Computational Aeroacoustics: A Wave Number Approach

✍ Scribed by Christopher K. W. Tam

• Publisher: Cambridge University Press
• Year: 2012
• Tongue: English
• Leaves: 497
• Series: Cambridge Aerospace Series
• Category: Library

✦ Synopsis

Computational Aeroacoustics (CAA) is a relatively new research area. CAA algorithms have developed rapidly and the methods have been applied in many areas of aeroacoustics. The objective of CAA is not simply to develop computational methods but also to use these methods to solve practical aeroacoustics problems and to perform numerical simulation of aeroacoustic phenomena. By analyzing the simulation data, an investigator can determine noise generation mechanisms and sound propagation processes. This is both a textbook for graduate students and a reference for researchers in CAA and as such is self-contained. No prior knowledge of numerical methods for solving PDE's is needed, however, a general understanding of partial differential equations and basic numerical analysis is assumed. Exercises are included and are designed to be an integral part of the chapter content. In addition, sample computer programs are included to illustrate the implementation of the numerical algorithms.

✦ Table of Contents

Computational Aeroacoustics: A Wave Number Approach......Page 6
Contents......Page 8
Preface......Page 12
1.1. Order of Finite Difference Equations: Concept of Solution......Page 16
1.2.1 Distinct Roots......Page 17
1.2.3 Complex Roots......Page 18
1.3 Finite Difference Solution as an Approximate Solution of a Boundary Value Problem......Page 19
1.3.2 Finite Difference Solution......Page 20
1.3.2.1 Second-Order Approximation......Page 21
1.3.2.2 Fourth-Order Approximation......Page 22
1.4 Finite Difference Model for a Surface of Discontinuity......Page 23
1.4.1 The Transmission Problem......Page 25
1.4.2 Finite Difference Model......Page 26
1.4.3 Boundary Modes......Page 30
Exercises......Page 32
2.1 Truncated Taylor Series Method......Page 36
2.2.1 Derivative Theorem......Page 37
2.2.2 Shifting Theorem......Page 38
2.3 Group Velocity Consideration......Page 40
2.4 Schemes with Large Stencils......Page 45
2.5 Backward Difference Stencils......Page 47
2.6 Coefficients of Several Large Optimized Stencils......Page 49
Exercises......Page 51
3.1 Single Time Step Method: Runge-Kutta Scheme......Page 53
3.2 Optimized Multilevel Time Discretization......Page 55
3.3 Stability Diagram......Page 56
Exercises......Page 58
4.1 Dispersive Waves of Physical Systems......Page 60
4.2 Group Velocity and Dispersion......Page 62
4.3 Origin of Numerical Dispersion......Page 63
4.4 Numerical Dispersion Arising from Temporal Discretization......Page 68
4.5 Origin of Numerical Dissipation......Page 72
4.6 Multidimensional Waves......Page 74
Exercises......Page 75
5.1 Dispersion Relations and Asymptotic Solutions of the Linearized Euler Equations......Page 76
5.1.1 The Entropy Wave......Page 78
5.1.2 The Vorticity Wave......Page 79
5.1.3 The Acoustic Wave......Page 80
5.2 Dispersion-Relation-Preserving (DRP) Scheme......Page 82
5.3 Numerical Stability......Page 84
5.4 Group Velocity for Finite Difference Schemes......Page 85
5.5 Time Step Δt: Accuracy Consideration......Page 87
5.6 DRP Scheme in Curvilinear Coordinates......Page 88
Exercises......Page 90
6. Radiation, Outflow, and Wall Boundary Conditions......Page 95
6.2 Outflow Boundary Conditions......Page 96
6.3 Implementation of Radiation and Outflow Boundary Conditions......Page 98
6.4 Numerical Simulation: An Example......Page 99
6.4.3 Vorticity Waves......Page 100
6.5 Generalized Radiation and Outflow Boundary Conditions......Page 103
6.6 The Ghost Point Method for Wall Boundary Conditions......Page 104
6.6.1 Concept of Ghost Points and Ghost Values......Page 105
6.6.2 Reflection of Acoustic Waves by a Plane Wall......Page 109
6.6.3 Numerical Examples......Page 112
