Theory of Lift: Introductory Computational Aerodynamics in MATLAB®/Octave -- Contents -- Preface -- Acknowledgements -- References -- Series Preface -- PART ONE: PLANE IDEAL AERODYNAMICS -- 1 Preliminary Notions -- 1.1 Aerodynamic Force and Moment -- 1.1.1 Motion of the Frame of Reference -- 1.1.2 Orientation of the System of Coordinates -- 1.1.3 Components of the Aerodynamic Force -- 1.1.4 Formulation of the Aerodynamic Problem -- 1.2 Aircraft Geometry -- 1.2.1 Wing Section Geometry -- 1.2.2 Wing Geometry -- 1.3 Velocity -- 1.4 Properties of Air -- 1.4.1 Equation of State: Compressibility and the Speed of Sound -- 1.4.2 Rheology: Viscosity -- 1.4.3 The International Standard Atmosphere -- 1.4.4 Computing Air Properties -- 1.5 Dimensional Theory -- 1.5.1 Alternative methods -- 1.5.2 Example: Using Octave to Solve a Linear System -- 1.6 Example: NACA Report No. 502 -- 1.7 Exercises -- 1.8 Further Reading -- References -- 2 Plane Ideal Flow -- 2.1 Material Properties: The Perfect Fluid -- 2.2 Conservation of Mass -- 2.2.1 Governing Equations: Conservation Laws -- 2.3 The Continuity Equation -- 2.4 Mechanics: The Euler Equations -- 2.4.1 Rate of Change of Momentum -- 2.4.2 Forces Acting on a Fluid Particle -- 2.4.3 The Euler Equations -- 2.4.4 Accounting for Conservative External Forces -- 2.5 Consequences of the Governing Equations -- 2.5.1 The Aerodynamic Force -- 2.5.2 Bernoulli's Equation -- 2.5.3 Circulation, Vorticity, and Irrotational Flow -- 2.5.4 Plane Ideal Flows -- 2.6 The Complex Velocity -- 2.6.1 Review of Complex Variables -- 2.6.2 Analytic Functions and Plane Ideal Flow -- 2.6.3 Example: the Polar Angle Is Nowhere Analytic -- 2.7 The Complex Potential -- 2.8 Exercises -- 2.9 Further Reading -- References -- 3 Circulation and Lift -- 3.1 Powers of z -- 3.1.1 Divergence and Vorticity in Polar Coordinates -- 3.1.2 Complex Potentials.

3.1.3 Drawing Complex Velocity Fields with Octave -- 3.1.4 Example: k = 1, Corner Flow -- 3.1.5 Example: k = 0, Uniform Stream -- 3.1.6 Example: k = -1, Source -- 3.1.7 Example: k = -2, Doublet -- 3.2 Multiplication by a Complex Constant -- 3.2.1 Example: w = const., Uniform Stream with Arbitrary Direction -- 3.2.2 Example: w = i/z, Vortex -- 3.2.3 Example: Polar Components -- 3.3 Linear Combinations of Complex Velocities -- 3.3.1 Example: Circular Obstacle in a Stream -- 3.4 Transforming the Whole Velocity Field -- 3.4.1 Translating the Whole Velocity Field -- 3.4.2 Example: Doublet as the Sum of a Source and Sink -- 3.4.3 Rotating the Whole Velocity Field -- 3.5 Circulation and Outflow -- 3.5.1 Curve-integrals in Plane Ideal Flow -- 3.5.2 Example: Numerical Line-integrals for Circulation and Outflow -- 3.5.3 Closed Circuits -- 3.5.4 Example: Powers of z and Circles around the Origin -- 3.6 More on the Scalar Potential and Stream Function -- 3.6.1 The Scalar Potential and Irrotational Flow -- 3.6.2 The Stream Function and Divergence-free Flow -- 3.7 Lift -- 3.7.1 Blasius's Theorem -- 3.7.2 The Kutta-Joukowsky Theorem -- 3.8 Exercises -- 3.9 Further Reading -- References -- 4 Conformal Mapping -- 4.1 Composition of Analytic Functions -- 4.2 Mapping with Powers of ζ -- 4.2.1 Example: Square Mapping -- 4.2.2 Conforming Mapping by Contouring the Stream Function -- 4.2.3 Example: Two-thirds Power Mapping -- 4.2.4 Branch Cuts -- 4.2.5 Other Powers -- 4.3 Joukowsky's Transformation -- 4.3.1 Unit Circle from a Straight Line Segment -- 4.3.2 Uniform Flow and Flow over a Circle -- 4.3.3 Thin Flat Plate at Nonzero Incidence -- 4.3.4 Flow over the Thin Flat Plate with Circulation -- 4.3.5 Joukowsky Aerofoils -- 4.4 Exercises -- 4.5 Further Reading -- References -- 5 Flat Plate Aerodynamics -- 5.1 Plane Ideal Flow over a Thin Flat Plate.

