Mathematical Methods in Aerodynamics

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Table of Contents\0......Page 6
Preface\0......Page 14
1.1.1 Elements of Kinematics\0......Page 18
1.1.2 The Equations of Motion\0......Page 20
1.2.1 Helmholtz\'s equation. Bernoulli\'s integral\0......Page 22
1.2.2 The Equation of the Potential\0......Page 23
1.2.3 The Linear Theory\0......Page 25
1.2.4 The Acceleration Potential\0......Page 26
1.3.1 The Jump Equations\0......Page 28
1.3.2 ilugoniot\'s Equation\0......Page 30
1.3.3 The Solution of the lump Equations\0......Page 32
1.3.4 Prandtl\'s Formula\0......Page 33
1.3.5 The Shock Polar\0......Page 34
10.1.3 The Integral Equation                        ie.\0......Page 36
2.1.1 The Fundamental Problem of Aerodynamics\0......Page 38
2.1.2 The Equations of Motion\0......Page 39
2.1.3 The Equations of Linear Aerodynamics\0......Page 41
2.1.4 The Equation of the Potential\0......Page 43
2.1.5 The Linear System\0......Page 45
2.1.6 The Uniform Motion in the Fluid at Rest\0......Page 46
2.2.1 The Steady Solutions\0......Page 47
2.2.2 Oscillatory Solutions\0......Page 49
2.2.3 Oscillatory Solutions for Al = 1\0......Page 51
2.2.4 The Unsteady Solutions\0......Page 53
2.2.5 The Unsteady Solutions for At = I\0......Page 58
2.2.6 The Fundamental Solutions for the Fluid at Rest\0......Page 59
2.2.7 On the Interpretation of the Fundamental Solution\0......Page 60
2.3.1 The Significance of the Fundamental Solution\0......Page 61
2.3.2 The General Form of the Fundeunental Solution\0......Page 62
2.3.3 The Subsonic Plane Solution\0......Page 63
2.3.4 The Three-Dimensional Subsonic Solution\0......Page 64
2.3.6 The Three-Dimensional Supersonic Solution\0......Page 65
2.4.1 The Determination of Pressure\0......Page 67
2.4.2 The Determination of the Velocity Field\0......Page 68
2.4.3 Other Fortes of the Components V and W\0......Page 70
2.4.5 The Fundamental Solutions in the Case Af = I\0......Page 72
2.5.1 Fundamental Solutions\0......Page 74
2.5.2 Fundamental Matrices\0......Page 75
2.3.3 Cauchy\'s Problem\0......Page 78
2.3.4 The Perturbation Produced by a Mobile Source\0......Page 79
2.6.1 The Fundamental Matrices\0......Page 81
2.6.2 The Method of the Minimal Polynomial\0......Page 83
3.1.1 The Statement of the Problem\0......Page 86
3.1.2 A Classical Method\0......Page 87
3.1.3 The Fundamental Solutions Method\0......Page 89
3.1.4 The Function f (z). The Complex Velocity in the Fluid\0......Page 92
3.1.5 The Calculation of the Aerodynamic Action\0......Page 93
3.1.6 Examples\0......Page 94
3.1.7 The General Case\0......Page 97
3.1.9 The Integration of the Thin Airfoil Equation with the Aid of Gauss-type Quadrature Formulas\0......Page 98
3.2.1 The Integral Equation\0......Page 99
3.2.3 The Flat Plate\0......Page 102
3.2.4 The Symmetric Airfoil\0......Page 103
3.3.1 The Integral Equation\0......Page 105
3.3.2 The Integration of the Equation (3.3.9)\0......Page 107
3.3.3 Numerical Results\0......Page 108
3.4.1 The Integral Equations\0......Page 109
3.4.2 The Numerical Integration\0......Page 112
3.5.1 The Integral Equation\0......Page 114
3.5.2 The Numerical Integration\0......Page 117
3.6.1 The Integral Equations\0......Page 118
3.6.2 The Determination of the Functions f, and ff\0......Page 120
3.6.3 The Lift and Moment Coefficients\0......Page 121
3.6.4 Numerical Values\0......Page 122
4.1.1 Introduction\0......Page 126
4.1.2 The Statement of the Problem\0......Page 127
4.1.3 The Fundamental Solutions\0......Page 129
