## 2 Numerical Methods
### Lagrangian Approach Based on Discrete Properties
#### 2.2.2 Bubble Dynamics and Noise Models  
The variations in the location and volume of a nucleus embedded in the background flow field are tracked by solving the bubble dynamics and trajectory equations. The Keller-Herring equation [42] is used as the bubble dynamics equation in the form:  
\[\left(1-\frac{\dot{R}}{c}\right)R\ddot{R}+\frac{3}{2}\!\left(1-\frac{1}{3}\frac {\dot{R}}{c}\right)\!\dot{R}^{2}=\left(1+\frac{\dot{R}}{c}+\frac{R}{c}\frac{d} {dt}\right)\!\left[h-\frac{p_{v}}{\rho_{\infty}}\right]+\frac{\left(\overline{ U}-\overline{U}_{b}\right)^{2}}{4} \tag{12}\]  
where \(R\) denotes the bubble radius, \(c\) is the speed of sound, \(\overline{U}\) is the flow velocity, and \(\overline{U}_{b}\) is the bubble moving velocity. The superscript '\(\cdot\)' and '\(\cdot\)' denote the single- and double-time derivatives, respectively, and the subscript '\(b\)' and '\(\infty\)' indicate the bubble and freestream flow properties, respectively. \(h\), \(p\), and \(\rho\) denote enthalpy, pressure, and density, respectively. The bubble radius is computed by solving Equation (12) in combination with the properties of the flow field surrounding the nucleus. The location of the nuclei (or bubble) in the flow field is computed by solving the bubble trajectory equation [43,44] in the following form:  
\[\frac{d\overline{U}_{b}}{dt}=-\frac{3}{\rho_{l}}\
abla p-2\vec{g}+\frac{3}{4} \frac{C_{D}}{R}\!\left(\overline{U}-\overline{U}_{b}\right)\!\left|\overline{ U}-\overline{U}_{b}\right|+\frac{3}{2\pi}\frac{C_{L}}{R}\sqrt{\frac{\mu_{l}}{ \rho_{l}}\!\left(\overline{U}-\overline{U}_{b}\right)\!\times\!\over\!\over \!\sqrt{\!\left|\!\over\!\left|\!\over\!\left|\!\over\!\left|\!\right|\!\right|} \right.}+\frac{3}{R}\!\left(\overline{U}-\overline{U}_{b}\right)\!\dot{R} \tag{13}\]  
where \(\mu_{l}\) is the liquid dynamic viscosity; \(\rho_{l}\) is the liquid density; \(\rho_{b}\) and \(\gamma\) represent the bubble density and surface tension, which are set to be 1.0 kg/m\({}^{3}\) and 0.0742 N/m, respectively. The speed of sound in water is 1450 m/s. The first term in the right side of Equation (13) denotes the pressure gradient on the bubble surface, and the second term, gravitational acceleration (\(\vec{g}\)), represents the buoyancy effect. The third term is the drag force, which is a function of bubble radius(\(R\)), bubble drag coefficient (\(C_{D}\)) and the slip velocity. The fourth term is the lifting force proportional to the bubble lift coefficient (\(C_{L}\)) and the slip velocity. The last term is the Bjerknes force accounting for the bubble radius variation. The bubble drag coefficient is calculated from the empirical equation by [PERSON] and [PERSON] [45], and the bubble lift coefficient is set to be 6.44 [44]. Equations (12) and (13) are numerically solved by using the 4 th order Runge-Kutta method with the time step size of 0.5 \(\upmu\)s.  
In the classical Rayleigh-Plesset equation, the liquid properties surrounding the bubble are extrapolated from the point properties of liquid at the position, which is the bubble center unless there is a bubble. However, the equation sometimes allows for the bubble volume to grow infinitely because it neglects the effects of the bubble volume. [PERSON] et al. [22] proposed to use the averaged liquid pressure on the bubble surface to avoid this problem, called the surface-averaged pressure (SAP) scheme. The same scheme is adopted in this study to identify the pressure (\(p\)) and velocity (\(\overline{U}\)) of liquid surrounding the bubble. To predict the flow noise due to cavitation, every single bubble is modeled as a point monopole noise source [46].