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 Circulation.- 4.3.4 The Discretization of the Equations.- 4.3.5 The Lifting Profile.- 4.3.6 The Local Pressure Coefficient.- 4.3.7 Appendix.- 4.3.8 Numerical Determinations.- 4.4 The Airfoil in Ground Effects.- 4.4.1 The Representation of the Solution.- 4.4.2 The Integral Equation.- 4.4.3 The Computer Implementation.- 4.4.4 The Treatment of the Method.- 4.4.5 The Circular Obstacle in a Compressible Fluid.- 4.4.6 Appendix.- 4.5 The Airfoil in Tunnel Effects.- 4.5.1 The Representation of the Solution.- 4.5.2 Green Functions.- 4.5.3 The Integral Equation.- 4.5.4 The Verification of the Method.- 4.5.5 Appendix.- 4.6 Other Methods. The Intrinsic Integral Equation.- 4.6.1 The Method of Regularization.- 5 The Theory of Finite Span Airfoil in Subsonic Flow. The Lifting Surface Theory.- 5.1 The Lifting Surface Equation.- 5.1.1 The Statement of the Problem.- 5.1.2 Bibliographical Comments.- 5.1.3 The General Solution.- 5.1.4 The Boundary Values of the Pressure.- 5.1.5 The Boundary Values of the Component ?.- 5.1.6 The Integral Equation.- 5.1.7 Other Forms of the Integral Equation.- 5.1.8 The Plane Problem.- 5.1.9 The Aerodynamic Action in the First Approximation.- 5.1.10 A More Accurate Calculation.- 5.1.11 Another Deduction of the Representation of the General Solution.- 5.2 Methods for the Numerical Integration of the Lifting Surface Equation.- 5.2.1 The General Theory.- 5.2.2 Multhopp's Method.- 5.2.3 The Quadrature Formulas Method.- 5.2.4 The Aerodynamic Action.- 5.2.5 The Third Method.- 5.3 Ground Effects in the Lifting Surface Theory.- 5.3.1 The General Solution.- 5.3.2 The Integral Equation.- 5.3.3 The Two-Dimensional Problem.- 5.4 The Wing of Low Aspect Ratio.- 5.4.1 The Integral Equation.- 5.4.2 The Case h = h(x).- 5.4.3 The General Case.- 6 The Lifting Line Theory.- 6.1 Prandtl's Theory.- 6.1.1 The Lifting Line Hypotheses. The Velocity Field.- 6.1.2 Prandtl's Equation.- 6.1.3 The Aerodynamic Action.- 6.1.4 The Elliptical Flat Plate.- 6.2 The Theory of Integration of Prandtl's Equation. The Reduction to Fredholm-Type Integral Equations.- 6.2.1 The Equation of Trefftz and Schmidt.- 6.2.2 Existence and Uniqueness Theorems.- 6.2.3 Foundation of Glauert's Method.- 6.2.4 Glauert's Approximation.- 6.2.5 The Minimal Drag Airfoil.- 6.3 The Symmetrical Wing. Vekua's Equation. A Larger Class of Exact Solutions.- 6.3.1 Symmetry Properties.- 6.3.2 The Integral Equation.- 6.3.3 Vekua's Equation.- 6.3.4 The Elliptical Wing.- 6.3.5 The Rectangular Wing.- 6.3.6 Extensions.- 6.4 Numerical Methods.- 6.4.1 Multhopp's Method.- 6.4.2 The Quadrature Formulas Method.- 6.4.3 The Collocation Method.- 6.5 Various Extensions of the Lifting Line Theory.- 6.5.1 The Equation of Weissinger and Reissner.- 6.5.2 Weissinger's Equation. The Rectangular Wing.- 6.6 The Lifting Line Theory in Ground Effects.- 6.6.1 The Integral Equation.- 6.6.2 The Elliptical Flat Plate.- 6.6.3 Numerical Solutions in the General Case.- 6.7 The Curved Lifting Line.- 6.7.1 The Pressure and Velocity Fields.- 6.7.2 The Integral Equation.- 6.7.3 The Numerical Method.- 7 The Application of the Boundary Integral Equations Method to the Theory of the Three-Dimensional Airfoil in Subsonic Flow.- 7.1 The First Indirect Method (Sources Distributions).- 7.1.1 The General Equations.- 7.1.2 The Integral Equation.- 7.1.3 The Integral Equation.- 7.1.4 The Discretization of the Integral Equation.- 7.1.5 The Singular Integrals.- 7.1.6 The Velocity Field. The Validation of the Method.- 7.1.7 The Incompressible Fluid. An Exact Solution.- 7.1.8 The Expression of the Potential.- 7.2 The Second Indirect Method (Doublet Distributions). The Incompressible Fluid.- 7.2.1 The Integral Equation.- 7.2.2 The Flow past the Sphere. The Exact Solution.- 7.2.3 The Velocity Field.- 7.2.4 The Velocity Field on the Body. N. Marcov's Formula.- 7.3 The Direct Method. The Incompressible Fluid.- 7.3.1 The Integral Representation Formula.- 7.3.2 The Integral Equation.- 7.3.3 Kutta's Condition.- 7.3.4 The Lifting Flow.- 7.3.5 The Discretization of the Integral Equation.- 8 The Supersonic Steady Flow.- 8.1 The Thin Airfoil of Infinite Span.- 8.1.1 The Analytical Solution.- 8.1.2 The Fundamental Solutions Method.- 8.1.3 The Aerodynamic Action.- 8.1.4 The Graphical Method.- 8.1.5 The Theory of Polygonal Profiles.- 8.1.6 Validity Conditions.