6.6.3.1 Reflection of a Transient Acoustic Pulse by a Wall......Page 113
6.6.3.2 Effect of Mean Flow......Page 114
6.6.4 Cartesian Boundary Treatment of Curved Walls......Page 117
6.7 Enforcing Wall Boundary Conditions on Curved Surfaces......Page 118
Exercises......Page 121
7.1 The Short Waves......Page 127
7.2 Artificial Selective Damping......Page 128
7.2.1 Basic Concept......Page 129
7.2.2 Numerical Example......Page 131
7.3 Excessive Damping......Page 133
7.3.1 Artificial Viscous Diffusion......Page 134
7.3.2 Damping-Induced Numerical Instability......Page 136
7.4 Artificial Damping at Surfaces of Discontinuity......Page 137
7.5 Aliasing......Page 139
7.6 Coefficients of Several Large Damping Stencils......Page 140
Exercises......Page 142
8.1.2 The C+ and C– Characteristic......Page 145
8.2 Spurious Oscillations: Origin and Characteristics......Page 147
8.3 Variable Artificial Selective Damping......Page 152
Exercises......Page 157
9.1 Boundaries with Incoming Disturbances......Page 159
9.2 Entrainment Flow......Page 161
9.3 Outflow Boundary Conditions: Further Consideration......Page 164
9.4.1 Linear Problem Involving a Single Azimuthal Fourier Component......Page 167
9.4.1.1 Scalar Potential Solutions......Page 168
9.4.1.3 Analytic Continuation into the r < 0 Region......Page 169
9.4.2.1 Directional Derivative......Page 170
9.4.2.4 r Derivative of the Velocity Field......Page 171
9.4.2.5 The Values of v and w at r = 0......Page 172
9.5 Perfectly Matched Layer as an Absorbing Boundary Condition......Page 173
9.5.1 Derivation of the PML Equation......Page 174
9.5.2 Perfectly Matching and Stability Consideration......Page 176
9.5.3 PML in Three Dimensions......Page 180
9.6 Boundaries with Discontinuities......Page 182
9.7 Internal Flow Driven by a Pressure Gradient......Page 185
Exercises......Page 186
10. Time-Domain Impedance Boundary Condition......Page 195
10.2 Stability of the Three-Parameter Time-Domain Impedance Boundary Condition......Page 197
10.3 Impedance Boundary Condition in the Presence of a Subsonic Mean Flow......Page 200
10.4 Numerical Implementation......Page 202
10.5 A Numerical Example......Page 204
10.6 Acoustic Wave Propagation and Scattering in a Duct with Acoustic Liner Splices......Page 207
10.6.1 Scattering of Acoustic Duct Mode at the Entrance and Exit of an Inlet Duct......Page 208
10.6.2 Scattering by Very Thin Axial Splices......Page 212
10.6.3 Scattering by Thin Circumferential Splices......Page 214
Exercises......Page 216
11.1 Extrapolation and Numerical Instability......Page 218
11.2 Wave Number Analysis of Extrapolation......Page 220
11.2.1 Extrapolation Error in Wave Number Space......Page 221
11.2.2 Additional Constraint on Optimized Extrapolation......Page 225
11.3 Optimized Interpolation Method......Page 228
11.4 A Numerical Example......Page 234
Exercises......Page 241
12. Multiscales Problems......Page 244
12.1 Spatial Stencils for Use in the Mesh-Size-Change Buffer Region......Page 246
12.2 Time Marching Stencil......Page 250
12.3 Damping Stencils......Page 253
12.4.1 First Example: The Sound Field of an Open Rotor......Page 256
12.4.1.2 Numerical Boundary Conditions......Page 258
12.4.1.5 Numerical Results......Page 260
12.4.2 Second Example: Acoustic Resonances Induced by Flow over an Automobile Door Cavity......Page 261
12.4.2.1 Computation Domain and Grid Design......Page 262
12.4.2.2 The Governing Equations and the Computational Algorithm......Page 263
12.4.2.5 Numerical Results......Page 264
12.5 Coefficients of Several Large Buffer Stencils......Page 267
12.6 Large Buffer Selective Damping Stencils......Page 271
Exercises......Page 276
13.1 Basic Concept of Overset Grids......Page 278
13.2 Optimized Multidimensional Interpolation......Page 281
13.2.1 Order Constraints......Page 285
13.2.2 Interpolation Errors in Wave Number Space......Page 287
13.2.3 Global Interpolation......Page 293
