mics N.F.Krasnov1 FundamentalsofTheory. Aerodynamicsofan AirfoilandaWing TranslatedfromtheRussianby G.Leib MirPublishersMoscow Firstpublished1985 Revisedfromthe1980Russianedition HaanMUUCKOMRablne MSp;aTeJIhCTBOBhICmaJIIDKOJIa,1980 Englishtranslation,MirPublishers.1985 ~ m C s 1 H.OJ.KpaCHOB AapOAVlHOMVlKO YaCTbI OCHOBblTeOpHH AspOAHHaMHKan p O ~ H n HHK p ~ n a j.13AaTenbCTBO BblcwaflWKOna MOCKBa Preface Aerodynamicsisthetheoreticalfoundationofaeronautical,rocket,space, andartilleryengineeringandthecornerstoneoftheaerodynamicdesignof modern craft.The fundamentals ofaerodynamics are usedin studyingthe exter-nalflowovervariousbodiesorthemotionofair(agas)insidevariousobjects. Engineeringsuccessinthefieldsofaviation,artillery,rocketry,spaceflight, motorvehicletransport,andsoon,i.e.fieldsthatpertaintotheflowofair or agasin someformorother,dependsonafirmknowledgeofaerodynamics. Thepresenttextbook,inadditiontothegenerallawsofflowofafluid, treatstheapplicationofaerodynamics,chieflyinrocketryandmodernhigh-,peed aviation.Thebookconsists oftwoparts,each formingaseparatevol ume. fhefirstofthemconcernsthefundamentalconceptsanddefinitionsofaero-dynamicsandthetheoryofflowoveranairfoilandawing,includinganun-steadyflow(Chapters1-9),whiletheseconddescribestheaerodynamicdesign ofcraftandtheirindividualparts(Chapters10-15).Thetwopartsaredesigned foruseinatwo-semestercourseofaerodynamics,althoughthefirstpart canbe usedindependentlybythoseinterestedinindividualproblemsoftheoretical aerodynamics. Asoundtheoreticalbackgroundisimportanttothestudyofanysubject becausecreativesolutionsofpracticalproblems,scientificresearch,anddis-coveriesareimpossiblewithoutit.Studentsshouldthereforedevotespecial attentiontothefirstfivechapters,whichdealwiththefundamentalconcepts anddefinitionsofaerodynamics,thekinematicsofafluid,thefundamentalsof fluiddynamics,thetheoryofshocks,andthemethodofcharacteristicsused widelyininvestigatingsuperO'onicflows.Chapters6and7,whichrelatetothe nowoverairfoils,arealsoimportanttoafundamentalunderstandingofthe subject.Thesechapterscontainafairlycompletediscussionofthegeneral theoryofnowofagasintwo-dimensionalspace(thetheoryoftwo-dimensional flow).TheinformationonthesupersonicsteadyflowoverawinginChapter8 relatesdirectly tothesematerials.The aerodynamic design ofmost modern craft isbasedonstudiesofsuchflow. OneofthemosttopicalareasofmOllernaerodynamicresearchisthestudy ofoptimalaerodynamicconfIgurationsofcraftandtheirseparate(isolated) parts(thefuselage,wing,empennage).Therefore,asmallsection(6.5\that defmesaflIlite-spanwing with the most advantageousplanformin anincompres-sibleflowhasbeenincludedhere.Thissectionpresentsimportantpractical andmethodologicalinformationontheconversionoftheaerodynamiccoeffic-ientsofawingfromoneaspectratiotoanother. Thestudyofnon-stationarygasflowsisaratherwelldevelopedfieldof moderntheoreticalandpracticalaerodynamics.Theresultsofthisstudyare widelyusedincalculatingtheeffectofaerodynamicforcesandmomentson 6Preface craftwhosemotionisgenerallycharacterizedbynon-uniformity,andthenou-stationary aerodynamic characteristicsthus calculated areusedin thedynamics ofcraftwhenstudyingtheirflightstability.Chapter9concernsthegeneral relationsoftheaerodynamiccoefficientsinunsteadyflow.Aerodynamicderiv-atives (stability derivatives) are analysed,as isthe concept ofdynamic stability. Unsteadyflowoverawingisalsoconsidered.Themostimportantsectionof thischapterisdevotedtonumericalmethodsofcalculatingthestabilityderiv-atives ofalifting surface ofarbitraryplanform,generally with acurved leading edge(i.e.withvariablesweepalongthespan).Bothexactandapproximate methodsofdeterminingthenon-stationaryaerodynamiccharacteristicsofa wingaregiven. Aspecialplaceinthebookisdevotedtothemostimportanttheoretical and applied problems ofhigh-speed aerodynamics,including thethermodynamic and kinetic parameters ofdissociating gases,the equations of motion and energy, andthetheoryofshocksanditsrelationtothephysicochemicalpropertiesof gasesatIitightemperatures.Considerableattentionisg i v l ~ ntoshockwaves (shocks),which areamanifestation ofthespecificpropertiesofsupersonicflows. The concept ofthe thickness of a shock is discussed, and thebook includes graphs ofthefunctionscharacterizingchangesintheparametersofagasasitpasses throughashock. Naturally,atElxtbookcannot reflecttheentirediversityofproblemsfacing thescienceofaerodynamics.Ihavetriedtoprovidethescientificinformation forspecialistsinthefieldofaeronauticandrocketenginering.Thisinform-ation,ifmasteredinitsentirety,shouldbesufficientforyoungspecialiststo copeindependently with other practical aerodynamicproblems that may appear. Among theseproblems, not reflectedin the book,are magnetogasdynamic invest-igations,theapplicationofthemethod ofcharacteristicstothree-dimensional gasflows,and experimental aerodynamics.Iwillbehappy if study ofthistext-bookleadsstudentstoamorecomprehensive,independentinvestigationot modernaerodynamics. Thebookistheresultofmyexperienceteachingcoursesinaerodynamics attheN.E.BaumanHigherEngineeringCollegeinl\[oscow,USSR.Intended forcollegeandjunior-collegestudents,it will alsobeausefulaidtospecialists inresearchinstitutions,designdepartments.andindustrialenterprises. AllphysicalquantitiesaregivenaccordingtotheInternationalSystemof Units(SI). In preparing thethird Russian edition ofthe book,which thepresent English editionhasbeentranslatedfrom,Itookaccountof readers'remarksandofthe valuable suggestions made by the reviewer,professor A.I\!.Mkhiteryan,to whom Iexpressmyprofoundgratitude. NikolaiF.Krasnov Preface Introduction Chapter1 BasicInformation fromAerodynamics Chapter2 KinematicsofaFluidr Contents 5 13 25 1.1.Forces Acting on a MovingBody25 SurfaceForce25 Property of Pressures in an Ideal Fluid26 InfluenceofViscosityontheFlow ofaFluid28 1.2.ResultantForceAction36 ComponentsofAerodynamicForces andMoments36 ConversionofAerodynamicForces and'\[omentsfromOneCoordinate SystemtoAnother40 1.3.DeterminationofAerodynamicFor-cesandMomentsAccordingtothe KnownDistributionofthePressure andShearStress.AerodynamicCoef-ficients41 Aerodynamic Forces and Moments and TheirCoefficients41 1.4.StaticEquilibriumandStaticStab-ility52 Concept ofEquilibrium and Stability52 StaticLongitudinalStability53 StaticLateralStability57 1.5.Features ofGasFlowatHighSpeeds58 CompressibilityofaGas58 HeatingofaGas59 StateofAiratHighTemperatures65 2.1.ApproachestotheKinematicInvest-igationofaFluid LagrangianApproach Eulerian Approach StreamlinesandPathlines '.2.AnalysisofFluidParticleMotion 2.3.Vortex-FreeMotionofaFluid 71 71 72 73 74 79 8Contents Chapter3 Fundamentals ofFluidDynamics Chapter4 ShockWaveTheory 2.4.ContinuityEquation80 GeneralFormoftheEquation80 CartesianCoordinateSystem81 CurvilinearCoordinateSystem82 ContinuityEquationofGasFlow along aCurvedSurface86 FlowRateEquation88 2.5.StreamFunction89 2.6.VortexLines90 2.7.VelocityCirculation91 Concept91 StokesTheorem92 Vortex-Induced94 2.8.ComplexPotential96 2.9.KindsofFluidFlows97 ParallelFlow98 Two-DimensionalPointSourceand Sink98 Three-DimensionalSourceandSink100 Doublet100 CirculationFlow(Vortex)103 3.1.EquationsofMotionofaViscous Fluid106 CartesianCoordinates106 VectorFormoftheEquationsof Motion113 Curvilinear Coordinates114 CylindricalCoordinates116 SphericalCoordinates118 EquationsofTwo-DimensionalFlow ofaGasNearaCurvedSurface120 3.2.EquationsofEnergyandDiffusion ofaGas121 DiffusionEquation121 EnergyEquation124 3.3.SystemofEquationsofGasDyna-mics.InitialandBoundaryCon-ditions129 3.4.Integrals of Motion for anIdeal Fluid134 3.5.AerodynamicSimilarity138 ConceptofSimilarity138 SimilarityCriteriaTakingAccount oftheViscosityandHeatCon-duction141 3.6.IsentropicGasFlows149 ConfigurationofGaslet149 FlowVelocity150 Pressure,Density,andTemperature152 FlowofaGasfromaReservoir154 4.1.PhysicalNatureof ShockWaveFor-mation159 4.2.GeneralEquationsforaShock162 ObliqueShock163 NormalShock168 4.3.ShockintheFlowofaGaswith Chapter5 Method of Characteristics Chapter6 Airfoiland Finite-SpanWing inanIncompressible Flow Chapter7 AnAirfoilina CompressibleFlow Contents9> ConstantSpecificHeats109 SystemofEquations169" FormulasforCalculatingtheParam-etersofaGasBehindaShock170 ObliqueShockAngle176 4.4.Hodograph179 4.5.ANormalShockintheFlowofa GaswithConstantSpecificHeats184 4.6.AShockatHypersonicVelocities andConstantSpecificHeatsofaGas186 4.7.A Shock in a Flow of a Gas with Vary-ing Specific Heats andwith Dissocia-tionandIonization188 4.8.RelaxationPhenomena193 Non-EquilibriumFlows193 EquilibriumProcesses195 RelaxationEffectsinShockWaves196 5.1.EquationsfortheVelocityPotential andStreamFunction2(10 5.2.TheCauchyProblem205 5.3.Characteristics209 Com pa tibili tyConui tions209 DeterminationofCharacteristics209 Orthogonali tyofCharacteristics213 TransformationoftheEquationsfor CharacteristicsinaHodograph214 EquationsforCharacteristicsina HodographforParticularCasesof GasFlow219 5.4.OutlineofSolutionofGas-Dynamic Problems Accordingtothe !\Iethod of Characteristics222 5.5.Application ofthe .'IIethodofCharac-teristicstotheSolutionoftheProb-lemon Shapingthe:'\JozzlesofSuper-sonicWindTunnels230 6.1.ThinAirfoilinallIncompressible Flow234 6.2.TransverseFlo\\'overaThinPlate240 6.3.ThinPlateatan,\ngleofAttack243 6.4.Finite-SpanWinginanIncompres-sibleFlow249 6.5.WingwithOptimalPlanform258 ConversionofCoefficientsclJandex,i fromOne\VingAspectRatioto Another258 7.1.SubsonicFlowoveraThin..\ irioil264 LinearizationoftheEquationfor theVelocityPotential264 RelationBetweentheParametersof aCompressibleandIncompres:-ible FluidFlowoveraThinAirfoil266 7.2.Khristianovich.'Ilethod269 Contentofthe!\Iethod269 10Contents Chapter8 AWingin aSupersonicFlow Conversion ofthePressureCoefficient foranIncompressibleFluidtothe NumberMoo> 0271 Conversion ofthe Pressure Coefficient fromM001>0toM002>0272 Determina tionoftheCri ticalNum-berM273 AerodynamicCoefficients274 7.3.FlowatSupercriticalVelocityover anAirfoil(Moo> M oo,cr)274 7.4.Supersonic Flow of aGaswithCon-stant Specific Heats over a ThinPlate278 7.5.Parameters of aSupersonic Flow over anAirfoilwithanArbitraryCon-figuration285 UseoftheMethodofCharacteristics285 HypersonicFlow overaThin Airfoil291 NearlyUniformFlowoveraThin Airfoil293 AerodynamicForcesandTheir Coefficients293 7.6.SideslippingWingAirfoil29g, DefinitionofaSideslippingWing299 AerodynamicCharacteristicsofa SideslippingWingAirfoil301 SuctionForce3(5 8.1.Linearized Theory of Supersonic Flow overaFinite-SpanWing308 Linearization oftheEquation forthe PotentialFunction308 BoundarvConditions310 ComponentsoftheTotalValuesof theVeloci tyPotentialsandAero-dynamicCoefficients313 FeaturesofSupersonicFlowover Wings315 8.2.:MethodofSources317 8.3.Wingwi thaSymmetricAirfoiland TriangularPlanform (a =0, cYa =0)321 Flow over aWingPanelwithaSub-sonicLeadingEdge321 TriangularWingSymmetricabout thex-AxiswithSubsonicLeading Edges326 Semi-InfiniteWing with aSupersonk Edge328 Triangular Wing Symmetric about the x-Axis with Supersonic Leading Edges330 8.4.FlowoveraTetragonalSymmetrie AirfoilWingwithSubsonicEdges ataZeroAngleof IAttack331 8.5.FlowoveraTetragonalSymmetrie Airfoil WingwithDifferent Kinds (SubsonicandSupersonic)343 Chapter9 Aerodynamic Characteristics of Craftin UnsteadyMotion Contents11 LeadingandEdgesareSub-sonic,TrailingEdgeisSupersonic343 LeadingEdgeisSubsonic,Middle andTrailingEdgesareSupersonic345 WingwithAllSupersonicEdges346 GeneralRelationforCalculating theDrag350 8.6.FieldofA pplica tionoftheSource 351 8.7.DoubletDistributionMethod353 8.8.FlowoveraTriangularWingwith SubsonicLeadingEdges355 8.9.FlolYoveraHexagonalWingwith SubsonicLeadingandSupersonic TrailingEdges366 8.10.FlowoveraHexagonalWingwith SupersonicLeadingandTrailing Edgl's372 !:I.l1.Dragof'>VingswithSubsonicLead-ingEdges381 8.12.AerodynamicCharacteristicsofa HectangularWing385 8.13.Heverse-Flow:\letllOd391 !1.1.GeneralBela tionsfortheAerodyna-micCoertkipnts395 9.2.AnalysisofStabilit\"Derivatives andAerod "namicCoe'flicients398 9.3.Conversion'ofStabilityDerivatives uponaChangeinthePositionof theForceHedllctionCentre404 9.4.ParticularCasesof!