Theory of vortex sound

Half-title......Page 3
Series-title......Page 5
Title......Page 7
Copyright......Page 8
Dedication......Page 9
Contents......Page 11
Preface......Page 15
1.1 What is Vortex Sound?......Page 17
1.2.1 Equation of Continuity......Page 18
1.2.2 Momentum Equation......Page 19
1.3 Equation of Linear Acoustics......Page 20
1.4.1 Pulsating Sphere......Page 23
1.4.2 Point Source......Page 25
1.5 Sound Produced by an Impulsive Point Source......Page 26
1.6 Free-Space Green’s Function......Page 28
1.7 Monopoles, Dipoles, and Quadrupoles......Page 29
1.7.1 The Point Dipole......Page 30
1.7.2 Quadrupoles......Page 31
1.7.3 Vibrating Sphere......Page 32
1.8 Acoustic Energy Flux......Page 34
1.9 Calculation of the Acoustic Far Field......Page 36
1.9.1 Dipole Source Distributions......Page 37
1.9.3 Example......Page 38
2.1 The Acoustic Analogy......Page 41
2.2 Lighthill’s v Law......Page 45
2.3.1 Volume and Surface Integrals......Page 48
2.3.2 Curle’s Equation......Page 50
2.4 Sound Produced by Turbulence Near a Compact Rigid Body......Page 52
2.5 Radiation from a Noncompact Surface......Page 53
3.1 The Influence of Solid Boundaries......Page 57
3.2.1 The Point Source......Page 60
3.2.2 Dipole and Quadrupole Sources......Page 61
3.3 The Reciprocal Theorem......Page 62
3.4 Time-Harmonic Compact Green’s Function......Page 65
3.5 Compact Green’s Function for a Rigid Sphere......Page 69
3.5.1 Radiation from a Dipole Adjacent to a Compact Sphere......Page 70
3.5.2 Sound Produced by a Vibrating Sphere......Page 72
3.6.1 Circular Cylinder......Page 74
3.6.2 Rigid Strip......Page 76
3.7 Symmetric Compact Green’s Function......Page 79
3.8 Low-Frequency Radiation from a Vibrating Body......Page 81
3.8.1 Far Field Pressure Produced by a Vibrating Body......Page 84
3.9.1 Compact Bodies and Cylindrical Bodies of Compact Cross Section......Page 86
3.9.2 Airfoil of Variable Chord......Page 87
3.9.3 Projection or Cavity on a Plane Wall......Page 88
3.9.4 Green’s Function for a Half-Plane (Howe, 1975a)......Page 90
3.9.6 Two-Dimensional Green’s Function for a Plane with an Aperture......Page 91
3.9.7 Green’s Function for Long Waves in a Rigid Walled Duct (Howe, 1975b)......Page 92
(i) Propagation within the Duct......Page 93
(ii) Propagation in Free Space (Fig.3.9.6b)......Page 94
4.1.1 Kelvin’s (1867) Definition......Page 98
4.2 The Vorticity Equation......Page 100
4.3 The Biot–Savart Law......Page 104
4.3.2 Incompressible Flow with an Internal Boundary......Page 106
4.3.3 Blowing Out a Candle (Lighthill 1963)......Page 107
4.4 Surface Force in Incompressible Flow Expressed in Terms of Vorticity......Page 109
4.4.1 Bound Vorticity and the Added Mass......Page 110
4.4.2 Force Exerted on an Incompressible Fluid by a Moving Body......Page 112
4.4.3 Stokes Drag on a Sphere......Page 114
4.5.1 Laplace’s Equation in Two Dimensions......Page 116
4.5.2 Hydrodynamics in Two Dimensions......Page 117
4.6 Motion of a Line Vortex......Page 122
4.6.1 Numerical Integration of the Vortex Path Equation......Page 124
5.1 The Role of Vorticity in Lighthill’s Theory......Page 130
5.2 The Equation of Vortex Sound......Page 132
5.2.1 Reformulation of Lighthill’s Equation......Page 133
5.2.2 Sound Waves in Irrotational Mean Flow......Page 134
5.2.3 Vortex Sound at Low Mach Numbers......Page 135
5.2.4 Example 1 (Powell 1963): Sound Generation by a Spinning Vortex Pair......Page 136
5.2.5 Example 2......Page 139
5.3 Vortex–Surface Interaction Noise......Page 140
5.4 Radiation from an Acoustically Compact Body......Page 144
5.5 Radiation from Cylindrical Bodies of Compact Cross Section......Page 146
5.6 Impulse Theory of Vortex Sound......Page 147
6.1 Compact Green’s Function in Two Dimensions......Page 152
6.2 Sound Generated by a Line Vortex Interacting with a Cylindrical Body......Page 155
6.2.1 Example 1: Sound Produced by Vortex Motion near a Circular Cylinder......Page 156
6.2.2 Example 2: Sound Produced by Vortex Motion near a Half-Plane (Crighton 1972)......Page 159
6.3 Influence of Vortex Shedding......Page 161
6.3.1 Example: Surface Force Produced by a Periodic Gust......Page 163
6.4 Blade–Vortex Interaction Noise in Two Dimensions......Page 166
7.1 Linear Theory of Vortex–Airfoil Interaction Noise......Page 172
7.2 Blade–Vortex Interactions in Three Dimensions......Page 174
7.3 Sound Produced by Vortex Motion near a Sphere......Page 178
7.4 Compression Wave Generated When a Train Enters a Tunnel......Page 182
7.4.1 Linear Theory......Page 185
8.1.1 Equation of Motion of the Vortex......Page 191
8.1.2 Formula for the Acoustic Pressure......Page 193
8.1.3 Linear Theory......Page 194
8.1.4 Nonlinear Theory......Page 196
8.1.5 Periodic Vortex Motion......Page 199
8.2 Parallel Blade–Vortex Interactions in Three Dimensions......Page 202
8.2.1 Linear Theory......Page 204
8.2.2 Nonlinear Theory......Page 205
8.3 Vortex Passing over a Spoiler......Page 207
8.4 Bluff Body Interactions: The Circular Cylinder......Page 210
8.4.1 The Acoustic Pressure......Page 211
8.4.2 Wall-Mounted Cylinder......Page 213
8.5 Vortex Ring and Sphere......Page 215
8.5.1 Acoustic Pressure......Page 218
8.6 Vortex Pair Incident on a Wall Aperture......Page 220
Bibliography......Page 225
Index......Page 229