4.2.1 The integral equation for the Distribution of Sources\0......Page 130
4.2.3 The Boundary Elements Method\0......Page 132
4.2.4 The Determination of the Unknowns\0......Page 134
4.2.5 The Circular Obstacle\0......Page 137
4.2.6 The Elliptical Obstacle\0......Page 138
4.3.1 The representation of the solution\0......Page 139
4.3.2 The Integral Equation\0......Page 140
4.3.3 The Circulation\0......Page 142
4.3.5 The Lifting Profile\0......Page 143
4.3.6 The Local Pressure Coefficient\0......Page 145
4.3.7 Appendix\0......Page 146
4.4.1 The Representation of the Solution\0......Page 148
4.4.2 The Integral Equation\0......Page 151
4.4.3 The Computer Implementation\0......Page 152
4.4.4 The Treatment of the Method\0......Page 153
4.4.5 The Circular Obstacle in a Compressible Fluid\0......Page 154
4.4.6 Appendix\0......Page 155
4.5.1 The Representation of the Solution\0......Page 157
4.5.2 Green Functions\0......Page 159
4.5.3 The Integral Equation\0......Page 161
4.5.4 The Verification of the Method\0......Page 163
4.5.5 Appendix\0......Page 166
4.6.1 The Method of Regularization\0......Page 167
5.1.1 The Statement of the Problem\0......Page 172
5.1.2 Bibliographical Comments\0......Page 175
5.1.3 The General Solution\0......Page 176
5.1.4 The Boundary Values of the Pressure\0......Page 178
5.1.5 The Boundary Values of the Component w\0......Page 180
5.1.6 The Integral Equation\0......Page 181
5.1.7 Other Forms of the Integral Equation\0......Page 183
5.1.8 The Plane Problem\0......Page 185
5.1.9 The Aerodynamic Action in the First Approximation\0......Page 186
5.1.10 A More Accurate Calculation\0......Page 188
5.1.11 Another Deduction of the Representation of the General Solution\0......Page 190
5.2.1 The General Theory\0......Page 192
5.2.2 Multhopp\'s Method\0......Page 195
5.2.3 The Quadratum Formulas Method\0......Page 196
5.2.4 The Aerodynamic Action\0......Page 197
5.2.5 The Third Method\0......Page 198
5.3.1 The General Solution\0......Page 201
5.3.2 The Integral Equation\0......Page 203
5.3.3 The Two-Dimensional Problem\0......Page 205
5.4.1 The Integral Equation\0......Page 206
5.4.2 The Case h = h(x)\0......Page 209
5.4.3 The General Case\0......Page 210
6.1.1 The Lifting Line Hypotheses. The Velocity Field\0......Page 214
6.1.2 Prandtl\'s Equation\0......Page 217
6.1.3 The Aerodynamic Action\0......Page 219
6.1.4 The Elliptical Flat Plate\0......Page 220
6.2.1 The Equation of Trefftz and Schmidt\0......Page 222
6.2.2 Existence and Uniqueness Theorems\0......Page 226
6.2.3 Foundation of Glauert\'s Method\0......Page 227
6.2.5 The Minimal Drag Airfoil\0......Page 229
6.3.1 Symmetry Properties\0......Page 231
6.3.2 The Integral Equation\0......Page 232
6.3.3 Vekua\'s Equation\0......Page 234
6.3.4 The Elliptical Wing\0......Page 237
6.3.5 The Rectangular Wing\0......Page 238
6.3.6 Extensions\0......Page 239
6.4.1 Multhopp\'s Method\0......Page 240
6.4.2 The Quadrature Formulas Method\0......Page 245
6.4.3 The Collocation Method\0......Page 248
6.5.1 The Equation of Weissinger and Reissner\0......Page 251
6.5.2 Weissinger\'s Equation. The Rectangular Wing\0......Page 253
6.6 The Lifting Line Theory in Ground Effects       2.\0......Page 35
6.6.1 The Integral Equation\0......Page 255
6.6.2 The Elliptical Flat Plate\0......Page 257
6.6.3 Numerical Solutions in the General Case\0......Page 258
6.7.1 The Pressure and Velocity Fields\0......Page 259
6.7.2 The Integral Equation\0......Page 263
6.7.3 The Numerical Method\0......Page 264