\Iotion406 LongitudinalandLateral:'Ilotions406 :\[ot ionoftheCentreof'\[assand HotationaboutIt407 Pi tching1\lotion408 9.5.1DynamicStabili ty410 Defini tion41U StabilityCharacteristics413 9.6.Basicl{elationsforUnsteadyFlowM6 AerorlynamicCoeflicient5'416 Cauch:v-LagrangeIntegral420 WaveEquation423 !l.7.BasicMethodsofSolvingNon-StationaryProblems425 !\lethoclofSources425 VortexTheory428 9.8.Numerical'\lethodofCalculatingthe StabilityDerivativesforaWingin anIncompressibleFlow431 VelocityFieldofanObliqueHorse-shoeVortex431 VortexModelofaWing436 Calcula tionofCirculaton"Flow439 AerodynamicCharacteristics446 DeformationofaWingSurface451 12Contents References SupplementaryReading NameIndex SubjectIndex InfluenceofCompressibility(the Number 211 00)on Non-Stationary Flow452 9.9.UnsteadySupersonicFlowovera Wing456 9.10.PropertiesofAerodynamicDeriv-atives478 9.11.A pproxima teMethodsforDetermin-ingtheNon-StationaryAerodynamic Characteristics488 HypothesesofHarmonicityandSta-tionari ty488 Tangent-Wedge:\Iethod489 493 494 495 497 Introduction AerodynamicsisacomplexwordoriginatingfromtheGreek wordsa'lP(air)and68vapw(power).Thisnamehasbeengivento asciencethat,beingapart ofmechanics-the scienceofthemotion ofbodiesingeneral-studies thelawsofmotionofairdependingon theactingforcesandontheirbasisestablishesspeciallawsofthe interactionbetweenairandasolidbodymoyinginit. ThepracticalproblemsconfrontiIlgmankindinconuectionwith flightsinheavier-than-air craftproYidedanimpetustothedeyelop-mentofaerodynamicsasascience.Theseproblems,vereassociated withthedeterminationoftheforcesandmoments(whatwecall theaerodynamicforcesandmoments)actingonmoyingbodies. Themaintaskininvestigatingtheactionofforceswascalculation ofthebuoyancy,orlift,force. Atthebeginningofitsdevelopment,aerodynamicsdealtwith theinvestigationofthemotionofairatquitelowspeedsbecanse aircraftatthattimehadalow flightspeed.It isquitenaturalthat aerodynamicswasfoundedtheoreticallyonhydrodynamics-the sciencedealingwiththemotionofadropping(inc am pressible) liquid.The cornerstonesof this science were laidinthe 18th century byL.Euler(1707-1783)andD.Bernoulli('1700-1782),membersof theRussianAcademvofSciences.Inhisscientifictreatise"The GeneralPrinciplesof l\IotionofFluids"(inRussian-1755),Euler forthefirsttimederivedthefundamentaldifferentialequationsof motionofideal(non-viscous)fluids.Thefundamentalequationof hydrodynamicsestablishingtherelationbetweenthepressureand speedinaflowofanincompressiblefhlidwasdiscoveredbyBer-noulli.Hepublishedthisequationin1738inhisworks"Fluid Mechanics"(inRussian). Atlowflightspeeds,theint1uenceonthenatureofmotionofair ofsuchitsimportantpropertyascompressibilityisnegligibly small.Buttherlevelopmentofartillery-rifle androcket-and high-speedaircraftmovedtotheforefrontthetaskofstudyingthelaws 14Introduction ofmotionofairoringeneralofagasathighspeeds.It wasfound thatiftheforcesactingonabodymovingatahighspeedarecal-culatedonthebasisofthelawsofmotionofairat lowspeeds,they maydiffergreatlyfromtheactualforces.Itbecamenecessaryto seektheexplanationofthisphenomenoninthenatureitselfof the motion ofair at high speeds.It consists inachange inits densi ty dependingonthepressure,whichmaybequite considerableat such speeds.It isexactlythischangethat underliesthepropertyofcom-pressibili tyofagas. Compressibilitycausesachangeintheinternalenergyofagas, whichmustbeconsideredwhencalculatingtheparametersdeter-miningthemotionofthefluid.Thechangeintheinternalenergy associatedwiththeparametersofstateandtheworkthatacom-pressedgascandouponexpansionisdeterminedbythefirstlawof thermodynamics.Hence,thermodynamicrelationswereusedinthe aerodynamicsofacompressiblegas. Aliquidandair(agas)differfromeachotherintheirphysical properties owing totheir molecular structure being different.Digres-singfromthesefeatures,wecantakeintoaccountonlythebasic differencebetweenaliquidandagasassociatedwiththedegreeof theircom pressibili ty.Accordingly,inaerohydromechanics,which dealswiththemotionofliquidsandgases,itiscustomarytouse thetermfluidtodesignatebothaliquidandagas,distinguishing betweenanincompressibleandacompressiblefluidwhennecessary. Aerohydromechanics treats laws ofmotion commontoboth liquids andgases,whichmadeitexpedientandpossibletocombine the studyingoftheselawswithintheboundsofasinglescienceofaero-dynamics(oraeromechanics).Inadditiontothe general laws charac-terizingthemotionoffluids,therearelawsobeyedonlybyagasor onlybyaliquid. Fluidmechanicsstudiesthemotionoffluidsatalowspeedat whichagasbehavespracticallylikeanincompressibleliquid. Intheseconditions,theenthalpyofagasislargeincomparison withitskineticenergy,andonedoesnothavetotakeaccountof thechangeintheenthalpywithachangeinthespeedoftheflow, i.e.withachangeinthekineticenergyofthefluid.Thisiswhy thereisnoneedtousethermodynamicconceptsandrelationsin low-speedaerodynamics(hydrodynamics).Themechanicsofagas differsfromthatofaliquidwhenthegashasahighspeed.Atsuch speeds,agasflowingoveracraft experiences not only achange inits density,butalsoanincreaseinitstemperaturethatmayresultin variousphysicochemicaltransformationsinit.Asubstantialpart of thekineticenergyassociatedwiththespeedofaflightisconverted intoheatandchemicalenergy. Allthesefeaturesofmotionofagasresultedintheappearance ofhigh-speedaerodynamicsorgasdynamics-aspecialbranchof Introduction aerodynamicsstudyingthelawsofmotionofair(agas)athigh subsonicandsupersonicspeeds,andalsothelawsofinteraction betweenagasandabody travelling init at suchspeeds. One of the foundersof gas dynamics was academician S.Chaplygin (1869-1942),whoin1902publishedanoutstandingscientificwork "OnGasJets"(inRussian).Equationsarederivedinthisworkthat formthetheoreticalfoundationofmoderngas dynamicsandentered theworld'sscienceunderthenameoftheChaplyginequations. Thedevelopmentoftheoreticalaerodynamicswasattendedby thecreationofexperimentalaerodynamicsdevotedtotheexperi-mentalinvestigationoftheinteractionbetweenabodyandagas flowpastit withtheaidofvarioustechnicalmeansS11Chasawind tunnelthatimitatetheflowofaircraft. Undertheguidanceofprofessor~ .Zhukovsky(1847-1921),the firstaerodynamiclaboratories inRllssiawereerected(atthe Moscow StateUniversity,theMoscowHigherTechnicalCollege,andat Kuchino,nearMosco",).In1918,theCentralAerohydrodynamic Institute(TsAGI)wasorganizedby Zhukovsky'sinitiative withthe direct aidof V.Lenin.Atpresent it is oneofthe major ~ w o r l dcentres fothescienceofaerodynamicsbearingthenameof~ .Zhukovsky. Thedevelopmentofaviation,artillery,androcketry,andthe maturingofthetheoreticalfundamentalsofaerodynamicschanged the nature ofaerodynamic installations, fromthe fjrst,comparatively smallandlow-speedwindtunnels uptothegiant high-speedtunnels of TsAGI(1940)and modern hypersonic installations, and also special facilitiesinwhichasupersonicflowofaheatedgasisartificially created (what we call high-temperature tunnels, shock tunnels, plasma installations,etc.). Thenatureoftheinteractionbetweenagasandabodymoving initmayvary.Atlowspeeds,theinteractionismainlyofaforce nature.'Vithagrowthinthe speed,the forceinteractionisattended by heatingofthesurfaceowingtoheattransferfromthegastothe body:thisgivesrisetothermalinteraction.Atveryhighspeeds, aerodynamicheatingissogreatthatitmayleadtofailureofthe materialofacraft wallbecauseofitsfusionorsublimationand.as aresult,totheentrainmentofthedestroyedmaterial(ablation) andtoachangeinthenatureofheatingofthewall.Aerodynamic heatingmayalsocausechemicalinteractionbetweenasolidwall andthegasflowingoverit,asaresultofwhichthesameeffectof ablationappears.Highflightspeedsmayalsocanseablationasa resultofmechanical interactionbetweenthegasandamovingbody consistinginerosionofthematerialofawallanddamagetoits structure. Theinvestigationofallkindsofinteractionbetweenagasand acraftallowsonetoperformaerodynamiccalculationsassociated withtheevaluationofthequantitativecriteriaofthisinteraction. 16Introduction namely,with determination ofthe aerodynamic forcesand moments, heattransfer,andablation.Asposedatpresent,thisproblemcon-sistsnotonlyindeterminingtheoverallaerodynamicquantities {thetotallift forceordrag,thetotal heat fluxfromthe gastoasur-face,etc.),but alsoin evaluating the distribution ofthe aerodynamic properties-dynamicandthermal-overasurfaceofanaircraft movingthroughagas(thepressureandshearingstressoffriction, localheatfluxes,localablation). Thesolutionofsuchaproblemrequiresadeeperinvestigationof the flowofagas than is neededto determine the overall aerodynamic action.Itconsistsindeterminingthepropertiesofthegascharac-terizingitsflowateachpointofthespaceitoccupiesandateach instant. Themodernmethodsofstudyingtheflowofagasarebasedon anumberofprinciplesandhypothesesestablishedinaerodynamics. Oneofthemisthecontinuum hypothesis-theassumptionofthe continuityofagasflowaccordingtowhichwemaydisregardthe intermoleculardistancesandmolecularmovementsandconsider thecontinuouschangesofthebasicpropertiesofagasinspaceand intime.Thishypothesisfollowsfromtheconditionconsistingin that the freepath of molecules andthe amplitude oftheir vibrational motionaresufficientlysmallincomparisonwiththelineardimen-sions characterizing flowaroundabody,forexamplethewingspan andthediameterorlengthofthefuselage(orbody). Theintroducedcontinuumhypothesisshouldnotcontradictthe conceptofthecompressibilityofagas,althoughthelattershould seem tobe incompressible in the absence of intermolecular distances. Therealityofacompressiblecontinuumfollowsfromthecircum-stancethattheexistenceofintermoleculardistancesmaybedis-regarded in many investigations, but at the same time one may assume thepossibilityoftheconcentration(density)varyingasaresultof achangeinthemagnitudeofthesedistances. Inaerodynamicinvestigations,theinteractionbetweenagas andabodymovinginitisbasedontheprincipleofinvertedflow accordingtowhichasystemconsistingofagas(air)atrestanda movingbodyisreplacedwithasystemconsistingofamovinggas andabodyat rest.Whenone systemis replacedwiththe other,the conditionmustbesatisfiedthatthefree-streamspeedofthegas relativetothebodyat restequalsthespeedofthisbodyinthegas atrest.Theprincipleofinvertedmotionfollowsfromthegeneral principleofrelativityofclassicalmechanicsaccordingtowhich forcesdonotdependonwhichoftwointeracting bodies(inour case the gas or craft) isat rest and which is performing uniform rectilinear motion. Thesystemofdifferentialequationsunderlyingthesolutionof problemsofflowoverobjectsiscustomarilytreatedseparatelyin Introduction17 