Hydrodynamics and sound

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Title:

Personal Author:

Howe, M. S.

Publication Information:

Cambridge, UK : Cambridge University Press, 2007

ISBN:

9780521868624

Subject Term:

Hydrodynamics -- Mathematical models

Fluid dynamics -- Mathematical models

Waves

Sound

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010155730	QC151 H68 2007	Open Access Book	Book	Searching... Unknown

There is a certain body of knowledge and methods that finds application in most branches of fluid mechanics. This book aims to supply a proper theoretical understanding that will permit sensible simplifications to be made in the formulation of problems, and enable the reader to develop analytical models of practical significance. Such analyses can be used to guide more detailed experimental and numerical investigations. As in most technical subjects, such understanding is acquired by detailed study of highly simplified 'model problems'. The first part (Chapters 1-4) is concerned entirely with the incompressible flow of a homogeneous fluid. It was written for the Boston University introductory graduate level course 'Advanced Fluid Mechanics'. The remaining Chapters 5 and 6 deal with dispersive waves and acoustics, and are unashamedly inspired by James Lighthill's masterpiece, Waves in Fluids.

Author Notes

Professor Howe is in the Department of Aerospace and Mechanical Engineering at Boston University

Preface	p. xv
1 Equations of Motion	p. 1
1.1 The fluid state	p. 1
1.2 The material derivative	p. 1
1.3 Conservation of mass: Equation of continuity	p. 2
1.4 Momentum equation	p. 3
1.4.1 Relative motion of neighbouring fluid elements	p. 3
1.4.2 Viscous stress tensor	p. 5
1.4.3 Navier-Stokes equation	p. 7
1.4.4 The Reynolds equation and Reynolds stress	p. 7
1.5 The energy equation	p. 8
1.5.1 Alternative treatment of the energy equation	p. 9
1.5.2 Energy equation for incompressible flow	p. 10
1.6 Summary of governing equations	p. 11
1.7 Boundary conditions	p. 12
Problems 1 p. 12
2 Potential Flow of an Incompressible Fluid	p. 14
2.1 Ideal fluid	p. 14
2.2 Kelvin's circulation theorem	p. 14
2.3 The velocity potential	p. 16
2.3.1 Bernoulli's equation	p. 16
2.3.2 Impulsive pressure	p. 18
2.3.3 Streamlines and intrinsic equations of motion	p. 18
2.3.4 Bernoulli's equation in steady flow	p. 20
2.4 Motion produced by a pulsating sphere	p. 21
2.5 The point source	p. 22
2.6 Free-space Green's function	p. 24
2.7 Monopoles, dipoles, and quadrupoles	p. 24
2.7.1 The vibrating sphere	p. 26
2.7.2 Streamlines	p. 28
2.7.3 Far field of a monopole distribution of zero strength	p. 29
2.8 Green's formula	p. 30
2.8.1 Volume and surface integrals	p. 30
2.8.2 Green's formula	p. 32
2.8.3 Sources adjacent to a plane wall	p. 34
2.9 Determinancy of the motion	p. 35
2.9.1 Fluid motion expressed in terms of monopole or dipole distributions	p. 37
2.9.2 Determinancy of cyclic irrotational flow	p. 39
2.9.3 Kinetic energy of cyclic irrotational flow	p. 40
2.10 The kinetic energy	p. 41
2.10.1 Converse of Kelvin's minimum-energy theorem	p. 43
2.10.2 Energy of motion produced by a translating sphere	p. 43
2.11 Problems with spherical boundaries	p. 45
2.11.1 Legendre polynomials	p. 45
2.11.2 Velocity potential of a point source in terms of Legendre polynomials	p. 50
2.11.3 Interpretation in terms of images	p. 52
2.12 The Stokes stream function	p. 53
2.12.1 Stream function examples	p. 55
2.12.2 Rankine solids	p. 56
2.12.3 Rankine ovoid	p. 58
2.12.4 Drag in ideal flow	p. 58
2.12.5 Axisymmetric flow from a nozzle	p. 60
