## 3 Governing Equations and Numerical Implementations  
### Governing Equations  
The numerical analysis is implemented by solving the full three-dimensional unsteady compressible RANS equations with homogeneous mixture approach. The accurate and effective numerical simulation of a multi-phase problem is challenging, as numerical instability issues arise from the large variations in the speed of sound with rapid local density changes [9; 11; 12; 16; 17; 18; 19]. To resolve these problems, the preconditioned and dual-time stepping numerical methods are employed. The governing equation in a generalized curvilinear coordinate is expressed as follows  
\[\mathbf{\Gamma}_{e}\frac{\partial\hat{Q}}{\partial t}+\mathbf{\Gamma}\frac{ \partial\hat{Q}}{\partial\tau}+\frac{\partial(\hat{E}-\hat{E}_{v})}{\partial \xi}+\frac{\partial(\hat{F}-\hat{F}_{v})}{\partial\eta}+\frac{\partial(\hat{G}- \hat{G}_{v})}{\partial\xi}=\hat{S} \tag{1}\]  
where \(\hat{Q}\) is the vector of primitive variables; \(\hat{E}\), \(\hat{F}\), \(\hat{G}\), \(\hat{E}_{v}\), \(\hat{F}_{v}\) and \(\hat{G}_{v}\) are the convective and viscous flux vectors; \(\hat{S}\) is the source vector; \(\mathbf{\Gamma}_{e}\) and \(\mathbf{\Gamma}\) are the flux Jacobian matrix and the preconditioning matrix, respectively; \(t\) and \(\tau\) denote the physical and pseudo times, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & \(\mathbf{Re}_{c}\) & \(\sigma\) & \(\mathbf{U}_{\infty}\) & \(\mathbf{C}\) **(m)** & **Angle of** \\  & & & **(m/s)** & **(m)** & **Attack (\({}^{\circ}\))** \\ \hline Values & \(7.0\times 10^{5}\) & 0.8 & 10 & 0.07 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational conditions. Figure 1: (**a**) Computational domain and (**b**) zoomed grid system. The fundamental hypothesis of the [PERSON]'s model is that the cavitation phenomena are occurred only due to the difference between the local static pressure, \(p\), and vapor pressure of liquid, \(p_{v}\). If the local static pressure decreases below the vapor pressure, evaporation occurs and thus the phase of fluid is changed from liquid water to water vapor with the amount of mass transfer rate of \(\dot{m}^{-}\). In contrast, if the local static pressure increases above the vapor pressure, the reversal phenomena occur with the amount of mass transfer rate of \(\dot{m}^{+}\). To close the system, the equation of state (EOS) for the liquid water, water vapor and mixture are required. In this study, the properties such as density, enthalpy, dynamic viscosity and thermal conductivity of liquid water, water vapor, mixture and its derivatives are considered as the function of pressure and temperature as reported by international standard IAPWS-IF97 [20]. \[\begin{array}{c}\rho_{l}=\rho_{l}(p,T),\,h_{l}=h_{l}(p,T),\,\,\mu_{l}=\mu_{l }(p,T),\,\,\lambda_{l}=\lambda_{l}(p,T)\\ \rho_{v}=\rho_{v}(p,T),\,  
As a result, Equation (12) now includes the mixed derivatives terms, such as \(\partial^{2}/\partial\xi\partial\eta\) and \(\partial^{2}/\partial\xi\partial\zeta\), that distort the diagonal matrix form. According to [PERSON] and [PERSON] [23], omitting these terms does not affect the accuracy of the solutions. Thus, in this study, these terms are neglected. Equation (12) then can be written as  
\[ \begin{split}\frac{\partial}{\partial\xi}\Big{(}A_{\flat} \Delta\hat{Q}^{n+1,k}+A_{\flat,\xi}\Delta\hat{Q}_{\xi}{}^{n+1,k}+A_{\flat, \eta}\Delta\hat{Q}_{\eta}{}^{n+1,k}+A_{\flat,\xi}\Delta\hat{Q}_{\xi}{}^{n+1,k }\Big{)}\\ \left[\left(\frac{\partial A_{\eta}}{\partial\xi}-\frac{\partial ^{2}A_{\flat,\xi}}{\partial\xi^{2}}\right)\Delta\hat{Q}^{n+1,k}+\frac{ \partial^{2}}{\partial\xi^{2}}\Big{(}A_{\flat,\xi}\Delta\hat{Q}^{n+1,k}\Big{)} \right]\end{split}  \tag{13}\]  
In addition, if the fluid obeys the [PERSON] hypothesis and the transport coefficients, such as \(\mu\), \(\kappa\), and \(\lambda\) are locally constant, i.e., slowly change with respect to temperature variations, Equation (13) can be reduced to the following form [24; 25; 26].