## 2 Numerical Simulation of Submerged Cavitating Jet
### Numerical Model  
The software Ansys Fluent (the version is 19.0) was utilized to simulate the submerged cavitation jet which was a two-phase flow. The mixture model was adopted here to simulate the cavitation jet due to its superior precision and computational efficiency. The continuity equation for the mixture model is:  
\[\frac{\partial}{\partial t}(\rho_{m})+\
abla\cdot(\rho_{m}\overline{v}_{m})= \dot{m}, \tag{1}\]  
The momentum equation of the mixture model is:  
\[\frac{\partial}{\partial t}\left(\rho_{m}\overrightarrow{v}_{m}\right)+\
abla \cdot\left(\rho_{m}\overrightarrow{v}_{m}\bigotimes\overrightarrow{v}_{m} \right)=-\
abla p+\
abla\cdot\left[\mu_{m}\left(\
abla\overrightarrow{v}_{m}+ \
abla\overrightarrow{v}_{m}^{\top}\right)\right]+\rho_{m}\overrightarrow{g}+ \overrightarrow{F}+\
abla\cdot\left(\sum_{k=1}^{n}\alpha_{k}\rho_{k} \overrightarrow{v}_{dr,k}\bigotimes\overrightarrow{v}_{dr,k}\right), \tag{2}\]  
where \(\dot{m}\) represents the mass source, \(\rho_{m}\) denotes the mixing density, \(\overrightarrow{v}_{m}\) signifies the average mass velocity, \(\overrightarrow{F}\) stands for the volume force, \(\alpha_{k}\) represents the volume fraction of the \(k\) phase, and \(\overrightarrow{v}_{dr,k}\) is identified as the drift velocity of the phase \(k\). \begin{table}
\begin{tabular}{c c c} \hline
**Nozzle Number** & **Structure Type** & **Outlet Shape** & **Profile** \\ \hline
3 & Shear nozzle with a pre-shrinking segment & Conical outlet with expand cylinder segment & \\ \hline
4 & Shear nozzle with no pre-shrinking segment & Conical outlet with expand cylinder segment & \\ \hline
5 & Shear nozzle with no pre-shrinking segment & Spherical outlet with expand cylinder & \\ \hline
6 & Self-resonance pulsed nozzle Helmholtz type & / & \\ \hline
7 & Self-resonance pulsed nozzle organ pipe type & / & \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._Water and water vapor were selected as the flow field materials for this study. The density of the water vapor was set as 0.02558 kg/m\({}^{3}\), with a viscosity of 1.26 \(\times\) 10\({}^{-6}\) kg/(m\(\cdot\)s). For water, the density was set as 1000 kg/m\({}^{3}\), with a viscosity of 0.001 kg/(m\(\cdot\)s). The conversion between the liquid phase (water) and vapor phase (water vapor) was simulated using the Schnerr and Sauer cavitation model. The transport equation of this model is as follows:  
\[\frac{\partial}{\partial t}(\alpha.\rho_{v})+\
abla\cdot\left(\alpha.\rho_{v }.\overset{\rightarrow}{V}_{V}\right)=\frac{\rho_{v}.\rho_{l}}{\rho}\frac{d \alpha}{dt} \tag{3}\]  
In Equation (3), the density of vapor is denoted as \(\rho_{v}\), while the density of liquid is represented by \(\rho_{l}\). The volume fraction of vapor in the system is defined as \(\alpha\), and the velocity of the vapor phase is denoted as \(\overset{\rightarrow}{V}_{V}\). Additionally, time is represented by the variable \(t\). The viscous model was selected as the Realizable model in the \(k\)-\(\varepsilon\) model. The Realizable \(k\)-\(\varepsilon\) model adheres to the constraint condition of Reynolds stress, ensuring consistency with real turbulence on this scale and providing a more accurate description of the diffusion velocity in circular jets. This model is comprised of two equations: the turbulent kinetic energy (\(k\)) equation and the dissipation rate (\(\varepsilon\)) equation.