Analytical scalings of the linear Richtmyer-Meshkov instability when a shock is reflected

When a planar shock hits a corrugated contact surface between two fluids, hydrodynamic perturbations are generated in both fluids that result in asymptotic normal and tangential velocity perturbations in the linear stage, the so called Richtmyer-Meshkov instability. In this work, explicit and exact analytical expansions of the asymptotic normal velocity (δv_{i}^{∞}) are presented for the general case in which a shock is reflected back. The expansions are derived from the conservation equations and take into account the whole perturbation history between the transmitted and reflected fronts. The important physical limits of weak and strong shocks and the high/low preshock density ratio at the contact surface are shown. An approximate expression for the normal velocity, valid even for high compression regimes, is given. A comparison with recent experimental data is done. The contact surface ripple growth is studied during the linear phase showing good agreement between theory and experiments done in a wide range of incident shock Mach numbers and preshock density ratios, for the cases in which the initial ripple amplitude is small enough. In particular, it is shown that in the linear asymptotic phase, the contact surface ripple (ψ_{i}) grows as ψ_{∞}+δv_{i}^{∞}t, where ψ_{∞} is an asymptotic ordinate different from the postshock ripple amplitude at t=0+. This work is a continuation of the calculations of F. Cobos Campos and J. G. Wouchuk, [Phys. Rev. E 90, 053007 (2014)PLEEE81539-375510.1103/PhysRevE.90.053007] for a single shock moving into one fluid.