modernaerodynamicsfortwobasickindsofmotion:free(in viscid) flowandflowinathin layerofthegasadjacenttoawallorbound-ary-in the boundary layer, where motion is considered with account takenofviscosity.This divisionofaflowisbasedonthehypothesis ofthe absenceofthe reVf'rseinfluenceofthe boundarylayer onthe freeflow.Accordingtothishypothesis,theparametersofinviscid flow,i.e.ontheoutersurfaceoftheboundary layer,are the sameas onawallintheabsenceofthislayer. Thelindingoftheaerodynamicparametersofcraftinunsteady motioncharacterizedbyachangeinthekinematicparameterswith timeisllsuallyaveryintricatetask.Simplifiedwaysofsolvingthis problemareusedforpracticalpurposes.Suchsimplificationispos-sible when the change occurs sufficiently slowly.This is characteristic ofmany craft.Whendeterminingtheiraerodynamiccharacteristics, wecanproceedfromthehypothesisofsteadinessinaccordance withwhichthesecharacteristicsinunsteadymotionaroassumedto be the same as illsteady motion,and are determined by the kinematic parametersofthismotionatagiveninstant. \Yhenperformingaerodynamicexperimentsandcalcul ations, accountmustbetakenofvariOllScircumstancesassociatedwiththe physicalsimilitudeoftheflowphenomenabeing studied.Aerodyna-miccalculationsoffull-scalecraft(rockets,airplanes)arebasedon preliminary widespreadinvestigations(theoreticalandexperimental) ofnowovermodels.TheconditionsthatmustbeobserYedinsuch investigations onmodelsarefoulldinthet h e o r ~ 'ofdynamicsimili-tude,andtypicalandconvenientparameters determiningthebasic conditionsoftheprocessesbeingstudiedareestablished.Theyare calleddimensionlessnumbersorsimilaritycritNia.Themodern problemsofsimilarityandalsothetheor;ofdimellsionswidely usedinaerodynamics are set out inafundamentalworkofacademi-cianL.Sedov titled "SimilarityandDimensionalMethodsinMech-anics"[1 J. Aerodynamics,figurativelyspeaking,isamulti branchscience.In accordance with the needs ofthe rapidly developing aviation, rocket, and cosmic engineering,more or less clearly expressed basic scientific trendshavetakenshapeinaerodynamics.Theyareassociatedwith theaerodynamicinvestigationsofcraftasawholeandtheirindivi-dualstructuralelements,andalsoofthemostcharacteristickinds ofgasflowsandprocessesattending the flowoverabody.It isquite naturalthatanyclassificationofaerodynamicsisconditionalto acertainextentbecauseallthesetrendsorpartofthemareinter-related.Nevertheless,sucha"branch"specializationoftheaero-dynamicscienceisofapracticalinterest. Thetwomainpathsalongwhichmodernaerodynamicsisdeve-lopingcanbedetermined.Thefirstofthemiswhatiscalledforce aerodynamicsoccupiedinsolvingproblemsconnectedwiththe 2-01715 18Introduction forceactionofafluid,i.e.in findingthe distributionofthepressure andshearingstressoverthesurfaceofacraft,andalsowiththe distributionoftheresultantaerodynamicforcesandmoments.The dataobtainedareusedforstrengthanalysisofacraftasawhole andofindividualelementsthereof,andalsofordeterminingits flightcharacteristics.Thesecondpathincludesproblemsofaero-thermodynamicsandaerodynamicheating-asciencecombining aerodynamics,thermodynamics,andheattransferandstudying flowoverbodiesinconnectionwiththermalinteraction.Asaresult oftheseinvestigations,wefindtheheatfluxesfromagastoawall anddetermineitstemperature.Thesedataareneededinanalysing thestrengthanddesigningthecoolingofcraft.Atthesametime, thetakingintoaccountofthechangesinthepropertiesofagas flowingoverabodyundertheint1uenceofhightemperaturesallows ustodeterminemorepreciselythequantitativecriteriaofforce interactionofboththeexternalflowandoftheboundarylayer. Alltheseproblemsareofaparamountimportanceforveryhigh airspeedsatwhichthethermalprocessesareveryintensive.Even greatercomplicationsareintroducedintothe solutionofsnchprob-lems,however,becauseitisassociatedwiththeneedtotakeinto considerationthechemicalprocessesoccurringinthegas,andalso the influence of chemical interaction between the gas and the material ofthewall. If wehaveinviewtherangeofairspeedsfromlowsubsonicto very high supersonic ones,then,asalready indicated, we can separate the followingbasicregionsinthe scienceofinvestigating flow:aero-dynamicsofanincompressiblefluid,orfluidmechanics(theMach numberoftheflowisM=0),andhigh-speedaerodynamics.The latter,inturn,isdividedinto subsonic(M 1)andhypersonic(M ~1)aerodynamics.Itmust benotedthateachofthesebranchesstudiesflowprocessesthatare characterizedbycertainspecificfeaturesofflowswiththeindicated Machnumbers.Thisiswhytheinvestigationofsuchflowscanbe basedonadifferentmathematicalfoundation. Wehavealreadyindicatedthataerodynamicinvestigationsare based onadivision ofthe flownear bodies into two kinds: free(exter-nal)inviscidflowandtheboundarylayer.Anindependentsection ofaerodynamicsisdevotedtoeachofthem. Aerodynamicsofanidealfluidstudies afreeflowandinvestigates thedistributionoftheparametersininviscidflowoverabodythat are treated as parameters on the boundary layer edge and, consequent-ly,aretheboundary conditions forsolving thedifferentialequations ofthislayer.Theinviscidparametersincludethepressure.If we knowitsdistribution,wecanfindtherelevantresultantforcesand moments.AerodynamicsofanidealfluidisbasedonEuler's fundamentalequations. Introduction19 Aerodynamicsofaboundarylayerisoneofthebroadestandmost developedsectionsofthescienceofafluidinmotion.Itstudies viscousgasflowinaboundary layer.Thesolutionoftheproblemof flowinaboundary layer makesit possibletofindthedistributionof the shearing stresses and,consequently,ofthe resnltant aerodynamic forcesandmomentscausedbyfriction.I talsomakesi Lpossibleto calculatethetransferofheatfromthegasflowingoverabodytoa boundary.Theconclusionsoftheboundarylayertheorycanalso DeusedforcorrectingthesoIl! tiononinviscidflow,partie ularl yfor fllldingthe correctionto the pressnre distribution due to the influence oftheboundarylayer. Themoderntheoryofthe boundary layerisbasedonfundamental investigations of A.~ a v i e r ,G.Stokes,O.Reynolds,L.Prandtl,and T.vonKarman.Asubstantialcontributiontothedevelopmentof theboundary layertheorywasmadebytheSoviet scientistsA.Do-roc!nitsyn,L.Loitsyansky,A.Melnikov,N.Kochin,G.Petrov, V.Strnminsky,andothers.Theycreatedaharmoniollstheoryof thehoundarylayerinacompressiblegas,workedontmethodsof calClllatingtheflowofaviscousfluidovervariousbodies(two- and three-d imensional),investigatedproblemsofthetransitionofa laminarboundarylayerintoaturbulentone,andstudiedthecom-plicatedproblemsofturbulentmotion. Inaerodynamicinvestigationsinvolvinglowairspeeds,the thermalprocessesintheboundarylayerdonothavetobetaken intoaccOlmtbecallseoftheirlowintensity.\Vhenhighspeedsare involved,however,accountmnstbetakenofheattransferandof theinfluenceofthehighboundarylayertemperaturesonfriction. Itisquitenatllralthatabundantattentionisbeinggiventothe solutionofsuchproblems,especiallyrecently.IntheSovietUnion, professors L.Kalikhman, I.KibeI, V.Iyevlev and others are develop-ingthegas-dynamic theory of heat transfer, stndying the viscous flow oyervariousbodies at high temperatures oftheboundary layer.Simi-larproblemsarealsobeingsoiYedbyanumberofforeignscien-tists. Athypersonicflowspeeds,theproblemsofaerodynamicheating arenottheonlyones.Thationizationoccursat sllchspeedsbecause ofthehightemperaturesandthegasbeginstocondnctelectricity causesnewproblemsassociatedwithcontroloftheplasmaflowwith theaidofamagneticfield.Whendescribingtheprocessesofinter-actionofamovingbodywithplasma,therelenHltaerodynamic calculations must take into account electromagneticforces in addition togas-dynamicones.Theseproblemsarestudiedinmagnetogas-dynamics . Themotionoffluidsinaccordancewiththecont in llUlllhypothe-sisf'etoutaboveisconsideredinaspecialbranch ofaerodYllamics-continuumaerodynamics.Manytheoreticalproblemsofthisbranch 2* 20Introduction ofaerodynamicsaretreatedinafundamentalworkofL.Sedov: "Continuum Mechanics"(inRussian-a textbook for universities)[21. It mustbenotedthatthecontinuumhypothesisholdsonly forcon-ditionsofflightatlowaltitudes,i.e.insufficientlydenselayersof theatmospherewherethemeanfreepathoftheairmoleculesis small. At high altitudes in conditions of a greatly rarefied atmosphere, the freepathofmoleculesbecomesquite signifIcant,andtheaircan nolongerbeconsideredasacontinuum.Thisiswhytheconclusions ofcontinuumaerodynamicsarenotvalidforsuchconditions. Theinteractionofararefiedgaswithabodymovinginitis studiedinaspecialbranchofaerodynamics-aerodynamicsof rarefiedgases.Therapiddevelopmentofthisscienceduringrecent yearsisduetotheprogressinspaceexplorationwiththeaidof artificialsatellitesoftheEarthandrocket-propelledvehicles,as well as in varioustypes of rocket systems(ballistic,intercontinental, globalmissiles,etc.)performingflightsneartheearthatyeryhigh altitudes. Theconditionsofflowoyercraftand,consequently,theiraero-dynamiccharacteristicsvarydependingonhowtheparametersof thegaschangeatfixedpointsonasurface.Abroadclassofflow problemsofapractical canbesolved,asalready noted, insteady-stateaerodynamics,presnmingthattheparameters are independent ofthe time at thesepoints.Whenstudying flight stabil-ity,however,accountmustbetakenoftheunsteadynatureofflow duetothenon-uniformairspeed,andofvibrations or rotationofthe craft, because in these conditions the flowover abody is characterized byalocalchangeinitsparameterswithtime.Theinvestigationof thiskindofflowrelatestounsteadyaerodynamics. \Vehaveconsideredaclassificationofmodernaerodynamicsby thekindsofgasflows.I tisobviousthat wi thintheconfmesofeach ofthesebranchesofaerodynamics,flowisstudiedasappliedto variousconfigurationsofcraftortheirparts.Inadditiontosuch aclassifIcation,ofinterestarethebranchesofmodernaerodynamics for which the confIguration ofacraft or its individual elements isthe determiningfactor. Asregardsitsaerodynamicscheme,amodernaircraftinthe generalizedformisacombinationofahull(fuselage),wings,atail unit(empennage),elevators,andrudders.Whenperformingaero-dynamiccalculationsofsuchcombinations,onemusttakeinto accounttheeffectsofaerodynamicintE'rference-theaerodynamic interactionbetweenalltheseelementsofanaircraft.Accordingly, inparticular,theoverallaerodynamiccharacteristicssuchasthe lift force,drag,ormomentmustbeevaluatedasthesumofsimilar characteristicsoftheisolatedhull,wings,tailunit,elevators,and rudderswithcorrectionsmadeforthisinteraction. Hence,this schemeofaerodynamiccalculations presumes a knowl-Introduction21 edgeoftheaerodynamiccharacteristicsoftheseparateconstituent partsofanaircraft. Aerodynamiccalculationsoftheliftingplanesofwingsisthe subjectofaspecialbranchoftheaerodynamicscience-wingaero-dynamics.The outstanding Russian scientists and mechanics N.Zhu-kovsky(J oukowski)andS.Cha pI yginarebyrightconsideredtobe thefoundersoftheaerodynamictheoryofawing. Thebeginningofthe20thcenturywasnotedbytheremarkable discoverybyZhukovskyofthenatureofthelift forceofawing;he derivedaformulaforcalculatingthisforcethatbearshisname. Hisworkontheboundvorticesthatareahydrodynamicmodelof awing was faraheadofhis time.The series of wing profiles(Zhukov-skywingprofiles)hedevelopedwerewidelyusedindesigningair-planes. Academician S.Chaplyginistheauthor ofmanyprominent works onwingaerodynamics.In1910inhiswork"OnthePressureofa ParallelFlowonObstacles"(inRussian),helaidthefoundationsof the theory of an infinite-span wing.In 1922, he published the scientific work"TheTheoryofaMonoplane'Wing"(inRussian)thatsetsout thetheoryofanumberofwingprofiles(Chaplyginwingprofiles) and also develops the theory of stability of a monoplane wing.Chaply-ginisthefounderofthetheoryofafinite-spanwing. The fundamental ideas of Zhukovsky and Chaplygin were developed in the works of Soviet scientists specializing in aerodynamics.Associ-ate member of the USSR Academy of Sciences V.Golubev(1884-1954) investigatedtheflowpastshort-spanwingsandvariouskindsof high-liftdevices.Importantresultsinthepotentialwingtheory 'wereobtainedbyacademicianM.Keldysh(1911-1978),andalsoby academiciansM.LavrentyevandL.Sedov.AcademicianA.Dorod-nitsynsummarizedthetheoryofthelifting(loaded)lineforaside-slipping wing. Considerableachievementsinthetheoryofsubsonicgasflows belongtoM.KeldyshandF.Frankl,whostrictlyformulatedthe problemofacompressibleflowpastawingandgeneralizedthe Kutta-Zhukovskytheoremforthiscase. AcademicianS.Khristianovichinhiswork"TheFlowofaGas PastaBodyatHighSubsonicSpeeds"(inRussian)[3]developed anoriginalandveryeffectivemethodfortakingintoaccountthe influenceofcompressibilityontheflowoverairfoilsofanarbitrary configuration. TheforeignscientistsL.Prandtl(Germany)andH.Glauert (Great Britain) studied the problem of the influence of compressibility onflowpastwings.Theycreatedanapproximatetheoryofathin winginasubsonicflowatasmallangleofattack.Theresultsthey obtainedcanbeconsideredasparticularcsesofthegeneraltheory offlowdevelopedbyKhristianovich. 