Theory of Lift

Dr. Geordie Drummond McBain, Australia Geordie McBain is an engineering consultant based in Sydney, Australia. In 1995 he graduated top of his class from James Cook University with first class honours in mechanical engineering, earning him the Faculty Medal, and went on to receive his PhD there in 1999. In 2002 he was awarded a Sesquicentennial Postdoctoral Fellowship at the University of Sydney, researching fluid dynamics. During this period, he taught aerodynamics to students on the Aeronautical and Aerospace Engineering degree programmes.

Preface xvii

Series Preface xxiii

Part One Plane Ideal Aerodynamics

1 Preliminary Notions 3

1.1 Aerodynamic Force and Moment 3

1.1.1 Motion of the Frame of Reference 3

1.1.2 Orientation of the System of Coordinates 4

1.1.3 Components of the Aerodynamic Force 4

1.1.4 Formulation of the Aerodynamic Problem 4

1.2 Aircraft Geometry 5

1.2.1 Wing Section Geometry 6

1.2.2 Wing Geometry 7

1.3 Velocity 8

1.4 Properties of Air 8

1.4.1 Equation of State: Compressibility and the Speed of Sound 8

1.4.2 Rheology: Viscosity 10

1.4.3 The International Standard Atmosphere 12

1.4.4 Computing Air Properties 12

1.5 Dimensional Theory 13

1.5.1 Alternative methods 16

1.5.2 Example: Using Octave to Solve a Linear System 16

1.6 Example: NACA Report No. 502 18

1.7 Exercises 19

1.8 Further Reading 22

References 22

2 Plane Ideal Flow 25

2.1 Material Properties: The Perfect Fluid 25

2.2 Conservation of Mass 26

2.2.1 Governing Equations: Conservation Laws 26

2.3 The Continuity Equation 26

2.4 Mechanics: The Euler Equations 27

2.4.1 Rate of Change of Momentum 27

2.4.2 Forces Acting on a Fluid Particle 28

2.4.3 The Euler Equations 29

2.4.4 Accounting for Conservative External Forces 29

2.5 Consequences of the Governing Equations 30

2.5.1 The Aerodynamic Force 30

2.5.2 Bernoulli’s Equation 33

2.5.3 Circulation, Vorticity, and Irrotational Flow 33

2.5.4 Plane Ideal Flows 35

2.6 The Complex Velocity 35

2.6.1 Review of Complex Variables 35

2.6.2 Analytic Functions and Plane Ideal Flow 38

2.6.3 Example: the Polar Angle Is Nowhere Analytic 40

2.7 The Complex Potential 41

2.8 Exercises 42

2.9 Further Reading 44

References 45

3 Circulation and Lift 47

3.1 Powers of z 47

3.1.1 Divergence and Vorticity in Polar Coordinates 48

3.1.2 Complex Potentials 48

3.1.3 Drawing Complex Velocity Fields with Octave 49

3.1.4 Example: k = 1, Corner Flow 50

3.1.5 Example: k = 0, Uniform Stream 51

3.1.6 Example: k =−1, Source 51

3.1.7 Example: k =−2, Doublet 52

3.2 Multiplication by a Complex Constant 53

3.2.1 Example: w = const., Uniform Stream with Arbitrary Direction 53

3.2.2 Example: w = i/z, Vortex 54

3.2.3 Example: Polar Components 54

3.3 Linear Combinations of Complex Velocities 54

3.3.1 Example: Circular Obstacle in a Stream 54

3.4 Transforming the Whole Velocity Field 56

3.4.1 Translating the Whole Velocity Field 56

3.4.2 Example: Doublet as the Sum of a Source and Sink 56

3.4.3 Rotating the Whole Velocity Field 56

3.5 Circulation and Outflow 57

3.5.1 Curve-integrals in Plane Ideal Flow 57

3.5.2 Example: Numerical Line-integrals for Circulation and Outflow 58

3.5.3 Closed Circuits 59

3.5.4 Example: Powers of z and Circles around the Origin 60

3.6 More on the Scalar Potential and Stream Function 61

3.6.1 The Scalar Potential and Irrotational Flow 61

3.6.2 The Stream Function and Divergence-free Flow 62

3.7 Lift 62

3.7.1 Blasius’s Theorem 62

3.7.2 The Kutta–Joukowsky Theorem 63

3.8 Exercises 64

3.9 Further Reading 65

References 66

4 Conformal Mapping 67

4.1 Composition of Analytic Functions 67

4.2 Mapping with Powers of ζ 68

4.2.1 Example: Square Mapping 68

4.2.2 Conforming Mapping by Contouring the Stream Function 69

4.2.3 Example: Two-thirds Power Mapping 69