2.12.6 Irrotational flow from a circular cylinder	p. 63
2.12.7 Borda's mouthpiece	p. 65
2.13 The incompressible far field	p. 67
2.13.1 Deductions from Green's formula	p. 68
2.13.2 Far field produced by motion of a rigid body	p. 69
2.13.3 Inertia coefficients	p. 70
2.13.4 Pressure in the far field	p. 70
2.14 Force on a rigid body	p. 71
2.14.1 Moment exerted on a rigid body	p. 73
2.15 Sources near solid boundaries	p. 75
2.15.1 The reciprocal theorem	p. 76
2.16 Far-field Green's function	p. 78
2.16.1 The Kirchhoff vector	p. 80
2.16.2 Far-field Green's function for a sphere	p. 80
2.17 Far-field Green's function for cylindrical bodies	p. 84
2.17.1 The circular cylinder	p. 85
2.17.2 The rigid strip	p. 86
2.18 Symmetric far-field Green's function	p. 89
2.18.1 Far field of an arbitrarily moving body	p. 90
2.19 Far-field Green's function summary and special cases	p. 91
2.19.1 General form	p. 91
2.19.2 Airfoil of variable chord	p. 92
2.19.3 Projection or cavity on a plane wall	p. 93
2.19.4 Rankine ovoid	p. 94
2.19.5 Circular aperture	p. 95
2.19.6 Circular disc	p. 96
Problems 2 p. 96
3 Ideal Flow in Two Dimensions	p. 102
3.1 Complex representation of fluid motion	p. 102
3.1.1 The stream function	p. 102
3.1.2 The complex potential	p. 104
3.1.3 Uniform flow	p. 104
3.1.4 Flow past a cylindrical surface	p. 105
3.2 The circular cylinder	p. 106
3.2.1 Circle theorem	p. 106
3.2.2 Uniform flow past a circular cylinder	p. 106
3.2.3 The line vortex	p. 109
3.2.4 Circular cylinder with circulation	p. 110
3.2.5 Equation of motion of a cylinder with circulation	p. 112
3.3 The Blasius force and moment formulae	p. 115
3.3.1 Blasius's force formula for a stationary rigid body	p. 116
3.3.2 Blasius's moment formula for a stationary rigid body	p. 117
3.3.3 Kutta-Joukowski lift force	p. 117
3.3.4 Leading-edge suction	p. 118
3.4 Sources and line vortices	p. 119
3.4.1 Line vorrtices	p. 122
3.4.2 Motion of a line vortex	p. 122
3.4.3 Karman vortex street	p. 127
3.4.4 Kinetic energy of a system of rectilinear vortices	p. 127
3.5 Conformal transformations	p. 128
3.5.1 Transformation of Laplace's equation	p. 129
3.5.2 Equation of motion of a line vortex	p. 132
3.5.3 Numerical integration of the vortex path equation	p. 133
3.6 The Schwarz-Christoffel transformation	p. 135
3.6.1 Irrotational flow from an infinite duct	p. 138
3.6.2 Irrotational flow through a wall aperture	p. 140
3.7 Free-streamline theory	p. 142
3.7.1 Coanda edge flow	p. 142
3.7.2 Mapping from the w plane to the t plane	p. 147
3.7.3 Separated flow through an aperture	p. 147
3.7.4 The wake of a flat plate	p. 151
3.7.5 Flow past a curved boundary	p. 152
3.7.6 The hodograph transformation formula	p. 158
3.7.7 Chaplygin's singular point method	p. 159
3.7.8 Jet produced by a point source	p. 160
3.7.9 Deflection of trailing-edge flow by a source	p. 161
3.8 The Joukowski transformation	p. 167
3.8.1 The flat-plate airfoil	p. 170
3.8.2 Calculation of the lift	p. 173
3.8.3 Lift calculated from the Kirchhoff vector force formula	p. 173
3.8.4 Lift developed by a starting airfoil	p. 174
3.9 The Joukowski airfoil	p. 175
3.9.1 Streamline flow past an airfoil	p. 176
3.10 Separation and stall	p. 179
3.10.1 Linear theory of separation	p. 180
3.11 Sedov's method	p. 183
3.11.1 Boundary conditions	p. 184
3.11.2 Sedov's formula	p. 185
3.11.3 Tandem airfoils	p. 187
3.11.4 High-lift devices	p. 190
3.11.5 Plain flap or aileron	p. 192
3.11.6 Point sources and vortices	p. 192
3.11.7 Flow through a cascade	p. 193
3.12 Unsteady thin-airfoil theory	p. 195
3.12.1 The vortex sheet wake	p. 195
3.12.2 Translational oscillations	p. 197