22Introduction Agreatcontributiontotheaerodynamicsofawingwasmadeby academicianA.Nekrasov(1883-1954),whodevelopedaharmonious theoryofaliftingplaneinanunsteadyflow.KeldyshandLav-rentyevsolvedtheimportantproblemontheflowoveravibrating airfoilbygeneralizingChaplygin'smethodforawingwithvarying circulation.AcademicianSedovestablishedgeneralformulasfor unsteadyaerodynamicforcesandmomentsactingonanarbitrarily movingwing. ProfessorsF.Frankl,E.Krasilshchikova,andS.Falkovichdevel-opedthetheoryofsteadyandunsteadysupersonicflowoverthin wingsofvariousconfigurations. Importantresultsinstudyingunsteadyaerodynamicsofawing wereobtainedbyprofessorS.Belotserkovsky,whowidelyused numericalmethodsandcomputers. Theresultsofaerodynamicinvestigationsofwingscanbeapplied tothe calculation ofthe aerodynamic characteristics ofthe tailunit, andalsoofelevatorsandruddersshapedlikeawing.Thespecific featuresofflowoverseparatekindsofaerodynamicelevatorsand ruddersandthepresenceofotherkindsofcontrolsresultedinthe appearanceofaspecialbranchofmodernaerodynamics-theaero-dynamicsofcontrols. Modernrocket-typecraftoftenhavetheconfigurationofbodies ofrevolutionorareclosetothem.Combinedrocketsystemsofthe type"hull-wing-tailunit"haveahull(bodyofrevolution)asthe maincomponentoftheaerodynamicsystem.Thisexplainswhythe aerodynamics ofhulls(bodiesofrevolution),whichhasbecomeone oftheimportantbranchesoftoday'saerodynamicscience,hasseen intensivedevelopmentinreeentyears. A major contribution to the development of aerodynamics ofbodies ofrevolutionwasmadebyprofessorsF.FranklandE.Karpovich, whopublishedaninterestingscientificwork"TheGasDynamics ofSlenderBodies"(inRussian). TheSovietscientists1.KibeIandF.Frankl,whospecialized inaerodynamics,developedthemethodofcharacteristicsthatmade itpossibletoperformeffectivecalculationsofaxisymmetricsuper-sonicflowpast pointedbodies ofrevolution ofanarbitrary thickness. Agroupofscientific workers oftheInstitute of Mathematics ofthe USSRAcademyofSciences(K.Babenko,G.Voskresensky.and others)developedamethodforthenumericalcalculationofthree-dimensionalsupersonicflowoverslenderbodiesinthegeneralcase whenchemicalreactionsintheflowaretakenintoaccount.The importantproblemonthesupersonieflo'."overaslenderconewas solvedbytheforeignspecialistsinaerodynamicsG.Taylor(Great Britain)andZ.Copal(USA). The intensive developmentofmodernmathematicsandcomputers andtheimprovementonthisbasisofthemethodsofaerodynamic Introduction23 investigationsleadtogreaterandgreatersuccessinsolvingmany complicatedproblemsofaerodynamicsincludingthedetermination oftheoverallaerodynamiccharacteristicsofacraft.Amongthem aretheaerodynamicderivativesatsubsonicspeeds,thefindingof whichawork of S.Belotserkovsky andB.Skripach [4]is devotedto. Inaddition,approximatemethodscameintouseforappraisingthe effectofaerodynamicinterferenceandcalculatingtherelevant correctionstoaerodvnamiccharacteristicswhenthelatterwere obtained in the formdfan additive sum ofthe relevant characteristics oftheindividual(isolated)elementsofacraft.Thesolutionofsuch problemsisthesubjectofaspecialbranchoftheaerodynamic science-interferenceaerodynamics. Atlowsupersonicspeeds,aerodynamicheatingiscomparatively smallandcannotleadtodestructionofacraftmember.Themain problemsolvedinthegivencaseisassociatedwiththe choiceofthe coolingformaintainingtherequiredboundarytemperature.More involvedproblemsappearforveryhighairspeedswhenamoving bodyhasatremendousstoreofkineticenergy.Forexample,ifa crafthasanorbitalorescapespeed,itissufficienttoconvertonly 25-300of thisenergy into heat for the entire material ofa structural membertoevaporate.Themainproblemthatappears,particularly, inorganizingthesafere-entryofacraftintothedenselayersofthe atmosphereconsistsindissipatingthisenergysothataminimum partofitKillbeabsorbedintheformofheatbythebody.It was foundthatblunt-nosedbodieshavesnchaproperty.Thisisexactly whatresultedinthedevelopmentofaerodynamicstudiesofsuch bodies. Animportantcontributiontoinvestigatingtheproblemsof aerodynamicsofblunt-nosedbodieswasmadebySovietscientists-academiciansA.Dorodnitsyn,G.Cherny,O.Belotserkovsky,and others.SimilarinvestigationswereperformedbyMLighthill (GreatBritain),P.Garabedian(USA),andotherforeignscientists. Billutingofthefrontsurfacemustbeconsideredinacertain sen8easawayofthermalprotectionofacraft.Theblnntednose experiencesthemostintensivethermalaction,thereforeitrequires thermalprotectiontoevenagreaterextentthantheperipheralpart ofthecraft.Themosteffectiveprotectionisassociatedwiththeuse ofyariouscoatingswhosematerialattherelevanttemperaturesis gradllallydestroyedandablated.Hereaconsiderablepartofthe energy 8uppliedby the heatedairto the craft isabsorbed.The devel-opmentofthetheoryandpracticalmethodsofcalculatingablation relates to amodern branch of the aerodynamic science-aerodynamics ofablatingsurfaces.. Abroadrangeofaerodynamicproblemsisassociatedwiththe determinationoftheinteractionofafluidwithacrafthayingan arbitrarypresetshapeillthegeneralcase.Theshapesofcraftsur-------------- ---, 24Introduction facescanalsobechosen forspecialpurposes ensuringadefiniteaero-dynamic effect.The shapeofblunt bodiesensuresaminimumtrans-ferofheattotheentirebody.Consequently,ablunt surfacecanbe consideredoptimal fromthe viewpointofheat transfer.In designing craft,theproblemappearsofchoosingashapewiththeminimum forceaction.Oneoftheseproblemsisassociated,particularly,with determination of the shape ofacraft headensuring the smallest drag atagivenairspeed.Problemsofthiskindaretreatedinabranchof aerodynamicscalledaerodynamicsofoptimalshapes. BasicInformation fromAerodynamics 1.1.ForcesActing ona MovingBody, SurfaceForce I Letusconsidertheforcesexertedbyagaseousviscouscontinuum onamovingbody.Thisactionconsistsintheuniformdistribution overthebody's surfaceoftheforcesPn producedbythenormaland the forcesP.,.producedbytheshear stresses(Fig.1.1.1).The surface' elementasbeingconsideredisacteduponbyaresultantforce calledasurfaceone.ThisforcePisdeterminedaccordingtothe' ruleofadditionoftwovectors:Pn andP.,..TheforcePn inaddition tothe forceproducedbythepressure,whichdoesnotdependonthe' viscosity,includesacomponentduetofriction(Maxwell'shypo-thesis). Inanidealfluidinwhichviscosityisassumedtobeabsent,the' action of a force on an area consists only in that ofthe forcesproduced bythenormalstress(pressure),Thisisobvious,becanseif theforce deviated froma normal to the area, its projection onto this area would appear,i.e.ashear stress wouldexist.The latter,however,isabsent inanidealfluid. Inaccordancewiththeprincipleofinverted' flow.theeffectof theforceswillbethesameifweconsiderabodvatrestandauni-formflowoverit havingavelocityatinfinityequaltothespeedof thebodybeforeinversion.Weshallcallthisvelocitythevelocity at infinity or the free-streamvelocity(the velocity ofthe undisturbed flow)andshalldesignateitby-V 0("incontrasttoV(thevelocity ofthebodyrelativetotheundisturbedflow),i.e.1 V1 =1 V 00I. Afreestreamischaracterizedbytheundisturbedparameters-thepressurep 00,densityp 00,andtemperatureToodifferingfrom theircounterpartsp,p,andToftheflowdisturbedbythebody (Fig.1.1. 2).The physical properties of a gas ( air) are also characterized bythefollowingkineticparameters:thedynamicviscosityItand thecoeffIcientofheatconductivity'A(theundisturbedparameters, are~ l00and'A00,respectively),as\',ellasbythermodynamic para-'26Pt.I.Theory.AerodynamicsofanAirfoilandaWing Fig.1.1.1 Forcesactingonasurfaceele-mentofamovingbody :Fig.1.1.2 Designa tionofparametersof ,disturbedand undisturbedflows ------------------------------------------~ - - - - - - - - - - - - - - - - - -meters:thespecificheatsat constantpressurecp(c p "")andconstant volumee"(el,"")andtheirratio(theadiabaticexponent)k=cplco (k""=cp""Ie,.(0). PropertyofPressures inanIdealFluid Todeterminethepropertyofpressuresinanidealfluid,letus takeanelementaryparticleofthefluidhavingthe shapeofatetra-hedron1110111111121113withedgedimensionsofL1.:r,L1y,andL1z (Fig.'1. '1.3)andcompileequationsofmotionfortheparticleby equatingtheproductofthemassofthiselementanditsacceleration tothesumoftheforcesactingonit.Weshall writetheseequations inprojectionsontothecoordinateaxes.'Veshalllimitourselvesto theequationsofmotionofthetetrahedronintheprojectionontothe x-axis,takingintoaccountthattheothertwohaveasimilarform. Theproductofthemassofanelementanditsaccelerationinthe directionofthex-axisisPaYL1W dV)dt,wherePavistheaverage densityofthefluidcontainedintheelementaryvolumeL1W,and dVxldtistheprojectionoftheparticle'saccelerationontothe x-axis. Theforcesactingonourparticlearedeterminedasfollows.Aswe havealreadyestablished,theseforcesincludewhatwecalledthe surfaceforce.Hereit isdeterminedbytheactionofthepressureon thefacesofourparticle,anditsprojectionontothex-axisis /\. PxL1Sx - PnL1Sn cos(n,x). Ch.1.BasicInformationfromAerodynamics27 fig.1.1.3 Normalstresses acting onaface ofanelementaryfluidparticle havingtheshape ofatetrahed-ron z Another forceacting on theisolated fluid volumeisthemass (body) forceproportionaltothemassoftheparticleinthisvolume.Mass forcesincludegravitationalones,andinparticulartheforceof gravity.Anotherexampleoftheseforcesisthemassforceofan electromagneticorigin,knownasaponderolllotiveforce,that appearsinagasifitisanelectricconductor(ionized)andisinan electromagneticfield.Hereweshallnotconsiderthemotionofa gasundertheactionofSHC hforces(seeaspecialcOllrseinmagneto-gasdynamics). Inthecasebeingconsidered,weshallwritetheprojectionofthe massforceontothex-axisintheformofdenotingbyX theprojectionofthemassforcerelatedtoaunitofmass.