Mathematical modeling of unsteady inviscid flows

Preface......Page 7
 Expected Level of Preparation......Page 8
 Other Sources......Page 9
Acknowledgments......Page 11
Contents......Page 12
 1.1 A Motivating Example Problem......Page 20
  1.1.1 What Would Viscosity Do?......Page 21
  1.1.2 The Inviscid Model......Page 22
 1.2 Organization of the Book......Page 24
 2.1 Reference Frames......Page 25
 2.2 Body Definitions......Page 26
 2.3 Vector Notation......Page 28
 2.4 Complex Notation......Page 34
 2.5 Plücker Notation......Page 35
   Note 2.5.1: Plücker Transform Matrices in Two Dimensions......Page 38
   Note 2.5.2: Setting the Pivot Axis of a Translating and Rotating Body......Page 39
3 Foundational Concepts......Page 42
  3.1.1 The Velocity Field and Flow Potentials......Page 43
   Result 3.1: Volume Flow Rate and Vector Potential......Page 44
  3.1.2 The No-Penetration Condition......Page 45
  3.1.3 Vorticity Transport Equation......Page 46
  3.1.4 Kinetic Energy and Rate of Work......Page 47
   Result 3.2: A Form of the Kinetic Energy in Three Dimensions......Page 48
   Note 3.1.2: Another Form of the Rotational Part of Kinetic Energy......Page 49
  3.1.5 Circulation and Its Invariance......Page 51
   Result 3.3: Kelvin's Circulation Theorem......Page 53
   Note 3.1.3: Other Invariants of Unbounded Inviscid Flow......Page 54
  3.1.6 Helmholtz' Theorems......Page 56
 3.2 Elements of Potential Flow......Page 60
  3.2.1 Pressure and the Bernoulli Equation......Page 61
   Note 3.2.1: Pressure in the Inertial and Windtunnel Frames......Page 62
  3.2.2 Two-Dimensional Flows and the Complex Potential......Page 63
   Note 3.2.2: Multi-Valuedness and the Branch Cut......Page 67
   Note 3.2.3: Velocity Behavior in Corner Flows......Page 70
  3.2.3 Three-Dimensional Flows......Page 73
 3.3 Vortex Structures......Page 75
  3.3.1 Point Vortex......Page 76
  3.3.2 Vortex Filament......Page 77
   Note 3.3.1: The Scalar Potential Field of a Closed Vortex Filament......Page 80
   Result 3.4: Volume Flow Rate Induced by a Closed Filament......Page 82
   Result 3.5: Strength of a Vortex Sheet......Page 83
   Note 3.3.2: Parameterization of a Vortex Sheet......Page 84
 3.4 Other Surface Distributions of Singularities......Page 89
  3.5.1 Two-Dimensional Vortex Sheet as a Set of Point Vortices......Page 91
  3.5.2 Vortex Filament or Vortex Pair as a Double-Layer Potential......Page 93
  3.5.3 Vortex Sheet as a Double-Layer Potential......Page 94
  3.5.4 Three-Dimensional Vortex Sheet as a Mesh of Vortex Filaments......Page 95
 3.6 Free and Bound Vortex Sheets......Page 97
  3.6.1 Free Vortex Sheets......Page 98
   Note 3.6.1: Parameterization of a Vortex Sheet by Circulation......Page 99
   Result 3.6: Characteristics of a Free Vortex Sheet......Page 101
   Note 3.6.2: The Surface Vortex Sheet and the No-Penetration Condition......Page 102
   Note 3.6.3: Complex Form of the No-Penetration Condition......Page 103
   Note 3.6.4: No-Penetration Condition on a Sheet Immersed in the Fluid......Page 104
   Note 3.6.5: A Note on Point Vortices Near Thin Plates......Page 106
4 General Results of Incompressible FlowAbout a Body......Page 108
 4.1 The Basic Potential Flow Problem......Page 109
  4.2.1 Vector Form......Page 111
   Result 4.1: Velocity Field of an Incompressible Flow in the Presence of a Body......Page 113
  4.2.2 Complex Form......Page 114
  4.2.3 Infinitely-Thin Plate......Page 115
  4.2.4 Alternative Surface Formulations: Source and Dipole Distributions......Page 116
