Self-Similar Shock Waves in Water

Shock wave phenomena in water have been extensively studied since the 1950s, with notable analytical investigations centering on implosions and cavity collapse conducted by several scientists including C. Hunter and R. Lazarus following in the tradition of Lord Rayleigh, who first examined the incompressible cavity collapse problem in 1917. A key feature of these and related studies is the construction of certain self-similar scaling solutions of the underlying one-dimensional Euler equations, wherein the structure of these solutions is intimately tied to the highly restrictive initial and boundary conditions assumed in the classical problem formulations. This work explores the relaxation of some of these assumptions and the attendant consequences for the structure of diverging and converging shock wave solutions. For example, the assumption of a linear-in-radius downstream velocity field corresponds to the so-called “singular case” of the classical Sedov-Taylor-von Neumann blast wave solution and ultimately leads a closed-form solution for a spherical piston-driven explosion or implosion process in a g = 7 polytropic fluid. A novel implosion solution which may be shown to be a special member of the broader class of “reverse blast wave” solutions is compared to the variety of existing results obtained for converging shock waves and collapsing cavities in water.