Method of discrete vortices

S. M. Belotserkovsky, I.K. Lifanov0849393078, 9780849393075

Method of Discrete Vortices presents a mathematical substantiation and in-depth description of numerical methods for solving singular integral equations with one-dimensional and multiple Cauchy integrals. The book also presents the fundamentals of the theory of singular equations and numerical methods as applied to solving problems in such branches of mechanics as aerodynamics, elasticity, and electrodynamics.

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Table of contents :
Author’s Preface to English Edition ……Page 4
Contents ……Page 6
An Introduction to Singular Integral Equations in Aerodynamics ……Page 10
Introduction ……Page 30
PART I: QUADRATURE FORMULAS FOR SINGULAR INTEGRALS ……Page 36
1.1. Some Definitions of the Theory of One-Dimensional Singular Integrals ……Page 37
1.2. Singular Integral over a Closed Contour ……Page 40
1.3. Singular Integral over a Segment ……Page 47
1.4. Singular Integral over a Piecewise Smooth Curve ……Page 59
1.5. Singular Integrals with Hilbert’s Kernel ……Page 65
1.6. Unification of Quadrature and Difference Formulas ……Page 66
2.1. Singular Integral with Hilbert’s Kernel ……Page 69
2.2. Singular Integral on a Circle ……Page 74
2.3. Singular Integral on a Segment ……Page 77
3.1. Multiple Cauchy Integrals ……Page 82
3.2. About Some Singular Integrals Frequently Used in Aerodynamics ……Page 90
3.3. Quadrature Formulas for Multiple Singular Integrals ……Page 99
3.4. Quadrature Formulas for the Singular Integral for a Finite-Span Wing ……Page 103
3.5. Quadrature Formulas for Multidimensional Singular Integrals ……Page 110
3.6. Examples of Calculating Singular Integrals ……Page 117
4.1. One-Dimensional Singular Integrals ……Page 121
4.2. Multiple Singular Cauchy Integrals ……Page 126
PART II: NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS ……Page 131
5.1. Characteristic Singular Equation on a Segment: Uniform Division ……Page 132
5.2. Characteristic Singular Equation on a Segment: Nonuniform Division ……Page 153
5.3. Full Equation on a Segment ……Page 159
5.4. Equation on a System of Nonintersecting Segments ……Page 167
5.5. Examples of Numerical Solution of the Equation on a Segment ……Page 173
6.1. Equation on a Circle ……Page 179
6.2. Equations with Hilbert’s Kernel ……Page 186
7.1. Equation on a Segment ……Page 195
7.2. Equation on a Circle ……Page 200
7.3. Equation with Hilbert’s Kernel ……Page 203
7.4. Equation on a Piecewise Smooth Curve with Variable Coefficients ……Page 207
7.5. Equation with Hilbert’s Kernel and Variable Coefficients ……Page 216
8.1. Analytical Solution to a Class of Characteristic Integral Equations ……Page 220
8.2. Numerical Solution of Singular Integral Equations with Multiple Cauchy Integrals ……Page 227
8.3. About an Integrodifferential Equation ……Page 242
PART III: APPLICATION OF THE METHOD OF DISCRETE VORTICES TO AERODYNAMICS: VERIFICATION OF THE METHOD ……Page 245
9.1. Formulation of Aerodynamic Problems ……Page 246
9.2. Fundamental Concepts of the Method of Discrete Vortices ……Page 249
9.3. Fundamental Discrete Vortex Systems ……Page 252
10.1. Steady Flow Past a Thin Airfoil ……Page 260
10.2. Airfoil Cascades ……Page 265
10.3. Thin Airfoil with Ejection ……Page 267
10.4. Finite-Thickness Airfoil with a Smooth Contour ……Page 272
10.5. Method of a “Slanting Normal” ……Page 277
10.6. A Permeable Airfoil ……Page 282
11.1. Flow with Circulation Past a Rectangular Wing ……Page 284
11.2. Flow without Circulation Past a Rectangular Wing ……Page 289
11.3. A Wing of an Arbitrary Plan Form ……Page 294
12.1. Linear Unsteady Problem for a Thin Airfoil ……Page 303
12.2. Numerical Solution of the Abel Equation ……Page 309
12.3. Some Examples of Numerical Solution of the Abel Equation ……Page 315
12.4. Nonlinear Unsteady Problem for a Thin Airfoil ……Page 316
13.1. Mathematical Formulation of the Problem ……Page 323
13.2. System of Integrodifferential Equations ……Page 324
13.3. Smooth Flow Past a Body: Virtual Inertia ……Page 328
13.4. Numerical Calculation of Virtual Inertia Coefficients: Some Calculated Data ……Page 331
13.5. Numerical Scheme for Calculating Separated Flows ……Page 335
14.1. Ill-Posedness of Equations Incorporating Singular Integrals ……Page 343
14.2. Regularization of Singular Integral Calculation ……Page 346
14.3. Method of Discrete Vortices and Regularization of Numerical Solutions to Singular Integral Equations ……Page 350
14.4. Regularization in the Case of Unsteady Aerodynamic Problems ……Page 357
PART IV: SOME PROBLEMS OF THE THEORY OF ELASTICITY, ELECTRODYNAMICS, AND MATHEMATICAL PHYSICS ……Page 359
15.1. Two-Dimensional Problems of the Theory of Elasticity ……Page 360
15.2. Contact Problem of Indentation of a Uniformly Moving Punch into an Elastic Half-Plane with Heat Generation ……Page 367
15.3. On the Indentation of a Pair of Uniformly Moving Punches into an Elastic Strip ……Page 379
16.1. Dual Equations for Solving Mixed Boundary Value Problems ……Page 386
16.2. Method for Solving Dual Equations ……Page 395
16.3. Diffraction of a Scalar Wave on a Plane Lattice: Dirichlet and Neumann Problems for the Helmholtz Equation ……Page 403
16.4. Application of the Method of Discrete Singularities to Numerical Solution of Problems of Electromagnetic Wave Diffraction on Lattices ……Page 413
17.1. Differentiation of Integral Equations of the First Kind with a Logarithmic Singularity ……Page 417
17.2. Dirichlet and Neumann Problems for the Laplace Equation ……Page 419
17.3. Mixed Boundary Value Problems ……Page 421
Conclusion ……Page 423
References ……Page 431
INDEX ……Page 436