 4.3 The Integral Equation for an Impenetrable Body......Page 118
  4.3.1 Vector Form......Page 119
   Result 4.3: General Integral Equation for Bound Vortex Sheet......Page 120
  4.3.2 Complex Form......Page 121
  4.3.3 Infinitely-Thin Plate......Page 122
  4.3.4 The Double Layer Formulation......Page 123
 4.4 Solution of Two-Dimensional Problem by Conformal Mapping......Page 124
   Note 4.4.1: Notational Convention for Functions and Mappings......Page 126
   Note 4.4.2: Solution for Rigid-Body Motion by Conformal Mapping......Page 128
  4.4.2 Stationary Body in a Known Potential Flow: The Circle Theorem......Page 130
   Note 4.4.3: Point Vortex Outside a Stationary Body......Page 131
   Note 4.4.4: Uniform Flow Past a Stationary Body......Page 132
  4.4.3 Solution via the Schwarz–Christoffel Transformation......Page 133
   Note 4.4.5: Velocity Field Near a Corner, Revisited......Page 134
   Note 4.4.6: Corner Intensity and the Bound Vortex Sheet Strength......Page 137
 4.5 The Non-uniqueness of Two-Dimensional Potential Flow......Page 138
 4.6 Decomposition of the Flow into Basis Fields......Page 142
  4.6.1 Complex Form, via Conformal Mapping Solution......Page 145
   Result 4.5: Two-Dimensional Velocity Field via Conformal Mapping......Page 146
  4.6.2 Vector Form......Page 152
   Note 4.6.1: Useful Properties of the Basis Scalar Potential Fields......Page 161
  4.6.3 Flow Decomposition and the Kinetic Energy: Added Mass......Page 162
   Result 4.6: Kinetic Energy of the Flow About a Rigid Body......Page 165
 4.7 Multipole Expansion of the Flow Field......Page 166
  4.7.1 Two-Dimensional Flow......Page 167
   Result 4.7: Multipole Expansion of the Two-Dimensional Flow Field......Page 169
  4.7.2 Three-Dimensional Flow......Page 170
   Result 4.8: Multipole Expansion of the Three-Dimensional Flow Field......Page 172
   Note 4.7.1: Multipole Coefficients in Terms of Body Vorticity......Page 173
  4.7.3 Two-Dimensional Expansion in Complex Form......Page 174
   Note 4.7.4: Multipole Coefficients Under Conformal Mapping......Page 176
5 Edge Conditions......Page 177
   Result 5.1: The Kutta Condition......Page 178
   Result 5.2: The Kutta Condition, Alternatively Stated......Page 179
  5.1.1 Steady Flow......Page 180
  5.1.2 Unsteady Flow......Page 181
   Result 5.3: Continuity of Vortex Sheet Strengths......Page 184
   Result 5.4: Giesing–Maskell Extension of the Kutta Condition......Page 186
   Note 5.1.2: The Kutta Condition and the Release of Vortex Elements......Page 188
 5.2 Application of the Kutta Condition......Page 189
   Result 5.5: The Constraint Form of the Kutta Condition......Page 190
 5.3 Traditional Enforcement of the Kutta Condition: A Contrast......Page 192
 5.4 Generalized Edge Conditions......Page 194
   Result 5.6: Critical Edge Suction Condition......Page 195
6 Force and Moment on a Body......Page 199
   Result 6.1: Surface Integral Form 1 of Force and Moment......Page 200
   Result 6.2: Surface Integral Form 2 of Force and Moment......Page 201
   Note 6.1.1: Force and Moment in the Windtunnel Frame......Page 202
 6.2 Force and Moment via Vorticity Impulse......Page 203
  6.2.1 Vectorial Forms......Page 204
   Result 6.3: Force as Rate of Change of Linear Impulse......Page 206
   Result 6.4: Moment as Rate of Change of Angular Impulse......Page 207
   Note 6.2.1: A Vorticity Form of the Impulse......Page 208
   Note 6.2.2: Another Form of the Moment's Relationship with Impulse......Page 209
