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

Cavitation is a process where bubbles form and collapse within a fluid with dynamic, spatially varying pressure. This phenomenon can concentrate energy density by 12 orders of magnitude, creating light-emitting plasma or damaging nearby surfaces. A key question in cavitation theory and experiments is: What are the upper limits of energy density achievable through this spontaneous multiscale process? 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. Sonoluminescence represents the extreme limit of cavitation. Here, 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. To capture such extreme motion, with Mach numbers relative to ambient sound speed greater than one during the final stages of implosion, require 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. Some of the reasons for its popularity include the fact that it is also expected to work for high Mach numbers, transport processes, nonspherical motion, shock wave capture, and involves low computational cost. Additionally, it has been tested for compressible motion at lower Mach numbers. Given its recent and extensive application, we endeavored to determine the specific solver's capability in resolving rapid bubble collapses. This solver was designed to handle the hydrodynamics of both the gas inside the bubble and the compressible fluid outside. However, we discovered that even at early times, when the Mach number is small and the fluid is approximately incompressible, the solver cannot accurately resolve the hydrodynamic motion within practical computational limits. 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. 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. This confirms our method's ability to capture high Mach number motion and suggests that the asymptotic regime could be experimentally observed. 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 for solutions based on characteristics or other approaches.