4.2.4 Branch Cuts 70

4.2.5 Other Powers 71

4.3 Joukowsky’s Transformation 71

4.3.1 Unit Circle from a Straight Line Segment 71

4.3.2 Uniform Flow and Flow over a Circle 72

4.3.3 Thin Flat Plate at Nonzero Incidence 73

4.3.4 Flow over the Thin Flat Plate with Circulation 74

4.3.5 Joukowsky Aerofoils 75

4.4 Exercises 75

4.5 Further Reading 78

References 78

5 Flat Plate Aerodynamics 79

5.1 Plane Ideal Flow over a Thin Flat Plate 79

5.1.1 Stagnation Points 80

5.1.2 The Kutta–Joukowsky Condition 80

5.1.3 Lift on a Thin Flat Plate 81

5.1.4 Surface Speed Distribution 82

5.1.5 Pressure Distribution 83

5.1.6 Distribution of Circulation 84

5.1.7 Thin Flat Plate as Vortex Sheet 85

5.2 Application of Thin Aerofoil Theory to the Flat Plate 87

5.2.1 Thin Aerofoil Theory 87

5.2.2 Vortex Sheet along the Chord 87

5.2.3 Changing the Variable of Integration 88

5.2.4 Glauert’s Integral 88

5.2.5 The Kutta–Joukowsky Condition 89

5.2.6 Circulation and Lift 89

5.3 Aerodynamic Moment 89

5.3.1 Centre of Pressure and Aerodynamic Centre 90

5.4 Exercises 90

5.5 Further Reading 91

References 91

6 Thin Wing Sections 93

6.1 Thin Aerofoil Analysis 93

6.1.1 Vortex Sheet along the Camber Line 93

6.1.2 The Boundary Condition 93

6.1.3 Linearization 94

6.1.4 Glauert’s Transformation 95

6.1.5 Glauert’s Expansion 95

6.1.6 Fourier Cosine Decomposition of the Camber Line Slope 97

6.2 Thin Aerofoil Aerodynamics 98

6.2.1 Circulation and Lift 98

6.2.2 Pitching Moment about the Leading Edge 99

6.2.3 Aerodynamic Centre 100

6.2.4 Summary 101

6.3 Analytical Evaluation of Thin Aerofoil Integrals 101

6.3.1 Example: the NACA Four-digit Wing Sections 104

6.4 Numerical Thin Aerofoil Theory 105

6.5 Exercises 109

6.6 Further Reading 109

References 109

7 Lumped Vortex Elements 111

7.1 The Thin Flat Plate at Arbitrary Incidence, Again 111

7.1.1 Single Vortex 111

7.1.2 The Collocation Point 111

7.1.3 Lumped Vortex Model of the Thin Flat Plate 112

7.2 Using Two Lumped Vortices along the Chord 114

7.2.1 Postprocessing 116

7.3 Generalization to Multiple Lumped Vortex Panels 117

7.3.1 Postprocessing 117

7.4 General Considerations on Discrete Singularity Methods 117

7.5 Lumped Vortex Elements for Thin Aerofoils 119

7.5.1 Panel Chains for Camber Lines 119

7.5.2 Implementation in Octave 121

7.5.3 Comparison with Thin Aerofoil Theory 122

7.6 Disconnected Aerofoils 123

7.6.1 Other Applications 124

7.7 Exercises 125

7.8 Further Reading 125

References 126

8 Panel Methods for Plane Flow 127

8.1 Development of the CUSSSP Program 127

8.1.1 The Singularity Elements 127

8.1.2 Discretizing the Geometry 129

8.1.3 The Influence Matrix 131

8.1.4 The Right-hand Side 132

8.1.5 Solving the Linear System 134

8.1.6 Postprocessing 135

8.2 Exercises 137

8.2.1 Projects 138

8.3 Further Reading 139

References 139

8.4 Conclusion to Part I: The Origin of Lift 139

Part Two Three-dimensional Ideal Aerodynamics

9 Finite Wings and Three-Dimensional Flow 143

9.1 Wings of Finite Span 143

9.1.1 Empirical Effect of Finite Span on Lift 143

9.1.2 Finite Wings and Three-dimensional Flow 143

9.2 Three-Dimensional Flow 145

9.2.1 Three-dimensional Cartesian Coordinate System 145

9.2.2 Three-dimensional Governing Equations 145

9.3 Vector Notation and Identities 145

9.3.1 Addition and Scalar Multiplication of Vectors 145

9.3.2 Products of Vectors 146

9.3.3 Vector Derivatives 147

9.3.4 Integral Theorems for Vector Derivatives 148

9.4 The Equations Governing Three-Dimensional Flow 149

9.4.1 Conservation of Mass and the Continuity Equation 149

9.4.2 Newton’s Law and Euler’s Equation 149

9.5 Circulation 150

9.5.1 Definition of Circulation in Three Dimensions 150

9.5.2 The Persistence of Circulation 151

9.5.3 Circulation and Vorticity 151