13.3.1 Transient Scattering Problem......Page 295
13.3.2 Plane Wave Scattering Problem......Page 297
13.4.1 Sliding Interface Problem in Two Dimensions......Page 299
13.4.2 Sliding Interface Problem in Three Dimensions......Page 304
Exercises......Page 309
Appendix 13B. Derivation of an Exact Solution for the Scattering of an Acoustic Pulse by a Circular Cylinder......Page 310
14. Continuation of a Near-Field Acoustic Solution to the Far Field......Page 313
14.1 The Continuation Problem......Page 314
14.2 Surface Green’s Function: Pressure as the Matching Variable......Page 316
14.3 Surface Green’s Function: Normal Velocity as the Matching Variable......Page 320
14.4 The Adjoint Green’s Function......Page 323
14.4.1 Computation of the Adjoint Green’s Function: Free Field Solution......Page 326
14.4.2 Adjoint Green’s Function for a Cylindrical Surface......Page 327
14.5 Adjoint Green’s Function for a Conical Surface......Page 328
14.5.1 Computation of the Adjoint Green’s Function for a Conical Surface......Page 331
14.5.2 Time Periodic Sources......Page 333
14.6 Generation of a Random Broadband Acoustic Field......Page 336
14.7 Continuation of Broadband Near Acoustic Field on a Conical Surface to the Far Field......Page 338
14.7.1 A Test Case: A Broadband Monopole Source......Page 342
Exercise......Page 343
15.1.1 Computational Model......Page 344
15.1.2 Computational Domain......Page 345
15.1.4 Computational Algorithm......Page 346
15.1.6 Distribution of Artificial Selective Damping......Page 347
15.2.1 Free Shear Flows......Page 348
15.2.1.1 Two-Dimensional Turbulent Mixing Layer......Page 351
15.2.1.2 Two-Dimensional Turbulent Jet......Page 352
15.2.2 Laminar Boundary Layer......Page 353
15.2.3 Turbulent Boundary Layer......Page 355
15.3.1 Conformal Transformation......Page 359
15.3.2 Inverse Mapping from the w Plane to the Physical Plane......Page 362
15.3.3 NACA Four-Digit Airfoils......Page 363
15.4 Example I: Direct Numerical Simulation of the Generation of Airfoil Tones at Moderate Reynolds Number......Page 365
15.4.1.2 Grid Design and Computational Domain......Page 369
15.4.1.3 Computational Algorithm and Boundary Conditions......Page 370
15.4.1.4 Distribution of Artificial Selective Damping......Page 372
15.4.1.5 Selection of Time Step......Page 374
15.4.1.7 Grid Refinement......Page 376
15.4.2 Numerical Results......Page 377
15.4.2.1 Flow Field......Page 378
15.4.2.2 Acoustics......Page 380
15.4.3 Energy Source of Airfoil Tones......Page 384
15.4.4 Sound Generation Processes......Page 387
15.5 Computation of Turbulent Flows......Page 394
15.5.1 Large Eddy Simulation......Page 395
15.5.2 RANS Computation: the k − ε Model......Page 397
15.6 Example II: Numerical Simulation of Axisymmetric Jet Screech......Page 400
15.6.1.1 Computational Model......Page 401
15.6.1.2 Grid Design and Computational Scheme......Page 403
15.6.1.3 Artificial Selective Damping......Page 404
15.6.2 Numerical Results and Comparisons with Experiments......Page 405
15.6.2.1 Mean Velocity Profiles and Shock Cell Structure......Page 406
15.6.2.2 Screech Tone Frequency and Intensity......Page 407
A.2 Laplace Transform......Page 410
Appendix B: The Method of Stationary Phase......Page 412
Appendix C: The Method of Characteristics......Page 413
Appendix D: Diffusion Equation......Page 415
Appendix E: Accelerated Convergence to Steady State......Page 418
Appendix F: Generation of Broadband Sound Waves with a Prescribed Spectrum by an Energy-Conserving Discretization Method......Page 422
G.1 Sample Program for Solving the One-Dimensional Convective Wave Equation......Page 425
G.2 Sample Program for Solving the Two-Dimensional Linearized Euler Equations......Page 435
G.3 Sample Program for Solving the Three-Dimensional Euler Equations......Page 452
References......Page 486
Index......Page 492

✦ Subjects

Транспорт;Авиационная техника;Аэродинамика в авиации;