HypersonicandHigh-TemperatureGasDynamicsSecondEditionThis page intentionally left blankHypersonicandHigh-TemperatureGasDynamicsSecondEditionJohnD.Anderson,Jr.NationalAirandSpaceMuseumSmithsonianInstitutionWashington,DCandProfessorEmeritus,AerospaceEngineeringUniversityofMarylandCollegePark,MarylandEDUCATIONSERIESJosephA.SchetzSeriesEditor-in-chiefVirginiaPolytechnicInstituteandStateUniversityBlacksburg,VirginiaPublishedbytheAmericanInstituteofAeronauticsandAstronautics,Inc.1801AlexanderBellDrive,Reston,Virginia20191-4344American InstituteofAeronautics andAstronautics, Inc.,Reston,Virginia1 2 3 4 5Library of Congress Cataloging-in-Publication DataAnderson, JohnDavid.Hypersonicandhigh-temperaturegasdynamics / John D.Anderson, Jr.- -2nded.p.cm.Includes bibliographicalreferences andindex.ISBN-13: 978-1-56347-780-5 (alk.paper)ISBN-10: 1-56347-780-7(alk.paper)1. Aerodynamics,Hypersonic. I. TitleTL571.5.A53 2006629.132306- -dc22 2006025724Copyright # 2006byJohnD. Anderson, Jr. All rightsreserved. PrintedintheUnitedStatesofAmerica.Nopartofthispublicationmaybereproduced,distributed,ortransmitted,inanyformorbyanymeans, orstoredinadatabaseorretrieval system, without thepriorwrittenpermissionofthepublisher.Data and information appearing in this book are for information purposes only. AIAA is not respon-sibleforanyinjuryordamageresultingfromuseorreliance, nordoesAIAAwarrant that useorreliancewillbefreefromprivately ownedrights.To Sarah-Allen, Katherine, and Elizabeth Anderson, for all theirlove and understandingThis page intentionally left blankAIAA Education SeriesEditor-in-ChiefJosephA.SchetzVirginiaPolytechnicInstituteandStateUniversityEditorial BoardTakahiraAokiUniversityofTokyoEdwardW.AshfordKarenD.BarkerBraheCorporationRobertH.BishopUniversityofTexasatAustinClaudioBrunoUniversityofRomeAaronR.ByerleyU.S.AirForceAcademyRichardColgrenUniversityofKansasKajalK.GuptaNASADrydenFlightResearchCenterRikardB.HeslehurstAustralianDefenceForceAcademyDavidK.HolgerIowaStateUniversityRakeshK.KapaniaVirginiaPolytechnicInstituteandStateUniversityBrianLandrumUniversityofAlabamaHuntsvilleTimothyC.LieuwenGeorgiaInstituteofTechnologyMichaelMohagheghTheBoeingCompanyConradF.NewberryNavalPostgraduateSchoolMarkA.PriceQueensUniversityBelfastJamesM.RankinOhioUniversityDavidK.SchmidtUniversityofColoradoColoradoSpringsDavidM.VanWieJohnsHopkinsUniversityThis page intentionally left blankForewordIt is a great personal pleasure for me towelcome the SecondEditionofHypersonicandHighTemperatureGasDynamicsbyJohnD. AndersontotheAIAAEducation Series. I have known John Andersonfor more years thaneither he or I are comfortable recalling, and I have always found him to be extre-mely articulate and insightful. The original edition published by McGraw-Hill in1989wasaverywellreceived,comprehensive,andin-depthtreatmentoftheseimportant topics, andit wasreprintedbyAIAAin2000. Thisneweditionhasupdatedthematerialandexpandedthecoverage, andweanticipatethatitwillbeequallywellreceived,especiallysincehypersonicsisnowenjoyingaresur-genceof interest. This editionhas 18chapters dividedintothreemainpartsandmorethan800pages.JohnAndersonisverywell-qualiedtowritethisbook, rstbecauseofhisbroadanddeepexpertiseinthearea. Second, hiscommandofthematerial isexcellent, andheisabletoorganizeandpresent it inaveryclearmanner. Inaddition,Johnwritesinaveryreadablestyle,whichhasmadeallofhisbookspopularwithbothstudentsandworkingprofessionals. Finally, JohnAndersonhas longplayedakeyroleinAIAApublications activities, includingbooksandjournal papers, aswell asleadershiproles, andthat makesusparticularlypleasedtohavethisbookundertheAIAAmasthead.The AIAA Education Series aims to cover a very broad range of topics in thegeneral aerospace eld, including basic theory, applications and design. A com-pletelistoftitlescanbefoundat http://www.aiaa.org. Thephilosophyoftheseries is to develop textbooks that can be used in a university setting, instructionalmaterialsfor continuingeducationandprofessional development courses, andalsobooksthat canserveasthebasisfor independent study. Suggestionsfornewtopicsorauthorsarealwayswelcome.JosephA. SchetzEditor-in-ChiefAIAA Education SeriesThis page intentionally left blankTable of ContentsPrefaceto theSecondEdition . .. . .. . . .. . .. . . .. . .. . . .. . .. . . xviiPrefaceto theFirstEdition . . .. . .. . . .. . .. . . .. . .. . . .. . .. . .. xixChapter 1. SomePreliminaryThoughts. . . . . . . . . . . . . . . . . . . . 11.1 HypersonicFlightSomeHistoricalFirsts . . . . . . . . . . . . . . . 21.2 HypersonicFlowWhyIsItImportant?. . . . . . . . . . . . . . . . . 51.3 HypersonicFlowWhatIsIt? . . . . . . . . . . . . . . . . . . . . . . 131.4 FundamentalSourcesofAerodynamicForceandAerodynamicHeating. . . . . . . . . . . . . . . . . . . . . . . . . . 231.5 HypersonicFlightPaths:Velocity-AltitudeMap . . . . . . . . . . . 271.6 SummaryandOutlook. . . . . . . . . . . . . . . . . . . . . . . . . . . 29Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Part1. InviscidHypersonicFlowChapter 2. HypersonicShockandExpansion-WaveRelations . . . . 352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.2 BasicHypersonicShockRelations . . . . . . . . . . . . . . . . . . . 362.3 HypersonicShockRelationsinTermsoftheHypersonicSimilarityParameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4 HypersonicExpansion-WaveRelations . . . . . . . . . . . . . . . . . 442.5 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . . 47Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Chapter 3. LocalSurfaceInclinationMethods . . . . . . . . . . . . . . 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 NewtonianFlow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 ModiedNewtonianLaw. . . . . . . . . . . . . . . . . . . . . . . . . 613.4 CentrifugalForceCorrectionstoNewtonianTheory . . . . . . . . 633.5 NewtonianTheoryWhatItReallyMeans . . . . . . . . . . . . . . 703.6 Tangent-WedgeTangent-ConeMethods . . . . . . . . . . . . . . . . 793.7 Shock-ExpansionMethod. . . . . . . . . . . . . . . . . . . . . . . . . 833.8 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . . 86xiDesignExample3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 87Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Chapter 4. HypersonicInviscidFlowelds:ApproximateMethods . 1034.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.2 GoverningEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3 Mach-NumberIndependence. . . . . . . . . . . . . . . . . . . . . . 1074.4 HypersonicSmall-DisturbanceEquations . . . . . . . . . . . . . . 1114.5 HypersonicSimilarity. . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.6 HypersonicSmall-DisturbanceTheory:SomeResults . . . . . . 1294.7 CommentonHypersonicSmall-DisturbanceTheory. . . . . . . 1454.8 HypersonicEquivalencePrincipleandBlast-WaveTheory. . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.9 ThinShock-Layer Theory. . . . . . . . . . . . . . . . . . . . . . . . 1674.10 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 173Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Chapter 5. HypersonicInviscidFlowelds:ExactMethods . . . . . . 1795.1 GeneralThoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1825.2 MethodofCharacteristics. . . . . . . . . . . . . . . . . . . . . . . . 1845.3 Time-MarchingFiniteDifferenceMethod:ApplicationtotheHypersonicBlunt-BodyProblem . . . . . . . . . . . . . . 1995.4 CorrelationsforHypersonicShock-Wave Shapes . . . . . . . . . 2225.5 ShockShockInteractions . . . . . . . . . . . . . . . . . . . . . . . . 2255.6 Space-MarchingFiniteDifferenceMethod:AdditionalSolutionsoftheEulerEquations . . . . . . . . . . . 2315.7 CommentsontheStateoftheArt . . . . . . . . . . . . . . . . . . . 2465.8 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 247DesignExample5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 247DesignExample5.2:HypersonicWaveridersPart1. . . . . . 250Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257Part2. ViscousHypersonicFlowChapter 6. ViscousFlow:BasicAspects,BoundaryLayerResults,andAerodynamicHeating . . . . . . . . . . . . . . 2616.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2626.2 GoverningEquationsforViscousFlow:Navier StokesEquations. . . . . . . . . . . . . . . . . . . . . . . 2666.3 SimilarityParametersandBoundaryConditions . . . . . . . . . . 2686.4 Boundary-LayerEquationsforHypersonicFlow. . . . . . . . . . 2726.5 HypersonicBoundary-Layer Theory:Self-SimilarSolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2786.6 NonsimilarHypersonicBoundaryLayers . . . . . . . . . . . . . . 3156.7 HypersonicTransition . . . . . . . . . . . . . . . . . . . . . . . . . . 327xii6.8 HypersonicTurbulentBoundaryLayer . . . . . . . . . . . . . . . . 3356.9 ReferenceTemperatureMethod . . . . . . . . . . . . . . . . . . . . 3416.10 HypersonicAerodynamicHeating:SomeCommentsandApproximateResultsAppliedtoHypersonicVehicles . . . . . . . . . . . . . . . . . . . . . . . . . . 3466.11 Entropy-LayerEffectsonAerodynamicHeating. . . . . . . . . . 3536.12 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 355DesignExample6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 358DesignExample6.2:HypersonicWaveridersPart2. . . . . . 361Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374Chapter 7. HypersonicViscousInteractions . . . . . . . . . . . . . . . . 3757.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3767.2 StrongandWeakViscousInteractions:DenitionandDescription. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3807.3 Roleofx inHypersonicViscousInteraction . . . . . . . . . . . . 3827.4 OtherViscousInteractionResults. . . . . . . . . . . . . . . . . . . 3897.5 HypersonicShock-Wave/Boundary-LayerInteractions . . . . . . 3957.6 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 407DesignExample7.1:HypersonicWaveridersPart3. . . . . . 409Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413Chapter 8. Computational-Fluid-DynamicSolutionsofHypersonicViscousFlows . . . . . . . . . . . . . . . . . . . . 4158.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4168.2 ViscousShock-Layer Technique. . . . . . . . . . . . . . . . . . . . 4188.3 ParabolizedNavier StokesSolutions . . . . . . . . . . . . . . . . . 4248.4 FullNavier StokesSolutions. . . . . . . . . . . . . . . . . . . . . . 4348.5 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 444Part3. High-TemperatureGasDynamicsChapter 9. High-TemperatureGasDynamics:SomeIntroductoryConsiderations . . . . . . . . . . . . . . 4499.1 ImportanceofHigh-TemperatureFlows . . . . . . . . . . . . . . . 4509.2 NatureofHigh-TemperatureFlows . . . . . . . . . . . . . . . . . . 4589.3 ChemicalEffectsinAir:TheVelocity-AltitudeMap. . . . . . . 4599.4 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 461Chapter10. SomeAspectsoftheThermodynamicsofChemicallyReactingGases(ClassicalPhysicalChemistry) . . . . . . 46310.1 Introduction:DenitionofRealGasesandPerfectGases . . . . 46410.2 VariousFormsofthePerfect-GasEquationofState. . . . . . . . 46610.3 VariousDescriptionsoftheCompositionofaGasMixture . . . 