Hydrodynamics and Sound / Edition 1
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Hydrodynamics and Sound / Edition 1

by M. S. Howe

ISBN-10: 0521868629

ISBN-13: 9780521868624

Pub. Date: 09/30/2006

Publisher: Cambridge University Press

This book is designed for a first graduate course in fluid dynamics. It focuses on knowledge and methods that find applications in most branches of fluid mechanics and aims to supply a 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


This book is designed for a first graduate course in fluid dynamics. It focuses on knowledge and methods that find applications in most branches of fluid mechanics and aims to supply a 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. The study of simplified model problems can be used to guide experimental and numerical investigations. The first part (Chapters 1-4) is concerned entirely with the incompressible flow of homogeneous fluid. Chapters 5 and 6 deal with dispersive waves and acoustics.

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

Product Details

Cambridge University Press
Publication date:
Edition description:
New Edition
Product dimensions:
6.97(w) x 9.96(h) x 1.18(d)

Table of Contents

Preface     xv
Equations of Motion     1
The fluid state     1
The material derivative     1
Conservation of mass: Equation of continuity     2
Momentum equation     3
Relative motion of neighbouring fluid elements     3
Viscous stress tensor     5
Navier-Stokes equation     7
The Reynolds equation and Reynolds stress     7
The energy equation     8
Alternative treatment of the energy equation     9
Energy equation for incompressible flow     10
Summary of governing equations     11
Boundary conditions     12
Problems 1     12
Potential Flow of an Incompressible Fluid     14
Ideal fluid     14
Kelvin's circulation theorem     14
The velocity potential     16
Bernoulli's equation     16
Impulsive pressure     18
Streamlines and intrinsic equations of motion     18
Bernoulli's equation in steady flow     20
Motion produced by a pulsating sphere     21
The point source     22
Free-space Green's function     24
Monopoles, dipoles, andquadrupoles     24
The vibrating sphere     26
Streamlines     28
Far field of a monopole distribution of zero strength     29
Green's formula     30
Volume and surface integrals     30
Green's formula     32
Sources adjacent to a plane wall     34
Determinancy of the motion     35
Fluid motion expressed in terms of monopole or dipole distributions     37
Determinancy of cyclic irrotational flow     39
Kinetic energy of cyclic irrotational flow     40
The kinetic energy     41
Converse of Kelvin's minimum-energy theorem     43
Energy of motion produced by a translating sphere     43
Problems with spherical boundaries     45
Legendre polynomials     45
Velocity potential of a point source in terms of Legendre polynomials     50
Interpretation in terms of images     52
The Stokes stream function     53
Stream function examples     55
Rankine solids     56
Rankine ovoid     58
Drag in ideal flow     58
Axisymmetric flow from a nozzle     60
Irrotational flow from a circular cylinder      63
Borda's mouthpiece     65
The incompressible far field     67
Deductions from Green's formula     68
Far field produced by motion of a rigid body     69
Inertia coefficients     70
Pressure in the far field     70
Force on a rigid body     71
Moment exerted on a rigid body     73
Sources near solid boundaries     75
The reciprocal theorem     76
Far-field Green's function     78
The Kirchhoff vector     80
Far-field Green's function for a sphere     80
Far-field Green's function for cylindrical bodies     84
The circular cylinder     85
The rigid strip     86
Symmetric far-field Green's function     89
Far field of an arbitrarily moving body     90
Far-field Green's function summary and special cases     91
General form     91
Airfoil of variable chord     92
Projection or cavity on a plane wall     93
Rankine ovoid     94
Circular aperture     95
Circular disc     96
Problems 2     96
Ideal Flow in Two Dimensions     102
Complex representation of fluid motion     102
The stream function     102
The complex potential     104
Uniform flow     104
Flow past a cylindrical surface     105
The circular cylinder     106
Circle theorem     106
Uniform flow past a circular cylinder     106
The line vortex     109
Circular cylinder with circulation     110
Equation of motion of a cylinder with circulation     112
The Blasius force and moment formulae     115
Blasius's force formula for a stationary rigid body     116
Blasius's moment formula for a stationary rigid body     117
Kutta-Joukowski lift force     117
Leading-edge suction     118
Sources and line vortices     119
Line vorrtices     122
Motion of a line vortex     122
Karman vortex street     127
Kinetic energy of a system of rectilinear vortices     127
Conformal transformations     128
Transformation of Laplace's equation     129
Equation of motion of a line vortex     132
Numerical integration of the vortex path equation     133
The Schwarz-Christoffel transformation      135
Irrotational flow from an infinite duct     138
Irrotational flow through a wall aperture     140
Free-streamline theory     142
Coanda edge flow     142
Mapping from the w plane to the t plane     147
Separated flow through an aperture     147
The wake of a flat plate     151
Flow past a curved boundary     152
The hodograph transformation formula     158
Chaplygin's singular point method     159
Jet produced by a point source     160
Deflection of trailing-edge flow by a source     161
The Joukowski transformation     167
The flat-plate airfoil     170
Calculation of the lift     173
Lift calculated from the Kirchhoff vector force formula     173
Lift developed by a starting airfoil     174
The Joukowski airfoil     175
Streamline flow past an airfoil     176
Separation and stall     179
Linear theory of separation     180
Sedov's method     183
Boundary conditions     184
Sedov's formula     185
Tandem airfoils     187
High-lift devices     190
Plain flap or aileron      192
Point sources and vortices     192
Flow through a cascade     193
Unsteady thin-airfoil theory     195
The vortex sheet wake     195
Translational oscillations     197
The unsteady lift     198
Leading-edge suction force     199
