## 2 Numerical Methods
### Spherical Bubble Dynamics
The Rayleigh-Plesset equation with the incompressibility assumption is as follows:  
\[R\ddot{R}+\frac{3}{2}\dot{R}^{2}=\frac{1}{\rho}\bigg{[}p_{v}+p_{g0}\bigg{(} \frac{R_{0}}{R}\bigg{)}^{3k}-p-\frac{2\gamma}{R}-\frac{4\mu}{R}\dot{R}\bigg{]} +\frac{\bigg{(}\overline{U}-\overline{U}_{b}\bigg{)}^{2}}{4} \tag{9}\]  
The Keller-Herring equation with the compressibility effects is as follows:  
\[\bigg{(}1-\frac{\dot{R}}{c}\bigg{)}R\ddot{R}+\frac{3}{2}\bigg{(}1-\frac{1}{3} \frac{\dot{R}}{c}\bigg{)}\dot{R}^{2}=\bigg{(}1+\frac{\dot{R}}{c}+\frac{R}{c} \frac{d}{dt}\bigg{)}\bigg{[}h-\frac{p_{v}}{\rho}\bigg{]}+\frac{\bigg{(} \overline{U}-\overline{U}_{b}\bigg{)}^{2}}{4} \tag{10}\]  
where \(h=(p_{B}-p)/\rho\), \(p_{B}=p_{g}-\frac{1}{R}\big{(}2\gamma+4\mu\dot{R}\big{)}\), \(R\) denotes the bubble radius, \(p_{v}\) denotes vapor pressure, \(p_{g0}\) denotes the initial internal gas pressure in the nuclei, \(R_{0}\) denotes initial nuclei radius, \(\gamma\) denotes surface tension, \(\mu\) denotes liquid dynamic viscosity, \(p\) denotes fluid pressure, and \(\overline{U}\) denotes the velocity on the bubble surface. Furthermore, \(\overline{U}_{b}\) denotes the moving velocity of a bubble, and the superscript '\(\cdot\)' denotes the time derivative term. Additionally, \(c\), \(h\), and \(k\) denote liquid sound speed, enthalpy, and polytropic gas constant, respectively. If the speed of sound becomes infinite, then Equation (10) is reduced to Equation (9). In this study, the Keller-Herring equation is used for predicting the variation in the volume of the bubble. However, in classical bubble dynamics, the external pressure on the bubble surface is extrapolated from the pressure at the position, which is the bubble center if there is no bubble. This approximation neglects the variation of the bubble surface pressure due to the change in bubble radius. Therefore, the external pressure in Equation (10) is determined using the surface-averaged pressure method [29; 41], where the external properties are computed by averaging the pressure and velocities in the flow field on the bubble surface. Gas in the nuclei or bubble is assumed to be an ideal gas. However, there is a significant difference in the time-scale between the growth and decrease phases of the bubble: the time-scale in the bubble growth phase is much higher than that in the bubble decrease phase. Therefore, the bubble growth process can be assumed as isothermal, and the bubble decrease process can be assumed as adiabatic. The polytropic exponent for isothermal and adiabatic processes is set as [32] follows:  
\[\textit{polytropic exponent},\ k= \begin{cases}1&\textit{Isothermal process}\big{(}\hat{R}>0\big{)}\\ 1.4&\textit{Adiabatic process}\big{(}\hat{R}<0\big{)}\end{cases} \]  
Nuclei containing non-condensable gas and liquid-vapor are in a dynamic equilibrium state with the surrounding fluid. The pressure due to non-condensable gas, \(p_{g}\), and the pressure due to vapor, \(p_{v}\), equilibrates with the surface tension on the nuclei surface and the pressure due to external liquid, \(p_{L}\), as shown in Figure 7.  
This equilibrium relationship can be written as follows:  
\[p_{L}=p_{v}+p_{g}-\frac{2\gamma}{R}\ \ \ \textit{where}\ p_{g}=p_{g0}\big{(} \frac{R_{0}}{R}\big{)}^{3k} \tag{11}\]  
Here, the initial gas pressure \(p_{g0}\) is determined from initial flow field information and initial nuclei radius [28]. Given that the bubble surface variation occurs very fast when compared with the time-scale of bubble dynamics simulation, the vapor can be assumed to be in equilibrium at each moment, and the vapor pressure remains constant. Conversely, the expansion of the gas occurs more slowly when compared with the time-scale of the bubble dynamics simulation. Therefore, it is necessary to consider the change in the internal gas pressure by changing the bubble radius at every moment. The gas pressure in Equation (11) is varied by changing the bubble radius.