Simulation of the high Mach number motion for bubble collapse in a compressible Euler fluid using Basilisk

Abstract

We examine an extreme case of experimentally realizable sonoluminescence, where spherical cavities have an initial radius that is $10$ to $20$ times their ambient radius and change their radius by a factor of over $100$ during the collapse. Among the many physical processes at play, we focus on fluid compressibility, modeled using the Tait-Murnaghan equation of state for a homentropic Euler fluid. To capture such extreme motion, with Mach numbers relative to ambient sound speed greater than one during the final stages of implosion, requires methods beyond the classic approaches of Rayleigh and Gilmore. In this direction, we applied an all-Mach solver developed in the Basilisk framework, actively used to model bubble dynamics. To capture high Mach number motion and resolve dynamics in the sonoluminescence regime, we employed the well-established uniform bubble approximation for the ideal gas inside the bubble. Within this approximation, the all-Mach solver achieved numerically converging results describing the evolution of the bubble wall $R(t)$. Although compressibility slows down the collapse, these bubbles reach velocities exceeding the ambient speed of sound of the surrounding fluid. Our method works for various fluids and is applied to liquid lithium as well as water. Our results reproduce the equation-of-state-dependent asymptotic power-law region predicted by analytic calculations for water and liquid lithium in the case of an empty cavity. When the cavity is filled with an ideal gas, the transition to Mach number greater than one in liquid lithium occurs later in the collapse than for water, making liquid lithium a possible candidate for achieving greater concentration of energy density. Furthermore, an outgoing shock wave, which can diagnose cavitation in opaque fluids such as liquid lithium, is captured without implementing an ad hoc construction algorithm.