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 Bernoullis 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 Blasiuss Theorem 62
3.7.2 The KuttaJoukowsky 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 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 Joukowskys 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 KuttaJoukowsky 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 Glauerts Integral 88
5.2.5 The KuttaJoukowsky 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 Glauerts Transformation 95
6.1.5 Glauerts 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 Newtons Law and Eulers 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 Eulers 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 Helmholtzs 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 BiotSavart 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 Glauerts Solution of the Lifting Line Equation 175
11.4.2 Wing Properties in Terms of Glauerts Expansion 176
11.5 The Elliptic Lift Loading 178
11.5.1 Properties of the Elliptic Lift Loading 179
11.6 LiftIncidence Relation 180
11.6.1 Linear LiftIncidence 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 LiftIncidence Relation 185
12.1.2 Linear Sectional LiftIncidence 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 Campbells Flat Swept Tapered Wing 216
14.3.2 Bertins 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 Cauchys 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 NavierStokes 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 Pohlhausens Quartic Profiles 252
17.3 Thwaitess Method for Laminar Boundary Layers 254
17.3.1 F() 0.45 6 255
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 PrandtlGlauert Transformation 276
19.5.1 Fundamental Linearized Compressible Flows 277
19.5.2 The Speed of Sound 278
19.6 Application of the PrandtlGlauert Rule 279
19.6.1 Transforming the Geometry 279
19.6.2 Computing Aerodynamical Forces 280
19.6.3 The PrandltGlauert 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