\Vithac-counttakenoftheseval liesfortheprojectionsofthesurfaceand massforces,weobtainanequationofmotion AWdFx_XAW'ASS(/'-PayUdt - (la"u,- Pxux- p"11 COSn,x) whereand aretheareasoffaces;110.1[2;1[3andJJ1J1[2 J13, ,'-respectively,cos(n,x),is the cosineoftheanglebetweenanormaln tofaceJ1f1Jf2Jl:J andthex-axis,andPx'andP1Iarethepressures actingonfaces;1roJ12JI3 allfi respectively. DividingtheequationobtaillerlhyandhaYingillviewthat cos(,;::r).letuspassovertothelimit \yiLltand tendingtozero.Consequently.thetermscon tainingW;willalsotelldtozerobecauseL1Wisasmallquantityofthethird order,whiletiS xisasmallqllantityofthesecondorderincom-parisonwiththe linear dimensions ofthe surface dement.Asaresult, wehavep"- p"=0,and,therefore,p" Pll' \Vhenconsideringtheequations ofmotioninprojectionsontothe y- anrlz-axes.findthatPy=p"anripz=fi,, Since0111'sllrface elemen twi t htIll'II ormal11isorientedarbi trarily, wecanarriveatthefollowingcondllsionfromtheresultsobtained. Thepressureatanypointofaflowofanidealt1uidisidenticalon 28Pt.I.Theory.AerodynamicsofanAirfoilandaWing allsurfaceelementspassingthroughthispoint,i.e.itdoesnot dependontheorientationoftheseelements.Consequently,the pressurecanbetreatedasascalarquantitydependingonlyonthe coordinatesofapointandthetime. InfluenceofViscosity ontheFlowof aFluid Laminar and TurbulentFlow.Twomodesofflowarecharacteristic ofaviscousfluid.Thefirstofthemislaminarflowdistinguished bytheorderlyarrangementoftheindividualfilamentsthat donotmixwithoneanother.Momentum,heat,andmatterare transferredinalaminarflowattheexpenseofmolecularprocesses offriction,heatconduction,anddiffusion.Suchaflowusually appearsandremainsstableatmoderatespeedsofafluid. If ingivenconditionsofflowoverasurfacethespeedoftheflow exceedsacertainlimiting(critical)valueofit,alaminarflowstops beingstableandtransformsintoanewkindofmotioncharacterized bylateralmixingofthefluidand,asaresult,bythevanishingof theordered,laminarflow.Suchaflowiscalledturbulent.Inatur-bulen tflow,themixingofmacroscopicparticleshavingvelocity componentsperpendiculartothedirectionoflongitudinalmotion isimposedonthemolecularchaoticmotioncharacteristicofa laminarflow.Thisisthebasicdistinctionofaturbulentflowfrom alaminarone.Anotherdistinctionisthatifalaminarflowmaybe eithersteadyorunsteady,aturbulentflowinitsessencehasan unsteadynaturewhenthevelocityandotherparametersatagiven poin tdependonthetime.Accordingtothenotionsofthekinetic theory of gases,random(disordered,chaotic)motionis characteristic oftheparticlesofafluid,asofmolecules. Whenstudyingaturbulentflow,itisconvenienttodealnot withtheinstantaneous(actual)velocity,but withits average(mean statistical)valueduringacertaintimeintervalt2For example,the componentoftheaveragevelocityalongthex-axisis11 x= t, =[1/(t2- t1)J~Vxdt,whereV xisthecomponentoftheactual t, velocity at thegivenpoint that isafunctionofthetime t.Thecom-ponentsVyand-v"alongthey- andz-axesareexpressedsimilarly. Using the concept ofthe average velocity, we can represent the actual velocity as the sum V x=V x+ V ~in whichV ~is a variable addition-alcomponentknownasthefluctuationvelocitycomponent(orthe velocityfluctuation).Thefluctuationcomponentsofthevelocity alongthey- andz-axesaredenotedbyV;andV ~ ,respectively. By placing a measuring instrument with a lowinertia (for example. ahot-wireanemometer)at the required point ofaflow,we can record Ch.1.BasicInformationfromAerodynamics29 or measurethe fluctuationspeed.Inaturbulent flow,theinstrument registersthedeviationofthespeedfromthemeanvalue-thefluc-tuationspeed. Thekineticenergyofaturbulentflowisdeterminedbythesum ofthekineticenergiescalculatedaccordingtothemeanandfluc-tuationspeeds.Forapointinquestion.thekineticenergyofa fluctuationflowcanbedeterminedasaquantityproportionaltothe meanvalueofthemeansquarefluctuationvelocities.If weresolve the nuctuation flowalong theaxes ofacoordinate system,the kinetic energyofeachofthecorn ponen tsofsuchanow willbeproportional tothe relevant meansqnare componentsofthefluctuationvelocities, designatedby l7. andV?anddeterminedfromtheexpression t2 1('V'"dt Vx(u,z)'=t2-t1 Jx(y,z) 11 The concepts ofaYerageandfluctuatingquantities canbeextended tothepressureandotherphysicalparameters.Theexistenceof fluctuationvelocitiesleadstoadditionalnormalandshearstresses andtothemoreintensivetransferofheatandmass.Allthishasto betakenintoaccountwhenrunningexperimentsillaerodynamic tunnels.Theturbulenceintheatmosphere wasfoundtoberelatively smalland,consequently,itshouldbejustassmallintheworking partofatunnel.Anincreasedturbulenceaffectstheresultsofan experimen tadversely.Thenatureofthisinfluellcedependsonthe turbulencelevel(ortheinitialturbulence).determinedfromthe expression =_1_V(V'2V'2-L V'2).'3 eVxv'z (1.1.1) whereVistheoverallaveragespeedoftheturbulentflowatthe pointbeingconsidered. Inmodernlow-turbulenceaerodynamictunnels,itispossiblein practicetoreachaninitialturbulenceclosetowhatisobservedin theatmosphere(e0.01-0.02). Theimportantcharacteristicsofturbulenceincludetheroot meansquare(rms)fluctuationsVandVThese quantities,relatedtotheoverallaveragespeed,areknownasthe turbulenceintensitiesinthecorrespondingdirectionsandarede-notedas (1.1.2) Usingthesecharacteristics,theinitialturbulence(1.1.1)can be expressedasfollows: (1.1.1') ---------------------. 30Pt.I.Theory.AerodynamicsofanAirfoilandaWing (IL) , ( b)

Vx D /X laminar l.a!Jer :t!Lx'('JTvrbulenty2 tayer2 Fig.1.1.4 Flowofaviscousfluidoverabody: a - schrmaticviewofflow;I-laminarboundarylayer; 2-viscoussublayrr;3-turbulent boundarylayer;4-surfacpintheflow;5-wake;6-free flow;7-wakevortex;b--velocityprofileintheboundarylayrr;c-diagramdefiningthe concept ofatwo-pointcorrelationcoefficirnt;Vbisthe velocitycomponent at the outer edge oftheboundarylayer Turbulenceisofavortexnature,i.e.mass,momentum,and energyaretransferredbyfluidparticlesofavortexorigin.Hence itfollowsthatfluctuationsarecharacterizedbyastatisticalasso-ciation.Thecorrelationcoefficientbetweenfluctuationsatpoints ofthe regionofadisturbedflowbeing studied is aquantitative meas-ureofthisassociation.Inthegeneralform,thiscoefficientbetween tworandomfluctuatingquantitiescpand 1jJis writtenas(see[51) (1.1.3) If thereisnostatisticalassociationbetweenthequantitiescpand 1jJ.thenR=0;if,conversely,thesequantitiesareregularlyasso-eiated,thecorrelationcoefficientR=1.Thischaracteristicof turbulence is calledatwo-point correlation coefficient.Its expression canbewritten(Fig.1.1.4c)fortwopoints 1and2ofafluidvolume withtherelevantfluctuationsandintheform II/'2'2r=v"= R=Vx1Vy2/(VVx1Vy2 )(1.1.3') Whenstudyingathree-dimensionalturbulentflow,oneusually considersalargenumberofsuchcoefficients.Theconceptofthe turbulence scaleisintroducedtocharacterizethisflow.Itisdeter-minedbytheexpression 00 L= i Rdr o (1.1.4) Theturbulencescaleisalineardimensioncharacterizingthe lengthofthesectionofaflowonwhichfluidparticlesmove"in Ch.1.BasicInformationfromAerodynamics311 flssociation",i.e.havestatisticallyassociatedfluctuations.By lOvingtogetherthepointsbeingconsideredinatur bulen tfI ow,in thelimitatr -- 0wecanobtainaone-pointcorrelationcoefficient. Withthiscondition,(1.1.3')acquirestheform (1.1.5) Thiscoefficientcharacterizesthestatisticalassociationbetween fluctuationsatapointand,aswillbeshownbelow,directlydeter-minestheshearstressinaturhllien tflow. Turbulencewillbehomogeneousifitsaveragedcharacteristics foundforapoint(thelevelandill tensityofturbulence,theone-point correlationcoefficient)arethe samefortheentireflow(illvari-anceofthecharacteristicsoftnrblilenceintransfers).Homogeneolls t1ll'bulenceisisotropicifitscharacteristicsdonotdepend011the direction for whichthey are caiclilated(inYariance ofthe characteris-ticsofturbulenceinrotationandreflection).Particularly.the followingconditionissatisl'ledforanisotropicflow: =T7'2V'2 xJt Y- z Ifthisconditioni",satisfIedforallpoiJlts,thet mbulenceis homogeneousandisotropic.Forsuchturbulence,theconstancy ofthetwo-pointcorrelationcoefficientisretainedwithvarioHs directionsofthelineconnectingthetwopointsinthellnidvolume-beingconsidered. Foranisotropicflow,thecorrelationcoeflic.ien t(1.1. 5)callbe expressedintermsofthetnrbulencelevelf=V(1. Ui) Theintroductionoftheconceptofaveragedparameters orprop-ertiesappreciablyfacilitatestheinvestigationoftnrbulentflows. Indeed,forpracticalpurposes,thereisnoneedtoknowtheinstan-taneousvaluesofthevelocities,pressures,orshearstresses,andwe canlimitourselvestotheirtime-averagedvalues.Theuseofaver-agedparameterssimplifiestherelevantequationsofmotion(the Reynoldsequations). Snchequations,althoughtheyaresimpler,includethepartial derivatives with respecttotimeofthoaveragedvelocity components Vx,Vy,andVz becausoillthegeneral case,theturbulentmotionis unsteady.Inpracticalcases,however,averagingisperformedfor asuffIcientlylongintervaloftime,andnowinvestigationofan uIlsteadyflowcanbereducedtotheinvestigationofsteadyflow (quasi-steadyturbulentflow). ShearStress.Letusconsidertheformulafortheshearstress inalaminarflow.HerefrictionappearsbecauseofdiffliSiollofthe 32Pt.I.Theory.AerodynamicsofanAirfoilandaWing moleculesattendedbytransferofthemomentumfromonelayer toanother.Thisleadstoachangeintheflowvelocity,i.e.tothe appearanceofthe relative motionofthe fluidparticles inthe layers. InaccordancewithahypothesisfirstadvancedbyI.Newton,the shearstressforgivenconditionsisproportionaltothevelocityof thismotionperunitdistancebetweenlayerswithparticlesmoving 'relativetooneanother.If thedistancebetweenthelayersisAn. andtherelativespeedoftheparticlesisAu,theratioAul Anatthe limitwhenAn-+ 0,i.e.whenthelayersareincontact,equalsthe ,derivative aulan knownasthenormal velocity gradient.Onthe basis ,ofthishypothesis.wecanwriteNewton'sfrictionlaw: (1.1. 7) where[tisaproportionalityfactordependingonthepropertiesof a fluid. its temperatnre andpressure; it is better known as the dynamic viscosity. Themagnitudeof~ tforagasinaccordancewiththeformulaof thekinetictheoryis [t=0.499pcl(1.1.8) Atagivendensityp,itdependsonkineticcharacteristicsofagas 'suchasthemeanfreepathlandthemeanspeed c ofitsmolecules. Letusconsider frictioninaturbulent flow.We shallproceedfrom thesimplifiedschemeoftheappearanceofadditionalfrictionforces inturbulentflowproposedbyL.Prandtlforanincompressible fluid.andfromthe semi-empiricalnature oftherelations introduced fortheseforces.Letustaketwolayersinaone-dimensionalflow ,characterizedby achangeintheaveragedvelocity only inonedirec-tion.\Viththisinview,weshallassumethatthevelocityinoneof thelayersissuchthatVx'4=0,Vy=Vz =O.Fortheadjacent layer at adistance ofAy=l'fromthe first one, the averaged velocity isVx. + (dVxldy)l'.AccordingtoPrandtl'shypothesis,aparticle movingfromthefirstlayerintothesecondoneretainsitsvelocity 17;"and,consequently,attheinstantwhenthisparticleappearsin thesecondlayer,thefluctuationvelocityV ~=(dVxldy)l'isob-served. Themomentumtransferredbythefluidmassp V ~dSthroughthe areaelementdSisp V ~(i'x+ V ~ )dS.Thismomentumdetermines theadditionalforceproducedbythestressoriginatingfromthe fluctuationvelocities.Accordingly,theshear(friction)stress(in magnitude)intheturbulentflowduetofluctuationsis I 'ttI =p V ~(V x+ V ~ ) Ch.1.BasicI nformationfromAerodynamics33 A veragingthisexpression,weobtain - 1212 I. V;dt+-P- \' t2-t1 Jt2-t1 .l 1111 whereistheaveragedvalueoftheprod\1ctofthefiuctuation velocities,and istheaveragedvalueofthefluctuationvelocity. Weshallshowthat thisvalue of the velocity equals zero.Integrat-ingtheequalityVy=V;+termwisewithrespecttotwithin thelimitsfromt1tot2andthendividingitbyt2- t1,wefind 12 Butsince,bydefinition,V,!