472xiii10.4 ClassicationofGases. . . . . . . . . . . . . . . . . . . . . . . . . . 47410.5 FirstLaw ofThermodynamics . . . . . . . . . . . . . . . . . . . . . 47810.6 SecondLaw ofThermodynamics . . . . . . . . . . . . . . . . . . . 48110.7 CalculationofEntropy. . . . . . . . . . . . . . . . . . . . . . . . . . 48310.8 GibbsFreeEnergyandtheEntropyProducedbyChemicalNonequilibrium . . . . . . . . . . . . . . 48510.9 CompositionofEquilibriumChemicallyReactingMixtures:TheEquilibriumConstant . . . . . . . . . . 48810.10 HeatofReaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49510.11 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 497Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Chapter11. ElementsofStatisticalThermodynamics . . . . . . . . . . 50111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50211.2 MicroscopicDescriptionofGases . . . . . . . . . . . . . . . . . . . 50311.3 CountingtheNumberofMicrostatesforaGivenMacrostate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51211.4 MostProbableMacrostate . . . . . . . . . . . . . . . . . . . . . . . . 51411.5 LimitingCase:BoltzmannDistribution. . . . . . . . . . . . . . . . 51611.6 EvaluationofThermodynamicPropertiesinTermsofthePartitionFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51811.7 EvaluationofthePartitionFunctioninTermsofTandV. . . . 52411.8 PracticalEvaluationofThermodynamicPropertiesforaSingleChemicalSpecies . . . . . . . . . . . . . . . . . . . . . . . 52811.9 CalculationoftheEquilibriumConstant . . . . . . . . . . . . . . . 53211.10 ChemicalEquilibriumSomeFurtherComments . . . . . . . . . 53711.11 CalculationoftheEquilibriumCompositionforHigh-TemperatureAir. . . . . . . . . . . . . . . . . . . . . . . 53811.12 ThermodynamicPropertiesofanEquilibriumChemicallyReactingGas . . . . . . . . . . . . . . . . . . . . . . . 54211.13 EquilibriumPropertiesofHigh-TemperatureAir . . . . . . . . . . 54711.14 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 557Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558Chapter12. ElementsofKineticTheory . . . . . . . . . . . . . . . . . . . 55912.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56012.2 Perfect-GasEquationofState(Revisited) . . . . . . . . . . . . . . 56012.3 CollisionFrequencyandMeanFreePath . . . . . . . . . . . . . . 56412.4 VelocityandSpeedDistributionFunctions:MeanVelocities . . 56712.5 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 571Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573Chapter13. ChemicalandVibrationalNonequilibrium. . . . . . . . . 57513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576xiv13.2 VibrationalNonequilibrium:TheVibrationalRateEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57713.3 ChemicalNonequilibrium:TheChemicalRateEquation. . . . . 58413.4 ChemicalNonequilibriuminHigh-TemperatureAir. . . . . . . . 59013.5 ChemicalNonequilibriuminH2-AirMixtures . . . . . . . . . . . 59513.6 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 597Chapter14. InviscidHigh-TemperatureEquilibriumFlows . . . . . . 59914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60014.2 GoverningEquationsforInviscidHigh-TemperatureEquilibriumFlow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60114.3 EquilibriumNormalandObliqueShock-WaveFlows. . . . . . . 60414.4 EquilibriumQuasi-One-DimensionalNozzleFlows . . . . . . . . 61714.5 FrozenandEquilibriumFlows:TheDistinction. . . . . . . . . . 62314.6 EquilibriumandFrozenSpecicHeats . . . . . . . . . . . . . . . . 62614.7 EquilibriumSpeedofSound . . . . . . . . . . . . . . . . . . . . . . 62914.8 EquilibriumConicalFlow . . . . . . . . . . . . . . . . . . . . . . . . 63314.9 EquilibriumBlunt-BodyFlows . . . . . . . . . . . . . . . . . . . . . 63614.10 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 643DesignExample14.1:HypersonicWaveridersPart4. . . . . 644Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646Chapter15. InviscidHigh-TemperatureNonequilibriumFlows . . . . 64715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64815.2 GoverningEquationsforInviscid,NonequilibriumFlows. . . . 64915.3 NonequilibriumNormalandObliqueShock-WaveFlows. . . . 65515.4 NonequilibriumQuasi-One-DimensionalNozzleFlows . . . . . 66315.5 NonequilibriumBlunt-BodyFlows . . . . . . . . . . . . . . . . . . 67115.6 BinaryScaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68015.7 NonequilibriumFlow overOtherShapes:NonequilibriumMethodofCharacteristics. . . . . . . . . . . . . . . . . . . . . . . 68315.8 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 687Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690Chapter16. KineticTheoryRevisited:TransportPropertiesinHigh-TemperatureGases . . . . . . . . . . . . . . . . . . . 69116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69216.2 DenitionofTransportPhenomena. . . . . . . . . . . . . . . . . . 69216.3 TransportCoefcients. . . . . . . . . . . . . . . . . . . . . . . . . . . 69616.4 MechanismofDiffusion. . . . . . . . . . . . . . . . . . . . . . . . . 70016.5 Energy TransportbyThermalConductionandDiffusion:TotalThermalConductivity . . . . . . . . . . . . . . 70216.6 TransportPropertiesforHigh-TemperatureAir . . . . . . . . . . . 70516.7 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 709xvChapter17. ViscousHigh-TemperatureFlows . . . . . . . . . . . . . . . 71117.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71217.2 GoverningEquationsforChemicallyReactingViscousFlow. . . . . . . . . . . . . . . . . . . . . . . . 71217.3 AlternateFormsoftheEnergyEquation . . . . . . . . . . . . . . . 71517.4 Boundary-LayerEquationsforaChemicallyReactingGas. . . 71917.5 BoundaryConditions:CatalyticWalls . . . . . . . . . . . . . . . . 72617.6 Boundary-LayerSolutions:Stagnation-PointHeatTransferforaDissociatingGas. . . . . . . . . . . . . . . . . . . 72917.7 Boundary-LayerSolutions:NonsimilarFlows . . . . . . . . . . . 73917.8 Viscous-Shock-LayerSolutionstoChemicallyReactingFlow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74117.9 ParabolizedNavier StokesSolutionstoChemicallyReactingFlows . . . . . . . . . . . . . . . . . . . . . 74917.10 FullNavier StokesSolutionstoChemicallyReactingFlows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75117.11 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 756Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756Chapter18. IntroductiontoRadiativeGasDynamics . . . . . . . . . . 75918.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76018.2 DenitionsofRadiativeTransferinGases. . . . . . . . . . . . . . 76018.3 Radiative-TransferEquation. . . . . . . . . . . . . . . . . . . . . . . 76318.4 SolutionsoftheRadiative-TransferEquation:TransparentGas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76518.5 SolutionsoftheRadiative-TransferEquation:AbsorbingGas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76818.6 SolutionsoftheRadiative-TransferEquation:EmittingandAbsorbingGas. . . . . . . . . . . . . . . . . . . . . 76918.7 RadiatingFlowelds:SampleResults. . . . . . . . . . . . . . . . . 77318.8 SurfaceRadiativeCooling . . . . . . . . . . . . . . . . . . . . . . . . 78018.9 SummaryandComments . . . . . . . . . . . . . . . . . . . . . . . . 781DesignExample18.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 782Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785Postface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805SupportingMaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813xviPreface to the Second EditionAlmost 20 years have passed since the publication of the rst edition of this book.During those 20 years, much progress has been made in hypersonic andhigh-temperaturegasdynamics, principallyintheextensiveuseofsophisticatedmodern computational uid dynamics and in the design of serious ighthardware.Thephysicalandmathematicalfundamentalsofhypersonicandhigh-temperature gas dynamics, however, have by their very nature remained the same.This bookis about thefundamentals, whichhavenot changed. Therefore,almost all of thecontent oftherst editionhasbeencarriedoverthepresentedition. Indeed, nowisagoodmoment topauseandreadtheprefaceof therst edition. Everything said there is appropriate to the second edition.For example, the book remains a self-contained teaching instrument for those stu-dents andreaders interestedinlearninghypersonicandhigh-temperaturegasdynamicsstartingwiththebasics. Thisbookassumesnopriorfamiliaritywitheither subject onthepart of thereader. If youhavenever studiedhypersonicand/or high-temperaturegas dynamics andif youhavenever workedinthearea, thenthisbookisforyou. Ontheotherhand, ifyouhaveworkedand/orare working in these areas and you want a cohesive presentation of the fundamen-tals, adevelopmentofimportanttheoryandtechniques, adiscussionofsalientresults with emphasis on the physical aspects, and a presentation of modern think-ingintheseareas,thenthisbookisalsoforyou.As with the rst edition, this second edition is written in an informal, conver-sational style. Thisbooktalkstoyou, just asif youandI weresittingdowntogether at a table discussing the subject matter. I want you to have fun learningthese topics. This is not difcult because the areas of hypersonic andhigh-temperaturegasdynamicsarefull of interestingandexcitingphenomenaandapplications.What is new and different about the second edition? A lot! Much new materialhas been added to accomplish two purposes, namely, to bring the book up to dateandto enhanceeven furtherthe pedagogical goalof helping the reader tolearn.Forexample:1) A lot of new literature has been published in the discipline over the past 20years.Thisneweditiondrawsfromthemodernliteratureinordertoupdatethepresentations. Thisisnot abookabout thestateoftheartit isabout funda-mentals.Butthestateoftheartisusedtoreinforcethefundamentals.2) Thetopicofshock-shockinteractions, particularlytheimportanttype-IVinteraction,hasbeenaddedasanewextensivesectioninChapter5.xvii3) Although this book emphasizes the fundamentals, modern hypersonic andhigh-temperature gas dynamics is moving more and more toward design of viablesystems. Therefore, this edition has a design avor not present in the rst edition.Atthebackofanumberofchapters, designexamplesthatillustratetheappli-cationofthefundamentalstomethodsofdesignareadded.Anumberofthesedesign examples focus on different aspect of hypersonic waverider design.Waveriderswerenottreatedintherstedition.Becausetheyareaninterestingconguration for possible future hypersonic vehicles they are extensivelytreatedhere.4) Chapter previews have been added at the beginning of most of the chapters.Thesearepedagogicaltoolstoprovidethereaderwithinsightaboutwhateachchapter isabout andwhythematerial is soimportant. Theyarewritteninaparticularlyinformal mannerplainspeakingtohelpturnthereader ontothecontent. InthesepreviewsIamunabashedlyadmittingtoprovidingsomefunforthereaders.5) Roadmapshavebeenplacedatthebeginningofalmosteverychaptertohelpguidethereaderthroughthe logicalow ofthe materialanotherpedago-gicaltooltoenhancetheself-learningnatureofthisbook.SpecialthanksgotoRodgerWilliamsoftheAIAAforsuggesting,encoura-ging, andessentiallycommissioningthissecondedition, andwithwhomit