Mathematical Methods in Aerodynamics

1 The Equations of Ideal Fluids.- 1.1 The Equations of Motion.- 1.2 The Potential Flow.- 1.3 The Shock Waves Theory.- 2 The Equations of Linear Aerodynamics and its Fundamental Solutions.- 2.1 The Equations of Linear Aerodynamics.- 2.2 The Fundamental Solutions of the Equation of the Potential.- 2.3 The Fundamental Solutions of the Steady System.- 2.4 The Fundamental Solutions of the Oscillatory System.- 2.5 Fundamental Solutions of the Unsteady System I.- 3 The Infinite Span Airfoil in Subsonic Flow.- 3.1 The Airfoil in the Unlimited Fluid.- 3.2 The Airfoil in Ground Effects.- 3.3 The Airfoil in Tunnel Effects.- 3.4 Airfoils Parallel to the Undisturbed Stream.- 3.5 Grids of Profiles.- 3.6 Airfoils in Tandem.- 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.- 4.2 Indirect Methods for the Unlimited Fluid Case.- 4.3 The Direct Method for the Unlimited Fluid Case.- 4.4 The Airfoil in Ground Effects.- 4.5 The Airfoil in Tunnel Effects.- 4.6 Other Methods. The Intrinsic Integral Equation.- 5 The Theory of Finite Span Airfoil in Subsonic Flow. The Lifting Surface Theory.- 5.1 The Lifting Surface Equation.- 5.2 Methods for the Numerical Integration of the Lifting Surface Equation.- 5.3 Ground Effects in the Lifting Surface Theory.- 5.4 The Wing of Low Aspect Ratio.- 6 The Lifting Line Theory.- 6.1 Prandtl’s Theory.- 6.2 The Theory of Integration of Prandtl’s Equation. The Reduction to Fredholm-Type Integral Equations.- 6.3 The Symmetrical Wing. Vekua’s Equation. A Larger Class of Exact Solutions.- 6.4 Numerical Methods.- 6.5 Various Extensions of the Lifting Line Theory.- 6.6 The Lifting Line Theory in Ground Effects.- 6.7 The Curved Lifting Line.- 7 The Application of the BoundaryIntegral Equations Method to the Theory of the Three-Dimensional Airfoil in Subsonic Flow.- 7.1 The First Indirect Method (Sources Distributions).- 7.2 The Second Indirect Method (Doublet Distributions). The Incompressible Fluid.- 7.3 The Direct Method. The Incompressible Fluid.- 8 The Supersonic Steady Flow.- 8.1 The Thin Airfoil of Infinite Span.- 8.2 Ground and Tunnel Effects.- 8.3 The Three-Dimensional Wing.- 8.4 The Theory of Integration of the H Equation.- 8.5 The Theory of Conical Motions.- 8.6 Flat Wings.- 9 The Steady Transonic Flow.- 9.1 The Equations of the Transonic Flow.- 9.2 The Plane Flow.- 9.3 The Three-Dimensional Flow.- 10 The Unsteady Flow.- 10.1 The Oscillatory Profile in a Subsonic Stream.- 10.2 The Oscillatory Surface in a Subsonic Stream.- 10.3 The Theory of the Oscillatory Profile in a Supersonic Stream.- 10.4 The Theory of the Oscillatory Wing in a Supersonic Stream.- 10.5 The Oscillatory Profile in a Sonic Stream.- 10.6 The Three-Dimensional Sonic Flow.- 11 The Theory of Slender Bodies.- 11.1 The Linear Equations and Their Fundamental Solutions.- 11.2 The Slender Body in a Subsonic Stream.- 11.3 The Thin Body in a Supersonic Stream.- A Fourier Transform and Notions of the Theory of Distributions.- A.1 The Fourier Transform of Functions.- A.3 Distributions.- A.4 The Convolution. Fundamental Solutions.- A.6 The Fourier Transform of the Temperate Distributions.- A.7 The Calculus of Some Inverse Fourier Transforms.- A.8 The Fourier Transform in Bounded Domains.- B Cauchy-type Integrals. Dirichlet’s Problem for the Half-Plane. The Calculus of Some Integrals.- B.1 Cauchy-type Integrals.- B.2 The Principal Value in Cauchy’s Sense.- B.3 Plemelj’s Formulas.- B.4 The Dirichlet’s Problem for the Half-Plane.- B.5 The Calculus of Certain Integralsin the Complex Plane.- B.6 Glauert’s Integral. Its Generalization and Some Applications.- B.7 Other Integrals.- C Singular Integral Equations.- C.1 The Thin Profile Equation.- C.2 The Generalized Equation of Thin Profiles.- C.3 The Third Equation.- C.4 The Forth Equation.- C.5 The Fifth Equation.- D The Finite Part.- D.1 Introductory Notions.- D.2 The First Integral.- D.3 Integrals with Singularities in an Interval.- D.4 Hadamard-Type Integrals.- D.5 Generalization.- E Singular Multiple Integrals.- F Gauss-Type Quadrature Formulas.- F.1 General Theorems.- F.2 Formulas of Interest in Aerodynamics.- F.3 The Modified Monegato’s Formula.- F.4 A Useful Formula.