==__1_ (V'I dt,itisobviousthat .t2-tlJ. 12 11 -,1', Vy=--- \Vy dt =.coO.Hence,theaveragedvalueoftheshear t2 - tlJ 11 stressduetofluctuationscanbeexpressedbytherelationI T I = =p thatisthegeneralizedReynoldsformula.Itsformdoes notdependonanyspecifICassumptionsonthestructureofthe turbulence. Theshearstressdeterminedbythisformulacanheexpressed directlyintermsofthecorrelationcoefficient.Inaccordancewith (1.1.5),wehave I T tl=pRVV2 V V;2 orforanisotropicflowforwhichwehaveV = V , ITt I = = pRe2J12 (1.1.9) (1.1.9') Accordingtothisexpression,anadditionalshearstressdueto Uuctuationsdoesnotnecessarilyappearinanyflowcharacterized by acertainturbulence level.Its magnitude dependsonthemeasure ofthestatisticalmutualassociationofthefluctuationsdetermined bythecorrelationcoefficientR. ThegeneralizedReynoldsformulafortheshearstressinaccord-ancewithPrandtl'shypothesisontheproportionalityofthefluc-tuation velocities==al'(dV,.idy),where a isacoefficientl 3-01715 ------_._----------_._-_.--34Pt.I.Theory.AerodynamicsofanAirfoilandaWing canbetransformedasfollows: _12_ (dVx)2I'l'2adt=Pl2(dVX)2 t2 - t1dyJdy 11 (1.1.10> Heretheproportionalitycoefficientahasbeenincludedinthe averagedvalueofl',designatedbyl. Thequantityliscalledthemixinglengthandis,asit were,an analogueofthemeanfreepath ofmoleculesinthekinetictheoryof gases.The sign of the shear stress is determined by that of the velocity gradient.Consequently, (1.1.10') Thetotalvalueoftheshearstressisobtainedif tothevalue'tJ duetotheexpenditureofenergybyparticlesontheircollisions and chaotic mixing weadd the shear stress occurring directly because oftheviscosityandduetomixingofthemoleculescharacteristic ofalaminarflow,i.e.thevalue'tl= dVxldy.Hence, (1.1.11) Prandtl'sinvestigationsshowthatthemixinglengthl=%y, where%isaconstant.Accordingly,atawallofthebodyinthe flow,wehave (1.1.12) It follows fromexperimental data that in aturbulent flowin direct proximitytoawall,wheretheintensityofmixingisverylow,the shear stress remains the same as in laminar flow,and relation(1.1.12) holds forit.Beyondthelimits ofthis flow,the stress't[willbevery small,andwemayconsiderthattheshearstressisdeterminedby thequantity(1.1.10'). BoundaryLayer.Itfollowsfromrelations(1.1.7)and(1.1.10) that forthe samefluidflowingoverabody,the shear stress at differ-entsections oftheflowisnotthesameandisdeterminedbythe magnitudeofthelocalvelocitygradient. Investigationsshowthatthevelocitygradientisthelargestnear awallbecauseaviscousfluidexperiencesaretardingactionowing toitsadheringtothesurfaceofthebodyinthefluid.Thevelocity ofthe flowiszeroat the wall(seeFig.1.1.4)andgradually increases withthedistancefromthesurface.Theshearstresschangesaccord-ingly-atthewallitisconsiderablygreaterthanfarfromit.The thinlayeroffluidadjacenttothesurfaceofthebodyinaflowthat ischaracterizedbylargevelocitygradientsalonganormaltoit and,consequently,byconsiderableshearstressesiscalledabound-arylayer.Inthislayer,theviscousforceshaveamagnitudeofthe Ch.1.BasicInformationfromAerodynamics35 sameorderasalltheotherforces(forexample,theforcesofinertia an dpressure)governingmotionand,therefore,takenin toaccount intheequationsofmotion. Aphysicalnotionoftheboundarylayercanbeobtainedifwe-imaginethesurfaceintheflowtobecoatedwithapigmentsoluble-inthefluid.Itisobviousthatthepigmentdiffusesintothei1uid andissimultaneouslycarrieddownstream.Consequently,thecolo-uredzoneisalayer graduallythickeningdownstream.Thecoloured regionofthei1uidapproximately coincides withthe boundary layer. Thisregionleavesthesurfaceintheformofacolouredwake(see-Fig.1.1.4a). Asshownbyobservations,foraturbulentflow,thedifferenceof thecolouredregionfromtheboundary layer iscomparatively small, whereasinalaminarflowthisdifferencemaybeverysignificant. Accordingtotheoreticalandexperimentalinvestigations,withan increase,inthevelocity,thethicknessofthelayerdiminishes,and thewakebecomesnarrower. Thenatureofthevelocitydistributionoverthecrosssectionof aboundarylayerdependsonwhetheritislaminarorturbulent. Owingtolateralmixing oftheparticlesandalsototheir collisions, thisdistributionofthevelocity,moreexactlyofitstime-averaged value,willbeappreciablymoreuniforminaturbulentflowthanin alaminarone(seeFig.1.1.4).Thedistributionofthevelocities nearthesurfaceofabodyinaflowalsoallowsustomakethecon-clusiononthehighershearstressinaturbulentboundarylayer determinedbytheincreasedvalueofthevelocitygradient. Beyondthelimitsoftheboundarylayer,thereisapartofthe flowwherethevelocitygradientsand,consequently,theforcesof frictionaresmall.Thispartoftheflowisknownastheexternal freeflow.Inin vestigationofanexternalflow,theinfluenceofthe viscous forcesis disregarded.Therefore,suchaflowis also considered tobeinviscid.Thevelocityintheboundarylayergrowswithan increasingdistancefromthewallandasymptoticallyapproaches atheoreticalvaluecorrespondingtotheflowoverthebodyofan inviscidfluid,i.e.tothevalueofthevelocityintheexternalflow attheboundaryofthelayer. We have already notedthat in direct proximity to it awall hinders mixing,and,consequently,wemayassumethatthepartofthe boundary layer adjacenttothe wallisin conditions closeto laminar ones.Thisthinsectionofaquasilaminarboundary layer iscalled3. viscoussuhlayer(itisalsosometimescalledalaminarsublayer). Laterinvestigationsshowthatfluctuationsareobservedinthe viscoussublayerthatpenetrateintoitfromaturbulentcore,but thereisnocorrelationbetweenthem(thecorrelationcoefficient R=0).Therefore,accordingtoformula(1.1.9),noadditionalshear-stressesappear. a 36Pt.I.Theory.AerodynamicsofanAirfoilandaWing Fig.U.5 Boundarylayer: J-wallofabodyintheflowl 2-outeredgeofthelayer.. ; y 2 .1._ x Themainpartoftheboundarylayeroutsideoftheviscollssub-layeriscalledtheturbulentcore.Thestudyingofthemotionin aboundarylayerisassociatedwiththesimultaneousinvestigation oftheflowofafluidinaturbulentcoreandaviscoussublayer. Thechangeinthevelocityoverthecrosssectionoftheboundary layerischaracterizedbyitsgraduallygrowingwiththedistance fromthewallandasymptoticallyapproachingthevalueofthe velocityintheexternalflow.Forpracticalpurposes,however,itis convenienttotakethepartoftheboundarylayerinwhichthis changeoccurssufficientlyrapidly,andthevelocityattheboundary ofthis layer differsonly slightly fromitsvalueintheexternalflow. The distance from the wall to this boundary is what is conventionally calledthethicknessoftheboundary layer6(Fig.1.1.5).This thick-nessisusuallydefinedasthedistancefromthecontourofabodyto apointintheboundarylayeratwhichthevelocitydiffersfromits valueintheexternallayerbynotoveronepercent. Theintroductionofthe conceptofaboundarylayermadepossible effective researchofthe frictionandheattransferprocessesbecause owing to the smallness ofits thickness in comparisonwith thedimen-sionsofabodyinaflow,it becamepossibletosimplify thedifferen-tial equationsdescribingthemotion ofagasinthisregionofaflow, whichmakestheirintegrationeasier. 1.2.ResultantForceAction ComponentsofAerodynamicForces andMoments The forcesproducedby thenormalandshearstressescontinuously distributedoverthesurfaceofabodyinaflowcanbereducedto asingle resultant vectorRaofthe aerodynamic forcesandaresultant vectorMofthemomentoftheseforces(Fig.1.2.1)relativetoa referencepointcalledthecentre ofmoments.Anypointofthebody Ch.1.BasicInformationfromAerodynamics37 Fig.1.2.t Aerodynamicforcesandmomentsactingonacraftintheflightpath(xa,Ya, andza)andbodyaxis(x,Y,andz)coordinatesystems canbeth iscentre.Particularly,whentestingcraftinwindtunnels, themomentisfoundaboutoneofthepointsofmountingofthe modelthatmaycoincidewiththenoseofthebody,theleading edgeofawing,etc.Whenstudyingrealcasesofthemotionofsuch craftintheatmosphere,onecandeterminetheaerodynamic moment abouttheircentreofmassorsomeotherpointthatisacentreof rotation. Inengineeringpractice,insteadofconsideringthevectorsRa andM,theirprojectionsontotheaxesofacoordinatesystemare usuallydealtwith.Letusconsidertheflightpathandfixedorbody axisorthogonalcoordinatesystems(Fig.1.2.1)encounteredmost ofteninaerodynamics.Intheflightpathsystem,theaerodynamic forcesandmomentsarellsuallygivenbecausetheinvestigationof manyproblemsofflightdynamicsisconnectedwiththeuseof coordinateaxesofexactlysuchasystem.Particularly,itiscon-venienttowritetheequationsofmotionofacraft'scentreof massinprojectionsontotheseaxes.TheflightpathaxisOXaof avelocitysystemisalwaysdirectedalongthevelocityvectorof acraft'scentreofmass.TheaxisOYaoftheflightpathsystem(the liftaxis)isintheplaneofsymmetryandisdirectedupward(its positivedirection).TheaxisOZa(thelateralaxis)isdirectedalong thespanoftheright(starboard)wing(aright-handedcoordinate system).In inverted flow,theflightpathaxisOXacoincides withthe directionoftheflowveloci t y,whiletheaxisOZaisdirectedalong 38Pt.I.Theory.AerodynamicsofanAirfoilandaWing the spanoftheleft(port)wingsoastoretainaright-handedcoordi-natesystem.Thelatteriscalledawindcoordinatesystem. Aerodynamiccalculationscanbeperformedinaflxedorbody axiscoordinatesystem.Inaddition,rotationofacraftisusually investigatedinthissystembecausetherelevantequationsare writteninbodyaxes.Inthissystem,rigidlyfixedtoacraft, the longitudinalbodyaxisOxisdirectedalongtheprincipalaxisof inertia.The normalaxisOyisintheplane of symmetryandis orient edtowardtheupperpartofthecraft.ThelateralbodyaxisOz isdirectedalong the span ofthe right wing andformsaright-handed coordinatesystem.ThepositivedirectionoftheOxaxisfromthe tailtothenosecorrespondstonon-invertedflow(Fig.1.2.1).The -originsofbothcoordinatesystems-theflightpath(wind)andthe bodyaxissystems-areatacraft'scentreofmass. TheprojectionsofthevectorRaontotheaxesofaflightpath ooordinatesystemarecalledthedragforceX a'andliftforceY a' andthesideforceZa'respectively.Thecorrespondingprojections -ofthesamevectorontotheaxesofabodycoordinatesystemare ~ a l l e dthelongitudinalX,thenormalY,andthelateralZforces. TheprojectionsofthevectorMontotheaxesinthetwocoordi-natesystemshavethesamename:thecomponentsrelativetothe longitudinalaxisare called the rolling moment(the relevant symbols areMXainaflightpathsystem andMxinabodyone),thecompo-nentsrelativetotheverticalaxisarecalledtheyawingmoment (MYa orMy),andthoserelativetothelateralaxisarecalledthe pitchingmoment(Mz orMz). a Inaccordancewiththeabove,thevectorsoftheaerodynamic forcesandmoment intheflightpathandbodyaxiscoordinate .systemsare: Ra=Xa+ Ya + Za=X+ Y+ Z M=Mx+ My+ Mz=Mx+ My+ Mz aaa (1.2.1) (1.2.2) Weshallconsideramomentaboutanaxistobepositiveifit tendstoturnthecraftcounterclockwise(whenwatchingthemotion fromthetipofthemomentvector).Inaccordancewiththeadopted arrangementofthecoordinateaxes,apositivemomentinFig.1.2.1 increasestheangleofattack,andanegativemomentreducesit. Themagnitudeanddirectionoftheforcesandmomentsata givenairspeedandaltitudedependontheorientationofthebody relativetothevelocityvectorV(orifinvertedflowisbeingcon-sidered,relativetothedirectionofthefree-streamvelocityV(0). This orientation,inturn,underliestherelevantmutualarrangement ()fthecoordinatesystemsassociatedwiththeflowandthebody. Thisarrangementisdeterminedbytheangleofattackexandthe sideslipangleP (Fig.1.2.1).Thefirstofthemistheanglebetween Ch.1.BasicInformationfromAerodynamics39 y* g

