Page 50 - Winter2014

book Reviews
“Special cases include the accurate capture of wave steepening and shock formation due to nonlinear propagation, the influence of variable mean flow and the application of dissipative wall boundary conditions”
capture of wave steepening and shock formation due to non- linear propagation, the influence of variable mean flow and the application of dissipative wall boundary conditions. The remainder of the book is devoted to illustrative applications, including wave scattering by solid bodies, rotor noise, wall cavity noise, the use of oversetting grids, and the prolongation of a numerical acoustic near field into the distant far field, culminating (Chapter 15) in an extensive discussion of airfoil noise and the sound generated by an imperfectly expanded supersonic jet. Each chapter ends with a series of problems for the reader. A set of useful sample Fortran codes is docu- mented in the appendix.
The book is described as both a professional research reference and a graduate level text. But CAA is a rapidly expanding and highly competitive subject, and Computational Aeroacoustics is more likely to be received as a substantive repository of one research group's approach to solving CAA problems. Prior exposure to the numerical treatment of partial differential equations is not required of the reader, although the text is difficult in parts and especially so for the uninitiated. The cur- rent trend encouraging the gradual demotion of undergradu- ate engineering mathematics in favor of commercial software tools could very possibly increase the difficulties faced by prospective student readers, who may not possess the ‘general understanding of partial differential equations’ assumed by the author.
The main thrust of this book is an account of how these problems are handled by clever modification of traditional finite difference approximations to the governing equations. Accurate resolution of small scale motions is achieved in CFD by requiring 18 - 25 mesh points per hydrodynamic wave- length. This is impracticably large for CAA, which usually requires much more extensive gridding to reach an acceptable approximation to the acoustic far field. The special schemes developed by Tam yield adequate resolutions of sound waves with about six or seven mesh points per wavelength. Roughly speaking, the secret lies in optimizing the coefficient values in a finite difference approximation to ensure that the dispersive characteristics -- such as ‘group velocity’ -- of the exact system of equations are replicated in the numerical model.
The first five or so chapters deal with the ‘wavenumber’ ap- proach to finite difference optimization, both for spatial and temporal variations. Essentially, because differentiation is equivalent to wavenumber multiplication of the Fourier trans- form, the selection of the finite difference coefficients is made by minimizing the averaged mean square difference between the exact wavenumber and its finite difference approximation, the average being over a wavenumber range that ensures a resolution of, say, one wavelength per 7 mesh points. The pro- cedure is actually a little more complicated, involving group velocity and some numerical experimentation, but the details are adequately explained by Tam. The purist might object
to such empiricism -- which, however, seems to pervade the world of CAA!
The application of non-reflective radiation conditions and mean outflow boundary conditions are discussed in Chapter 6, and the following four chapters review subsidiary mate- rial and special cases, including the introduction of artificial damping to remove ‘numerical noise’ and associated instabili- ties from small scale variations not fully resolved by the opti- mized difference equations. Special cases include the accurate
48 | Acoustics Today | Winter 2014