isalwaysapleasuretowork,andtotheentireAIAApublicationsteamwhohavealwaysmademefeellikeoneofthem.ThanksalsotoSusanCunninghamwhotyped the original manuscriptof therstedition20 years agoandwho receivedthecallagainfortheaddedmaterialforthesecondedition.Finally,IwanttoacknowledgeinaveryspecialwaythelateRudolphEdse,mymentor andadvisor at TheOhioStateUniversity, whogavemethetrueappreciationfor the fundamentals, andDr. JohnD. Lee andthe rest of thefacultyof the Department of Aeronautical andAstronautical EngineeringatOhio State during the 1960s who taught me all there was to knowabouthypersonicow.JohnD. Anderson,Jr.July 2006xviii PREFACETOTHESECONDEDITIONPreface to the First EditionThisbookisdesignedtobeaself-containedteachinginstrumentforthosestu-dentsandreadersinterestedinlearninghypersonicowandhigh-temperaturegas dynamics. It assumes noprior familiaritywitheither subject onthepartof thereader. If youhavenever studiedhypersonicand/or high-temperaturegas dynamics before, andif youhavenever workedextensivelyinthearea,thenthisbookisforyou. Ontheother hand, if youhaveworkedand/or areworkinginthese areas, andyouwant a cohesive presentationof the funda-mentals,adevelopmentofimportanttheoryandtechniques,adiscussionofthesalient results withemphasis onthe physical aspects, anda presentationofmodernthinkingintheseareas, thenthisbookisalsoforyou. Inotherwords,thisbookisaimedfortworoles: 1)asaneffectiveclassroomtext, whichcanbeusedwitheasebytheinstructorandwhichcanbeunderstoodwitheasebythe student; and 2) as a viable, professional working tool on the desk of all engin-eers, scientists, and managers who have any contact in their jobs with hypersonicand/orhigh-temperatureow.The only background assumed on the part of the reader is a basic knowledge ofundergraduate uid dynamics, including a basic introductory course on compres-sible ow; that is, the reader is assumed to be familiar with material exempliedbytwooftheauthorspreviousbooks,namely,FundamentalsofAerodynamics(McGraw-Hill, 1984), andtherst half of ModernCompressibleFlow: withHistorical Perspective (McGraw-Hill, 1982). Indeed, throughout the presentbook, frequent reference is made to basic material presented in these twobooks.Finally,thepresentbookispitchedattheadvancedseniorandrst-yeargraduatelevelsandisdesignedtobeusedintheclassroomasthemaintextfor courses at these levels in hypersonic ow and high-temperature gas dynamics.Homework problems are given at the ends of most chapters in order to enhance itsuseasateachinginstrument.Hypersonicaerodynamicsisanimportantpart oftheentireightspectrum,representingthesegmentattheextremehighvelocityofthisspectrum.Interestinhypersonicaerodynamicsgrewinthe1950sand1960swiththeadvent ofhypersonicatmosphericentryvehicles, especiallythemannedspaceprogramas representedbyMercury, Gemini, andApollo. Today, manynew, excitingvehicleconceptsinvolvinghypersonicightaredrivingrenewedand, insomecases, frenzied interest in hypersonics. Such newconcepts are described inChapter1. Thisbookisaresponsetotheneedtoprovideabasiceducationinhypersonic and high-temperature gas dynamics for a new generation of engineersxixandscientists, aswell astoprovideabasicdiscussionof theseareasfromamodernperspective. Sixtextsinhypersonicowwerepublishedbefore1966;the present bookis the rst basic classroomtext tobecome available sincethen. Therefore, the present book is intended to make up for this 20-yearhiatusandtoprovideamoderneducationinhypersonicandhigh-temperaturegasdynamics,whilediscussingatlengththebasicfundamentals.Toenhancethereadersunderstandingandtopeakhisor her interest, thepresentbookiswritteninthestyleoftheauthorspreviousones, namely, itisintentionallywritteninaninformal,conversationalstyle. Theauthorwantsthereadertohavefunwhilelearningthesetopics. Thisisnotdifcultbecausetheareasofhypersonicandhigh-temperaturegasdynamicsarefull ofinterestingandexcitingphenomenaandapplications.The present book is divided into three parts. Part 1 deals with inviscidhypersonic ow, emphasizing purely the uid-dynamic effects of the Machnumber becoming large. High-temperature effects are not included. Part 2deals with viscous hypersonic ow, emphasizing the purely uid-dynamiceffects of including the transport phenomena of viscosity and thermal conductionat the same time that the Mach number becomes large. High-temperature effectsare not included. Finally, Part 3 deals with the inuence of high temperatures onboth inviscid and viscous ows. In this fashion, the reader is led in an organizedfashion through the various physical phenomena that dominate high-speedaerodynamics. Tofurther enhancetheorganizationofthematerial, thereaderis given a road map in Fig. 1.24 to help guide his or her thoughts as we progressthroughourdiscussions.Whenthisbookwasrst started, theauthorsintent wastohaveaPart 4,which would cover the miscellaneous but important topics of low-densityows, experimental hypersonics, and applied aerodynamics associated withhypersonicvehicledesign. Duringthecourseof writingthis book, it quicklybecame apparent that including Part 4 would vastly exceed the length constraintsallotted to this book. Therefore, the preceding matters are not considered in anydetail here. This is not because of a lack of importance of such material, but ratherbecause of an effort to emphasize the basic fundamentals in the presentbook. Therefore, Parts1, 2, and3aresufcient;theyconstitutetheessenceofanecessaryfundamental backgroundinhypersonicandhigh-temperaturegasdynamics. The material of the missing Part 4 will have to wait for another time.Thecontent of this bookis inuencedinpart bytheauthors experiencein teaching such material in courses at the University of Maryland. It is also inu-encedbytheauthorsthree-dayshortcourseontheintroductiontohypersonicaerodynamics, which he has had the privilege to give at 10 different laboratories,companies, and universities over the past year. These experiences have ne tunedthe present material in favor of what the reader wants to know and what he or sheisthinking.Several organizations and people are owed the sincere thanks of the author inaiding the preparation of this book. First, the author is grateful to the National Airand Space Museum of the Smithsonian Institution where he spent an enlighteningsabbatical year during1986l987as theCharles LindberghProfessor intheAeronautics Department. A substantial portion of this book was writtenduringthat sabbatical year at themuseum. Secondly, theauthorisgrateful toxx PREFACETOTHEFIRSTEDITIONtheUniversityofMaryland forprovidingthe intellectualatmosphereconducivetoscholarlyprojects.Also,manythanksgototheauthorsgraduatestudentsinthe hypersonic aerodynamics program at Marylandthanks for the many enligh-tening discussions on the nature of hypersonic and high-temperature ows.Forthemechanicalpreparationofthismanuscript,theauthorhasusedhisownwordprocessornamedSusanO. Cunninghamatrulyhumanhumanbeingwhohastypedthemanuscript withthehighestprofessionalstandards. Finally,once again the author is grateful for the support at home provided by theAndersonfamily, whoallowedhimtoundertakethisproject intherst place,andforjoininghiminthecollectivesighofreliefuponitscompletion.Iwouldliketoexpressmythanksforthemanyuseful commentsandsug-gestionsprovidedbycolleagueswhoreviewedthistextduringthecourseofitsdevelopment, especially to Judson R. Baron, Massachusetts Institute of Technol-ogy; Daniel Bershader, Stanford University; John D. Lee, Ohio State University;andMauriceL.Rasmussen,UniversityofOklahoma.John D. Anderson, Jr.October 1987PREFACETOTHEFIRSTEDITION xxiThis page intentionally left blank1Some Preliminary ThoughtsAlmost everyone has their own denition of the term hyper-sonic. If we were to conduct something like a public opinionpoll amongthosepresent, andaskedeveryonetonameaMach number above which the ow of a gas should properlybedescribedas hypersonictherewouldbeamajorityofanswers round about ve or six, but it would be quite poss-ible for someone to advocate, and defend, numbers as smallasthree,orashighas12.P.L.Roe,commentmadeinalectureattheVonKarmanInstitute.Belgium,January1970Chapter PreviewThis is the rst of many short preview boxes in this bookone for each of thechapters. Their purpose is to tell you in plain language right up front what toexpect fromeachchapter andwhythematerial isimportant andexciting.Theyareprimarilymotivational; theyareherefor youtoencourageyoutoactuallyenjoyreadingthechapter.Partofthesuccessoflearninganewsubjectissimplytogetgoing.Thepurposeof thischapter istoget goingonour journeythroughhypersonicandhigh-temperaturegasdynamics. It startswithsomehistory. Whenandhowdidtherst human-madeobject achievehypersonicight?Whenwastherst timethat ahumanewhypersonically?What issospecial abouthypersonicight anyway?Whyis it important?Whydoaerodynamicistssingle out hypersonic as a special ight regime separate fromthe moregeneral regime of supersonic ight? After all, both supersonic and hypersonicight deal with speeds above Mach 1. So what is so special about hypersonicspeedsthatwecanwriteabookaboutthesubject?Thesearemorethanjustinterestingquestionstheyareimportant questions that demandanswers.Thischaptersuppliestheanswers. Thepreliminarythoughtssuppliedhere,moreover, set thestageandthetonefortherest of ourdiscussionsinthisbook. Astudyofhypersonicowisfunit canbetechnicallydemanding1at times as we will see, but it can also be fun. So sit back, relax, and especiallyenjoy this rst chapter. Let it give you a smooth takeoff on our ight throughthe worlds of hypersonic and high-temperature gas dynamicsa ight that isbothintellectuallyandprofessionallyrewarding.1.1 Hypersonic FlightSome Historical FirstsThedayisThursday, 24February1949; thepensontheautomaticplottingboardsat SouthStationarebusytrackingthealtitudeandcourseof arocket,whichjustmomentsbeforehadbeenlaunchedfromasitethreemilesawayonthetestrangeoftheWhiteSandsProvingGround.TherocketisaV-2,oneofmanybroughttotheUnitedStatesfromGermanyafterWorldWarII. Bythistime,launchingV-2shadbecomealmostroutineforthecrewsatWhiteSands,butonthisdayneitherthelaunchnortherocketareroutine.Mountedontopof this V-2is aslender, needle-likerocket calledtheWACCorporal, whichservesasasecondstagetotheV-2. Thistest ringof thecombinationV-2/WAC Corporal is the rst meaningful attempt to demonstrate the use of a multi-stage rocket for achieving high velocities and high altitudes and is part of a largerprogram labeled Bumper by the U.S. Army. All previous rocket launchings ofany importance, both in the United States and in Europe, had utilized the single-stage V-2by itself. Figure 1.1 shows a photograph of the Bumper rocket as itlifts off the New Mexico desert on this clear, February day. The pen plotters trackthe V-2 to an altitude of 100 miles at a velocity of 3500 mph, at which point theWACCorporal isignited. Theslender upper stageacceleratestoamaximumvelocityof 5150 mphandreaches analtitude of 244miles, exceedingbyahealthy130milespreviousrecordsetbyaV-2alone.Afterreachingthispeak,theWACCorporal noses over andcareers backintotheatmosphereat