Numerical Study of Shock Induced Mixing in a Cylindrical Shell

Abstract

The Richtmyer-Meshkov instability (RMI) develops when a shock wave traverses a density interface separating two gases. The miss-alignment of the pressure gradient across the shock and the local density gradient at the contact during shock passage leads to vorticity production at the interface. Subsequently the flow driven by the deposited vorticity leads to interfacial instability growth and eventually to turbulence mixing. RMI is important in many areas of physics, from geophysical and astrophysical problems to industrial applications. In particular, attention has recently focused on RMI and RM mixing in the converging geometry such as that occurs in an imploding inertial confinement fusion (ICF) capsule. When a stratified cylindrical shell with initial perturbations is driven by a convergent shock wave, the effect of convergence tends to enhance the perturbation growth compared with that in a planar geometry. The convergent incident shock wave reflects at the cylinder center and the succedent reflected shock waves move to and fro within the whole region. Besides, the Rayleigh-Taylor instability (RTI) also occurs whenever the light fluid accelerates the heavy one during the evolution. All these factors make the mixing procedure in a stratified cylindrical shell driven by shock wave much complex than that in the planar geometry. Although many models have been proposed to predict the instability growth in the linear, weakly nonlinear, and turbulent regimes, each of these models has limitations and a restricted domain of applicability. For this complex mixing process with strongly nonlinear transition stage, the direct numerical simulation with high resolution is the common way to study its evolution. In this paper a hybrid scheme combined with the finite-difference and the weighted essentially non-oscillatory (WENO) method, combined with high order strong-stability preserving Runge-Kutta scheme for the time integration, is used to simulate the mixing due to the interfacial instability in a stratified cylindrical shell driven by convergent shock wave. Growth and mixing properties of the turbulent mixing zone (TMZ) have been investigated using the simulation results. And the effect of initial perturbation on the mixing has been discussed.