Nonlinear regime of a multimode Richtmyer–Meshkov instability: A simplified perturbation theory

In this paper we present a drastic simplification of the perturbation method for the Richtmyer–Meshkov instability developed by Zhang and Sohn [Phys. Fluids 9, 1106 (1997)]. This theory is devoted to the calculus of the growth rate of the perturbation of the interface in the weakly nonlinear stage. In the standard approach, expansions appear to be power series in time. We build accurate approximations by retaining only the terms with the highest power in time. This simplifies and accelerates the solution. High-order expressions are then easily reachable. Furthermore, computations for multimode interfaces become tractable. The accuracy of this approach is checked against two-dimensional numerical simulations. The selection mode process is studied and the phase between modes is shown to be as important as the wave number or the amplitude. Inferences for the intermediate nonlinear regime are also proposed. In particular, a class of homothetic configurations is inferred; its validity is verified with numerical simulations even as vortex structures appear at the interface.