over5000 mph. Insodoing, it becomestherst object of humanorigintoachievehypersonic ightthe rst time that any vehicle has own faster than vetimesthespeedof sound. Inspiteof thepenplotterschartingitscourse, theWAC Corporal cannot be found in the desert after the test. Indeed, the only rem-nants to be recovered later are a charred electric switch and part of the tail section,andthesearefoundmorethanayearlater,inApril1950.Thesceneshifts tothesmall villageof SmelookaintheTernovDistrict,Saratov region of Russia. The time is now1055 hrs (Moscowtime) on 12April 1961. A strange, spherical object has just landed under the canopy of a para-chute. Thesurfaceofthiscapsuleischarredblack, andit containsthreesmallviewingports coveredwithheat-resistant glass. Inside this capsule is FlightMajor Yuri Gagarin, whojust 108 minearlier hadbeensittingontopof arocketattheRussiancosmodromeatBaikonurneartheAralSea. Whatpartlytranspiredduringthose108 minisannouncedtotheworldbyabroadcastfromtheSovietnewsagencyTassat9 : 59a.m.:Theworldsrstspaceship,Vostok(East),withamanonboardwaslaunchedintoorbitfromtheSovietUniononApril12,1961.Thepilotspace-navigator2 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSof the satellite-spaceshipVostok is a citizen of the U.S.S.R., Flight Major YuriGagarin.The launching of the multistage space rocket was successful and, after attain-ingtherst escapevelocityandtheseparationofthelast stageofthecarrierrocket, the spaceship went into freeight on around-the-earth orbit. Accordingto preliminary data, the period of the revolution of the satellite spaceship aroundthe earth is 89.1 min. The minimum distance from the earth at perigee is 175 km(108.7 miles) and the maximum at apogee is 302 km (187.6 miles), and the angleof inclination of the orbit plane to the equator is 658 40. The spaceship with thenavigator weighs 4725 kg (10,418.6 lb), excluding the weight of the nal stageofthecarrierrocket.After this announcement is made, Major Gagarins orbital craft, called Vostok I,is slowed at 10 : 25 a.m. by the ring of a retrorocket and enters the atmosphere ata speed in excess of 25 times the speed of sound. Thirty minutes later, Major YuriGagarin becomes the rst man to y in space, to orbit the Earth, and safely return.Fig. 1.1 V-2/WAC Corporal liftoff on 24 Feb. 1949, the rst object of human originto achievehypersonicight(NationalAir and SpaceMuseum).SOMEPRELIMINARYTHOUGHTS 3Moreover, onthat day, 12April 1961, Yuri Gagarinbecomestherst humanbeinginhistorytoexperiencehypersonicight. AphotographoftheVostokIcapsuleisshowninFig.1.2.Later, 1961 becomes a bumper year for manned hypersonic ight. On 5 May,Alan B. Shepard becomes the second man in space by virtue of a suborbital ightovertheAtlanticOcean, reachinganaltitudeof115.7miles, andenteringtheatmosphereat aspeedaboveMach5. Then, on23June, U.S. Air Forcetestpilot Major Robert Whiteies theX-15airplaneat Mach5.3, therst X-15ight toexceedMach5. (Insodoing, Whiteaccomplishestherst mile-persecondightinanairplane,reachingamaximumvelocityof3603 mph.)ThisrecordisextendedbyWhiteon9November,yingtheX-15atMach6.Theprecedingeventsarehistoricalrstsintheannualsofhypersonicight.They represent certain milestones and examples of the application of hypersonicaerodynamictheoryandtechnology.Thepurposeofthisbookistopresentanddiscussthistheoryandtechnology, withthehopethat thereader, asastudentand professional, will be motivated and prepared to contribute to the hypersonicmilestonesofthefuture.Fig. 1.2 Vostok I, in which Russian Major Yuri Gagarin became the rst human toy at hypersonic speed, during the worlds rst manned, orbital ight. 12 April 1961(NationalAirand SpaceMuseum).4 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICS1.2 Hypersonic FlowWhy Is It Important?The development of aeronautics and spaceight, from its practical beginningswiththeWrightBrothersrstairplaneighton17 December1903andRobertH. Goddards rst liquid-fueled rocket launch on 16 March 1926, has been drivenby one primary urgethe urge to always y faster and higher. Anyone who hastracedadvancements inaircraft inthe 20thcenturyhas seenanexponentialgrowthinbothspeedandaltitude,startingwiththe35-mphWrightyeratsealevel in 1903, progressing to 400-mph ghters at 30,000 ft in World War II, tran-sitioningto1200-mphsupersonicaircraft at 60,000ft inthe1960sand1970s,highlighted by the experimental X-15 hypersonic airplane, which achievedMach7andanaltitudeof 354,200ft on22August 1963, andnallycappedby the space shuttlethe ultimate in manned airplanes with its Mach 25reentryintotheEarthsatmospherefroma200-milelowEarthorbit. (See[1]for graphs whichdemonstratetheexponential increaseinbothaircraft speedand altitude over the past 100 years.) Superimposed on this picture is theadventofhigh-speedmissilesandspacecraft:forexample, thedevelopmentofthe Mach 25 intercontinental ballistic missile in the 1950s; the Mach 25Mercury, Gemini, andVostokmannedorbital spacecraft of the1960; andofcoursethehistoricMach36Apollospacecraft, whichreturnedmenfromthemoonstartingin1969. Thepoint hereis that theextremehigh-speedendoftheightspectrumhasbeenexplored,penetrated,andutilizedsincethe1950s.Moreover,ightatthisendofthespectrumiscalledhypersonicight,andtheaerodynamicandgasdynamiccharacteristicsofsuchightareclassiedunderthe label of hypersonic aerodynamicsone of the primary subjects of this book.Hypersonic aerodynamics is different than the more conventional and experi-enced regime of supersonic aerodynamics. These differences will be discussed atlength in Sec. 1.3, along with an in-depth denition of just what hypersonic aero-dynamics really means. However, we can immediately see that such differencesmust exist just bycomparingtheshapesof hypersonicvehicleswiththoseofmore commonplace supersonic aircraft. For example, Fig. 1.3 shows a LockheedF-104, the rst ghter aircraft designed for sustained supersonic ight at Mach 2.This aircraft embodies principles for good supersonic aerodynamic design:Fig. 1.3 Lockheed F-104, a supersonic airplane designed in the early l950s (NationalAir and SpaceMuseum).SOMEPRELIMINARYTHOUGHTS 5asharp, needle-likenoseandslenderfuselage,very thin wingsandtail surfaces(3.36percent thickness-to-chordratio) withverysharpleadingedges (almostsharpenoughtoposeahazardduringgroundhandling),andwithalowaspectratioof2.45forthestraight wingitselfall designedtominimizewavedragatsupersonicspeeds.TodesignahypersonicairplaneforightatmuchhigherMach numbers, it is tempting toutilize these same designprinciplesonlymore so. Indeed, such was the case for an early hypersonic aircraft concept con-ceived by Robert Carman and Hubert Drake of the NACA (now NASA) in 1953.OneoftheirhanddrawingsfromaninternalNACAmemorandumisshowninFig. 1.4 (see [2] for more details). Here we see an early concept for a hypersonicbooster/orbiter combination, where each aircraft has a sharp nose, slender fuse-lage, and thin, low-aspect-ratio straight wingsthe same features that are seen inthe F-104except the aircraft in Fig. 1.4 is designed for Mach 25. However, in1953 hypersonic aerodynamics was in its infancy. Contrast Fig. 1.4 with anotherhypersonic airplane designed just seven years later, the X-20 Dynasoar shown inFig. 1.5. Hereweseeacompletelydifferent-lookingaircraftoneembodyingnewhypersonicprinciples that werenot fullyunderstoodin1953. TheX-20designutilizedasharplysweptdeltawingwithablunt, roundedleadingedge,and a rather thick fuselage with a rounded (rather than sharp) nose. The fuselagewas placed on top of the wing, so that the entire undersurface of the vehicle wasat. TheX-20wasintendedtobeanexperimental aircraft forrocket-poweredight at Mach 20. Eclipsed by the Mercury, Gemini, and Apollo manned space-ight program, the X-20 project was cancelled in 1963 without the production ofa vehicle. However, the X-20 reected design features that were uniquelyFig. 1.4 DrakeCarmanhypersonicaircraft/orbiter,proposedin 1953(from [2]).6 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICShypersonicandthatwerelatercontainedinthespaceshuttle.Indeed,thespaceshuttleis showninFig. 1.6for further comparisonwiththeearlier conceptsshowninFigs.1.3and1.4.Clearly,hypersonicvehiclesaredifferentcongur-ationsfromsupersonicvehicles, andhencewemight conclude(correctly)thathypersonic aerodynamics is different from supersonic aerodynamics. This differ-enceis dramaticallyreinforcedwhenweexamineFig. 1.7, whichshows theApollospacevehicle, designedtoreturnhumansfromthemoon, andtoentertheEarthsatmosphereat theextremehypersonicspeedofMach36. Hereweseeaverybluntbodywithnowingsatall.Tobeobjective,wehavetorealizethat manyconsiderationsbesideshigh-speedaerodynamicsgointothedesignof the vehicles shown in Figs. 1.31.7; however, to repeat once again, the import-antpointhereisthathypersonicvehiclesaredifferentthansupersonicvehicles,and this is in part because hypersonic aerodynamics is different from supersonicaerodynamics.Hypersonic ight, both manned and unmanned, has been successfullyachieved. However, at the timeofthiswriting,it is by no means commonplace.The era of practical hypersonic ight is still ahead of us, and it poses many exci-ting challenges to the aerodynamicist. Let us briey examine some new ideas formodern hypersonic vehicles. For example, airbreathing hypersonic vehiclesdesignedforsustainedight intheatmospherehavecapturedtheimaginationof aerospaceengineersandmissionplannersalike. Oneconcept isthat of anFig. 1.5 BoeingX-20A Dynasoarorbital hypersonicaircraft,1963(from [2]).SOMEPRELIMINARYTHOUGHTS 7Fig.1.6 Spaceshuttle(NationalAir and SpaceMuseum).Fig. 1.7 Artists conception of the atmosphere entry of the Apollo spacecraft(NationalAirand SpaceMuseum).8 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSaerospaceplaneideallyanaircraft designedtotakeoff horizontallyfromarunway and then accelerate into orbit around the Earth, for the most partpowered byanairbreathing supersonic combustion ramjet engine (scramjet)with most likely a rocket assist for nal insertion into orbit. It will subsequentlycarryoutamissioninorbit,orwithintheouterregionsoftheatmosphere,andthen reenter the atmosphere at Mach 25, nally landing under power on a conven-tional runway. This idea was rst seriously examined by the U.S. Air Force in theearly 1960s, and a combination of airbreathing and rocket propulsion wasintended to power the vehicle. Work on the early aerospace plane was cancelledin October 1963 mainly because of the design requirements exceeding the state ofthe art at that time. The idea was resurrected in the middle 1980s by both NASAandtheU.S. Department of Defense, as well as byaerospace companies inEngland and Germany. A generic single-stage-to-orbit aerospace plane isshowninFig. 