3.12.3 The unsteady lift	p. 198
3.12.4 Leading-edge suction force	p. 199
3.12.5 Energy dissipated by vorticity production	p. 201
3.12.6 Hankel function formulae	p. 202
Problems 3 p. 203
4 Rotational Incompressible Flow	p. 211
4.1 The vorticity equation	p. 211
4.1.1 Vortex lines	p. 212
4.1.2 Vortex tubes	p. 212
4.1.3 Movement of vortex lines: Helmholtz's vortex theorem	p. 213
4.1.4 Crocco's equation	p. 214
4.1.5 Convection and diffusion of vorticity	p. 215
4.1.6 Vortex sheets	p. 218
4.2 The Biot-Savart law	p. 221
4.2.1 The far field	p. 223
4.2.2 Kinetic energy	p. 227
4.2.3 The Biot-Savart formula in the presence of an internal boundary	p. 228
4.2.4 The Biot-Savart formula for irrotational flow	p. 229
4.3 Examples of axisymmetric vortical flow	p. 232
4.3.1 Circular vortex filament	p. 232
4.3.2 Rate of production of vorticity at a nozzle	p. 233
4.3.3 Blowing out a candle	p. 235
4.3.4 Axisymmetric steady flow of an ideal fluid	p. 236
4.3.5 Hill's spherical vortex	p. 237
4.4 Some viscous flows	p. 239
4.4.1 Diffusion of vorticity from an impulsively started plane wall	p. 239
4.4.2 Diffusion of vorticity from a line vortex	p. 240
4.4.3 Creeping flow	p. 242
4.4.4 Motion of a sphere at very small Reynolds number	p. 242
4.4.5 The Oseen approximation	p. 245
4.4.6 Laminar flow in a tube (Hagen-Poiseuille flow)	p. 247
4.4.7 Boundary layer on a flat plate; Karman momentum integral method	p. 249
4.5 Force on a rigid body	p. 253
4.5.1 Surface force in terms of the impulse	p. 254
4.5.2 The Kirchhoff vector force formula	p. 256
4.5.3 The Kirchhoff vector force formula for irrotational flow	p. 258
4.5.4 Arbitrary motion in a viscous fluid	p. 258
4.5.5 Body moving without rotation	p. 239
4.5.6 Surface force in two dimensions	p. 261
4.5.7 Bluff body drag at high Reynolds number	p. 261
4.5.8 Modelling vortex shedding from a sphere	p. 265
4.5.9 Force and impulse in fluid of non-uniform density	p. 270
4.5.10 Integral identities	p. 271
4.6 Surface moment	p. 273
4.6.1 Moment for a non-rotating body	p. 273
4.6.2 Airfoil lift, drag, and moments	p. 274
4.7 Vortex-surface interactions	p. 276
4.7.1 Pressure expressed in terms of the total enthalpy	p. 276
4.7.2 Equation for B	p. 277
4.7.3 Solution of the B equation	p. 278
4.7.4 The far field	p. 279
Problems 4 p. 281
5 Surface Gravity Waves	p. 286
5.1 Introduction	p. 286
5.1.1 Conditions at the free surface	p. 286
5.1.2 Wave motion within the fluid	p. 287
5.1.3 Linearised approximation	p. 288
5.1.4 Time harmonic, plane waves on deep water	p. 288
5.1.5 Water of finite depth	p. 290
5.2 Surface wave energy	p. 291
5.2.1 Wave-energy density	p. 293
5.2.2 Wave-energy flux	p. 294
5.2.3 Group velocity	p. 295
5.3 Viscous damping of surface waves	p. 297
5.3.1 The interior damping	p. 297
5.3.2 Boundary-layer damping	p. 298
5.3.3 Comparison of boundary-layer and internal damping for long waves	p. 299
5.4 Shallow-water waves	p. 299
5.4.1 Waves on water of variable depth	p. 300
5.4.2 Shallow-water Green's function	p. 301
5.4.3 Waves generated by a localised pressure rise	p. 302
5.4.4 Waves approaching a sloping beach	p. 307
5.5 Method of stationary phase	p. 309
5.5.1 Formulation of initial-value dispersive-wave problems	p. 309
5.5.2 Evaluation of Fourier integrals by the method of stationary phase	p. 311
5.5.3 Numerical results for the surface displacement	p. 313
5.5.4 Conservation of energy	p. 315
5.5.5 Rayleigh's proof that energy propagates at the group velocity	p. 317
5.5.6 Surface wave-energy equation	p. 318
5.5.7 Waves generated by a submarine explosion	p. 319
5.6 Initial-value problems in two surface dimensions	p. 321