x* g Fig.t.2.2 Determiningthepositionofacraftinspace theaxisOxandtheprojectionofthevectorVontotheplanexOy, and thesecondistheanglebetweenthevectorVandtheplanexOy. Theangleofattackisconsideredtobepositiveif theprojection oftheairvelocityontothenormalaxisisnegative.Thesideslip angleispositiveifthisprojectionontothelateralaxisispositive. Whenstudyingaflight,anormalearth-fixedcoordinatesystemis usedrelativetowhichthepositionofabodymovinginspaceis determined.Theoriginofcoordinatesofthissystem(Fig.1.2.2) coincideswithapointontheEarth'ssurface,forexamplewiththe launchingpoint.TheaxisOoygisdirectedupwardalongalocal vertical,whiletheaxesOOXgandOOZgcoincidewithahorizontal plane.TheaxisOOXgisusuallyorientedinthedirectionofflight, whilethedirectionoftheaxisOOZgcorrespondstoaright-handed coordinatesystem. If theoriginofanearth-fixedsystemofcoordinatesismadeto coincide with the centre ofmass ofacraft, weobtainanormal earth-fixedcoordinatesystemalsoknownasalocalgeographicalcoordi-nate system(Fig.1.2.2).Thepositionofacraftrelativeto thiscoordinatesystemisdeterminedbythreeangles:theyawing (course)anglethepitchingangle&,andtherolling(banking) angle"(. Theangleisformedbytheprojectionofthelongitudinalbody axisOxontothehorizontalplane(Ox * ) andtheaxisthisangleispositiveiftheaxiscoincideswiththeprojection ofOx*byclockwiserotationabouttheaxisTheangle&isthatbetweentheaxisOxandthehorizontalplane andwillbepositiveifthisplaneisbelowthelongitudinal -------- ------..40Pt.I.Theory.AerodynamicsofanAirfoilandaWing bodyaxis.Theangleyisformedupontherotation(rolling)ofa craftaboutthelongitudinalaxisOxandismeasuredinmagnitude asthe anglebetweenthelateral body axisandthe axis OZgdisplaced toapositioncorrespondingtoazeroyawingangle(orastheangle betweentheaxisOzanditsprojectionontoahorizontalplane--theaxisOz;).If displacementoftheaxisOzZwithrespecttothe lateralaxisoccursclockwise,theangleyispositive. Thepitchingangledeterminestheinclinationofacrafttothe horizon,andtheyawingangle-thedeviationofthedirectionof itsflightfromtheinitialone(foranaircraftthisisthedeviation fromitscourse,foraprojectileorrocketthisisthedeviation from theplaneoflaunching). ConversionofAerodynamicForces andMomentsfromOneCoordinateSystem toAnother Knowingtheanglesaandp,wecanconvertthecomponentsof theforceandmomentinonecoordinatesystemt()componentsin anothersysteminaccordancewiththerulesofanalytical geometry. Particularly,thecomponentsoftheaerodynamicforceandmoment inabodyaxissystemareconvertedtothedrag forceandthe rolling moment,respectively,inaflightpathsystemofcoordinatesbythe formulas X a=Xcos( ~ a )+ ycos(Yxa)+ Zcos(?xa)(1.2.3) MXa=Mxcos(x;a)+ Mycos(;;;:a)+ Mz cos(;;a)(1.2.3') wherecos(;;a),cos(;;;:a),cos(;;a)arethecosinesoftheangles betweentheaxisOXaandtheaxesOx,Oy,andOz,respectively. Theexpressionsfortheothercomponentsoftheforcevector,and alsoforthecomponentsofthemomentvector,arewrittenina similarway.Thevaluesofthedirectioncosinesusedforconverting forcesandmomentsfromonecoordinatesystemtoanotherare giveninTable1.2.1. Table1.2.1 Flightpathsystem Bodyaxis II sy,tem QXaoVa oZa Oxcos a cos Bsina-cos asin B Oy-sin a cosBcos asin asin B Ozsin B0cos B Ch.1.BasicInformationfromAerodynamics41 InaccordancewiththedataofTable1.2.1,Eqs.(1.2.3)and (1.2.3')acquirethefollowingform: X a=Xcosex.cosB - Ysin ex.cosB + ZsinB(1.2.4) Mx=Mxcosex.cosB - lvlysinex.cosB + Mz SillB(1.2.4') a Forexample,forthemotionoftheaircraftshowninFig.1.2.1, Eq.(1.2.4)yields,withtherelevantsigns: -X a=-X cosex.cosB - Ysinex.cosB + ZsinB Theforceandmomentcomponents*areconvertedinasimilar way fromaflightpath to abody axis coordinate system.For example, byusingthedataof Table1.2.1,weobtainthe followingconversion formulasfortholongitudinalforceandtherollingmoment: X=Xacosex.cosB + Ya sinex.- Zacosex.sinB(1.2.5) JlJx=J1xacosex.cosBJly"sinex.- JlJz"cosex.sinB(1.2.5') Wecangoover fromalocal geographicalcoordinate system(anor-malsystem)toabodyaxisorflightpathone,orviceversa,if we knowthecosines oftheanglesbetweenthecorrespondingaxes.Their valuescanbedeterminedfromFig.1.2.2thatshowsthemutual arrangementoftheaxesofthesecoordinatesystems. 1.3.Determination ofAerodynamicForces andMomentsAccordingto theKnownDistribution of thePressureandShearStress. AerodynamicCoefficients AerodynamicForces andMomentsandTheirCoefficients Assumethatforacertainangleofattackandsideslipangle,and alsoforgivenparametersofthefreestream(thespeedV 00,static pressure p 00,densityp 00,andtemperatureToo),weknowthedistrib-ution ofthepressure pandshear stress 'Lover the surfaceofthebody intheflow.Wewanttodeterminetheresultantvaluesoftheaero-dynamicforcesandmoments. TheisolatedsurfaceelementdSofthebodyexperiencesanormal forceproduced by the excess pressure(p- p 00)dSand the tangential *Weshallomittheword"components"belowforbrevity,but shallmean itanduseformulasforscalarquantities. 42Pt.I.Theory.AerodynamicsofanAirfoilandaWing Fig.t.l.t Actionofpressureandfriction '(shear)forcesonan elementary ..area SIJ force1:dS.Thesumoftheprojeetionsofthese forcesontothex-axis ,ofawind(flightpath)coordinatesystemis(Fig.1.3.1) /\/\ [(p - poo)cos (n,xa) + 1:cos (t,xa)]dS(1.3.1) wherenandtareanormalandatangenttotheelementofarea, respectively. TheothertwoprojectionsontotheaxesYaandZaareobtainedby asimilar formulawith thecorrespondingcosines.Tofindtheresul-tantforces,wehavetointegrateexpression(1.3.1)overtheentire surfaceS.Introducingintotheseequationsthepressurecopfficient p=(p- p oo)/q 00andthelocalfrictionfactorCj,x=1:/q 00,where q 00=p 00 V!,/2isthevelocityhead,weobtainformulasforthedrag force,theliftforce,andthe side force,respectively: r- /\/\ Xa =qooSrI[pcos (n,xa) +Ct, x cos (t,Xa)]dS/Sr (8) r_/\/\ Ya=qooSrJ [-pcos(n,Ya)+Cr,xCOS(t,Ya)]dS/Sr (8) ,- /\/\ Za=-qooSr)[pcos(n,Za)+Ct,xCOS(t,za)]dS/Sr (8) (1.3.2) (1.3.3) (1.3.4) Wecanchoosearandomsurfaceareasuchasthatofawingin ..planviewortheareaofthelargestcrosssection(themid-section)of thefuselageasthecharacteristicareaS rintheseformulas.The integralsinformulas(1.3.2)-(1.3.4)aredimensionlessquantities takingintoaccounthowtheaerodynamicforcesareaffectedbythe natureofthe flowoverabody ofagivengeometricconfigurationand bythedistributionofthedimensionlesscoefficientsofpressureand jfrictionduetothisflow. Informulas(1.3.2)fortheforceXa,thedimensionlessquantity ,isusuallydesignatedbyCx andisknownasthedragcoefficient. a Ch.1.BasicI nformationfromAerodynamics43 Intheothertwoformulas,thecorrespondingsymbolscYa andcZa areintroduced.Therelevantquantitiesareknownastheaerodyna-micliftcoefficientandtheaerodynamicside-forcecoefficient.With ,itviewtotheabove,wehave Xa=Cx qooSnYa =cY (jooSr,Za=Cz qooSr aaa (1.3.5) \Vecanobtaingeneralrelationsforthemomentsinthesameway as formulas(1.3.2)-(1.3.4) for the forces.Let us consider as anexample 'sucharelationforthepitchingmomentJVfz".It isevidentthatthe elementaryvalueofthismomentd}1f zisdeterminedbythesumof a themomentsabouttheaxisZaoftheforcesactingonanareadS in aplaneatrightanglestotheaxisZa'If thecoordinatesofthearea dSareYaandXa,theelementaryvalueofthemomentis - /\./\. dMza =qooSr{[p cos (n,Ya) - Cr.xcos (t,Ya)lxa - /\/\. - [pcos (n,xa) + Cr.xcos (t,xa)Ya}dS/Sr IntegratingthisexpressionoverthesurfaceSandintroducing thedimensionlessparameter Mza\- /"'- I', mZa=SL={lpcos (n,Ya)-cr,xcos(t,Ya)] Xa qoor.. (s) - 1"- /"'- dS - [p cos (n,xa) + Ct,x cos (t,xa)]Ya}SrL ( 1.3.6) in whichLisacharacteristicgeometriclength,weobtaina formula forthepitchingmoment: Mz =mz qooS rL aa (1.3.7) TheparametermZaiscalledtheaerodynamicpitching-moment coefficient.Theformulasfortheothercomponentsofthemoment arewrittensimilarly: JVf x=tnx q ooS rLand111y =tny q ooS rL aaaa (U3.8) Thedimensionlessparameterstnx andtny arecalledtheaerody-aa Ilamicrolling-momentandyawing-moment coefficients,respectively. TherelevantGoefficientsoftheaerodynamicforcesandmoments canalsobeintroducedinabodyaxiscoordinatesystem.Theuseof thesecoefficientsallowstheforcesandmomentstobewrittenas follows: x =cxq ooS r,M x=mxq ooS rL Y=cyqooSr,My=7nyqooSrL Z=czqooSr,Mz =7nzqooSrL (1.3.9) 44Pt.I.Theory.AerodynamicsofanAirfoilandaWing Thequantitiescx,cy,andCz arecalledtheaerodynamiclongitu-dinal-force,normal-force,andlateral-forcecoefficients,andthe parametersmx,my,andmz-theaerodynamicbodyaxisrolling-moment,yawing-moment,andpitching-moment coefficients,respec-tively. Ananalysisoftheexpressionsfortheaerodynamicforces(1.3.2)-(1.3.4)allowsustoarriveattheconclusionthateachoftheseforces can be resolved into a component due to the pressure and acomponent associatedwiththeshearstressesappearinguponthemotionofa viscousfluid.Forexample,thedragX a=X a,p+ X a.f'where X a.pisthep r e s s u n ~dragandX a.fisthehictiondrag.Accordingly, theoverallcoefficientofdragequalsthe sumofthecoefficientsof pressureandfrictiondrags:Cx =Cx + Cx aa,pa,1 Similarly,theaerodynamiclift- andside-forcecoefficients,and alsothemoments,canberepresentedasthesumoftwocomponents. Theforces,moments,andtheircoefficientsarewritteninthesame wayinabodyaxes.Forexample,thelongitudinal-forcecoeffIcient Cx=cx,p+ cx,f'wherecx,pandXx,rarethecoefficientsofthe longitudinalforcesduetopressureandfriction,respectively. The components ofthe aerodynamic forcesand momentsdepending on frictionare not always the same as those depending on the pressure asregardstheirorderofmagnitude.Investigationsshowthatthe influenceoffrictionismoreappreciableforflowoverlongandthin bodies.Inpractice,itisgoodtotakethisinfluenceintoaccount mainlywhendeterminingthedragorlongitudinalforce. Whenasurfaceinaflowhasaplane areaatits tail part(abottom cutofthefuselageorablunttrailingedgeofawing),thepressure dragisusuallydividedintotwomorecomponents,namely,the pressure drag on aside surface (the nose drag), and the drag due to the pressureonthe base cut or section (the base drag).Hence,the overall dragandtherelevantaerodynamiccoefficientare Xa=Xa.n + Xa.b+ Xa.f andcXa =cxan+ Cxa.b+ cxa.t Whendeterminingthelongitudinalforceanditscoefficient,we obtain X=Xn+ X b+ Xf andCx=Cx,n+ cx.b+ Cx .! InaccordancewithFig.1.3.1,wehave Xb =- goor Pb dSb andCx,b--=Xhs Jqoor Sb wherePb=(P.b- P oo)/g 00(thisquantityisnegativebecausea rarefactionappearsafterabottomcut,i.e.Pb< P 00)' CharacteristicGeometricDimensions.Theabsolutevalueofan aerodynamiccoefficient,whichisarbitrarytoacertainextent, Ch.1.BasicInformationfromAerodynamics45 Fig.1.3.2 Schematicviewofawing: bo-centrechord.bt-tipchord. andb-localchord z x dependsonthechoiceofthecharacteristicgeometricdimension s S rand L.To facilitatepractical calculations, however,acharacteris-tic geometric quantity is chosenbeforehand.In aerospacetechnology, theareaofthemid-section(thelargestcrosssection)ofthebody S r=S mirlisusuallychosenasthecharacteristicarea,andthe length of the rocket istakenasthe characteristiclineardimensionL. Inaerodynamiccalculationsofaircraft,thewingplanareaS r= =Sw,thewingspanl(thedistancebetweenthewingtips)orthe wingchordbareadoptedasthecharacteristicdimensions.Bythe chordofawingismeantalengthequalc'lthedistancebetweenthe farthestpointsofanairfoil(section).Forawingwitharectangular planform,the chordequals the width ofthe wing.In practice,awing usually has a chord varying along its span.Eitherthe mean geometric chordb=bm equaltobm =Swllorthemeanaerodynamicchord b=bA istaken asthe characteristicdimensionforsuchawing.The meanaerodynamicchordisdeterminedasthechordoftheairfoil