9.5.4 Rotational Form of Euler’s Equation 153

9.5.5 Steady Irrotational Motion 153

9.6 Exercises 154

9.7 Further Reading 155

References 155

10 Vorticity and Vortices 157

10.1 Streamlines, Stream Tubes, and Stream Filaments 157

10.1.1 Streamlines 157

10.1.2 Stream Tubes and Stream Filaments 158

10.2 Vortex Lines, Vortex Tubes, and Vortex Filaments 159

10.2.1 Strength of Vortex Tubes and Filaments 159

10.2.2 Kinematic Properties of Vortex Tubes 159

10.3 Helmholtz’s Theorems 159

10.3.1 ‘Vortex Tubes Move with the Flow’ 159

10.3.2 ‘The Strength of a Vortex Tube is Constant’ 160

10.4 Line Vortices 160

10.4.1 The Two-dimensional Vortex 160

10.4.2 Arbitrarily Oriented Rectilinear Vortex Filaments 160

10.5 Segmented Vortex Filaments 161

10.5.1 The Biot–Savart Law 161

10.5.2 Rectilinear Vortex Filaments 162

10.5.3 Finite Rectilinear Vortex Filaments 164

10.5.4 Infinite Straight Line Vortices 164

10.5.5 Semi-infinite Straight Line Vortex 164

10.5.6 Truncating Infinite Vortex Segments 165

10.5.7 Implementing Line Vortices in Octave 165

10.6 Exercises 166

10.7 Further Reading 167

References 167

11 Lifting Line Theory 169

11.1 Basic Assumptions of Lifting Line Theory 169

11.2 The Lifting Line, Horseshoe Vortices, and the Wake 169

11.2.1 Deductions from Vortex Theorems 169

11.2.2 Deductions from the Wing Pressure Distribution 170

11.2.3 The Lifting Line Model of Air Flow 170

11.2.4 Horseshoe Vortex 170

11.2.5 Continuous Trailing Vortex Sheet 171

11.2.6 The Form of the Wake 172

11.3 The Effect of Downwash 173

11.3.1 Effect on the Angle of Incidence: Induced Incidence 173

11.3.2 Effect on the Aerodynamic Force: Induced Drag 174

11.4 The Lifting Line Equation 174

11.4.1 Glauert’s Solution of the Lifting Line Equation 175

11.4.2 Wing Properties in Terms of Glauert’s Expansion 176

11.5 The Elliptic Lift Loading 178

11.5.1 Properties of the Elliptic Lift Loading 179

11.6 Lift–Incidence Relation 180

11.6.1 Linear Lift–Incidence Relation 181

11.7 Realizing the Elliptic Lift Loading 182

11.7.1 Corrections to the Elliptic Loading Approximation 182

11.8 Exercises 182

11.9 Further Reading 183

References 183

12 Nonelliptic Lift Loading 185

12.1 Solving the Lifting Line Equation 185

12.1.1 The Sectional Lift–Incidence Relation 185

12.1.2 Linear Sectional Lift–Incidence Relation 185

12.1.3 Finite Approximation: Truncation and Collocation 185

12.1.4 Computer Implementation 187

12.1.5 Example: a Rectangular Wing 187

12.2 Numerical Convergence 188

12.3 Symmetric Spanwise Loading 189

12.3.1 Example: Exploiting Symmetry 191

12.4 Exercises 192

References 192

13 Lumped Horseshoe Elements 193

13.1 A Single Horseshoe Vortex 193

13.1.1 Induced Incidence of the Lumped Horseshoe Element 195

13.2 Multiple Horseshoes along the Span 195

13.2.1 A Finite-step Lifting Line in Octave 197

13.3 An Improved Discrete Horseshoe Model 200

13.4 Implementing Horseshoe Vortices in Octave 203

13.4.1 Example: Yawed Horseshoe Vortex Coefficients 205

13.5 Exercises 206

13.6 Further Reading 207

References 207

14 The Vortex Lattice Method 209

14.1 Meshing the Mean Lifting Surface of a Wing 209

14.1.1 Plotting the Mesh of a Mean Lifting Surface 210

14.2 A Vortex Lattice Method 212

14.2.1 The Vortex Lattice Equations 213

14.2.2 Unit Normals to the Vortex-lattice 215

14.2.3 Spanwise Symmetry 215

14.2.4 Postprocessing Vortex Lattice Methods 215

14.3 Examples of Vortex Lattice Calculations 216

14.3.1 Campbell’s Flat Swept Tapered Wing 216

14.3.2 Bertin’s Flat Swept Untapered Wing 218

14.3.3 Spanwise and Chordwise Refinement 219

14.4 Exercises 220

14.5 Further Reading 221

14.5.1 Three-dimensional Panel Methods 222

References 222

Part Three Nonideal Flow in Aerodynamics

15 Viscous Flow 225

