Kinetic energy of the rotational flow behind an isolated rippled shock wave

The kinetic energy of the perturbation velocity field generated by a corrugated isolated shock is analyzed as a function of the shock Mach number (M-s) and the fluid compressibility (characterized here with the ideal gas ratio of specific heats.) within the framework of a linear theory. The shock front dynamics is analyzed, evaluating the zeros and critical points of the temporal evolution of the pressure perturbations behind the shock front. These characteristic times are essential in determining the spatial distribution of the velocity fluctuations in the inhomogeneously compressed fluid. The velocity perturbations are followed in time and space behind the corrugated shock front until an asymptotic stage emerges. Graf's addition theorem of the Bessel functions is seen to be an adequate mathematical tool with which to evaluate the detailed temporal evolution. The kinetic energy density is analyzed in time and space well into the asymptotic stage. Exact criteria are given, based on the properties of the asymptotic velocity field, in order to determine the critical points of the kinetic energy density. The space integral of the kinetic energy density is studied, and an explicit analytical formula can be written in finite form and in terms of known functions. The integrated kinetic energy (KE) is analyzed at the limits of very weak and very strong shocks. For very weak shocks the kinetic energy scales as KE similar to u(i)(2), where u(i) is the asymptotic normal velocity perturbation at the surface x = 0. For very strong fronts, the scaling changes to KE similar to u(i)(2) lnM(s). The logarithmic factor is due to contribution of the bulk vorticity profile, which becomes important at high compressions.