5.1.1 Stagnation Points -- 5.1.2 The Kutta-Joukowsky Condition -- 5.1.3 Lift on a Thin Flat Plate -- 5.1.4 Surface Speed Distribution -- 5.1.5 Pressure Distribution -- 5.1.6 Distribution of Circulation -- 5.1.7 Thin Flat Plate as Vortex Sheet -- 5.2 Application of Thin Aerofoil Theory to the Flat Plate -- 5.2.1 Thin Aerofoil Theory -- 5.2.2 Vortex Sheet along the Chord -- 5.2.3 Changing the Variable of Integration -- 5.2.4 Glauert's Integral -- 5.2.5 The Kutta-Joukowsky Condition -- 5.2.6 Circulation and Lift -- 5.3 Aerodynamic Moment -- 5.3.1 Centre of Pressure and Aerodynamic Centre -- 5.4 Exercises -- 5.5 Further Reading -- References -- 6 Thin Wing Sections -- 6.1 Thin Aerofoil Analysis -- 6.1.1 Vortex Sheet along the Camber Line -- 6.1.2 The Boundary Condition -- 6.1.3 Linearization -- 6.1.4 Glauert's Transformation -- 6.1.5 Glauert's Expansion -- 6.1.6 Fourier Cosine Decomposition of the Camber Line Slope -- 6.2 Thin Aerofoil Aerodynamics -- 6.2.1 Circulation and Lift -- 6.2.2 Pitching Moment about the Leading Edge -- 6.2.3 Aerodynamic Centre -- 6.2.4 Summary -- 6.3 Analytical Evaluation of Thin Aerofoil Integrals -- 6.3.1 Example: the NACA Four-digit Wing Sections -- 6.4 Numerical Thin Aerofoil Theory -- 6.5 Exercises -- 6.6 Further Reading -- References -- 7 Lumped Vortex Elements -- 7.1 The Thin Flat Plate at Arbitrary Incidence, Again -- 7.1.1 Single Vortex -- 7.1.2 The Collocation Point -- 7.1.3 Lumped Vortex Model of the Thin Flat Plate -- 7.2 Using Two Lumped Vortices along the Chord -- 7.2.1 Postprocessing -- 7.3 Generalization to Multiple Lumped Vortex Panels -- 7.3.1 Postprocessing -- 7.4 General Considerations on Discrete Singularity Methods -- 7.5 Lumped Vortex Elements for Thin Aerofoils -- 7.5.1 Panel Chains for Camber Lines -- 7.5.2 Implementation in Octave -- 7.5.3 Comparison with Thin Aerofoil Theory.