   Note 6.2.3: Impulse Defined About Points Other than the Origin......Page 210
   Note 6.2.4: Impulse Formulas in the Presence of a Uniform Flow......Page 211
   Result 6.5: Complex Form of Force as Rate of Change of Linear Impulse......Page 212
   Result 6.6: Complex Form of Moment as Rate of Change of Angular Impulse......Page 213
   Note 6.2.5: Impulses Obtained Through Conformal Mapping......Page 214
 6.3 Reconciliation of Force via Traction and via Impulse......Page 215
  6.3.1 The Force and Moment on a Region of Vorticity......Page 216
   Result 6.7: The Force and Moment on a Vortex......Page 218
  6.3.2 The Spurious Force and Moment on Vorticity......Page 220
  6.3.3 Spurious Force and Moment on a Vortex in the Presence of a Body......Page 223
   Result 6.8: Spurious Force and Moment on a Vortex Element Near a Body......Page 226
  6.3.4 Revisiting the Traction Force and Moment on the Body......Page 227
   Result 6.9: Force and Moment on a Body via Traction over Its Surface......Page 228
 6.4 Edge Suction......Page 229
   Result 6.10: Edge Suction and the Suction Coefficient......Page 231
 6.5 Decomposition of the Force into Contributors......Page 232
  6.5.1 Complex Form, via Conformal Mapping Solution......Page 233
  6.5.2 Decomposed Force and Moment, Generalized......Page 236
   Result 6.11: Decomposed Force and Moment on a Rigid Body......Page 238
   Note 6.5.1: Plücker Added Mass Matrices in Two Dimensions......Page 239
 6.6 The Contribution of Fluid Vorticity to Force and Moment......Page 242
  6.6.1 Basic Definitions of the Vorticity-Induced Impulses......Page 243
  6.6.2 The Rates of Change of the Vorticity-Induced Impulses......Page 244
   Result 6.12: Vorticity-Induced Impulse and the Basis Vector Potentials......Page 245
   Note 6.6.1: The Elemental Contribution to Vorticity-Induced Impulses......Page 248
   Result 6.13: Time Derivative of the Vorticity-Induced Impulses......Page 249
   Result 6.14: The Force and Moment on a Rigid Body......Page 251
  6.6.3 The Rate of Change of Impulse for Singular Vortex Elements......Page 253
  6.6.4 Alternative Derivation of the Force and Moment......Page 255
 6.7 The Mechanical Energy Equation, Revisited......Page 259
7 Transport of Vortex Elements......Page 261
 7.1 Planar Vortex Elements......Page 262
   Result 7.1: Transport of a Point Vortex in the Physical Plane......Page 263
  7.1.1 Justification for Ignoring the Self-induction of a Point Vortex......Page 264
  7.1.2 Vortex Transport in the Conformal Mapping Plane: The Routh Correction......Page 265
   Result 7.2: The Routh Correction......Page 266
  7.1.3 Vortex Clouds and Sheets......Page 268
   Note 7.1.1: Efficient Calculation of Vortex Cloud Transport......Page 269
   Result 7.4: Evolution Equation for a Free VortexSheet in Two Dimensions......Page 270
  7.1.4 Variable-Strength Point Vortices......Page 271
  7.1.5 Blob Regularization......Page 276
 7.2 Vortex Filaments......Page 279
   Result 7.6: Evolution Equation for a Vortex Filament......Page 281
8 Flow About a Two-Dimensional Flat Plate......Page 284
 8.1 Basic Notation......Page 285
 8.2 Three Approaches to Solving for the Flow......Page 288
  8.2.1 Approach 1: Conformal Transformation from the Circle Plane......Page 289
  8.2.2 Approach 2: Inversion of the Cauchy Integral......Page 295
   Result 8.1: Cauchy Integral Equation for a Rigid Flat Plate......Page 296
  8.2.3 Approach 3: Fourier–Chebyshev Expansion......Page 302
   Result 8.2: Fourier–Chebyshev Form of the Bound Vortex Sheet Strength......Page 304
