Theory of Lift: Introductory Computational Aerodynamics in MATLAB®/OCTAVE

325 Pages·2012·4.17 MB·English

by  G. D. McBain(auth.)

THEORY OF LIFT
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THEORY OF LIFT
INTRODUCTORY COMPUTATIONAL
AERODYNAMICS IN
MATLAB®/OCTAVE
G.D.McBain,
SchoolofAerospace,Mechanical,&MechatronicEngineering
TheUniversityofSydney,Australia
Thiseditionfirstpublished2012
©2012,JohnWiley&Sons,Ltd
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LibraryofCongressCataloging-in-PublicationData
McBain,G.D.(GeordieG.)
Theoryoflift:introductorycomputationalaerodynamicswithMATLAB®/
OCTAVE/G.D.McBain.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-119-95228-2(hardback)
1. Lift(Aerodynamics)–Mathematicalmodels. 2. Aerodynamics–Data
processing. 3. MATLAB. I. Title.
TL574.L5M332012
629.132(cid:1)33028553–dc23 2012004735
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
PrintISBN:9781119952282
Setin10/12pt,TimesRomanbyThomsonDigital,Noida,India
InmemoryofDrJonathanHarris
Contents
Preface xvii
SeriesPreface xxiii
PARTONE PLANEIDEALAERODYNAMICS
1 PreliminaryNotions 3
1.1 AerodynamicForceandMoment 3
1.1.1 MotionoftheFrameofReference 3
1.1.2 OrientationoftheSystemofCoordinates 4
1.1.3 ComponentsoftheAerodynamicForce 4
1.1.4 FormulationoftheAerodynamicProblem 4
1.2 AircraftGeometry 5
1.2.1 WingSectionGeometry 6
1.2.2 WingGeometry 7
1.3 Velocity 8
1.4 PropertiesofAir 8
1.4.1 EquationofState:CompressibilityandtheSpeedofSound 8
1.4.2 Rheology:Viscosity 10
1.4.3 TheInternationalStandardAtmosphere 12
1.4.4 ComputingAirProperties 12
1.5 DimensionalTheory 13
1.5.1 Alternativemethods 16
1.5.2 Example:UsingOctavetoSolveaLinearSystem 16
1.6 Example:NACAReportNo.502 18
1.7 Exercises 19
1.8 FurtherReading 22
References 22
2 PlaneIdealFlow 25
2.1 MaterialProperties:ThePerfectFluid 25
2.2 ConservationofMass 26
2.2.1 GoverningEquations:ConservationLaws 26
2.3 TheContinuityEquation 26
2.4 Mechanics:TheEulerEquations 27
2.4.1 RateofChangeofMomentum 27
viii Contents
2.4.2 ForcesActingonaFluidParticle 28
2.4.3 TheEulerEquations 29
2.4.4 AccountingforConservativeExternalForces 29
2.5 ConsequencesoftheGoverningEquations 30
2.5.1 TheAerodynamicForce 30
2.5.2 Bernoulli’sEquation 33
2.5.3 Circulation,Vorticity,andIrrotationalFlow 33
2.5.4 PlaneIdealFlows 35
2.6 TheComplexVelocity 35
2.6.1 ReviewofComplexVariables 35
2.6.2 AnalyticFunctionsandPlaneIdealFlow 38
2.6.3 Example:thePolarAngleIsNowhereAnalytic 40
2.7 TheComplexPotential 41
2.8 Exercises 42
2.9 FurtherReading 44
References 45
3 CirculationandLift 47
3.1 Powersofz 47
3.1.1 DivergenceandVorticityinPolarCoordinates 48
3.1.2 ComplexPotentials 48
3.1.3 DrawingComplexVelocityFieldswithOctave 49
3.1.4 Example:k =1,CornerFlow 50
3.1.5 Example:k =0,UniformStream 51
3.1.6 Example:k =−1,Source 51
3.1.7 Example:k =−2,Doublet 52
3.2 MultiplicationbyaComplexConstant 53
3.2.1 Example:w=const.,UniformStreamwithArbitraryDirection 53
3.2.2 Example:w=i/z,Vortex 54
3.2.3 Example:PolarComponents 54
3.3 LinearCombinationsofComplexVelocities 54
3.3.1 Example:CircularObstacleinaStream 54
3.4 TransformingtheWholeVelocityField 56
3.4.1 TranslatingtheWholeVelocityField 56
3.4.2 Example:DoubletastheSumofaSourceandSink 56
3.4.3 RotatingtheWholeVelocityField 56
3.5 CirculationandOutflow 57
3.5.1 Curve-integralsinPlaneIdealFlow 57
3.5.2 Example:NumericalLine-integralsforCirculationandOutflow 58
3.5.3 ClosedCircuits 59
3.5.4 Example:PowersofzandCirclesaroundtheOrigin 60
3.6 MoreontheScalarPotentialandStreamFunction 61
3.6.1 TheScalarPotentialandIrrotationalFlow 61
3.6.2 TheStreamFunctionandDivergence-freeFlow 62
Contents ix
3.7 Lift 62
3.7.1 Blasius’sTheorem 62
3.7.2 TheKutta–JoukowskyTheorem 63
3.8 Exercises 64
3.9 FurtherReading 65
References 66
4 ConformalMapping 67
4.1 CompositionofAnalyticFunctions 67
4.2 MappingwithPowersofζ 68
4.2.1 Example:SquareMapping 68
4.2.2 ConformingMappingbyContouringtheStreamFunction 69
4.2.3 Example:Two-thirdsPowerMapping 69
4.2.4 BranchCuts 70