1.8, representativeofthissecondgenerationofaerospaceplanetechnology.In the United States,thiseffortwaslabeledthe National AerospacePlane program (NASP). It too was terminated in the mid-1990s after a decade ofintensive research and development. The reason was the samethe designrequirementsexceededthestateoftheart.Thisworkliveson,however,intheformof more modest but focused research and technology advancementefforts. Arecent success has been the X-43 Hyper-Xunmanned researchvehicle shown in Fig. 1.9. In November 2004 the X-43 made aeronautical engin-eeringhistorybyachievingsustainedightatMach10poweredbyascramjetengineforaperiodofabout 10 s; it wastherst sustainedatmosphericightbyascramjet-poweredvehicle. Of moreimportance, theight datafromtheX-43veriedthe predictions basedonwind-tunnel andcomputational-uid-dynamicdataandprovedthebasicmethodologyusedfor thedesignof suchvehicles. Thedoor tosuccessful sustainedhypersonic atmospheric ight hasbeenslightlycrackedopen.Fig. 1.8 Genericrepresentationof a nominalsingle-stage-to-orbitvehicle(NASA).SOMEPRELIMINARYTHOUGHTS 9Otherideasfornewhypersonicvehiclesincludescramjet-poweredmissilestoyintheMach68rangefor militaryuse, bothtactical andstrategic. Inthe United States, at the time of writing, both the U.S. Air Force and theU.S. Navy have active research and development programs for hypersonicatmospheric ight vehicles. In terms of getting payloads into orbit, thesingle-stage-to-orbit vehiclerepresentedinFig. 1.8hasastrongcompetitorinthe formof a two-stage-to-orbit vehicle combination shown generically inFig. 1.10. Here, the rst stage is a hypersonic ramjet/scramjet-powered airbreath-ingvehicle,andthesecondstageisarocket-poweredorbiterridingpiggyback.Finally,darewementionthepossibilityofhypersoniccommercialairtranspor-tation?At thetimeof writing, thereis noactiveprogramtodevelopevenasecond-generation supersonic transport to replace the retired AngloFrench Con-corde SST. So, is a hypersonic transport only a pipe dream? This author thinksnot. Figure 1.11 shows an artists rendering, circa 1985, of a futuristic hypersonictransport. Sometimeinthe21st century, but well after thedevelopment of asecond-generation SST, such an airplane as shown in Fig. 1.11 will be streakingFig.1.9 X-43AHyper-Xtestvehicle(NASA):a) pictorial viewand b)three view.10 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSthrough the sky, carrying passengers and cargo at Mach numbers of 5 and larger.Thetechnical,economic,andenvironmentalchallengeswill beenormous,butIcannotbelieveitwillnothappen.It is important to mention an aspect that distinguishes hypersonic atmosphericight vehicles from conventional subsonic and supersonic airplane design philos-ophy. For subsonic and supersonic aircraft, the components for providing lift (thewings), propulsion (the engines and nacelles), and volume (the fuselage) are notstronglycoupledwitheachother. Theyareseparateanddistinct components,easilyidentiablebylookingat theairplane; moreover, theycanbetreatedasFig.1.10 Genericrepresentationof atwo-stage-to-orbitvehicle(from [270]).Fig. 1.11 Concept for a hypersonic transport (McDonnell-Douglas AircraftCorporation).SOMEPRELIMINARYTHOUGHTS 11separateaerodynamicbodieswithonlyamoderateinteractionwhentheyarecombined in the total aircraft. Modern hypersonic aerodynamic design isexactly the opposite. Figure 1.12 is an example of an integrated airframe-propulsionconcept for ahypersonicairplane, whereintheentireundersurfaceof the vehicle is part of the scramjet engine. Initial compressionof the airtakesplacethroughthebowshockfromthenoseoftheaircraft; furthercom-pressionandsupersoniccombustiontakeplaceinsideaseriesofmodulesnearthe rear of the aircraft, and then expansion of the burned gases is partially realizedthroughnozzlesintheenginemodules,butmainlyoverthebottomrearsurfaceof the aircraft, which is sculptured to a nozzle-like shape. Hence, the propulsionmechanism is intimately integrated with the airframe. Moreover, most of the liftis produced by high pressure behind the bow shock wave and exerted on the rela-tivelyat undersurfaceof thevehicle; theuseof large, distinct wings is notnecessary for the production of high lift. Finally, the fuel for airbreathing hyper-sonicairplanesshowninFigs. 1.81.12isliquidH2, whichoccupiesalargevolume. Alloftheseconsiderationscombineinahypersonicvehicleinsuchafashion that the components to generate lift, propulsion, and volume are not sep-aratefromeachother; rather, theyarecloselyintegratedinthesameoveralllifting shape, in direct contrast to conventional subsonic and supersonicvehicledesign.Finally, return to the question asked at the beginning of this section: Hyperso-nic ow, why is it important? We now have a feeling for the answer. Hypersonicowisimportantbecauseofthefollowing:1) Itisphysicallydifferentfromsupersonicow.2) It is the ow that will dictate many of the new exciting vehicle designs forthe21stcentury.Recognizingthisimportance,thepurposeofthepresentbookistointroducethe reader to the basic fundamentals of hypersonic ow, including an emphasis onFig.1.12 Hypersonic vehiclewith integratedscramjet(NASA).12 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICShigh-temperature gas dynamics, which, as we will see, is an important aspect ofhigh-speedowsingeneral. Whereverpertinent, wewill alsodiscussmodernexperimental andcomputational-uid-dynamicapplications inhypersonicandhigh-temperatureow, aswell ascertainrelatedaspectsofhypersonicvehicledesign. Suchmaterial isanintegral part of modernaerodynamics. Moreover,theimportanceofthismaterialwill growsteadilyintothe21st century, aswecontinuetoextendtheboundariesofpracticalight.1.3 Hypersonic FlowWhat Is It?There is a conventional rule of thumb that denes hypersonic aerodynamics asthoseowswheretheMachnumber Misgreater than5. However, thisisnomorethanjust aruleof thumb; whenaowisacceleratedfromM 4.99to5.01, thereisnoclashofthunder,andtheowdoesnot instantlyturnfromgreen to red. Rather, hypersonic owis best dened as that regime wherecertainphysical owphenomenabecomeprogressivelymoreimportant as theMachnumberisincreasedtohighervalues.Insomecases,oneormoreofthesephenomenamightbecomeimportantaboveMach3,whereasinothercasestheymaynotbecompellinguntilMach7orhigher.Thepurposeofthissectionistobrieydescribethesephysicalphenomena;insomesensethisentiresectionwillconstituteadenitionofhypersonicow. Formoredetailsofanintroductorynature, see [3].1.3.1 Thin Shock LayersRecall fromobliqueshocktheory(forexample, see[4]and[5])that, foragiven ow deection angle, the density increase across the shock wavebecomes progressively larger as the Mach number is increased. At higherdensity, themass owbehindtheshockcanmoreeasilysqueezethroughsmaller areas. For owover ahypersonicbody, this means that thedistancebetweenthebodyandtheshockwavecanbesmall.Theoweldbetweentheshockwave andthe bodyis denedas the shock layer, andfor hypersonicspeedsthisshocklayercanbequitethin. Forexample, considertheMach36owofacaloricallyperfectgaswitharatioofspecicheats,g cp/cv 1.4,over awedgeof 15-deghalf-angle. Fromstandardobliqueshocktheorytheshock-wave angle will be only 18 deg, as shown in Fig. 1.13. If high-temperature,Fig. 1.13 Thinhypersonicshocklayer.SOMEPRELIMINARYTHOUGHTS 13chemicallyreactingeffects are included, the shock-wave angle will be evensmaller. Clearly, this shock layer is thin. It is a basic characteristic of hypersonicowsthatshockwaveslieclosetothebodyandthattheshocklayeristhin.Inturn, thiscancreatesomephysical complications, suchasthemergingof theshockwaveitselfwithathick, viscousboundarylayergrowingfromthebodysurfacea problem that becomes important at low Reynolds numbers.However,athighReynoldsnumbers,wheretheshocklayerisessentiallyinvis-cid, its thinness can be used to theoretical advantage, leading to a general analyti-cal approach called thin shock-layer theory (to be discussed in Chapter 4). In theextreme, athinshocklayerapproachestheuid-dynamicmodelpostulatedbyIssacNewtonin1687; suchNewtoniantheoryis simpleandstraightforwardand is frequently used in hypersonic aerodynamics for approximate calculations(tobediscussedinChapter3).1.3.2 Entropy LayerConsider thewedgeshowninFig. 1.13, except nowwithablunt nose, assketched in Fig. 1.14. At hypersonic Mach numbers, the shock layer overthe blunt nose is alsoverythin, witha small shock-detachment distance d.Inthenoseregion,theshockwaveishighlycurved. Recallthattheentropyoftheowincreasesacrossashockwave, andthestrongertheshock, thelargertheentropyincrease. Astreamlinepassingthroughthestrong, nearlynormalportionof thecurvedshocknear thecenterlineof theowwill experiencealargerentropyincreasethananeighboringstreamline, whichpassesthroughaweakerportionoftheshockfurtherawayfromthecenterline.Hence,therearestrongentropygradientsgeneratedinthenoseregion; thisentropylayerowsdownstreamandessentiallywetsthebodyforlargedistancesfromthenose,asshowninFig. 1.14. The boundarylayer alongthesurfacegrows inside thisFig. 1.14 Entropylayer.14 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSentropylayer andis affectedbyit. Becausethe entropylayeris alsoaregionofstrong vorticity, as related through Croccos theorem from classical compressibleow (for example, see [4]), this interaction is sometimes called a vorticity inter-action. The entropy layer causes analytical problems when we wish to perform astandard boundary-layer calculation on the surface because there is a question asto what the proper conditions should be at the outer edge of the boundary layer.1.3.3 Viscous InteractionConsider a boundary layer on a at plate in a hypersonic ow, as sketched inFig. 1.15. Ahigh-velocity, hypersonicowcontainsalargeamount ofkineticenergy;whenthisowisslowedbyviscouseffectswithintheboundarylayer,thelostkineticenergyistransformed(inpart)intointernalenergyofthegasthisiscalledviscousdissipation. Inturn, thetemperatureincreaseswithintheboundarylayer;atypical temperatureprolewithintheboundarylayerisalsosketchedinFig. 1.15. The characteristics of hypersonic boundarylayers aredominated by such temperature increases. For example, the viscosity coefcientincreases withtemperature, andthis byitself will make the boundarylayerthicker. Inaddition, becausethepressurepisconstant inthenormal directionthroughaboundarylayer,theincreaseintemperatureTresultsinadecreaseindensity r through the equation of state r p/RT, where R is the specic gas con-stant. Topass therequiredmass owthroughtheboundarylayer at reduceddensity,theboundary-layerthicknessmustbelarger.Bothofthesephenomenacombine to make hypersonic boundary layers growmore rapidly than atslower speeds. Indeed, the at-plate compressible laminar boundary-layerthickness dgrowsessentiallyasd /M21Rexpwhere M1is the freestream Mach number and Rex is the local Reynolds number.(Thisrelationwill bederivedinChapter 6.) Clearly, becausedvariesasthesquareofM1,itcanbecomeinordinatelylargeathypersonicspeeds.Fig. 1.15 Temperatureproleina hypersonicboundary layer.SOMEPRELIMINARYTHOUGHTS 15Thethickboundarylayerinhypersonicowcanexertamajordisplacementeffect ontheinviscidowoutsidetheboundarylayer, causingagivenbodyshapeto appear much thickerthan it really is. Because of the extreme thicknessof the boundary-layer ow, the outer inviscid owis greatly changed; thechanges in the inviscid ow in turn feed back to affect the growth of the boundarylayer.Thismajorinteractionbetweentheboundarylayerandtheouterinviscidowis called viscous interaction. Viscous interactions can have importanteffects on the surface-pressure distribution, hence lift, drag, and stability on hyper-sonic vehicles. Moreover, skin friction and heat transfer are increased by viscousinteraction. For example, Fig. 1.16 illustrates the viscous interaction on a sharp,right-circular cone at zero degree of angle of attack. Here, the pressure distributionontheconesurfacepisgivenasafunctionofdistancefromthetip.Theseareexperimentalresultsobtainedfrom[6].Iftherewerenoviscousinteraction,theinviscid surface pressure would be constant, equal to pc (indicated by the horizon-tal dashedlineinFig. 