Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified cylindrical shells

We study the linear stability of an arbitrary number N of cylindrical concentric shells undergoing a radial implosion or explosion.We derive the evolution equation for the perturbation ηi at interface i; it is coupled to the two adjacent interfaces via ηi±1. For N=2, where there is only one interface, we verify Bell’s conjecture as to the form of the evolution equation for arbitrary ρ1 and ρ2, the fluid densities on either side of the interface. We obtain several analytic solutions for the N=2 and 3 cases. We discuss freeze-out, a phenomenon that can occur in all three geometries (planar, cylindrical, or spherical), and “critical modes” that are stable for any implosion or explosion history and occur only in cylindrical or spherical geometries. We present numerical simulations of possible gelatin-ring experiments illustrating perturbation feedthrough from one interface to another. We also develop a simple model for the evolution of turbulent mix in cylindrical geometry and define a geometrical factor G as the ratio hcylindrical∕hplanar between the cylindrical and planar mixing layers. We find that G is a decreasing function of R∕R0 implying that in our model hcylindrical evolves faster (slower) than hplanar during an implosion (explosion).