4.2.5 OtherPowers 71
4.3 Joukowsky’sTransformation 71
4.3.1 UnitCirclefromaStraightLineSegment 71
4.3.2 UniformFlowandFlowoveraCircle 72
4.3.3 ThinFlatPlateatNonzeroIncidence 73
4.3.4 FlowovertheThinFlatPlatewithCirculation 74
4.3.5 JoukowskyAerofoils 75
4.4 Exercises 75
4.5 FurtherReading 78
References 78
5 FlatPlateAerodynamics 79
5.1 PlaneIdealFlowoveraThinFlatPlate 79
5.1.1 StagnationPoints 80
5.1.2 TheKutta–JoukowskyCondition 80
5.1.3 LiftonaThinFlatPlate 81
5.1.4 SurfaceSpeedDistribution 82
5.1.5 PressureDistribution 83
5.1.6 DistributionofCirculation 84
5.1.7 ThinFlatPlateasVortexSheet 85
5.2 ApplicationofThinAerofoilTheorytotheFlatPlate 87
5.2.1 ThinAerofoilTheory 87
5.2.2 VortexSheetalongtheChord 87
5.2.3 ChangingtheVariableofIntegration 88
5.2.4 Glauert’sIntegral 88
5.2.5 TheKutta–JoukowskyCondition 89
5.2.6 CirculationandLift 89
5.3 AerodynamicMoment 89
5.3.1 CentreofPressureandAerodynamicCentre 90
5.4 Exercises 90
5.5 FurtherReading 91
References 91
x Contents
6 ThinWingSections 93
6.1 ThinAerofoilAnalysis 93
6.1.1 VortexSheetalongtheCamberLine 93
6.1.2 TheBoundaryCondition 93
6.1.3 Linearization 94
6.1.4 Glauert’sTransformation 95
6.1.5 Glauert’sExpansion 95
6.1.6 FourierCosineDecompositionoftheCamberLineSlope 97
6.2 ThinAerofoilAerodynamics 98
6.2.1 CirculationandLift 98
6.2.2 PitchingMomentabouttheLeadingEdge 99
6.2.3 AerodynamicCentre 100
6.2.4 Summary 101
6.3 AnalyticalEvaluationofThinAerofoilIntegrals 101
6.3.1 Example:theNACAFour-digitWingSections 104
6.4 NumericalThinAerofoilTheory 105
6.5 Exercises 109
6.6 FurtherReading 109
References 109
7 LumpedVortexElements 111
7.1 TheThinFlatPlateatArbitraryIncidence,Again 111
7.1.1 SingleVortex 111
7.1.2 TheCollocationPoint 111
7.1.3 LumpedVortexModeloftheThinFlatPlate 112
7.2 UsingTwoLumpedVorticesalongtheChord 114
7.2.1 Postprocessing 116
7.3 GeneralizationtoMultipleLumpedVortexPanels 117
7.3.1 Postprocessing 117
7.4 GeneralConsiderationsonDiscreteSingularityMethods 117
7.5 LumpedVortexElementsforThinAerofoils 119
7.5.1 PanelChainsforCamberLines 119
7.5.2 ImplementationinOctave 121
7.5.3 ComparisonwithThinAerofoilTheory 122
7.6 DisconnectedAerofoils 123
7.6.1 OtherApplications 124
7.7 Exercises 125
7.8 FurtherReading 125
References 126
8 PanelMethodsforPlaneFlow 127
8.1 DevelopmentoftheCUSSSPProgram 127
8.1.1 TheSingularityElements 127
8.1.2 DiscretizingtheGeometry 129
8.1.3 TheInfluenceMatrix 131
8.1.4 TheRight-handSide 132
Contents xi
8.1.5 SolvingtheLinearSystem 134
8.1.6 Postprocessing 135
8.2 Exercises 137
8.2.1 Projects 138
8.3 FurtherReading 139
References 139
8.4 ConclusiontoPartI:TheOriginofLift 139
PARTTWO THREE-DIMENSIONALIDEALAERODYNAMICS
9 FiniteWingsandThree-DimensionalFlow 143
9.1 WingsofFiniteSpan 143
9.1.1 EmpiricalEffectofFiniteSpanonLift 143
9.1.2 FiniteWingsandThree-dimensionalFlow 143
9.2 Three-DimensionalFlow 145
9.2.1 Three-dimensionalCartesianCoordinateSystem 145
9.2.2 Three-dimensionalGoverningEquations 145
9.3 VectorNotationandIdentities 145
9.3.1 AdditionandScalarMultiplicationofVectors 145
9.3.2 ProductsofVectors 146
9.3.3 VectorDerivatives 147
9.3.4 IntegralTheoremsforVectorDerivatives 148
9.4 TheEquationsGoverningThree-DimensionalFlow 149
9.4.1 ConservationofMassandtheContinuityEquation 149
9.4.2 Newton’sLawandEuler’sEquation 149
9.5 Circulation 150
9.5.1 DefinitionofCirculationinThreeDimensions 150
9.5.2 ThePersistenceofCirculation 151
9.5.3 CirculationandVorticity 151
9.5.4 RotationalFormofEuler’sEquation 153
9.5.5 SteadyIrrotationalMotion 153
9.6 Exercises 154
9.7 FurtherReading 155
References 155
10 VorticityandVortices 157
10.1 Streamlines,StreamTubes,andStreamFilaments 157
10.1.1 Streamlines 157
10.1.2 StreamTubesandStreamFilaments 158
10.2 VortexLines,VortexTubes,andVortexFilaments 159
10.2.1 StrengthofVortexTubesandFilaments 159
10.2.2 KinematicPropertiesofVortexTubes 159
10.3 Helmholtz’sTheorems 159
10.3.1 ‘VortexTubesMovewiththeFlow’ 159
10.3.2 ‘TheStrengthofaVortexTubeisConstant’ 160