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.

Figures (10)

• Figure 1: Implosion of an ideal gas xenon bubble in water at ambient pressure $p_{\infty} = 1\textrm{ atm}$ and ambient temperature $T_{\infty} = 293\textrm{ K}$, with $R_m/R_0 = 20$ and $R_0 = 2.2\textrm{ }\upmu\textrm{m}$. The red dot-dashed line represents the incompressible solution for the ambient density of water, $\rho_{\infty} = 998\textrm{ kg/}\mathrm{m^3}$. The black solid line is obtained using our modified all-Mach solver with the uniform bubble approximation. The compressibility of water is given by Eq. \ref{['Tait_Murnaghan_EoS']} with $\rho_0 = 998\textrm{ kg/}\mathrm{m^3}$, $\Gamma = 7$, and $B = 3.12 \times 10^8\textrm{ Pa}$. The black dots indicate the point at which the Mach number $\textrm{M} = \dot{R}/c_{\infty} = 1$ relative to water is reached. The dashed green line is a power-law fit of the form as in Eq. \ref{['incompressible_Rayleigh_R_t']} with $n = 0.56$, as predicted in Ref. Hunter_1960. The comparison is shown from time $t_0$, where $R(t_0) = R_m/5$, as before this time the compressible and incompressible results agree. In (b), the solutions shown in (a) are zoomed in near the minimum radii of the curves.
• Figure 2: Implosion of an ideal gas xenon bubble in liquid lithium at ambient pressure $p_{\infty} = 1\textrm{ atm}$ and ambient temperature $T_{\infty} = 454\textrm{ K}$, with $R_m/R_0 = 20$ and $R_0 = 2.2\textrm{ }\upmu\textrm{m}$. The red dot-dashed line represents the incompressible solution for the ambient density of liquid lithium, $\rho_{\infty} = 516\textrm{ kg/}\mathrm{m^3}$. The black solid line is obtained using our modified all-Mach solver with the uniform bubble approximation. The compressibility of liquid lithium is given by Eq. \ref{['Tait_Murnaghan_EoS']} with $\rho_0 = 516\textrm{ kg/}\mathrm{m^3}$, $\Gamma = 3.75$, and $B = 2.83 \times 10^9\textrm{ Pa}$. The black dots indicate the point at which the Mach number $\textrm{M} = \dot{R}/c_{\infty} = 1$ relative to liquid lithium is reached. The dashed green line is a power-law fit of the form as in Eq. \ref{['incompressible_Rayleigh_R_t']} with $n = 0.65$, as predicted in Ref. 10.1063/5.0160469. The comparison is shown from time $t_0$, where $R(t_0) = R_m/5$, as before this time the compressible and incompressible results agree. In (b), the solutions shown in (a) are zoomed in near the minimum radii of the curves.
• Figure 3: Implosion of an ideal gas xenon bubble surrounded by compressible water with $R_m/R_0 = 10$ and ambient parameters $R_0 = 2.2\textrm{ }\upmu\textrm{m}$, $p_{\infty} = 1\textrm{ atm}$, and $T_{\infty} = 293\textrm{ K}$. The full hydrodynamic equations given in Eqs. \ref{['Euler_equation']} and \ref{['mass_conservation']} are solved for both the gas and the surrounding fluid, starting from rest at the maximum radius $R_m$. The numerical parameters are $\lambda = 4$, $\Delta r = R_m/2^{15} = 10R_0/2^{15}$, and $\Delta t_{\textrm{max}} = 1.0 \times 10^{-6}$$R_0 \sqrt{\rho_{\infty}/p_{\infty}}$. The figure shows the solution at three times $t = 0.0, 1.0, 1.6$, in units of $R_0 \sqrt{\rho_{\infty}/p_{\infty}}$, corresponding to (I), (II), (III), respectively. Figures (a), (d), (g) show profiles of the average mass density, $\overline{\rho}$; (b), (e), (h) show the average speed of sound, $\overline{c}$; and (f) and (i) show the average pressure, $\overline{p}$, with the red dashed line indicating the location of the cavity wall at each time. In (c), $R(t)$ is shown. As observed, shortly after the collapse is initiated, while $R(t)$ remains close to $R_m$, a region of low mass density and high speed of sound develops in the gas near the wall.
• Figure 4: Implosion of an ideal gas xenon bubble surrounded by compressible water with $R_m/R_0 = 10$ and ambient parameters $R_0 = 2.2\textrm{ }\upmu\textrm{m}$, $p_{\infty} = 1\textrm{ atm}$, and $T_{\infty} = 293\textrm{ K}$. The uniform bubble approximation for the motion of the gas is assumed, and the motion starts from rest at the maximum radius $R_m$. The numerical parameters are $\lambda = 4$, $\Delta r = R_m/2^{15} = 10R_0/2^{15}$, and $\Delta t_{\textrm{max}} = 1.0 \times 10^{-6}$$R_0 \sqrt{\rho_{\infty}/p_{\infty}}$. The figure shows the solution at three times $t = 0.0, 1.0, 1.6$, in units of $R_0 \sqrt{\rho_{\infty}/p_{\infty}}$, corresponding to (I), (II), (III), respectively. Figures (a), (d), (g) show profiles of the average mass density, $\overline{\rho}$; (b), (e), (h) show the average speed of sound, $\overline{c}$; and (f) and (i) show the average pressure, $\overline{p}$, with the red dashed line indicating the location of the cavity wall at each time. In (c), $R(t)$ is shown.
• Figure 5: The dependence of the strong collapse on the size of the numerical domain, parametrized by $\lambda$, for a case with $R_m/R_0 = 20$ involving an ideal xenon gas cavity surrounded by compressible water. The uniform bubble approximation for the gas motion is assumed, with the motion starting from $R(t_0) = R_m/5$. The ambient parameters are $R_0 = 2.2\textrm{ }\upmu\textrm{m}$, $p_{\infty} = 1\textrm{ atm}$, and $T_{\infty} = 293\textrm{ K}$. The numerical parameters are $\Delta r = R_0/2^{9}=R_0/512$ and $\Delta t_{\textrm{max}} = 5.9 \times 10^{-7}$$R_0 \sqrt{\rho_{\infty}/p_{\infty}}$. In (b), the results shown in (a) are zoomed in near the minimum radius of the collapse.