A shock driven material interface between two fluids of different density is unstable. This instability is known as Richtmyer-Meshkov (RM) instability. In this paper, we present a quantitative nonlinear theory of compressible Richtmyer-Meshkov instability in two dimensions. Our nonlinear theory contains no free parameter and provides analytical predictions for the overall growth rate, as well as the growth rates of the bubble and spike, from early to later times for fluids of all density ratios. The theory also includes a general formulation of perturbative nonlinear solutions for incompressible fluids (evaluated explicitly through the fourth order). Our theory shows that the RM unstable system goes through a transition from a compressible and linear one at early times to a nonlinear and incompressible one at later times. Our theoretical predictions are in excellent agreements with the results of full numerical simulations from linear to nonlinear regimes.

The vortex method is applied to simulations of Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities. The numerical results from the vortex method agree well with analytic solutions and other numerical results. The bubble velocity in the RT instability converges to a constant limit, and in the RM instability, the bubble and spike have decaying growth rates, except for the spike of infinite density ratio. For both RT and RM instabilities, bubbles attain constant asymptotic curvatures. It is found that, for the same density ratio, the RT bubble has slightly larger asymptotic curvature than the RM bubble. The vortex sheet strength of the RM interface has different behavior than that of the RT interface. We also examine the validity of theoretical models by comparing the numerical results with theoretical predictions.

We generalize the Layzer-type model for unstable interfaces to the system of arbitrary density ratio. The predictions from the generalized model for bubble growth rates of Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities are in good agreement with numerical results. We present the theoretical prediction for asymptotic growth rates for RT and RM bubbles for finite density ratios in two and three dimensions.

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