Linearized analysis of Richtmyer–Meshkov flow for elastic materials

Abstract

We present a study of Richtmyer–Meshkov flow for elastic materials. This flow, in which a material interface is struck by a shock wave, was originally investigated for gases, where growth of perturbations of the interface is observed. Here we consider two elastic materials in frictionless contact. The governing system of equations comprises conservation laws supplemented by constitutive equations. To analyse it, we linearize the equations around a one-dimensional background solution under the assumption that the perturbation is small. The background problem defines a Riemann problem that is solved numerically; its solution contains transmitted and reflected shock waves in the longitudinal modes. The linearized Rankine–Hugoniot condition provides the interface conditions at the longitudinal and shear waves; the frictionless material interface conditions are also linearized. The resulting equations, a linear system of partial differential equations, is solved numerically using a finite-difference method supplemented by front tracking. In verifying the numerical code, we reproduce growth of the interface in the gas case. For the elastic case, in contrast, we find that the material interface remains bounded: the non-zero shear stiffness stabilizes the flow. In particular, the linear theory remains valid at late time. Moreover, we identify the principal mechanism for the stability of Richtmyer–Meshkov flow for elastic materials: the vorticity deposited on the material interface during shock passage is propagated away by the shear waves, whereas for gas dynamics it stays on the interface.