7.6 Disconnected Aerofoils -- 7.6.1 Other Applications -- 7.7 Exercises -- 7.8 Further Reading -- References -- 8 Panel Methods for Plane Flow -- 8.1 Development of the CUSSSP Program -- 8.1.1 The Singularity Elements -- 8.1.2 Discretizing the Geometry -- 8.1.3 The Influence Matrix -- 8.1.4 The Right-hand Side -- 8.1.5 Solving the Linear System -- 8.1.6 Postprocessing -- 8.2 Exercises -- 8.2.1 Projects -- 8.3 Further Reading -- References -- 8.4 Conclusion to Part I: The Origin of Lift -- PART TWO: THREE-DIMENSIONAL IDEAL AERODYNAMICS -- 9 Finite Wings and Three-Dimensional Flow -- 9.1 Wings of Finite Span -- 9.1.1 Empirical Effect of Finite Span on Lift -- 9.1.2 Finite Wings and Three-dimensional Flow -- 9.2 Three-Dimensional Flow -- 9.2.1 Three-dimensional Cartesian Coordinate System -- 9.2.2 Three-dimensional Governing Equations -- 9.3 Vector Notation and Identities -- 9.3.1 Addition and Scalar Multiplication of Vectors -- 9.3.2 Products of Vectors -- 9.3.3 Vector Derivatives -- 9.3.4 Integral Theorems for Vector Derivatives -- 9.4 The Equations Governing Three-Dimensional Flow -- 9.4.1 Conservation of Mass and the Continuity Equation -- 9.4.2 Newton's Law and Euler's Equation -- 9.5 Circulation -- 9.5.1 Definition of Circulation in Three Dimensions -- 9.5.2 The Persistence of Circulation -- 9.5.3 Circulation and Vorticity -- 9.5.4 Rotational Form of Euler's Equation -- 9.5.5 Steady Irrotational Motion -- 9.6 Exercises -- 9.7 Further Reading -- References -- 10 Vorticity and Vortices -- 10.1 Streamlines, Stream Tubes, and Stream Filaments -- 10.1.1 Streamlines -- 10.1.2 Stream Tubes and Stream Filaments -- 10.2 Vortex Lines, Vortex Tubes, and Vortex Filaments -- 10.2.1 Strength of Vortex Tubes and Filaments -- 10.2.2 Kinematic Properties of Vortex Tubes -- 10.3 Helmholtz's Theorems -- 10.3.1 'Vortex Tubes Move with the Flow'.

10.3.2 'The Strength of a Vortex Tube is Constant' -- 10.4 Line Vortices -- 10.4.1 The Two-dimensional Vortex -- 10.4.2 Arbitrarily Oriented Rectilinear Vortex Filaments -- 10.5 Segmented Vortex Filaments -- 10.5.1 The Biot-Savart Law -- 10.5.2 Rectilinear Vortex Filaments -- 10.5.3 Finite Rectilinear Vortex Filaments -- 10.5.4 Infinite Straight Line Vortices -- 10.5.5 Semi-infinite Straight Line Vortex -- 10.5.6 Truncating Infinite Vortex Segments -- 10.5.7 Implementing Line Vortices in Octave -- 10.6 Exercises -- 10.7 Further Reading -- References -- 11 Lifting Line Theory -- 11.1 Basic Assumptions of Lifting Line Theory -- 11.2 The Lifting Line, Horseshoe Vortices, and the Wake -- 11.2.1 Deductions from Vortex Theorems -- 11.2.2 Deductions from the Wing Pressure Distribution -- 11.2.3 The Lifting Line Model of Air Flow -- 11.2.4 Horseshoe Vortex -- 11.2.5 Continuous Trailing Vortex Sheet -- 11.2.6 The Form of the Wake -- 11.3 The Effect of Downwash -- 11.3.1 Effect on the Angle of Incidence: Induced Incidence -- 11.3.2 Effect on the Aerodynamic Force: Induced Drag -- 11.4 The Lifting Line Equation -- 11.4.1 Glauert's Solution of the Lifting Line Equation -- 11.4.2 Wing Properties in Terms of Glauert's Expansion -- 11.5 The Elliptic Lift Loading -- 11.5.1 Properties of the Elliptic Lift Loading -- 11.6 Lift-Incidence Relation -- 11.6.1 Linear Lift-Incidence Relation -- 11.7 Realizing the Elliptic Lift Loading -- 11.7.1 Corrections to the Elliptic Loading Approximation -- 11.8 Exercises -- 11.9 Further Reading -- References -- 12 Nonelliptic Lift Loading -- 12.1 Solving the Lifting Line Equation -- 12.1.1 The Sectional Lift-Incidence Relation -- 12.1.2 Linear Sectional Lift-Incidence Relation -- 12.1.3 Finite Approximation: Truncation and Collocation -- 12.1.4 Computer Implementation -- 12.1.5 Example: a Rectangular Wing.

12.2 Numerical Convergence.