandexpansionfans.PrandtlMeyerfansalsocrossaboundarylayer(i.e.flowingandsolid)whichreactsindifferentchangesaswell.Whenashockwavehitsasolidsurfacetheresultingfanreturnsasonefromtheoppositefamilywhilewhenonehitsafreeboundarythefanreturnsasafanofoppositetype.

PrandtlMeyerExpansionFans

Tothispoint,theonlyflowphenomenathathavebeendiscussedareshockwaves,whichslowtheflowandincreaseitsentropy.ItispossibletoacceleratesupersonicflowinwhathasbeentermedaPrandtlMeyerexpansionfan,afterLudwigPrandtlandTheodoreMeyer.Themechanismfortheexpansionisshowninthefigurebelow.

Asopposedtotheflowencounteringaninclinedobstructionandforminganobliqueshock,theflowexpandsaroundaconvexcornerandformsanexpansionfanthroughaseriesofisentropicMachwaves.TheexpansionfaniscomposedofMachwavesthatspanfromtheinitialMachangletothefinalMachangle.Flowcanexpandaroundeitherasharporroundedcornerequally,astheincreaseinMachnumberisproportionaltoonlytheconvexangleofthepassage().TheexpansioncornerthatproducesthePrandtlMeyerfancanbesharp(asillustratedinthefigure)orrounded.Ifthetotalturningangleisthesame,thenthePMflowsolutionisalsothesame.

ThePrandtlMeyerexpansioncanbeseenasthephysicalexplanationoftheoperationoftheLavalnozzle.ThecontourofthenozzlecreatesasmoothandcontinuousseriesofPrandtlMeyerexpansionwaves.

PrandtlMeyerCompressionFans

APrandtlMeyercompressionistheoppositephenomenontoaPrandtlMeyerexpansion.Iftheflowisgraduallyturnedthroughanangleof,acompressionfancanbeformed.ThisfanisaseriesofMachwavesthateventuallycoalesceintoanobliqueshock.Becausetheflowisdefinedbyanisentropicregion(flowthattravelsthroughthefan)andananisentropicregion(flowthattravelsthroughtheobliqueshock),asliplineresultsbetweenthetwoflowregions.

Applications

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SupersonicWindTunnels

Supersonicwindtunnelsareusedfortestingandresearchinsupersonicflows,approximatelyovertheMachnumberrangeof1.2to5.Theoperatingprinciplebehindthewindtunnelisthatalargepressuredifferenceismaintainedupstreamtodownstream,drivingtheflow.

SupersonicWindTunnelClassificationList

Windtunnelscanbedividedintotwocategories:continuousoperatingandintermittentoperatingwindtunnels.Continuousoperatingsupersonicwindtunnelsrequireanindependentelectricalpowersourcethatdrasticallyincreaseswiththesizeofthetestsection.Intermittentsupersonicwindtunnelsarelessexpensiveinthattheystoreelectricalenergyoveranextendedperiodoftime,thendischargetheenergyoveraseriesofbrieftests.Thedifferencebetweenthesetwoisanalogoustothecomparisonbetweenabatteryandacapacitor.

Blowdownsupersonicwindtunnelschematic

Langleyindraftsupersonicwindtunnelvacuumsphere

BlowdowntypesupersonicwindtunnelsofferhighReynoldsnumber,asmallstoragetank,andreadilyavailabledryair.However,theycauseahighpressurehazard,resultindifficultyholdingaconstantstagnationpressure,andarenoisyduringoperation.

Indraftsupersonicwindtunnelsarenotassociatedwithapressurehazard,allowaconstantstagnationpressure,andarerelativelyquiet.Unfortunately,theyhavealimitedrangefortheReynoldsnumberoftheflowandrequirealargevacuumtank.Thereisnodisputethatknowledgeisgainedthroughresearchandtestinginsupersonicwindtunnelshowever,thefacilitiesoftenrequirevastamountsofpowertomaintainthelargepressureratiosneededfortestingconditions.Forexample,ArnoldEngineeringDevelopmentComplexhasthelargestsupersonicwindtunnelintheworldandrequiresthepowerrequiredtolightasmallcityforoperation.Forthisreason,largewindtunnelsarebecominglesscommonatuniversities.

