The GRP algorithm was developed in Part I and tested against a variety of analytical examples. The next challenge is to adapt it to the needs of “Scientific Computing,” that is, to implement it in the simulation of problems of physical significance. This is a formidable task for any numerical algorithm. The problems encountered in applications usually are multidimensional and involve complex geometries and additional physical phenomena. In this chapter we first describe in more detail some of these problems, as a general background for the following chapters. We then proceed to the main goal of the chapter, namely, the upgrading of the GRP algorithm (which is essentially one dimensional) to a two-dimensional, second-order scheme. To this end we recall (Section 7.2) Strang's method of “operator splitting” and then (Section 7.3) apply it to the (planar) fluid-dynamical case. Although the method can be further extended to the three-dimensional case, we confine our presentation to the two-dimensional setup, which serves in our numerical examples (Chapters 8 and 10).

General Discussion

In Part I we studied the mathematical basis of the GRP method. We also considered a variety of numerical examples for which analytic (or at least asymptotic) solutions were available. This provided us with some measure of the accuracy of the computational results, and helped identifiy potential difficulties (such as the resolution of contact discontinuities).

The GRP method was developed (Chapter 5) for compressible, unsteady flow in a duct of varying cross section. In the case of a planar two-dimensional duct, the quasi-1-D formulation is taken to be a reasonable approximation of the actual (2-D) flow. In this chapter we study a duct flow where an incident wave interacts with a short converging segment, producing interesting wave structures. An illustrative case is that of a rarefaction wave propagating through a “converging corridor,” producing (at later times) a complex “reflected” wave pattern. Such a case is studied (numerically) in this chapter, using (a) the quasi-1-D approach of Chapter 5 and (b) the full two-dimensional computation as described in Section 8.3 (with the duct contour taken as a stationary boundary). The comparison between the two computations reveals some bounds of validity of the quasi-1-D approximation. We conclude the chapter by listing (Remark 10.1) several articles describing the application of the GRP to diverse fluid-dynamical problems, including well-known test cases, shock wave reflection phenomena compared to experimental observation, and even a case where a “moving boundary” experiment is favorably compared to the corresponding GRP solution.

Consider a centered rarefaction wave that propagates in a planar duct comprising two long segments of uniform cross-sectional area joined by a smooth converging nozzle.

What is Vortex Sound?

Vortex sound is the sound produced as a by-product of unsteady fluid motions (Fig. 1.1.1). It is part of the more general subject of aerodynamic sound. The modern theory of aerodynamic sound was pioneered by James Lighthill in the early 1950s. Lighthill (1952) wanted to understand the mechanisms of noise generation by the jet engines of new passenger jet aircraft that were then about to enter service. However, it is now widely recognized that any mechanism that produces sound can actually be formulated as a problem of aerodynamic sound. Thus, apart from the high speed turbulent jet – which may be regarded as a distribution of intense turbulence velocity fluctuations that generate sound by converting a tiny fraction of the jet rotational kinetic energy into the longitudinal waves that constitute sound – colliding solid bodies, aeroengine rotor blades, vibrating surfaces, complex fluid–structure interactions in the larynx (responsible for speech), musical instruments, conventional loudspeakers, crackling paper, explosions, combustion and combustion instabilities in rockets, and so forth all fall within the theory of aerodynamic sound in its broadest sense.

In this book we shall consider principally the production of sound by unsteady motions of a fluid. Any fluid that possesses intrinsic kinetic energy, that is, energy not directly attributable to a moving boundary (which is largely withdrawn from the fluid when the boundary motion ceases), must possess vorticity. We shall see that in a certain sense and for a vast number of flows vorticity may be regarded as the ultimate source of the sound generated by the flow.

Vortex sound is the branch of fluid mechanics concerned with the conversion of hydrodynamic (rotational) kinetic energy into the longitudinal disturbances we call sound. The subject is itself a subsection of the theory of aerodynamic sound, which encompasses a much wider range of problems also involving, for example, combustion and ‘entropy’ sources of sound. The book is based on an introductory one-semester graduate level course given on several occasions at Boston University. Most students at this level possess an insufficient grasp of basic principles to appreciate the subtle coupling of the hydrodynamic and acoustic fields, and many are ill-equipped to deal with the novel analytical techniques that have been developed to investigate the coupling. Great care has therefore been taken to discuss underlying fluid mechanical and acoustic concepts, and to explain as fully as possible the steps in a complicated derivation.

A considerable number of practical problems occur at low Mach numbers (say, less than about 0.4). It seems reasonable, therefore, to confine an introductory discussion specifically to low Mach number flows. It is then possible to investigate a number of idealized hydrodynamic flows involving elementary distributions of vorticity adjacent to solid boundaries, and to analyze in detail the sound produced by these vortex–surface interactions. For a broad range of such problems, and a corresponding broad range of noise problems encountered in industrial applications, the effective acoustic sources turn out to be localized to one or more regions that are small compared to the acoustic wavelength.