Testing an analytic model for Richtmyer–Meshkov turbulent mixing widths

Abstract

We discuss a model for the evolution of the turbulent mixing width $h(t)$ after a shock or a reshock passes through the interface between two fluids of densities $\rho _A$ and $\rho _B$ inducing a velocity jump $\Delta V$. In this model, the initial growth rate is independent of the surface finish or initial mixing width $h_0$, but its duration $t^{*}$ is directly proportional to it: $h(t)=h_0 +2\alpha A\Delta Vt$ for $0\le t\le t^{*}$, and $h(t)=h^{*}({1+(\dot{h}^{*}/\theta h^{*})(t-t^{*})})^{\theta }$ for $t\ge t^{*}$. Here $A$ is the Atwood number $(\rho _B -\rho _A)/(\rho _B +\rho _A), \alpha$ and $\theta$ are dimensionless, $A$-dependent parameters measured in past Rayleigh–Taylor experiments, and $\beta$ is a new dimensionless parameter we introduce via $t^{*}=(h_0 /\Delta V)\beta$. The mixing width $h$ and its derivative $\dot{h}$ remain continuous at $t=t^{*}$ since $h^{*}=h_0 +2\alpha A\Delta Vt^{*}$ and $\dot{h}^{*}=2\alpha A\Delta V$. We evaluate $\beta \sim 6$ at $A\approx 0.7$ from air/SF$_{6}$ experiments and propose that the transition at $t=t^{*}$ signals isotropication of turbulence. We apply this model to the recent experiments of Jacobs et al. (Shock Waves 23:407–413, 2013) on shock and reshock, and discuss briefly the third wave causing an unstable acceleration of the interface. We also consider the experiments of Weber et al. (Phys Fluids 24:074105, 2012) and argue that their smaller growth rates reflect density gradient stabilization.