- 8.2 Ground and Tunnel Effects.- 8.2.1 The General Solution.- 8.2.2 The Aerodynamic Coefficients.- 8.3 The Three-Dimensional Wing.- 8.3.1 Subsonic and Supersonic Edges.- 8.3.2 The Representation of the General Solution.- 8.3.3 The Influence Zones. The Domain D1.- 8.3.4 The Boundary Values of the Pressure.- 8.3.5 The First Form of the Integral Equation.- 8.3.6 The Equation D in Coordinates on Characteristics.- 8.3.7 The Plane Problem.- 8.3.8 The Equation of Heaslet and Lomax (the HL Equation).- 8.3.9 The Deduction of HL Equation from D Equation.- 8.3.10 The Equation of Homentcovschi (H Equation).- 8.4 The Theory of Integration of the H Equation.- 8.4.1 Abel's Equation.- 8.4.2 The Solution of the H Equation in the Domain of Influence of the Supersonic Trailing Edge.- 8.4.3 The Solution in the Domains of Influence of the Subsonic Leading Edge.- 8.4.4 The Wing with Dependent Subsonic Leading Edges and Independent Subsonic Trailing Edges.- 8.4.5 The Wing with Dependent Subsonic Trailing Edges.- 8.4.6 The Solution in the Zone of Influence of the Subsonic Edges under the Hypothesis that the Subsonic Leading Edges are Independent.- 8.4.7 The Wing with Dependent Subsonic Trailing Edges.- 8.5 The Theory of Conical Motions.- 8.5.1 Introduction.- 8.5.2 The Wing with Supersonic Leading Edges.- 8.5.3 The Wing with a Supersonic Leading Edge and with Another Subsonic Leading or Trailing Edge.- 8.5.4 The Wing with Subsonic Leading Edges.- 8.6 Flat Wings.- 8.6.1 The Angular Wing with Supersonic Leading Edges.- 8.6.2 The Triangular Wing. The Calculation of the Aerodynamic Action.- 8.6.3 The Trapezoidal Wing with Subsonic Lateral Edges.- 8.6.4 The Trapezoidal Wing with Lateral Supersonic Edges.- 9 The Steady Transonic Flow.- 9.1 The Equations of the Transonic Flow.- 9.1.1 The Presence of the Transonic Flow.- 9.1.2 The Equation of the Potential.- 9.1.3 The System of Transonic Flow.- 9.1.4 The Shock Equations.- 9.2 The Plane Flow.- 9.2.1 The Fundamental Solution.- 9.2.2 The General Solution.- 9.2.3 The Lift Coefficient.- 9.2.4 The Symmetric Wing.- 9.2.5 The Solution in Real.- 9.2.6 The Symmetric Wing.- 9.3 The Three-Dimensional Flow.- 9.3.1 The Fundamental Solution.- 9.3.2 The Study of the Singular Integrals.- 9.3.3 The General Solution.- 9.3.4 Flows with Shock Waves.- 9.4 The Lifting Line Theory.- 9.4.1 The Velocity Field.- 9.4.2 The Integral Equations.- 10 The Unsteady Flow.- 10.1 The Oscillatory Profile in a Subsonic Stream.- 10.1.1 The Statement of the Problem.- 10.1.2 The Fundamental Solution.- 10.1.3 The Integral Equation.- 10.1.4 Considerations on the Kernel.- 10.2 The Oscillatory Surface in a Subsonic Stream.- 10.2.1 The General Solution.- 10.2.2 The Integral Equation.- 10.2.3 Other Expressions of the Kernel Function.- 10.2.4 The Structure of the Kernel.- 10.2.5 The Sonic Flow.- 10.2.6 The Plane Flow.- 10.3 The Theory of the Oscillatory Profile in a Supersonic Stream.- 10.3.1 The General Solution.- 10.3.2 The Integral Equation and its Solution.- 10.3.3 Formulas for the Lift and Moment Coefficients.- 10.3.4 The Flat Plate.- 10.3.5 The Oscillatory Profile in the Sonic Flow.- 10.4 The Theory of the Oscillatory Wing in a Supersonic Stream.- 10.4.1 The General Solution.- 10.4.2 The Boundary Values of the Pressure.- 10.4.3 The Boundary Values of the Velocity. The Integral Equation.- 10.4.4 Other Expressions of the Kernel.- 10.4.5 A New Form.- 10.4.6 The Plane Problem.- 10.5 The Oscillatory Profile in a Sonic Stream.- 10.5.1 The General Solution. The Integral Equation.- 10.5.2 Some Formulas for the Lift and Moment Coefficients.- 10.6 The Three-Dimensional Sonic Flow.- 10.6.1 The General Solution.- 10.6.2 The Integral Equation.- 10.6.3 The Plane Problem.- 10.6.4 Other Forms of the Kernel.- 11 The Theory of Slender Bodies.- 11.1 The Linear Equations and Their Fundamental Solutions.- 11.1.1 The Boundary Condition. The Linear Equations.- 11.1.2 Fundamental Solutions.- 11.2 The Slender Body in a Subsonic Stream.- 11.2.1 The Solution of the Problem.- 11.2.2 The Calculus of Lift and Moment Coefficients.- 11.3 The Thin Body in a Supersonic Stream.- 11.3.1 The General Solution.- 11.3.2 The Pressure on the Body. The Lift and Moment Coefficients.- 11.3.3 The wing at zero angle of attack.- 11.3.4 Applications.- 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 Integrals in 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.