Mach number effect on the instability of a planar interface subjected to a rippled shock

The Richtmyer-Meshkov (RM) instability of a planar interface (${\mathrm{N}}_{2}\ensuremath{-}{\mathrm{SF}}_{6}$) subjected to a sinusoidal rippled shock, as the variant of a sinusoidal interface impinged by a planar shock, is investigated through high-order compressible multicomponent hydrodynamic simulations. The rippled shock is generated by a planar shock penetrating through a single-mode interface ($\mathrm{He}\ensuremath{-}{\mathrm{N}}_{2}$), and its propagation characteristic agrees reasonably with Bates' analytical solution. Evolution of the flat contact surface impacted by the rippled shock is found to be heavily dependent on the rippled shock phase, and it can be well explained by the impulsive perturbation and continuous perturbation regimes. Various rippled shocks with different Mach numbers ranging from 1.15 to 1.80 are considered. It is found that the influence of the shock strength on the instability growth behaves differently for rippled shocks at different phases. In the case that the shock-interface collision happens when the rippled shock amplitude vanishes for the first time, as the shock strength increases, the impulsive perturbation (i.e., amplitude growth caused by the impulsive shock impact) plays an increasingly more important role in the instability growth than the continuous perturbation (i.e., amplitude growth induced by the disturbed postshock pressure field). In contrast, in the case that the impingement occurs when the rippled shock amplitude becomes zero for the second time, the instability development contributed by the impulsive perturbation is a certain percentage of the total instability growth regardless of the shock strength. The role of the impulsive perturbation in the present nonstandard RM instability within the single-mode framework can be reasonably predicted by an empirical formula combined with the model of Ishizaki et al. [Phys. Rev. E 53, R5592 (1996)].