SupersonicAircraftInlets

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McDonnellDouglasF15C ConcordeonBristol LockheedSR71Blackbird

Constructingnaturalgaslineinwinter,Finland

Perhapsthemostcommonapplicationforobliqueshocksisinhighspeedaircraftinlets.Thepurposeoftheinletistoslowincomingsupersonicflowtothesubsonicregimebeforeitenterstheturbojetengine,withthecaveatofminimizinglossesacrosstheshock.Knowledgeofnormalandobliqueshockssuggeststhatthisbeaccomplishedwithaseriesofweakeningobliqueshocksfollowedbyaveryweaknormalshock,usuallylessthanM=1.4.Thismaysoundrelativelystraightforward,butthereisoneratherlargeissuetobedealtwithwhendesigningasupersonicaircraftinlet:acceleration.Betweentakingoff,maneuvering,andcruising,anaircrafttravelsatarangeofMachnumbers.Inordertoensureefficientflight,theaircraftintakemustbecapableofvariablegeometry.Ifitisnot,theshockwaveswillnotreflectproperlythroughtheinletandnegativelyaffectperformance.

AlthoughvariablegeometryisauniversallyrecognizedapproachtoimproveaircraftefficiencyandperformanceoverarangeofMachnumbers,thereisnoonemethodtoachievevariablegeometry.TheF15Eagleemployswedgeinletswithadjustableflapstocontroltheflow.Forsubsonicflow,theflapsarecompletelyclosedandforsupersonicflow,theflapsareopen.TheConcordeemployedanexternalcompressioninlet,usingaseriesofobliqueshocksfollowedbyanormalshocktoslowtheflowsufficientlyfortheturbojetengine.Perhapsthemostrecognizablesupersonicaircraft,theSR71,usedahydraulicallyactuatedconetoreducethespeedofthesupersonicflowthroughtheaircraftinlet.

NaturalGasPipeline

Naturalgaspipelinesareusedtotransportnaturalgasfromextractionsitestorefinementorchemicalprocessingfacilities.IntheUnitedStatestherearemorethan210naturalgaspipelinesystemswithmorethan305,000milesofintrastatetransmissionpipelines.[4]Twocompressibleflowphenomenoncharacterizetheflowthroughthesepipelines:friction(Fannoflow)and(Rayleighflow)andheattransfer.Naturalgaspipelinesareburiedinthegroundataconstanttemperatureof15C.However,thefrictiongeneratedbytheflowoffsetstheheatlosstotheEarth,thusresultinginanisothermalflow.

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FrictionfactorasafunctionofMachnumberinFannoFlow

TherelationshipbetweenfLmax/DandMachnumberforFannoflowsuggeststhatonlysubsonicflowcanbeusedinthelongpipesusedtotransportnaturalgas(eventhesepipesmustbebrokenintoshortersegmentswithcompressorstationsatthediscontinuitiesinthepipeline).Additionallyusingconservation,anequationcanbederivedtodescribetheflow.

ThisequationdescribesflowthatchokesatM*=0.87fornaturalgas=1.32howeverchokingrequiresaninfiniteheatflux.Therefore,acombinationofintuitionandmathematicsexplainswhyitismosteconomicallyfeasiblethatsubsonicnaturalgasispumpedthroughlongsectionsofpipetoreachitsintendeddestination.

Seealso

ConservationlawsEquationofstateThermodynamicsespeciallyCommonlyConsideredThermodynamicProcessesandLawsofThermodynamicsEnthalpyEntropyLagrangianandEulerianspecificationoftheflowfieldHeatcapacityratioChokedflowHypersonicflowTransonicflowIsothermalflowPrandtlMeyerfunctionIsentropicnozzleflow

References

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1. Butseecompressibilitywhichlistscompressibilitiesforwaterandsomeotherliquids2. [1](http://www.ibiblio.org/potto/text.pdf)3. [2](http://history.nasa.gov/SP4219/Chapter3.html)4. http://www.eia.gov/pub/oil_gas/natural_gas/analysis_publications/ngpipeline/index.html

Externallinks

NASABeginner'sGuidetoCompressibleAerodynamics(http://www.grc.nasa.gov/WWW/K12/airplane/bgc.html)VirginiaTechCompressibleFlowCalculators(https://engineering.purdue.edu/~wassgren/applet/java/comp_calculator/Index.html)[3](http://www.dept.aoe.vt.edu/~devenpor/aoe3114/calc.html)

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