ofanequivalentrectangularwing forwhichwithanidenticalwing plan area the momentaerodynamic characteristics areapproximately thesameasofthegivenwing. Thelengthofthemeanaerodynamicchordandthecoordinateof itsleadingedgearedeterminedasfollows(Fig.1.3.2): 1/2 XAO=_2_('bxdz Swj o Whencalculatingforcesandmomentsaccordingtoknownaero-dynamiccoefficients,thegeometricdimensionsmustbeusedfor whichthesecoefficientswereevaluated.Shouldsuchcalculations havetobeperformedforothergeometricdimensions,theaero-dynamiccoefficientsmustbepreliminarilyconvertedtothe relevant geometricdimension.Forthispurpose,onemustusetherelations CISI =C2S2 (fortheforcecoeffIcients),andm1S1L1 =m2S2L2 (for the moment coefficients) obtained fromthe conditions of the constancy (a) 46Pt.I.Theory.AerodynamicsofanAirfoilandaWing (b) CI CYomox !fa,,---------- __ CYan Fig.t.3.3 Constructingapolarofthefirstkindofacraft: a-cxa vs.a;b-CYa VS.a;c-polaroffirstkind (c) ton-' ffmax: /---oftheforcesandmomentsactingonthe samecraft.Theserelations areusedtofindthecoefficientsC2 andm2,respectively,convertedto thenewcharacteristicdimensions82 andL2: c2 =c1 (81/82),m2 =m1 (81L1/82L2) wherethepreviousdimensions81,L1andaerodynamiccoefficients c1'm1,aswellasthenewdimensions82,L2areknown. PolarofaCraft.Averyimportantaerodynamiccharacteristic thathasfoundwidespreaduseiswhatisknownasthepolarof acraft.Itestablishesarelationbetweentheliftanddragforcesor, whichisthesame,betweentheliftanddragcoefficientsinaflight pathcoordinatesystem.Thiscurve,calledapolarofthefirstkind (Fig.1.3.3c)is the locus of the tips of the resultant aerodynamic force vectorsRaactingonacraftatvariousanglesofattack[orofthe vectorsofthecoefficientCRofthisforcedeterminedinaccordance a withthe relationCR=Ra/(8 rq 00)]. a Apolarofthefirstkindisconstructedwiththeaidofgraphsof Cx versusexandcyversusexsothatthevaluesofCx andcy are aaaa laidoffalongtheaxesofabscissasandordinates,respectively.The relevantangleofattackex,whichisaparameterofthepolarinthe givencase,iswrittenateachpointofthecurve. Apolarofthefirstkindisconvenientforpracticalusebecauseit allows oneto readily findforany angle ofattack suchavery import-antaerodynamiccharacteristicofacraftasitslift-to-dragratio (1.3.10) If the scales ofcYa (orYa)andcXa (orX a)are the same,thequan-tity Kequalsthe slopeofavectordrawnfromtheoriginofcoordi-Ch.1.BasicInformationfromAerodynamics47 Fig.f.l.4 Dragpolarofthesecondkind nates(thepole)to thepoint of thepolardiagramcorresponding tothe chosenangleofattack. Wecanuseapolartodeterminetheoptimalangleofattackaopt correspondingtothemaximumlift-to-dragratio: K max=tanaopt (CLiO'}> ifwedrawatangenttothepolarfromtheoriginofcoordinates. ThecharacteristicpointsofapolarincludethepointcYamax correspondingtothemaximumliftforcethatisachievedatthe criticalangleofattack aeroWecan markapointonthecurvedeter-miningtheminimumdragcoefficientCx minandthecorresponding a valuesoftheangleofattackandtheliftcoefficient. Apolarissymmetricabouttheaxisofabscissasifacrafthas, horizontalsymmetry.Forsuchacraft,thevalueofexmincorre-a spondsto a zeroliftforce,cYa =O. Inadditiontoapolarofthefirstkind.apolarof the second kind; issometimesused.Itdiffersinthatitisplottedinabodyaxis coordinatesystemalongwhoseaxisofabscissasthevaluesofthe longitudinal-forcecoefficientCx arelaidoff,andalongtheaxisof ordinates-thenormal-forcecoefficientscY (Fig.1.3.4).Thiscurve isused,particularly,inthestrengthanalysisofcraft. Theoreticalandexperimentalinvestigationsshowthatinthe mostgeneralcase,theaerodynamiccoefficientsdependforagiven bodyconfIgurationandangleofattackondimensionlessvariables suchastheMachnumberMoo=V oofa ooandtheReynoldsnumber Re 00=V ooLp oo/ftoo.Intheseexpressions,a 00isthespeedofsound intheoncomingflow,p 00andft00arethedensityanddynamicvis-cosityofthegas,respectively,andListhelengthofthebody. Hence,amultitudeofpolarcurvesexistsforeachgivencraft.For example,foradefinitenumberRe 00,wecanconstructafamilyof sllchcurveseachofwhichcorrespondstoadefinitevalueofthe ._--------_.. _---_. 48Pt.I.Theory.AerodynamicsofanAirfoilandaWing (a) !:I (b) Mz(mz}Y(e",) !:i Y CP eM/'--F----0'- x X(Cx) Xp I xc ..xn(XFo0.ithasstaticinstability,andwhenmr=0,Ileutrality. Theconceptofdirectionalstabilityisassociatedwi tIltheprop-ertyofacrafttoeliminateanappearingsideslipAtthe sametime,acraftdoesnotmaintainthestabilityofitsownflight directionbecauseafterchangingitsdirectionofmotionUlHjCl'the actionofvariousdisturbances,itdoesnotretmrttoitsprevious flBPt.I.Theory.AerodynamicsofanAirfoilandaWing direction,butlikeaweathercock,turnswithitsnosepartinthe directionofthenewvleocityvectorV. Similartotheaerodynamiccentrebasedontheangleofattack, wecanintroducetheconceptoftheaerodynamiccentrebasedon thesideslipanglewhosecoordinateweshalldesignatebyXFj3' Usingthisconcept,wecanrepresentthedegreeofstaticdirectional stabilityintheform =-(XFf;- XCM)(1.4.7) where XFfj=xF(/Zand =xcMIl(lisacharacteristicgeometric dimensionthat candeterminethewingspan,fuselagelength,etc.). Hence,thestaticdirectionalstabilityorinstabilitydependson the mutual position ofthe centre of mass and the aerodynamic centre. Areararrangementoftheaerodynamiccentre(XFj3>deter-minesstaticdirectionalstability < 0),whileitsfrontarrange-ment(XF0).When 11thetwocentrescoincide(xF=XCM),the craft isneutralasregards tl staticdirectionalstability =0). Aparticularcaseofmotionofacraftintheplaneoftheangleof attackcanbecharacterizedbyaconstantangularvelocity(Qy = =const)andstabilizationwithrespecttorollingwiththeaidof anautomaticpilot(Qx ;:::::;0).Theconditionforsuchsteadymotion islateraltrimofthecraftwhentheyavdngmomentvanishes,i.e. my=myo+ +r+ m;yQy=0 Allcraftcustomarilyhavelongitudinalsymmetry,therefore myo=O.Whenthisconditionholds,theequationobtainedallows ustodeterminethebalancerudderangleI')r=I')r.balcorresponding tothe given values ofthe balance sideslipangleBba)andthe angular velocityQy.Mostcrafthaveasufficientlyhighdegreeofstatic 10001\.,lhi:,;exponelltcallben";;II111edtobe constantandequnlto0.1.ForToo=288E.thesppcitiC'heat cl' 00= =1000J/(kg. K)fur1000FOJ'llllrln11.5.1)mayheIlsed forofTupto:2200-250uK.iltwhiciltheyibralioJlaldegrees offreedomareclose10the"laIcof:olllpleteexcitation. \Vhen setsin,the heatdepel1(lsJlotonlyon thetemperature.butalsoonthepr8ssure.The heatsand thendia haLicexponen to'k=(I"'(/'\\-el'ec akula tedfo),cond i ti011S ofthermodynamiceCfl1ilibril1l11athiglltemperatlll'esoncompllters byagr'ollpofSo\"iet!:'cienti.'' , .... ,-c-.... fOOO2000JOOO'1000T,K /. % f-+-+-+-I-+-+-+-1I-+-l-+-+-+--+-+-+-I '\, '2,35r---,Ii!' "'::;.'" :;, ' at which the pressureat the exit reaches thesamevalueasp(:).Thesubsonicflowformedinthiscaseis characterizedbycurves2showninFig.3.6.2.Wefindthevelocity ratio"A(M)ofthisflowbyEq.(3.6.23),choosingfromthetwosol-utionsfor"Atheonethatislessthanunity. Uponafurtherincreaseinthebackpressuretothevaluep(i). conditionsappearinwhichthepressure attheexitwillequal thebackpressure,i.e. =p(i),whilethepressureinthethroat willexceedthecriticalpressurep*;therelevantspeedinthiscross sectionissubsonic(V1-1. 4.2.GeneralEquations foraShock Weshallconsiderthemoregeneralcasewhenthegasbehind ashock,owingtosubstantialheating,experiencesphysicochemical transformationsandchangesitsspecificheat.Ofmajorsignificance whenstudyingshocksbehindwhichoscillationsaregeneratedand dissociation,ionization,andchemicalreactionsoccuraretherates ofthephysicochemicaltransformations. Processesbehindshockwavesarecharacterizedbyafractionof thekineticenergyofthemovinggasvirtuallyinstantaneously transforming intotheinternalenergy ofthe gas.In these conditions, wecannotignorethefactthatthermodynamicequilibriumisach-ievedafteracertaintimeelapsesonlyinconditionsofsuchequi-librium do all the parameters experiencing discontinuities (the preSSlll'e, density,temperature)becometime-independent.Theanalysisof thesephenomenaisamoreinvolvedproblemandisassoeiated primarilywithstudyingofthemechanismofnon-equilibriumpro-cesses,andwithaknowledge,particularly,oftheratesofchemical reactionsintheair. Thesimplestcaseischaracterizedbyaninfinitelyhighrateof thephysicochemicaltransformationsand,consequently,bythe instantaneoussettinginofthermodynamicequilibrium.Suchpro-cessesbehind shock waves arepossiblephysically, which is confirmed byexperimentalstudies. Letusconsiderthebasictheoreticalrelationsallowingoneto evaluatethe equilibriumparametersbehindashockwave. Ch.4.ShockWaveTheory163 ObliqueShock Ashockformedinrealconditionsischaracterizedbyacertain thiclmess.Theparametersofthegasinsuchashockchangenot instantaneously,butduringacertaintimeinterval.Asshownby theoreticalandexperimentalinvestigations,ho\yever,thethickness ofashockisyerysmallandisoftheorderofthemeanfreepath ofthemolecules. Foratmosphericconditions,calculationsyieldedthefollowing valuesofthethicknessofashockmeasuredinthedirectionofthe-free-streamvelocity: ;\IachnumberMoo. Thickness,mm 1.53 4.5X10-41.2xto-4 O.7X10-4 1l.2X10-' For Moo=2,thethicknessofashockequalsabout fourmolecular freepaths,andforMoo=3-about three.Therefore,whenstudying ashockinanidealfluid,thisthicknessmaybedisregardedandthe shockrepresentedintheformofageometricdiscontinuitysurface forthe gasparameters,assumingtheseparametersto change instan-taneollsly. Omtaskcons-istsindeterminingthe unknownparametersofagas behindashockaccordingtothepresetparameterscharacterizing theflowofthegasaheadoftheshock. Foranobliqueshockformedinadissociatingandionizinggas, thereareninelmknownparameters:thepressureP2,densityP2, temperatmeT2,velocityV2,enthalpyi2,entropyS2,speedof sounda2'themeanmolarmassandthesllOckangleOs(orthe flowdeviationanglePs).Consequently,itisnecessarytocompile-ninesimultaneousequations.Theparametersaheadofashockwill belheknownonesintheseequations,namely,thepressurePIt densityPI'velocity VI' etc.Insteadofthevelocity V2 behind a shock. \vecandetermineitscomponentsalonganormalVn2 andatangent. VT?totheshock.Thiswillincreasethenumberofequationsneeded to "ten.TheseequationsincludethefundamentalequationsofgaS' dynamics(ofmotion,continuity,energy,andstate),anumberof kinematicrelationsforthevelocities,andalsothermodynamic relationscharacterizing thepropertiesofagas.Let us each equationofthissystem. Figure 4.2.1showstriangles oftheflowvelocitiesaheadofashock (theparameters withthe subscript1)andbehindit (the subscript 2). Weshallusethefiguretodeterminethefollowingrelationsfor thesecomponents: =V 2cos(Ss- Ps),V n2=V 2sin(Sa- Ps)(4.2.1) Thisyieldsthefirstequationofthesystemofsimultaneousones: Vn21V-r2=tan(Ss- Ps)(4.2.2)-11* 164Pt.I.Theory.AerodynamicsofanAirfoilandaWing Fig.".2.1 Obliqueshock

Vi Shockline P,Pz T,Tz Vrze '.Bs.-[3.{3s "'.!:.L__ _____-=___ S(S2 a,az The continuityequation(orthemassflowequation)isthesecond one.It determinestheamountoffluidpassingthroughunitsurface ofashockinunittime: PIVIO =pzVoz (4.2.3) hereVnl =VIsin asisthenormalcomponentofthevelocityahead oftheshock(Fig.4.2.1). Let us usethe equationofmotion reducedtothe formofanequa-tion ofmomentum for the conditions ofthepassage throughashock. This is the third equation ofthe system.We shall obtain it by assum-ingthatthechangeinthemomentumofthefluidpassinginunit timethroughunitsurfaceareaoftheshockinthedirectionofa normaltothissurfaceequalstheimpulseofthepressureforces: -=pz - PI(4.2.4) Wi