## 2 Materials and Methods
### Numerical Models, Solution Methods and Boundary Conditions
#### 2.2.2 Turbulence Model  
Cavitation flow is a complex gas-liquid two-phase flow process that satisfies the law of mass and momentum conservation. The Mixture model [26] is a simplified Eulerian model that can be used both for two-phase fluids with velocity differences between phases and for simulating vesicular flows. The simulated results match the experimental results. The model is based on the finite volume method to solve the mass and momentum conservation equations in the three-dimensional Cartesian coordinate system to couple the gas-liquid two-phase flow. The continuity equation and momentum equation [27] are as follows. However, when calculating the short-time cavitation flow, it must maintain the equilibrium on locally very small spatial scales due to the strong coupling between the liquid and gas phases, and Equation (6) can be satisfied. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Maximum Mesh Size** & **Meshes** & **Cavitation Vesicle** & **Calculate Time** \\
**(mm)** & **Meshes** & **Volume Fraction** & **(Days)** \\ \hline
2.2 & 875913 & 0.35 & 6.5 \\ \hline
2.1 & 966335 & 0.36 & 7 \\ \hline
2.0 & 1125457 & 0.36 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mesh number independent verification. Figure 2: Schematic diagram of the hydrodynamic cavitation generator mesh model. 1. Continuity equation: \[\frac{\partial}{\partial T}(\rho_{m})+\
abla\cdot\left(\rho_{m}\overset{\to}{V}_{m} \right)=0\] (1) \[\rho_{m}=\sum_{k=1}^{n}\alpha_{k}\rho_{k}\] (2) \[\overset{\to}{V}_{m}=\frac{\sum_{k=1}^{n}\alpha_{k}\rho_{k} \overset{\to}{V}_{k}}{\rho_{m}}\] (3) 2. Momentum equation: \[\frac{\partial}{\partial T}\left(\rho_{m}\overset{\to}{V}_{m} \right)+\
abla\cdot\left(\rho_{m}\overset{\to}{V}_{m}\overset{\to}{V}_{m} \right)=-\
abla\text{p}+\
abla\cdot\left[\mu_{m}\left(\
abla\overset{\to}{V}_ {m}+\
abla\overset{\to}{V}_{m}^{T}\right)\right]\] \[+\rho_{m}\overset{\to}{g}+\overset{\to}{F}+\
abla\cdot\left( \sum_{k=1}^{n}\alpha_{k}\rho_{k}\overset{\to}{V}_{dr,k}\overset{\to}{V}_{dr,k }\right)\] (4) \[\mu_{m}=\sum_{k=1}^{n}\alpha_{k}\mu_{k}\] (5) \[\overset{\to}{V}_{dr,k}=\overset{\to}{V}_{k}-\overset{\to}{V}_{m}\] (6)  
where \(t\) is the time, \(\rho_{m}\) is the mixed-phase density, \(\rho_{k}\) is the kth-phase density, \(\overset{\to}{V}_{m}\) is the weighted mass average velocity, \(n\) is the number of phases, \(\
abla\text{p}\) is the pressure gradient acting on a unit volume of fluid, \(\overset{\to}{V}_{k}\) is the kth-phase velocity, \(\mu_{m}\) is the viscosity of the mixture, \(\overset{\to}{g}\) is the acceleration of gravity, \(\overset{\to}{F}\) is the volume force, \(\mu_{k}\) is the kth-phase viscosity, and \(\alpha_{k}\) is the kth-phase volume fraction. \(\overset{\to}{V}_{dr,k}\) is the drift velocity for phase \(k\), and is defined as the relative velocity between \(k\) phase velocity and the velocity of center of mass. Turbulence is the main reason for transient cavitation [28]. In the shear hydrodynamic cavitation generators, the high-speed rotation of the rotated teeth will cause a change in the turbulent kinetic energy of the fluid inside the cavity. The Reynolds-averaged Navier-Stokes [29] (RANS) method is chosen for solving turbulence models, where turbulent motion is considered as a synthesis of time-averaged flow and instantaneous pulsating flow. This computational approach encompasses the standard k-\(\varepsilon\) model, the RNG k-\(\varepsilon\) model, and the realizable k-\(\varepsilon\) model. The standard k-\(\varepsilon\) model is only suitable for simulating single-phase flow fields in a fully turbulent state, not for simulating cavitation with gas-liquid interactions. Although the RNG k-\(\varepsilon\) model considers gas-liquid two-phase flow, it cannot simulate the complete turbulent process and thus does not applicable to the cavitation model in this study. The fluid inside a shear hydrodynamic cavitation generator has rotational shear, high turbulence, and complex pressure variations.