Mathematical Methods in Aerodynamics

1 The Equations of Ideal Fluids.- 1.1 The Equations of Motion.- 1.1.1 Elements of Kinematics.- 1.1.2 The Equations of Motion.- 1.2 The Potential Flow.- 1.2.1 Helmholtz's equation. Bernoulli's integral.- 1.2.2 The Equation of the Potential.- 1.2.3 The Linear Theory.- 1.2.4 The Acceleration Potential.- 1.3 The Shock Waves Theory.- 1.3.1 The Jump Equations.- 1.3.2 Hugoniot's Equation.- 1.3.3 The Solution of the Jump Equations.- 1.3.4 Prandtl's Formula.- 1.3.5 The Shock Polar.- 1.3.6 The Compression Shock past a Concave Bend.- 2 The Equations of Linear Aerodynamics and its Fundamental Solutions.- 2.1 The Equations of Linear Aerodynamics.- 2.1.1 The Fundamental Problem of Aerodynamics.- 2.1.2 The Equations of Motion.- 2.1.3 The Equations of Linear Aerodynamics.- 2.1.4 The Equation of the Potential.- 2.1.5 The Linear System.- 2.1.6 The Uniform Motion in the Fluid at Rest.- 2.2 The Fundamental Solutions of the Equation of the Potential.- 2.2.1 The Steady Solutions.- 2.2.2 Oscillatory Solutions.- 2.2.3 Oscillatory Solutions for M = 1.- 2.2.4 The Unsteady Solutions.- 2.2.5 The Unsteady Solutions for M = 1.- 2.2.6 The Fundamental Solutions for the Fluid at Rest.- 2.2.7 On the Interpretation of the Fundamental Solution.- 2.3 The Fundamental Solutions of the Steady System.- 2.3.1 The Significance of the Fundamental Solution.- 2.3.2 The General Form of the Fundamental Solution.- 2.3.3 The Subsonic Plane Solution.- 2.3.4 The Three-Dimensional Subsonic Solution.- 2.3.5 The Two-Dimensional Supersonic Solution.- 2.3.6 The Three-Dimensional Supersonic Solution.- 2.4 The Fundamental Solutions of the Oscillatory System.- 2.4.1 The Determination of Pressure.- 2.4.2 The Determination of the Velocity Field.- 2.4.3 Other Forms of the Components V and W.- 2.4.4 The Incompressible Fluid.- 2.4.5 The Fundamental Solutions in the Case M = 1.- 2.5 Fundamental Solutions of the Unsteady System I.- 2.5.1 Fundamental Solutions.- 2.5.2 Fundamental Matrices.- 2.5.3 Cauchy's Problem.- 2.5.4 The Perturbation Produced by a Mobile Source.- 2.6 Fundamental Solutions of the Unsteady System II.- 2.6.1 The Fundamental Matrices.- 2.6.2 The Method of the Minimal Polynomial.- 3 The Infinite Span Airfoil in Subsonic Flow.- 3.1 The Airfoil in the Unlimited Fluid.- 3.1.1 The Statement of the Problem.- 3.1.2 A Classical Method.- 3.1.3 The Fundamental Solutions Method.- 3.1.4 The Function f(x). The Complex Velocity in the Fluid.- 3.1.5 The Calculation of the Aerodynamic Action.- 3.1.6 Examples.- 3.1.7 The General Case.- 3.1.8 Numerical Integrations.- 3.1.9 The Integration of the Thin Airfoil Equation with the Aid of Gauss-type Quadrature Formulas.- 3.2 The Airfoil in Ground Effects.- 3.2.1 The Integral Equation.- 3.2.2 A Numerical Method.- 3.2.3 The Flat Plate.- 3.2.4 The Symmetric Airfoil.- 3.3 The Airfoil in Tunnel Effects.- 3.3.1 The Integral Equation.- 3.3.2 The Integration of the Equation (3.3.9).- 3.3.3 Numerical Results.- 3.4 Airfoils Parallel to the Undisturbed Stream.- 3.4.1 The Integral Equations.- 3.4.2 The Numerical Integration.- 3.5 Grids of Profiles.- 3.5.1 The Integral Equation.- 3.5.2 The Numerical Integration.- 3.6 Airfoils in Tandem.- 3.6.1 The Integral Equations.- 3.6.2 The Determination of the Functions f1 and f2.- 3.6.3 The Lift and Moment Coefficients.- 3.6.4 Numerical Values.- 4 The Application of the Boundary Element Method to the Theory of the Infinite Span Airfoil in Subsonic Flow.- 4.1 The Equations of Motion.- 4.1.1 Introduction.- 4.1.2 The Statement of the Problem.- 4.1.3 The Fundamental Solutions.- 4.2 Indirect Methods for the Unlimited Fluid Case.- 4.2.1 The integral equation for the Distribution of Sources.- 4.2.2 The Integral Equation for the Distribution of Vortices.- 4.2.3 The Boundary Elements Method.- 4.2.4 The Determination of the Unknowns.- 4.2.5 The Circular Obstacle.- 4.2.6 The Elliptical Obstacle.- 4.3 The Direct Method for the Unlimited Fluid Case.- 4.3.1 The representation of the solution.- 4.3.2 The Integral Equation.- 4.3.3 The