1.16). However, becauseoftheviscousinteraction, thepressurenearthenoseisconsiderablygreater;thesurface-pressuredistributiondecays further downstream, ultimately approaching the inviscid value far down-stream.Theseandmanyotheraspectsofviscousinteractionswillbediscussedin Chapter 7.The boundary layer on a hypersonic vehicle can become so thick that it essen-tiallymergeswiththeshockwaveamergedshocklayer.Whenthishappens,the shock layer must be treated as fully viscous, and the conventional boundary-layer analysis must be completely abandoned. Such matters will be discussed inChapter9.1.3.4 High-Temperature FlowsAs discussed earlier, the kinetic energy of a high-speed, hypersonic ow is dis-sipated by the inuence of friction within a boundary layer. The extreme viscousFig.1.16 Viscousinteractioneffect.InducedpressureonasharpconeatM5 11andRe 5 1.88 3 105per foot.16 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSdissipationthatoccurswithinhypersonicboundarylayerscancreateveryhightemperatureshighenoughtoexcitevibrational energyinternallywithinmol-ecules and to cause dissociation and even ionization within the gas. If thesurfaceofahypersonicvehicleisprotectedbyanablativeheatshield,thepro-ductsofablationarealsopresentintheboundarylayer,givingrisetocomplexhydrocarbonchemicalreactions.Onbothaccounts,weseethatthesurfaceofahypersonicvehiclecanbewettedbyachemicallyreactingboundarylayer.Theboundarylayerisnot theonlyregionofhigh-temperatureowoverahypersonicvehicle. Consider thenoseregionof ablunt body, assketchedinFig. 1.17. The bowshock wave is normal, or nearly normal, in the noseregion, and the gas temperature behind this strong shock wave can be enormousathypersonicspeeds. Forexample, Fig. 1.18isaplotoftemperaturebehindanormalshockwaveasafunctionoffreestreamvelocity, foravehicleyingata standardaltitude of 52 km; this gure is takenfrom[4]. Twocurves areshown: 1) the upper curve, whichassumes a caloricallyperfect nonreactinggaswiththeratioofspecicheatsg 1.4, andwhichgivesanunrealisticallyhigh value of temperature; and 2) the lower curve, which assumes an equilibriumchemicallyreactinggasandwhichisusuallyclosertotheactualsituation.Thisgureillustratestwoimportantpoints:1) By any account, the temperature in the nose region of a hypersonic vehiclecan be extremely high, for example, reaching approximately 11,000 K at a Machnumberof36(Apolloreentry).Fig.1.17 High-temperatureshocklayer.SOMEPRELIMINARYTHOUGHTS 172) Theproperinclusionofchemicallyreactingeffectsisvital tothecalcu-lationofanaccurateshock-layertemperature;theassumptionthatgisconstantandequalto1.4isnolongervalid.Soweseethat, for ahypersonicow, not onlycantheboundarylayer bechemicallyreacting,buttheentireshocklayercanbedominatedbychemicallyreactingow.For a moment, let us examine the physical nature of a high-temperature gas. Inintroductory studies of thermodynamics and compressible ow, the gas isassumedtohaveconstantspecicheats;hence,theratiog cp/cvisalsocon-stant. Thisleadstosomeideal resultsfor pressure, density, temperature, andMach-number variations in a ow. However, when the gas temperature isincreasedtohighvalues, thegasbehavesinanonidealfashion, specicallyasfollows:1) The vibrational energyof the molecules becomes excited,and this causesthe specic heats cp and cv to become functions of temperature. In turn, the ratioof specic heats, g cp/cv, also becomes a function of temperature. For air, thiseffectbecomesimportantaboveatemperatureof800 K.2) As the gas temperature is further increased, chemical reactions can occur.For an equilibrium chemically reacting gas, cp and cv are functions of both temp-erature and pressure, and hence g f(T, p). For air at 1 atmpressure, O2Fig. 1.18 Temperaturebehindanormal shockwaveasafunctionof freestreamvelocityata standardaltitudeof52 km(from [4]).18 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSdissociation(O2!2O) beginsat about 2000 K, andthemolecular oxygenisessentiallytotallydissociatedat 4000 K. At this temperatureN2dissociation(N2!2N) begins andis essentiallytotallydissociatedat 9000 K. Above atemperatureof 9000 K, ionsareformed(N !Ne2, andO !Oe2),andthegasbecomesapartiallyionizedplasma.All of thesephenomenaarecalledhigh-temperatureeffects. (Theyarefre-quentlyreferredtointheaerodynamicliteratureasreal-gaseffects, but therearegoodtechnical reasonstodiscouragetheuseof that label, aswewill seelater.) If the vibrational excitation and chemical reactions take place veryrapidly in comparison to the time it takes for a uid element to movethroughtheoweld, wehavevibrational andchemical equilibriumow. Ifthe opposite is true, we have nonequilibriumow, which is considerablymore difcult toanalyze. All of these effects will be discussedat lengthinChapters1018.High-temperaturechemicallyreactingows canhaveaninuenceonlift,drag, andmoments onahypersonicvehicle. For example, sucheffects havebeenfoundtobeimportant for estimatingtheamount of body-apdeectionnecessarytotrimthespaceshuttleduringhigh-speedreentry. However, byfarthemost dominant aspect ofhightemperaturesinhypersonicsistheresultanthigh heat-transfer rates to the surface. Aerodynamic heating dominates thedesignof all hypersonic machinery, whether it be a ight vehicle, a ramjetenginetopowersuchavehicle,orawindtunneltotestthevehicle.Thisaero-dynamicheatingtakestheformofheattransferfromthehotboundarylayertothecooler surfacecalledconvectiveheatinganddenotedbyqcinFig. 1.17.Moreover, iftheshock-layertemperatureishighenoughthethermal radiationemittedbythegasitselfcanbecomeimportant, givingrisetoaradiativeuxtothesurfacecalledradiativeheatinganddenotedbyqRinFig.1.17.(Inthewinter, when you warmyourself beside a roaring re in the replace, thewarmthyoufeelisnothotairblowingoutofthereplace,butratherradiationfromthehotbricksorstoneofthewallsofthereplace.Imaginehowwarmyouwouldfeel standingnext tothegasbehindastrongshockwaveat Mach36,wherethetemperatureis11,000 Kabouttwicethesurfacetemperatureofthesun.)Forexample,forApolloreentryradiativeheattransferwasmorethan30%ofthetotalheating.ForaspaceprobeenteringtheatmosphereofJupiter,theradiativeheatingwillbemorethan95%ofthetotalheating.Another consequenceof high-temperatureowover hypersonicvehiclesisthecommunicationsblackoutexperiencedat certainaltitudesandvelocitiesduring atmospheric entry, where it is impossible to transmit radio waveseither toor fromthevehicle. This is causedbyionizationinthechemicallyreactingow, producingfreeelectrons that absorbradio-frequencyradiation.Therefore, the accurate predictionof electrondensitywithintheoweldisimportant.Clearly, high-temperature effects canbe a dominant aspect of hypersonicaerodynamics, andbecauseof thisimportancePart 3of thisbookisdevotedentirelytohigh-temperaturegas dynamics. (Part 3is self-containedandrep-resentsastudyofhigh-temperaturegasdynamicsingeneral,aeldwithappli-cations that go far beyond hypersonics, such as combustion, high-energylasers,plasmas,andlaser-matterinteraction,tonamejustafew.)SOMEPRELIMINARYTHOUGHTS 191.3.5 Low-Density FlowConsider for a moment the air around you; it is made up of individual molecules,principallyoxygenandnitrogen,whichareinrandommotion.Imaginethatyouisolate one of these molecules, and watch its motion. It will move a certain distanceandthencollidewithoneofitsneighboringmolecules,afterwhichitwillmoveanother distance, and collide again with another neighboring molecule, and it willcontinue this molecular collision process indenitely. Although the distancebetweencollisionsisdifferentforeachoftheindividual collisions,overaperiodof time there will be some average distance the molecule moves between successivecollisions. This average distance is dened as the mean free path, denoted by l. Atstandardsea-levelconditionsforair, l 2.176 1027ft,averysmalldistance.Thisimpliesthat,atsealevel,whenyouwaveyourhandthroughtheairthegasitself feels like a continuous mediuma so-called continuum. Most aerodynamicproblems (more than 99.9%of all applications) are properly addressed by assumingacontinuous medium; indeed, all of ourpreceding discussion has so far assumedthat the ow is a continuum.Imagine now that we are at analtitudeof342,000 ft, where theair density ismuch lower, and consequently the mean free path is much larger than at sea level;indeed, at 342,000 ft, l 1 ft. Now, whenyou wave your hand through the air,you are more able to feel individual molecular impacts; the air no longer feels likea continuous substance, but rather like an open region punctuated by individual,widely spaced particles of matter. Under these conditions, the aerodynamic con-cepts, equations, andresultsbasedontheassumptionofacontinuumbegintobreak down; when this happens, we have to approach aerodynamics from a differ-ent point of view, usingconcepts fromkinetic theory. This regimeof aero-dynamicsiscalledlow-densityow.There are certain hypersonic applications that involve low-density ow, gen-erally involving ightathighaltitudes.Forexample,asnotedin[7] theowinthenoseregionofthespaceshuttlecannotbeproperlytreatedbypurelyconti-nuumassumptionsforaltitudesabove92 km(about300,000 ft). Foranygivenight vehicle, as the altitude progressively increases (hence the density decreasesand l increases), the assumption of a continuum ow becomes tenuous. An alti-tude can be reached where the conventional viscous ow no-slip conditions begintofail. Specically, at lowdensitiestheowvelocityat thesurface, whichisnormallyassumedtobezerobecauseoffriction, takesonanitevalue. Thisiscalledthevelocity-slipcondition. Inanalogousfashion, thegastemperatureat thesurface, whichisnormallytakenasequal tothesurfacetemperatureofthematerial,nowbecomessomethingdifferent.Thisiscalledthetemperature-slipcondition.Attheonsetoftheseslipeffects,thegoverningequationsoftheowarestillassumedtobethefamilarcontinuum-owequations,exceptwiththeproper velocityandtemperature-slipconditionsutilizedasboundarycon-ditions. However, as the altitude continues toincrease, there comes a pointwhere the continuum-ow equations themselves are no longer valid, andmethodsfromkinetictheorymustbeusedtopredict theaerodynamicbehavior.Finally, the air density can become low enough that only a few molecules impactthe surface per unit time, and after these molecules reect from the surface theydo not interact with the incoming molecules. This is the regime of free molecule20 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSow. For the space shuttle, the free molecular regime begins about 150 km(500,000 ft). Therefore, inasimpliedsense, wevisualizethat ahypersonicvehiclemovingfromaveryrariedatmospheretoadenser atmospherewillshift fromthefreemolecularregime, whereindividual molecular impactsonthe surface are important, to the transition regime, where slip effects are import-ant,andthentothecontinuumregime.ThesimilarityparameterthatgovernsthesedifferentregimesistheKnudsennumber, dened as Kn l/L, where L is a characteristic dimension of the body.ThevaluesofKninthedifferentregimesarenotedinFig.1.19,takenfrom[7].Note theregion wherethecontinuumNavierStokes equationshold is describedbyKn , 0.2. However, slipeffects must beincludedintheseequations whenKn . 