5.6.1 Waves generated by a surface elevation symmetric about the origin	p. 322
5.6.2 The energy equation in two dimensions	p. 324
5.7 Surface motion near a wavefront	p. 325
5.7.1 One-dimensional waves	p. 325
5.7.2 Waves generated by motion of the seabed	p. 328
5.7.3 Tsunami produced by an undersea earthquake	p. 332
5.8 Periodic wave sources	p. 333
5.8.1 One-dimensional waves	p. 334
5.8.2 Periodic sources in two surface dimensions	p. 336
5.8.3 The surface wave power	p. 339
5.8.4 Surface wave amplitude	p. 340
5.9 Ship waves	p. 341
5.9.1 Moving line pressure source	p. 342
5.9.2 Wave-making resistance	p. 343
5.9.3 Moving point-like pressure source	p. 345
5.9.4 Plotting the wave crests	p. 349
5.9.5 Behaviour at the caustic	p. 351
5.9.6 Wave-making power	p. 352
5.9.7 Wave amplitude calculated from the power	p. 354
5.10 Ray theory	p. 354
5.10.1 Kinematic theory of wave crests	p. 354
5.10.2 Ray tracing in an inhomogeneous medium	p. 357
5.10.3 Refraction of waves at a sloping beach	p. 357
5.11 Wave action	p. 364
5.11.1 Variational description of a fully dispersed wave group	p. 365
5.11.2 Fully dispersed waves in a non-uniformly moving medium	p. 366
5.11.3 General wave-bearing media	p. 369
5.12 Diffraction of surface waves by a breakwater	p. 373
5.12.1 Diffraction by a long, straight breakwater	p. 373
5.12.2 Solution of the diffraction problem	p. 374
5.12.3 The surface wave pattern	p. 377
5.12.4 Uniform asymptotic approximation: Method of steepest descents	p. 379
Problems 5 p. 384
6 Introduction to Acoustics	p. 390
6.1 The wave equation	p. 390
6.1.1 The linear wave equation	p. 391
6.1.2 Plane waves	p. 392
6.1.3 Speed of sound	p. 393
6.2 Acoustic Green's function	p. 395
6.2.1 The impulsive point source	p. 395
6.2.2 Green's function	p. 396
6.2.3 Retarded potential	p. 397
6.2.4 Sound from a vibrating sphere	p. 397
6.2.5 Acoustic energy flux	p. 399
6.2.6 Green's function in one space dimension: Method of descent	p. 400
6.2.7 Waves generated by a one-dimensional volume source	p. 401
6.3 Kirchhoff's formula	p. 401
6.4 Compact Green's function	p. 403
6.4.1 Generalized Kirchhoff formula	p. 403
6.4.2 The time harmonic wave equation	p. 404
6.4.3 The compact approximation	p. 404
6.4.4 Rayleigh scattering: Scattering by a compact body	p. 407
6.5 One-dimensional propagation through junctions	p. 409
6.5.1 Continuity of volume velocity	p. 410
6.5.2 Continuity of pressure	p. 410
6.5.3 Reflection and transmission at a junction	p. 411
6.6 Branching systems	p. 413
6.6.1 Fundamental formula	p. 414
6.6.2 Energy transmission	p. 415
6.6.3 Acoustically compact cavity	p. 416
6.6.4 The Helmholtz resonator	p. 417
6.6.5 Acoustic filter	p. 418
6.6.6 Admittance of a narrow constriction	p. 419
6.7 Radiation from an open end	p. 421
6.7.1 Rayleigh's method for low-frequency sound	p. 421
6.7.2 The reflection coefficient	p. 423
6.7.3 Admittance of the open end	p. 423
6.7.4 Open-end input admittance	p. 424
6.7.5 Flanged opening	p. 426
6.7.6 Physical significance of the end correction	p. 428
6.7.7 Admittance of a circular aperture	p. 431
6.8 Webster's equation	p. 432
6.9 Radiation into a semi-infinite duct	p. 435
6.9.1 The compact Green's function	p. 435
6.9.2 Wave generation by a train entering a tunnel	p. 439
6.10 Damping of sound in a smooth-walled duct	p. 445
6.10.1 Time harmonic propagation in a duct	p. 446
6.10.2 The viscous contribution	p. 447
6.10.3 The thermal contribution	p. 449
6.10.4 The thermo-viscous damping coefficient	p. 450
Problems 6 p. 450
Bibliography	p. 455
Index	p. 457

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