Effect of viscosity on Rayleigh-Taylor and Richtmyer-Meshkov instabilities.

Abstract

We consider the effect of viscosity on Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities by deriving a moment equation for fluids with arbitrary density and viscosity profiles, including surface tension. We apply our result to the classical case of two semi-infinite fluids with densities ${\mathrm{\ensuremath{\rho}}}_{1}$ and ${\mathrm{\ensuremath{\rho}}}_{2}$ and viscosities ${\mathrm{\ensuremath{\mu}}}_{1}$ and ${\mathrm{\ensuremath{\mu}}}_{2}$. Treating a shock as an instantaneous acceleration we find that perturbations at the interface undergo damped oscillations when viscosity and surface tension are both present. For pure viscosity the amplitude \ensuremath{\eta}(t) evolves according to \ensuremath{\eta}(t)/\ensuremath{\eta}(0)=1+(\ensuremath{\Delta}vA/2k\ensuremath{\nu})(1-${\mathit{e}}^{\mathrm{\ensuremath{-}}2\mathit{k}2}$\ensuremath{\nu}t) where \ensuremath{\Delta}v is the jump velocity imparted by the shock, A=(${\mathrm{\ensuremath{\rho}}}_{2}$-${\mathrm{\ensuremath{\rho}}}_{1}$)/(${\mathrm{\ensuremath{\rho}}}_{2}$+${\mathrm{\ensuremath{\rho}}}_{1}$), \ensuremath{\nu}=(${\mathrm{\ensuremath{\mu}}}_{1}$+${\mathrm{\ensuremath{\mu}}}_{2}$)/(${\mathrm{\ensuremath{\rho}}}_{1}$+${\mathrm{\ensuremath{\rho}}}_{2}$), k=2\ensuremath{\pi}/\ensuremath{\lambda} is the wave number of the perturbation, and t is time. We also consider the turbulent energy in accelerating fluids and calculate the reduction in ${\mathit{E}}_{\mathrm{turbulent}}$ as a function of \ensuremath{\nu}, and propose experiments to measure the effect of viscosity on RT and RM instabilities.