Startup process in the Richtmyer-Meshkov instability

An analytical model for the initial growth period of the planar Richtmyer-Meshkov instability is presented for the case of a reflected shock, which corresponds in general to light-to-heavy interactions. The model captures the main features of the interfacial perturbation growth before the regime with linear growth in time is attained. The analysis provides a characteristic time scale tau for the startup phase of the instability, expressed explicitly as a function of the perturbation wavenumber k, the algebraic transmitted and reflected shock speeds U-S1 < 0 and U-S2>0 (defined in the frame of the accelerated interface), and the postshock Atwood number A(+): tau=[(1-A(+))/U-S2+(1+A(+))/(-U-S1)]/(2k). Results are compared with computations obtained from two-dimensional highly resolved numerical simulations over a wide range of incident shock strengths S and preshock Atwood ratios A. An interesting observation shows that, within this model, the amplitude of small perturbations across a light-to-heavy interface evolves quadratically in time (and not linearly) in the limit A -> 1(-).