Energy dissipated by vorticity production     201
Hankel function formulae     202
Problems 3     203
Rotational Incompressible Flow     211
The vorticity equation     211
Vortex lines     212
Vortex tubes     212
Movement of vortex lines: Helmholtz's vortex theorem     213
Crocco's equation     214
Convection and diffusion of vorticity     215
Vortex sheets     218
The Biot-Savart law     221
The far field     223
Kinetic energy     227
The Biot-Savart formula in the presence of an internal boundary     228
The Biot-Savart formula for irrotational flow     229
Examples of axisymmetric vortical flow     232
Circular vortex filament     232
Rate of production of vorticity at a nozzle     233
Blowing out a candle      235
Axisymmetric steady flow of an ideal fluid     236
Hill's spherical vortex     237
Some viscous flows     239
Diffusion of vorticity from an impulsively started plane wall     239
Diffusion of vorticity from a line vortex     240
Creeping flow     242
Motion of a sphere at very small Reynolds number     242
The Oseen approximation     245
Laminar flow in a tube (Hagen-Poiseuille flow)     247
Boundary layer on a flat plate; Karman momentum integral method     249
Force on a rigid body     253
Surface force in terms of the impulse     254
The Kirchhoff vector force formula     256
The Kirchhoff vector force formula for irrotational flow     258
Arbitrary motion in a viscous fluid     258
Body moving without rotation     239
Surface force in two dimensions     261
Bluff body drag at high Reynolds number     261
Modelling vortex shedding from a sphere     265
Force and impulse in fluid of non-uniform density     270
Integral identities     271
Surface moment     273
Moment for a non-rotating body     273
Airfoil lift, drag, and moments      274
Vortex-surface interactions     276
Pressure expressed in terms of the total enthalpy     276
Equation for B     277
Solution of the B equation     278
The far field     279
Problems 4     281
Surface Gravity Waves     286
Introduction     286
Conditions at the free surface     286
Wave motion within the fluid     287
Linearised approximation     288
Time harmonic, plane waves on deep water     288
Water of finite depth     290
Surface wave energy     291
Wave-energy density     293
Wave-energy flux     294
Group velocity     295
Viscous damping of surface waves     297
The interior damping     297
Boundary-layer damping     298
Comparison of boundary-layer and internal damping for long waves     299
Shallow-water waves     299
Waves on water of variable depth     300
Shallow-water Green's function     301
Waves generated by a localised pressure rise     302
Waves approaching a sloping beach     307
Method of stationary phase      309
Formulation of initial-value dispersive-wave problems     309
Evaluation of Fourier integrals by the method of stationary phase     311
Numerical results for the surface displacement     313
Conservation of energy     315
Rayleigh's proof that energy propagates at the group velocity     317
Surface wave-energy equation     318
Waves generated by a submarine explosion     319
Initial-value problems in two surface dimensions     321
Waves generated by a surface elevation symmetric about the origin     322
The energy equation in two dimensions     324
Surface motion near a wavefront     325
One-dimensional waves     325
Waves generated by motion of the seabed     328
Tsunami produced by an undersea earthquake     332
Periodic wave sources     333
One-dimensional waves     334
Periodic sources in two surface dimensions     336
The surface wave power     339
Surface wave amplitude     340
Ship waves     341
Moving line pressure source     342
Wave-making resistance     343
Moving point-like pressure source     345
Plotting the wave crests      349
Behaviour at the caustic     351
Wave-making power     352
Wave amplitude calculated from the power     354
Ray theory     354
Kinematic theory of wave crests     354
Ray tracing in an inhomogeneous medium     357
Refraction of waves at a sloping beach     357
Wave action     364
Variational description of a fully dispersed wave group     365
Fully dispersed waves in a non-uniformly moving medium     366
General wave-bearing media     369
Diffraction of surface waves by a breakwater     373
Diffraction by a long, straight breakwater     373
Solution of the diffraction problem     374
The surface wave pattern     377
Uniform asymptotic approximation: Method of steepest descents     379
Problems 5     384
Introduction to Acoustics     390
The wave equation     390
The linear wave equation     391
Plane waves     392
Speed of sound     393
Acoustic Green's function     395
The impulsive point source     395
Green's function     396
Retarded potential     397
Sound from a vibrating sphere     397
Acoustic energy flux     399
Green's function in one space dimension: Method of descent     400
Waves generated by a one-dimensional volume source     401
Kirchhoff's formula     401
Compact Green's function     403
Generalized Kirchhoff formula     403
The time harmonic wave equation     404
The compact approximation     404
Rayleigh scattering: Scattering by a compact body     407
One-dimensional propagation through junctions     409
Continuity of volume velocity     410
Continuity of pressure     410
Reflection and transmission at a junction     411
Branching systems     413
Fundamental formula     414
Energy transmission     415
Acoustically compact cavity     416
The Helmholtz resonator     417
Acoustic filter     418
Admittance of a narrow constriction     419
Radiation from an open end     421
Rayleigh's method for low-frequency sound     421
The reflection coefficient     423
Admittance of the open end     423
Open-end input admittance     424
Flanged opening     426
Physical significance of the end correction     428
Admittance of a circular aperture     431
Webster's equation     432
Radiation into a semi-infinite duct     435
The compact Green's function     435
Wave generation by a train entering a tunnel     439
Damping of sound in a smooth-walled duct     445
Time harmonic propagation in a duct     446
The viscous contribution     447
The thermal contribution     449
The thermo-viscous damping coefficient     450
Problems 6     450
Bibliography     455
Index     457

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