7.1.1 The General Equations\0......Page 268
7.1.3 The Integral Equation\0......Page 270
7.1.4 The Discretixation of the Integral Equation\0......Page 272
7.1.6 The Velocity Field. The Validation of the Method\0......Page 275
7.1.7 The Incompressible Fluid. An Exact Solution\0......Page 276
7.1.8 The Expression of the Potential\0......Page 280
7.2.1 The Integral Equation\0......Page 282
7.2.2 The Flow past the Sphere. The Exact Solution\0......Page 284
7.2.4 The Velocity Field on the Body. N. Marcov\'s Formula\0......Page 285
7.3.1 The Integral Representation Formula\0......Page 288
7.3.2 The Integral Equation\0......Page 291
7.3.3 Kutta\'s Condition\0......Page 292
7.3.4 The Lifting Flow\0......Page 293
7.3.5 The Discretization of the Integral Equation\0......Page 296
8.1.1 The Analytical Solution\0......Page 300
8.1.2 The Fundamental Solutions Method\0......Page 303
8.1.3 The Aerodynamic. Action\0......Page 304
8.1.4 The Graphical Method\0......Page 306
8.1.5 The Theory of Polygonal Profiles\0......Page 307
8.1.6 Validity Conditions\0......Page 311
8.2.1 The General Solution\0......Page 312
8.2.2 The Aerodynamic Coefficients\0......Page 315
8.3.1 Subsonic and Supersonic Edges\0......Page 317
8.3.2 The Representation of the General Solution\0......Page 319
8.3.3 The Influence Zones. The Domain Di\0......Page 321
8.3.4 The Boundary Values of the Pressure\0......Page 322
8.3.5 The First Form of the Integral Equation\0......Page 323
8.3.6 The Equation D in Coordinates on Characteristics\0......Page 325
8.3.7 The Plane Problem\0......Page 327
8.3.8 The Equation of Heaslet and Lomax (the 11L Equation)\0......Page 328
8.3.9 The Deduction of HL Equation from D Equation\0......Page 330
8.3.10 The Equation of Homentcovschi (II Equation)\0......Page 335
8.4.1 Abel\'s Equation\0......Page 337
8.4.2 The Solution of the H Equation in the Domain of Influence of the Supersonic Trailing Edge\0......Page 338
8.4.3 The Solution in the Domains of Influence of the Subsonic Leading Edge\0......Page 340
8.4.4 The Wing with Dependent Subsonic Leading Edges and Independent Subsonic Trailing Edges\0......Page 341
8.4.5 The Wing with Dependent Subsonic Trailing Edges\0......Page 343
8.4.6 The Solution in the Zone of Influence of the Subsonic Edges under the Hypothesis that the Subsonic Leading Edges are let-dependent\0......Page 344
8.4.7 The Wing with Dependent Subsonic Trailing Edges\0......Page 354
8.5.1 Introduction\0......Page 356
8.5.2 The Wing with Supersonic Leading Edges\0......Page 357
8.5.3 The Wing With a Supersonic Leading FAlge and with Another  Subsonic Leading or Trailing Edge\0......Page 359
8.5.4 The Wing with Subsonic Leading Edges\0......Page 360
8.6.1 The Angular Wing with Supersonic Leading Edges\0......Page 364
8.6.2 The Triangular Wing. The Calculation of the Aerodynamic Action\0......Page 366
8.6.3 The Trapezoidal Wing with Subsonic Lateral Edges\0......Page 369
8.6.4 The Trapezoidal Wing with Lateral Supersonic Edges\0......Page 372
9.1.1 The Presence of the Transonic Flow\0......Page 376
9.1 The Equations of the Transonic Flow              :3.\0......Page 76
9.1.2 The Equation of the Potential\0......Page 378
9.1.3 The System of \'transonic Flow\0......Page 381
9.1.4 The Shock Equations\0......Page 385
9.2.1 The Fundamental Solution\0......Page 386
9.2.2 The General Solution\0......Page 388
9.2.4 The Symmetric Wing\0......Page 393
9.2.5 The Solution in Real\0......Page 394
9.2.6 The Symmetric Wing\0......Page 397