0.03. Theeffectsoffreemolecular owbeginaroundavalueorKn land extend out to the limit of Kn becoming innite. Hence, the transitionalregimeisessentiallycontainedwithin0.03 , Kn , 1.0.Inagivenproblem,theKnudsennumber is thecriteriontoexamineinorder todecideif low-densityeffectsareimportant andtowhat extent. Forexample, ifKnisverysmall, wehave continuumow; if Kn is very large, we have free molecular ow, and so forth.Ahypersonicvehicleenteringtheatmospherefromspacewillencounterthefullrange of these low-density effects, down to an altitude below which the full conti-nuumaerodynamics takes over. Because Kn l/L is the governing parameter, thataltitude below which we have continuum ow is greater or lesser as the character-istic length L is larger or smaller. Hence, large vehicles experience continuum owto higher altitudes than small vehicles. Moreover, if we let the characteristic lengthbe a running distance x from the nose or leading edge of the vehicle then Kn l/xbecomes innitewhenx 0. Hence, for anyvehicleat anyaltitudetheowimmediatelyat theleadingedgeis governedbylow-densityeffects. For mostpractical applications in aerodynamics, this leading-edge region is very small andis usuallyignored. However, for high-altitude hypersonic vehicles the propertreatment of the leading-edge ow by low-density methods can be important.Fig. 1.19 Regimesofapplicabilityofvariousowequationsforlow-densityows(from [7]).SOMEPRELIMINARYTHOUGHTS 21To consider low-density effects as part of the denition of hypersonic aero-dynamics might be stretching that denition too much. Recall that we are den-ing hypersonic aerodynamics as that regime where certain physical owphenomena become progressively more important as the Mach number isincreased to high values. Low-density effects are not, per se, high-Mach-numbereffects. However, low-densityeffects are includedinour discussionbecausesomeclasses of hypersonicvehicles, as aresult of their highMachnumber,will y at or through the outer regions of the atmosphere and hence will experi-encesucheffectstoagreaterorlesserextent.1.3.6 RecapitulationTo repeat, hypersonic ow is best dened as that regime where all or some oftheprecedingphysical phenomenabecomeimportant as theMachnumber isincreased to high values. To help reinforce this denition, Fig. 1.20 summarizesFig.1.20 Physicaleffectscharacteristicof hypersonicow.22 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSthe important physical phenomena associated with hypersonic ight. Throughoutthis book, the fundamental aspects and practical consequences of these phenom-enawillbeemphasized.1.4 Fundamental Sources of Aerodynamic Force andAerodynamic HeatingAnyobjectimmersedinaowinggasexperiencesaforceasaresultoftheinteractionofthebodywiththegastheaerodynamicforce,illustratedbythevectorRinFig. 1.21. ThetwocomponentsofRperpendicularandparallel totherelativewindfaraheadofthebodyarelift LanddragD, respectively, asalsoshowninFig.1.21.Similarly, Rcanberesolvedintotwocomponents per-pendicular and parallel to the chord (or axis) of the body, the normal force N andtheaxial forceA, respectively, showninFig. 1.21. Theseforces areusuallyFig. 1.21 a)Resultantaerodynamicforceandthecomponentsintowhichitsplitsand b)illustrationof pressureand shearstress onanaerodynamicsurface.SOMEPRELIMINARYTHOUGHTS 23couchedintermsofforcecoefcients.Deningthedynamicpressureq1asq1 12r1V21where r1 and V1 are the freestream density and velocity respectively, and choos-ingsomereferenceareaS,wehavethefollowingbydenition.Liftcoefcient:CL Lq1SDragcoefcient:CD Dq1SNormal-forcecoefcient:CN Nq1SAxial-forcecoefcient:CA Aq1SInaddition, consideranypointonthebody.TheresultantforceRcancreateamomentMaboutthispoint,andwedeneMomentcoefcient:Mq1Scwhere c is a characteristic length of the body, such as the chord length shown inFig.1.21a.How does nature exert an aerodynamic force on the body? To help answer thisquestion, pick up this book, and hold it stationary in the palm of your hand. Youare exerting a force on the book, equal to its weight. This force is communicatedto the book by the skin of your hand being in contact with the bottom cover of thebook. This is the only waythatthe bookknows that you are exerting aforce onitby contact of the books surface with your hand. Similarly, the surface of thebody in Fig. 1.21a is in contact with the airthe molecular layer of air immedi-ately adjacent to the body surface. The only way that the body feels the presenceof the air is through the pressure distribution and shear-stress distribution exertedon the body surface that is in contact with the air. As illustrated in Fig. 1.21b, thepressure acts locally normal to the body surface, and shear stress, which is causedbyfrictionbetweenthebodyandthemolecularlayerofairjustadjacenttothebodysurface, actslocallytangentialtothesurface.Letsbethedistancealongthebodysurfacemeasuredfromtheleadingedge(nose)ofthebody,asshown24 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSin Fig. 1.21b. The variation of pressure with distance along the surface, denotedby p(s), is the pressure distribution, and the variation of shear stress with distancealongthesurface, denotedbytw(s), is theshear-stress distribution. Clearly,pressure and shear stress are distributed loads acting on the body and when inte-grated over the complete surface area of the body give rise to the resultant aero-dynamic force vector R shown in Fig. 1.21a. The two hands of nature that reachout and grab hold of the body, exerting a force on the body, are the pressure dis-tribution and shear-stress distribution exerted on the body surface. These are thetwofundamentalsourcesofaerodynamicforceandthatisallthereis.Except in the cases where viscous interaction is dominant and/or ow separ-ation takes place, the surface-pressure distribution is essentially, by nature,an inviscid ow phenomena. Part 1 (Chapters 25) of this book deals with hyper-sonic inviscid ow. Much of the material in Part 1 is focused on the calculation ofpressuredistributionsoverthesurfaceofhypersonicaerodynamicshapes, thusleadingtothepredictionofliftandwavedrag(apressureeffect).Shear stress is a viscous ow phenomena. Part 2 (Chapters 68) of this bookdealswithviscousow.SomeofthematerialinPart2isfocusedonthecalcu-lationofshear-stressdistributionsoverthesurfaceofhypersonicaerodynamicshapes, thusleadingtothepredictionof skin-frictiondrag. Thesurfaceshearstressisdictatedbythevelocitygradientatthesurfacetw mw@u@y w(1:1)where (@u=@y)wis the velocity gradient at the wall and mwis the viscosity coef-cientatthewall.Anyobject immersedinaowinggasalsoexperiencesheat transferat thebodysurface. Forthemajorityofpractical casesinhigh-speedaerodynamics,heatistransferredfromthegasintothebodyaerodynamicheating.Forlow-speed subsonic ows, aerodynamic heating, though present, is small and not gen-erallyaplayer.Forsupersonicows,however,aerodynamicheatingbecomesaconsideration.Forhypersonicows,aerodynamicheatingisamajorplayer,somuchsothatitdictatesthecongurationdesignofmosthypersonicvehicles.The sources of aerodynamic heating to a body are related to the hot gas in theoweld around the body. The boundary layer adjacent to the body surface is hotbecausethehigh-kinetic-energyhypersonicowenteringtheboundarylayerisslowedbyviscous effects withinthe boundarylayer, dissipatingthe kineticenergy,andtransformingit(inpart)intointernalenergyofthegas.Thegasintheinviscidportionoftheshocklayerbetweentheshockwaveandthebodyishotbecauseithascomethroughastrongshockwave.Shock-waveheatingandintenseviscousdissipationarethecausesof thehypersonicowover abodybeinghot.Nature,inturn,pumpssomeofthisheatenergyintothebodyintheformofaerodynamicheating. Therearetwophysicalsources, ormechanisms,thatnatureusestotransfertheheatintothebody:thermalconductionandradi-ation.Letusexamineeachoneofthesemechanismsinturn.Thermal conductiontakesplacewhenatemperaturegradient exists inthegas, andfromFourierslawofheatconductiontheheattransfertothesurfaceSOMEPRELIMINARYTHOUGHTS 25qwisgivenbyqw kw@T@y w(1:2)wherekwisthethermalconductivityofthegasatthewalland(@T=@y)wisthetemperaturegradientinthegasatthewall.Theminussigndenotesthatheatisconductedintheoppositedirectionofthetemperaturegradient. Forexample,reect againontheboundary-layer temperatureprolesketchedinFig. 1.15.The temperature gradient at the wall is clearly seen; at the wall, the gas tempera-tureincreaseswithdistanceabovethewall;hence,(@T=@y)wisapositivequan-tity. Consequently, qwowsintothewall inthenegativeydirection. ThermalconductionasquantiedbyEq.(1.2)isoneofnatureswaystotransferheattothesurfaceofabody. Thehandofnaturethatdoesthisisnotthetemperatureof thegas at thewall, but rather thetemperaturegradient inthegas at thewall. Part 2 of this book deals in part with the calculation of temperature gradientsatthewallandhencetheconductiveheattransferatthesurface.Thermal radiation is the other hand that nature uses to heat a body (or to cool abody). If the temperature of the oweld around a body is hot enough, signicantthermal radiation fromthe gas to the body takes place. Howhot is hot enough? Forair, if the shock-layer temperature is about 10,000 Kor higher radiative heat trans-ferfromthehotshocklayertothecoolerbodybecomesaplayerintheoverallheating of the body. Examples are space vehicles entering the Earths atmosphereat superorbital velocities, suchas theApollolunar returncapsule, wheretheshock-layertemperaturereachedashighas11,000 K. Almost30%ofthetotalheat transfer totheApolloduringatmosphericentrywascausedbyradiativeheating. For the Galileo probe entering the Jovian atmosphere at 15 km/s,wheretheshock-layertemperatureexceeded15,000 K, morethan95%of thetotal heat transfer was radiative heating. Radiative heating is treated in Chapter 18.Parenthetically, wenotethat thesolidsurfaceof thebodyradiatesenergyawayfromthe surface. This is a formof coolingof the surfaceradiativecooling. Solid bodies emit radiation primarily in the infrared part of the spectrum.