Mathematical Methods in Aerodynamics

Mathematical Methods in Aerodynamics
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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.
The researchers in Aerodynamics know that there is not a unitary method of investigation in this field. The first mathematical model of the air­ plane wing, the model meaning the integral equation governing the phe­ nomenon, was proposed by L. Prandtl in 1918. The integral equation deduced by Prandtl, on the basis of some assumptions which will be specified in the sequeL furnishes the circulation C(y) (see Chapter 6). U sing the circulation, one calculates the lift and moment coefficients, which are very important in Aerodynamics. The first hypothesis made by Prandtl consists in replacing the wing by a distribution of vortices on the plan-form D of the wing (i. e. the projection of the wing on the plane determined by the direction of the uniform stream at infinity and t he direction of the span of the wing). Since such a distribution leads to a potential flow in the exterior of D and the experiences show that downstream the flow has not this character, Prandtl introduces as a sup­ plementary hypothesis another vortices distribution on the trace of the domain D in the uniform stream. The first kind of vortices are called tied vortices and the second kind of vortices are called free vortices.

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