Numerical study of axisymmetric Richtmyer-Meshkov instability and azimuthal effect on spherical mixing

In this paper, we present a numerical study of the axisymetric Richtmyer-Meshkov instability in converging spherical geometry by the front tracking method for the first time. The front tracking method has been successfully used in solving fluid instability problems in both rectangular and curved geometry. The central issue for axisymmetric flows is the absence of the rotational symmetry in the (r, z) plane, although the perturbed shape of the initial contact interface appears to have it. The cause of the asymmetry is somewhat obvious. The sinusoidal perturbations appear symmetric only in the cross-sectional view; in actuality they are not symmetric because they represent rings around the z-axis and hence the perturbed mass at the equator, for example, is different from the perturbed mass at the pole. The first purpose of this paper is to quantify the effect of this inherited asymmetry on the growth of the spherical mixing. We find this asymmetry drives the original structure to some degree so that the mixing radius at the north pole is noticeably larger than at the equator during the evolution of chaotic mixing. We also study quantitatively the azimuthal dependence of the mixing statistics, such as the mixing edges, the growth rate and volume fraction. Richtmyer-Meshkov (RM) instabilities in spherical geometry have been a challenge due to the inherent difficulty of their accurate simulation. Our second purpose is to demonstrate that our Front Tracking method can describe the Richtmyer-Meshkov instability growth in a complex flow involving multiple reshocks. We have successfully displayed the converging geometry, reshock process, asymmetry phenomenon through the density and pressure color plots. The quantitative growth rate analysis is also presented.