   Note 8.2.1: Fourier–Chebyshev for Wavy Plate or Sinusoidal Gust......Page 306
   Result 8.3: Fourier–Chebyshev Form of the Velocity and Potential Fields......Page 308
 8.3 Force and Moment......Page 310
   Result 8.4: Force and Moment on a Flat Plate......Page 311
   Result 8.5: Local Pressure Jump Across a Flat Plate......Page 314
   Note 8.3.1: Edge Suction on a Two-Dimensional Flat Plate......Page 315
  8.4.1 At the Trailing Edge: Thin Airfoil Theory......Page 317
 8.5 Classical Results......Page 324
  8.5.1 Steady Flow at Fixed Small Angle of Attack......Page 325
   Note 8.5.1: Alternative Enforcement of the Trailing-Edge Kutta Condition......Page 326
  8.5.2 General Background on Classical Unsteady Results......Page 328
  8.5.3 Oscillatory Motion: Theodorsen......Page 331
   Result 8.6: Low-Amplitude Oscillatory Motion of a Flat Plate......Page 335
  8.5.4 Impulsive Change of Motion: Wagner......Page 336
   Result 8.7: Force and Moment due to a Sudden Change of Motion......Page 338
   Note 8.5.2: The Wagner Function......Page 340
  8.5.5 Sinusoidal Gust Response......Page 342
  8.5.6 Sharp-Edge Gust......Page 345
 8.6 Generalized Edge Conditions and Their Interpretation......Page 346
 8.7 A Deforming Plate......Page 349
9 Examples of Two-Dimensional FlowModeling......Page 355
 9.1 Time Marching......Page 356
   Result 9.1: State Update......Page 357
 9.2 Co-rotating Vortex Patches......Page 358
 9.3 Interaction of Vortex Patches with a Bluff Body......Page 362
  9.4.1 Free Vortex Sheet Model......Page 366
   Result 9.2: The Minimum Physical Length Scale in an Inviscid Flow......Page 373
  9.4.2 Variable-Strength Vortex Model......Page 375
 9.5 Flow Past a NACA Airfoil......Page 378
 10.1 Ellipsoidal Coordinates and Harmonics......Page 382
 10.2 Translation of a General Ellipsoidal Body......Page 385
 10.3 Rotational Motion of a General Ellipsoidal Body......Page 392
 10.4 Added Mass......Page 393
   Result 10.1: The Added Mass of an Ellipsoid......Page 395
  A.1.1 Cartesian Index Notation......Page 401
  A.1.2 Derivatives of Fundamental Solutions of Laplace's Equation......Page 404
  A.1.3 The Divergence Theorem and Some Relevant Uses......Page 405
   Result A.1: Scalar Form of Green's Theorem......Page 412
   Result A.2: Vector Form of Green's Theorem......Page 413
   Result A.3: Scalar Form of Extended Green's Theorem......Page 414
   Result A.4: Vector Form of Extended Green's Theorem......Page 415
  A.1.4 Stokes' Theorem and Some Relevant Uses......Page 417
   Result A.5: Continuity of Tangent Component of Vector Fields......Page 420
   Result A.6: Equivalence of Two Forms of Surface Singularity Distributions......Page 421
  A.1.6 Field Quantities and Their Rate of Change......Page 423
  A.1.7 Time Differentiation of Spatial Integrals......Page 424
 A.2 Useful Tools from Complex Analysis......Page 428
  A.2.1 Basic Properties......Page 429
  A.2.2 The Cauchy Integral and Residue Theorem......Page 432
  A.2.3 Conformal Mapping......Page 437
  A.2.4 The Joukowski and Kármán–Trefftz Airfoils......Page 443
  A.2.5 The Schwarz–Christoffel Transformation......Page 448
   Result A.7: Power Series Representation of Schwarz–Christoffel Map......Page 450
   Note A.2.1: Schwarz–Christoffel Transformation for Rectangular Bodies......Page 453
  A.3.1 Notes on an Important Factor......Page 455
  A.3.2 Properties of Chebyshev Polynomials......Page 456
  A.3.3 Contour Integrals of Interest......Page 464
References......Page 467
Index......Page 471