9.:3.1 The Fundamental Solution\0......Page 400
9.3.2 The Study of the Singular Integrals\0......Page 403
9.3.3 The General Solution\0......Page 404
9.3.4 Flows with Shock Waves                 :\0......Page 406
9.4.1 The Velocity Field              :\0......Page 409
9.4.2 The Integral Equations\0......Page 411
10.1.2 The Fundamental Solution\0......Page 414
10.1.4 Considerations on the Kernel\0......Page 419
10.2.1 The General Solution\0......Page 421
10.2.2 The Integral Equation\0......Page 422
10.2.3 Other Expressions of the Kernel Function\0......Page 426
10.2.4 The Structure of the Kernel\0......Page 429
10.2.5 The Sonic Flow\0......Page 430
10.2.6 The Plane Flow\0......Page 431
10.3.1 The General Solution\0......Page 432
10.3.2 The Integral Equation and its Solution\0......Page 435
10.3,3 Formulas for the Lift and Moment Coefficients\0......Page 438
10.3.4 The Flat Plate\0......Page 440
10.3.5 The Oscillatory Profile in the Sonic Flow\0......Page 441
10.4.1 The General Solution\0......Page 443
10.4.2 The Boundary Values of the Pressure\0......Page 445
10.4.3 The Boundary Values of the Velocity, The Integral Equation\0......Page 447
10.4.4 Other Expressions of the Kernel\0......Page 450
10.4.5 A New Form\0......Page 452
10.4.6 The Plane Problem\0......Page 453
10.5.1 The General Solution. The Integral Equation\0......Page 455
10.5.2 Some Formulas for the Lift and Moment Coefficients\0......Page 458
10.6.1 The General Solution\0......Page 459
10.6.2 The Integral Equation\0......Page 460
10.6.3 The Plane Problem\0......Page 463
10.6.4 Other Forms of the Kernel\0......Page 464
11.1.1 The Boundary Condition. The Linear Equations\0......Page 466
11.1.2 Fundamental Solutions\0......Page 469
11.2.1 The Solution of the Problem\0......Page 471
11.2.2 The Calculus of Lift and Moment Coefficients\0......Page 473
11.3.1 The General Solution\0......Page 475
11.3.2 The Pressure on the Body. The Lift and Moment Coefficients\0......Page 478
11.3.4 Applications\0......Page 480
A.1 The Fourier Transform of Functions\0......Page 482
A.2 The Spaces V and S\0......Page 483
A.3 Distributions\0......Page 484
A.4 The Convolution. Fundamental Solutions\0......Page 487
A.5 The Fourier Transform of the Functions from S\0......Page 489
A.6 The Fourier Transform of the Temperate Distributions\0......Page 490
A.7 The Calculus of Some Inverse Fourier Transforms\0......Page 492
A.8 The Fourier Transform in Bounded Domains\0......Page 495
B.1 Cauchy-type Integrals\0......Page 498
B.2 The Principal Value in Cauchy\'s Sense\0......Page 499
B.4 The Dirichlet\'s Problem for the Half-Plane\0......Page 500
B.5 The Calculus of Certain Integrals in the Complex Plane\0......Page 502
B.6 Glauert\'s Integral. Its Generalization and Some Applications\0......Page 506
B.7 Other Integrals\0......Page 508
C. 1 The Thin Profile Equation\0......Page 510
C.2 The Generalized Equation of Thin Profiles\0......Page 513
C.3 The Third Equation\0......Page 515
C.4 The Forth Equation\0......Page 519
C.5 The Fifth Equation\0......Page 521
D.2 The First Integral\0......Page 526
D.3 Integrals with Singularities in an interval\0......Page 527
D.4 Hadamard-Type Integrals\0......Page 530
D.5 Generalization\0......Page 532
E Singular Multiple Integrals\0......Page 534
F.1 General Theorems\0......Page 538
F.2 Formulas of Interest in Aerodynamics\0......Page 542
F.3 The Modified Monegato\'s Formula\0......Page 546
F.4 A Useful Formula\0......Page 547
Bibliography\0......Page 550
Index\0......Page 588