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.