Vorticity generation by shock propagation through bubbles in a gas

Abstract

We present a new theoretical model of ‘late-time’ phenomena related to the interaction of a planar shock with a local, discrete inhomogeneity in an ambient gas. The term ‘late-time’ applies to the evolution of the inhomogeneity and the flow field after interaction with the incident shock has ceased. Observations of a shock propagating through a bubble or a spherical flame have exhibited or implied the formation of vortex structures and have showed continual distortion of the bubble or flame. Our theory shows that this is due to the generation of long-lived vorticity at the edge of the discrete inhomogeneity. The vorticity interacts with itself through the medium of the fluid, and, depending on the geometry of the discrete inhomogeneity, can roll up into vortex filaments or vortex rings. To verify and amplify this theoretical description, we use numerical solutions of the fluid equations for conservation of mass, momentum, and energy to study the interaction of a weak shock with a cylindrical or spherical bubble. The simulated bubble has either a higher or lower density than the ambient gas. In this way, the calculations provide insights into the effects of both geometry and distortion of the local sound speed. The Mach number of the shock is 1.2, the ambient gas is air, and the pressure is 1 atmosphere. Because of the simple geometry of each bubble, the vorticity generated at the boundary rolls up into a vortex filament pair (cylindrical bubble) or a vortex ring (spherical bubble). The structural features and timescales of the phenomena observed in the calculations agree closely with recent experiments of Haas & Sturtevant, in which helium and Freon bubbles were used to provide the local departures from ambient density. The discussion of results includes a survey of alternative numerical methods, sources of uncertainty in velocities of interfaces or structures, as derived from the laboratory and numerical experiments, and the relationship of our analysis to other theories.