Book contents
- Frontmatter
- Contents
- Preface
- 1 Finite Difference Equations
- 2 Spatial Discretization in Wave Number Space
- 3 Time Discretization
- 4 Finite Difference Scheme as Dispersive Waves
- 5 Finite Difference Solution of the Linearized Euler Equations
- 6 Radiation, Outflow, and Wall Boundary Conditions
- 7 The Short Wave Component of Finite Difference Schemes
- 8 Computation of Nonlinear Acoustic Waves
- 9 Advanced Numerical Boundary Treatments
- 10 Time-Domain Impedance Boundary Condition
- 11 Extrapolation and Interpolation
- 12 Multiscales Problems
- Chapter 13 Complex Geometry
- 14 Continuation of a Near-Field Acoustic Solution to the Far Field
- 15 Design of Computational Aeroacoustic Codes
- Appendix A Fourier and Laplace Transforms
- Appendix B The Method of Stationary Phase
- Appendix C The Method of Characteristics
- Appendix D Diffusion Equation
- Appendix E Accelerated Convergence to Steady State
- Appendix F Generation of Broadband Sound Waves with a Prescribed Spectrum by an Energy-Conserving Discretization Method
- Appendix G Sample Computer Programs
- References
- Index
12 - Multiscales Problems
Published online by Cambridge University Press: 05 October 2012
- Frontmatter
- Contents
- Preface
- 1 Finite Difference Equations
- 2 Spatial Discretization in Wave Number Space
- 3 Time Discretization
- 4 Finite Difference Scheme as Dispersive Waves
- 5 Finite Difference Solution of the Linearized Euler Equations
- 6 Radiation, Outflow, and Wall Boundary Conditions
- 7 The Short Wave Component of Finite Difference Schemes
- 8 Computation of Nonlinear Acoustic Waves
- 9 Advanced Numerical Boundary Treatments
- 10 Time-Domain Impedance Boundary Condition
- 11 Extrapolation and Interpolation
- 12 Multiscales Problems
- Chapter 13 Complex Geometry
- 14 Continuation of a Near-Field Acoustic Solution to the Far Field
- 15 Design of Computational Aeroacoustic Codes
- Appendix A Fourier and Laplace Transforms
- Appendix B The Method of Stationary Phase
- Appendix C The Method of Characteristics
- Appendix D Diffusion Equation
- Appendix E Accelerated Convergence to Steady State
- Appendix F Generation of Broadband Sound Waves with a Prescribed Spectrum by an Energy-Conserving Discretization Method
- Appendix G Sample Computer Programs
- References
- Index
Summary
Many aeroacoustics problems involve multiple length and time scales. This should not be difficult to understand. For, in addition to the intrinsic sizes and scales of the noise sources, the acoustic wavelength is an inherent length scale of the problem. In many instances, the length scale of the noise source differs greatly from the acoustic wavelength. This leads to a large disparity in length scales as in classical multiscales problems. For example, in supersonic jet noise, Mach wave radiation is generated by the instability waves of the jet flow. The instability waves are supported by the thin shear layer of the jet. In the region near the nozzle exit, the averaged shear layer thickness is about 0.1D, where D is the jet diameter. The acoustic wavelength, on the other hand, is two or more jet diameters long. Thus, there is an order of magnitude difference between those characteristic lengths. In sound scattering problems, the length scale of the surface geometry of the scatterers may be much smaller than the acoustic wavelength. This occurs very often in edge scattering and diffraction problems. A concrete example is the radiation of fan noise from a jet engine inlet. The acoustic wavelength could be much longer than the radius of the lip of the engine inlet. To obtain an accurate numerical solution of the inlet diffraction problem, a fine mesh is needed around the lip region. Oftentimes, an aeroacoustics problem becomes a multiscales problem because of the change in the physics governing the different parts of the computational domain. An example is the shedding of vortices at the edge of a resonator or a sharp edge of a solid body induced by high-intensity incident sound waves. Away from the solid surface, the fluid is nearly inviscid, but close to the wall, the viscosity effect dominates. The oscillatory motion of the incident sound waves induces a very thin Stokes layer on the solid surface. The Stokes layer rolls up at the corner of a solid surface to form vortices that shed periodically. To simulate the vortex shedding process, therefore, it is necessary to use very fine mesh close to the solid surface and around the corner to resolve the Stokes layer. But away from the solid surface, a coarse mesh with 7 mesh points per acoustic wavelength is all that is needed to capture the sound waves accurately using the 7-point stencil dispersion-relation-preserving (DRP) scheme.
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- Computational AeroacousticsA Wave Number Approach, pp. 229 - 262Publisher: Cambridge University PressPrint publication year: 2012