15.1 Cauchy’s First Law of Continuum Mechanics 225

15.2 Rheological Constitutive Equations 227

15.2.1 Perfect Fluid 227

15.2.2 Linearly Viscous Fluid 227

15.3 The Navier–Stokes Equations 228

15.4 The No-Slip Condition and the Viscous Boundary Layer 228

15.5 Unidirectional Flows 229

15.5.1 Plane Couette and Poiseuille Flows 229

15.6 A Suddenly Sliding Plate 230

15.6.1 Solution by Similarity Variable 230

15.6.2 The Diffusion of Vorticity 233

15.7 Exercises 234

15.8 Further Reading 234

References 235

16 Boundary Layer Equations 237

16.1 The Boundary Layer over a Flat Plate 237

16.1.1 Scales in the Conservation of Mass 237

16.1.2 Scales in the Streamwise Momentum Equation 238

16.1.3 The Reynolds Number 239

16.1.4 Pressure in the Boundary Layer 239

16.1.5 The Transverse Momentum Balance 239

16.1.6 The Boundary Layer Momentum Equation 240

16.1.7 Pressure and External Tangential Velocity 241

16.1.8 Application to Curved Surfaces 241

16.2 Momentum Integral Equation 241

16.3 Local Boundary Layer Parameters 243

16.3.1 The Displacement and Momentum Thicknesses 243

16.3.2 The Skin Friction Coefficient 243

16.3.3 Example: Three Boundary Layer Profiles 244

16.4 Exercises 248

16.5 Further Reading 249

References 249

17 Laminar Boundary Layers 251

17.1 Boundary Layer Profile Curvature 251

17.1.1 Pressure Gradient and Boundary Layer Thickness 252

17.2 Pohlhausen’s Quartic Profiles 252

17.3 Thwaites’s Method for Laminar Boundary Layers 254

17.3.1 F(λ) ≈ 0.45 − 6λ 255

17.3.2 Correlations for Shape Factor and Skin Friction 256

17.3.3 Example: Zero Pressure Gradient 256

17.3.4 Example: Laminar Separation from a Circular Cylinder 257

17.4 Exercises 260

17.5 Further Reading 261

References 262

18 Compressibility 263

18.1 Steady-State Conservation of Mass 263

18.2 Longitudinal Variation of Stream Tube Section 265

18.2.1 The Design of Supersonic Nozzles 266

18.3 Perfect Gas Thermodynamics 266

18.3.1 Thermal and Caloric Equations of State 266

18.3.2 The First Law of Thermodynamics 267

18.3.3 The Isochoric and Isobaric Specific Heat Coefficients 267

18.3.4 Isothermal and Adiabatic Processes 267

18.3.5 Adiabatic Expansion 268

18.3.6 The Speed of Sound and Temperature 269

18.3.7 The Speed of Sound and the Speed 269

18.3.8 Thermodynamic Characteristics of Air 270

18.3.9 Example: Stagnation Temperature 270

18.4 Exercises 270

18.5 Further Reading 271

References 271

19 Linearized Compressible Flow 273

19.1 The Nonlinearity of the Equation for the Potential 273

19.2 Small Disturbances to the Free-Stream 274

19.3 The Uniform Free-Stream 275

19.4 The Disturbance Potential 275

19.5 Prandtl–Glauert Transformation 276

19.5.1 Fundamental Linearized Compressible Flows 277

19.5.2 The Speed of Sound 278

19.6 Application of the Prandtl–Glauert Rule 279

19.6.1 Transforming the Geometry 279

19.6.2 Computing Aerodynamical Forces 280

19.6.3 The Prandlt–Glauert Rule in Two Dimensions 282

19.6.4 The Critical Mach Number 284

19.7 Sweep 284

19.8 Exercises 285

19.9 Further Reading 285

References 286

Appendix A Notes on Octave Programming 287

A. 1 Introduction 287

A. 2 Vectorization 287

A.2. 1 Iterating Explicitly 288

A.2. 2 Preallocating Memory 288

A.2. 3 Vectorizing Function Calls 288

A.2. 4 Many Functions Act Elementwise on Arrays 289

A.2. 5 Functions Primarily Defined for Arrays 289

A.2. 6 Elementwise Arithmetic with Single Numbers 289

A.2. 7 Elementwise Arithmetic between Arrays 290

A.2. 8 Vector and Matrix Multiplication 290

A. 3 Generating Arrays 290

A.3. 1 Creating Tables with bsxfun 290

A. 4 Indexing 291

A.4. 1 Indexing by Logical Masks 291

A.4. 2 Indexing Numerically 291

A. 5 Just-in-Time Compilation 291

A. 6 Further Reading 292

References 292

Glossary 293

Nomenclature 305

Index 309