(When you stand in front of a replace on a cold winter day, and enjoy the heatfromthe re, most of the heatyou are feeling is coming from infrared radiationemitted from the hot bricks or stone of the walls of the replace.) Moreover, theradiativeenergyemittedfromthesurfaceofasolidbodyisafunctionofthesurface temperature, following the T4variation of a blackbody. Surface radiation,therefore, can be used effectively as part or all of the thermal protection mechan-ismforahypersonicvehicle.In summary, the two sources of aerodynamic heating are the temperature gra-dient in the gas at the body surface, adding heat to the body by means of thermalconduction, andthehot temperaturethat existsgloballythroughout theshocklayer, addingheat tothebodybymeans of thermal radiationfromeachhotuidelementintheoweld.Frequently,theformermodeofheatingiscalledconvectiveheating,althoughthermalconductionatthesurfaceisthehandthatnatureusestotransfertheenergytothesurface. Thelattermodeofheatingissimplycalledradiativeheating.26 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICS1.5 Hypersonic Flight Paths: Velocity-Altitude MapAlthough this is a book on hypersonic and high-temperature gas dynamics, wemustkeepinmindthatthefrequentapplicationofthismaterialistothedesignandunderstandingof hypersonicight vehicles. Inturn, it ishelpful tohavesomeknowledgeof theight pathsof thesevehiclesthroughtheatmosphereandtheparameters that governsuchight paths. This is thepurposeof thepresent section. Inparticular, we will examine the ight pathof liftingandnonliftinghypersonicvehiclesduringatmosphericentryfromspace.Consideravehicleyingat avelocityValongaight pathinclinedat theangleubelowthelocalhorizontal,asshowninFig.1.22.Theforcesactingonthevehiclearelift L, dragD, andweight W;thethrust isassumedtobezero;hence, weareconsideringahypersonicglidevehicle. Summingforces alongand perpendicular to the curvilinear ight path, we obtain the following equationsofmotionfromNewtonssecondlaw.Alongightpath:W sin u D mdVdt(1:3)Perpendiculartoightpath:L W cos u mV2R(1:4)Fig.1.22 Forcediagramforreentry body.SOMEPRELIMINARYTHOUGHTS 27InEq.(1.4),Risthelocalradiusofcurvatureoftheightpath.Formostentryconditions, uis small; hence, we assume sinu0andcos u1. For thiscase,Eqs.(l.3)and(l.4)become,notingthatm W/gD WgdVdt(1:5)L W WgV2R(1:6)The drag can be expressed in terms of the drag coefcient CD as D 12rV2SCD,whereristhefreestreamdensityandSisareferencearea. Hence, Eq. (1.5)becomes12rV2SCD WgdVdtRearranging,weobtain1gdVdt WCDS 1rV22(1:7)In Eq. (1.7), W=CDS is dened as the ballistic parameter; it clearly inuences theight path of the entry vehicle via the solution of Eq. (1.7). For a purely ballisticreentry(nolift), W=CDSistheonlyparametergoverningtheight pathforagivenentryangle.ReturningtoEq. (l.6)andexpressingthelift intermsofthelift coefcientCLasL 12rV2SCL,weobtain12rV2SCLW WgV2RRearranging,1 1gV2R WCLS 1rV22(1:8)In Eq. (1.8), W/CL S is the lift parameter; it clearly inuences the ight path of alifting-entryvehicleviathesolutionofEq.(1.8).Equations (1.7) and(1.8) illustratetheimportanceof W=CDSandW=CLSindeterminingtheight paththroughtheatmosphereof avehiclereturningfromspace. Suchight pathsarefrequentlyplottedonagraphof altitudevsvelocityavelocity-altitudemap, anexampleofwhichisshowninFig. 1.23.Here, twoclassesofight pathsareshown: 1)liftingentry, governedmainlybyW=CLS, and2) ballistic entry, governedmainlybyW=CDS. The vehicleenters the atmosphere at either satellite velocity (such as fromorbit) or at28 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSescape velocity (such as a return from a lunar mission). As it ies deeper into theatmosphere,itslowsasaresultofaerodynamicdrag,givingrisetoightpathsshowninFig. 1.23. Notethat vehicles withlarger values of W=CLSand/orW=CDSpenetratedeeperintotheatmospherebeforeslowing. Thelifting-entrycurveforW/CLS 100 lb/ft2pertainsapproximatelytothespaceshuttle; thecurveinitiatedat escapevelocitywithW/CDS 100 lb/ft2pertains approxi-mately to the Apollo entry capsule. Velocity-altitude maps are convenientdiagramstoillustratevariousaerothermodynamicregimesofsupersonicight,andtheywillbeusedassuchinsomeofoursubsequentdiscussion.1.6 Summary and OutlookThemajorpurposesofthischapterhavebeenmotivationandorientationmotivationastotheimportance, interest, andchallengeassociatedwithhyper-sonic aerodynamics, andorientationas towhat hypersonics entails. For theremainder of this book, our purpose is topresent anddiscuss theimportantfundamental aspectsofhypersonicandhigh-temperaturegasdynamicsandtohighlight various practical applications as appropriate. Towards this end, thebook is organized into three major parts, as diagramed inFig. 1.24. Thesethreepartsareasfollows.Part1InviscidFlow:Here, thepurelyuid-dynamiceffect oflargeMachnumber isemphasized, without theaddedcomplicationsof viscousandhigh-temperature effects.In this part, we examine what happens when the freestreamMach number M1 becomes large and how this inuences aerodynamic theory athighMachnumbers. Herewecalculatepressuredistributionsonsurfacesandobtaintheaerodynamicliftandwavedragonhypersonicbodies.Fig. 1.23 Atmosphericentryightpaths onavelocity-altitudemap.SOMEPRELIMINARYTHOUGHTS 29Fig.1.24Roadmapforourstudyofhypersonicandhigh-temperatureows.30 HYPERSONICANDHIGH-TEMPERATUREGASDYNAMICSPart2ViscousFlow:Here,thecombinedeffectofhighMachnumberandniteReynoldsnumber will beexamined. Thepurelyuid-dynamiceffect ofhypersonicowwithfrictionandthermalconductionwillbepresented;again,high-temperature effects will not be included. Here, we calculate the shearstressat thesurfaceandtheresultingskinfrictiondragonhypersonicbodies.Inadditionwecalculatethetemperaturegradientsinthegasat asurfaceandtheresultingaerodynamicheatingtohypersonicbodiesasaresult of thermalconduction(convectiveaerodynamicheating).Part 3High-Temperature Flow: Here, the important aspects of high-temperaturegas dynamicswill bepresented. Emphasiswill beplacedonthedevelopment of basicphysical chemistryprinciplesandhowtheyaffect bothinviscidandviscous ows. High-temperatureows ndapplicationinmanyeldsinadditiontohypersonicaerodynamics, suchascombustionprocesses,explosions, plasmas, high-energylasers, etc. Therefore, Part 3will beaself-contained presentation of high-temperature gas dynamics in general, alongwith pertinent applications to hypersonic ow. Here we also calculate the radia-tiveheatingofahypersonicbodycausedbyintenseradiationfromveryhigh-temperatureshocklayers.Figure1.24isablockdiagramshowingeachoneofthethreepartsjustdis-cussed,alongwiththemajoritemstobediscussedundereachpart.Inessence,thisgureisaroadmapforourexcursionsinhypersonicandhigh-temperaturegas dynamics. Figure 1.24 is important, and we will refer to it often in order to seewhere we are, where we have been, and where we are going in our presentation.Problems1.1 Considerthesupersonicandhypersonicowofair(withconstantratioofspecicheats,g 1.4)overa20-deghalf-anglewedge.Letudenotethewedgehalf-angleandbtheshock-waveangle. Thenb-uisameasureofthe shock-layer thickness. Make a plot of b-uvs the freestreamMachnumber M1fromM1 2.0to20.0. Makesomecomments as towhatMach-numberrangeresultsinathinshocklayer.1.2 Theliftingparameter W/CDSis givenFig. 1.23inunits of lb/ft2. Fre-quently, theanalogousparameter m=CDSisused, whenmisthevehiclemass; the units of m=CDS are usually given in kg/m2. Derive the appropri-ateconversionbetweenthesetwosetsofunits,thatis,whatnumbermustW=CDSexpressedinlb/ft2bemultipliedbytoobtainm=CDSinkg/m2?(Comment: Evenat thegraduatelevel, it is useful nowandthentogothroughthistypeofexercise.)SOMEPRELIMINARYTHOUGHTS 31This page intentionally left blankPart 1Inviscid Hypersonic FlowIn Part 1 we emphasize the purely uid-dynamic effects of highMach number; the complicating effects of transport phenomena(viscosity, thermal conduction, and diffusion) and high-temperature phenomena will be treated in Parts 2 and 3, respec-tively. In dealing with inviscid, hypersonic ow in Part 1, we aresimply examining the question: what happens to the uiddynamicsofaninviscidowwhentheMachnumberismadeverylarge?Wewill seethatsuchanexaminationgoesalongwaytowardtheunderstandingof manypractical hypersonicapplications.This page intentionally left blank2Hypersonic Shock and Expansion-WaveRelationsItisclearthatthethoroughstudyofgas-dynamicdisconti-nuities andtheir structures combines inanessential waytheelds of hydrodynamics, physics, andchemistry, andthatthereisnolackofproblemswhichdeserveattention.WallaceD.Hayes,PrincetonUniversity,1958Chapter PreviewThis chapter is short and sweet. Shock waves and expansion waves are ubiqui-tousfeaturesinsupersonicandhypersonicowelds. Thedetailedanalysisand calculation of shock and expansion wave properties are bread-and-buttertopics in the study of compressible ow. This chapter goes a step further; here,theassumptionofveryhighMachnumbersaheadofthewavesallowstherather elaborate, but exact, shock- and expansion-wave relations to bereducedtomuchsimpler,albeitapproximate,equationsthatholdforhyper-sonicow.Youmightasksowhat?Indeed,foranaccuratecalculationofthedetailsofahypersonicowyoushouldalwaysusetheexactshock-andexpansion-waveequations. Thevalueofthischapterlieselsewhereitliesin the world of hypersonic aerodynamic theory. This chapter is the front endof the beautiful intellectual triumphs that are the foundation of the hypersonicaerodynamic theories to be developed in subsequent chapters. One such theoryis hypersonic similarity, to be explained and developed in Chapter 4. But rightaway, in the present chapter, the approximate hypersonic shock- andexpansion-waverelationsexposeoneof themost important parametersinhypersonicaerodynamictheorythehypersonicsimilarityparameter. Butwearegettingaheadofourselves.Forrightnow,justsitbackandenjoytheclean and neat development of the simplied equations for shock and expan-sionwavesaffordedbytheassumptionof highMachnumbersintheowahead of the waves. Then store these hypersonic shock- and expansion-waverelations on your mental bookshelf for future use in subsequent chapters.352.1 IntroductionConsider anairplaneyingat Mach28at theouter regionsof theEarthsatmosphere, say at an altitude of 120 km(approximately 400,000 ft). Upondescentintothelowerregionsoftheatmosphere,theaircraftcanfollowoneoftheliftingtrajectoriesshownonthealtitude-velocitymapinFig. 2.1.Super-imposedonthisgurearelinesofconstantMachnumber. Thepurposeofthisgureistoemphasizetheobviousfactthatsuchhypersonicvehiclesencounterexceptionally high Mach numbers. Moreover, the ight path remains hypersonicover most of its extent. Figure 2.1 justies the study of high-Mach-number owsand underscores the question: what happens in a purely uid-dynamic sense whenthe Mach number becomes very large? This question has particular relevance inregardtothe basicshock-andexpansion-waverelations. Inthepresent chapter,wewillobtainandexaminethelimitingformsofboththeconventionalshock-wave equations and the Prandtl Meyer expansion-wave relations when theupstream Mach number increases toward innity. These limiting forms are inter-estingintheir ownright; however, of more importance, theyare absolutelynecessaryfor the development of various inviscidhypersonic theories tobediscussedinsubsequentchapters.2.2 Basic Hypersonic Shock RelationsAnytime a supersonic ow is turned into itself (such as owing over a wedge,cone, or compression corner), a shock wave is created. Also, if a sufciently highbackpressure is created downstream of a supersonic ow, a standing shock wavecan be established. Such shock waves are extremely thin regions (on the order of1025cminair)acrosswhichlargechangesindensity, pressure, velocity, etc.occur.Thesechangestakeplaceinacontinuousfashionwithintheshockwaveitself, where viscosity and thermal conduction are important mechanism