## 2 Materials and Methods  
### Governing Equation and Turbulence Model  
The governing equation can be utilized to elucidate the liquid flow behavior in a Venturi tube. In the presence of cavitation, the flow in a Venturi tube is a combination of vapor and liquid phases. To describe the two-phase flow of vapor and liquid in a Venturi tube, the mixture model is applied [7; 15; 21; 26]. Based on this, a hypothesis is proposed, which assumes that the vapor and liquid phases are homogeneously mixed and able to permeate each other, regardless of their relative velocities. Moreover, it is assumed that the fluid is continuous during the flow process and the impact of volume force on fluid movement is negligible. The continuity and momentum equations of the mixture phase are provided as follows:  
\[\
abla\cdot\left(\rho_{\mathrm{m}}v_{\mathrm{m}}\right)=0 \tag{1}\]  
\[\
abla\cdot\left(\rho_{\mathrm{m}}v_{\mathrm{m}}v_{\mathrm{m}}\right)=-\
abla p +\
abla\cdot\left[\mu_{\mathrm{m}}\left(\
abla v_{\mathrm{m}}+\
abla v_{ \mathrm{m}}^{T}\right)\right] \tag{2}\]  
where \(\rho_{\mathrm{m}}\), \(\
u_{\mathrm{m}}\), and \(\mu_{\mathrm{m}}\) denote the mixed phase's density, mass-averaged velocity, and viscosity, respectively. \(p\) is pressure; \(\
abla\) is the nabla symbol. The phenomenon of cavitation is often accompanied by significant turbulence and the formation of cavities within the fluid flow, which grow, merge, collapse, and ultimately affect the performance and durability of engineering devices. The complex nature of cavitation events, which exhibit different time-scales and are influenced by various flow parameters, poses significant challenges for accurate and efficient numerical simulation. Although Direct Numerical Simulation (DNS) can be used to simulate the details of cavitation flows, it is computationally expensive and impractical for most actual engineering applications. Consequently, the Reynolds-averaged Navier-Stokes (RANS) equations are widely accepted to model the time-averaged variables of such a turbulent flow in the Venturi device. According to the calculations performed using the Reynolds number formula,  
\[\mathrm{Re}=\frac{\rho_{\mathrm{w}}\
u d}{\mu_{\mathrm{w}}} \tag{3}\]  
where \(\rho_{\mathrm{w}}\) denotes water density, \(\
u\) represents the average flow velocity at the inlet of the Venturi tube, \(d\) is the diameter at the inlet of the Venturi tube, and \(\mu_{\mathrm{w}}\) represents the viscosity of water. The Reynolds numbers for all investigated operating conditions in this study fall within the range of 34,174 to 141,767. These values significantly exceed the widely accepted critical Reynolds number threshold of 5000 for lamina flow in pipes. Hence, it can be inferred that the flow inside the Venturi tube exhibits turbulent characteristics. To account for the turbulence, the standard k-\(\epsilon\) model is employed, which has been extensively vali dated by previous researchers [20; 27; 28]. The transport equations for the turbulence kinetic energy (\(k\)) and its dissipation rate (\(\varepsilon\)) are expressed as follows:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{i} )=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k }}\bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+G_{k}+G_{b}-\rho\varepsilon -Y_{M}+S_{K} \tag{4}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}}( \rho\varepsilon u_{i})=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+ \frac{\mu_{t}}{\sigma_{\varepsilon}}\bigg{)}\frac{\partial\varepsilon}{ \partial x_{j}}\Bigg{]}+\mathrm{C_{1}}_{t}\frac{\varepsilon}{\bar{k}}(G_{k}+ \mathrm{C_{3}}_{t}G_{b})-\mathrm{C_{2}}_{t}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{5}\]  
where \(\rho\) denotes fluid density; \(k\) is turbulent kinetic energy; \(\varepsilon\) is turbulent kinetic energy dissipation rate.