## 2 Governing Equation and Numerical Method  
### Governing Equations  
In this study, we identified the dimensionless variables that are needed for the related studies of the cavitation shape around the underwater vehicle. To confirm the characteristics of the cavitation according to the change of the dimensionless variable such as cavitation number (\(\sigma\)) and ventilated gas flux coefficient (Cq), the computational analysis using the multiphase flow analysis program was performed. The program solves a multiphase flow field problem based on a three-dimensional Navier-Stokes equation. In the case of continuity equations, each phase is solved individually. In the case of the momentum equations, a homogeneous mixture model using a mixture equation is used. Continuous equations, momentum equations, and energy equations can be expressed in vector form and represented again as a governing equation for a generalized coordinate system in the analysis space [3; 11; 13; 14; 15]:  
\[\frac{\partial\hat{Q}}{\partial t}+\frac{\partial(\hat{E}-\hat{E}_{V})}{ \partial\xi}+\frac{\partial(\hat{F}-\hat{F}_{V})}{\partial\eta}+\frac{ \partial(\hat{G}-\hat{G}_{V})}{\partial\zeta}=\hat{S}, \tag{1}\]  
where, \(\hat{Q}\) is the vectors of the conserved variables; \(\hat{E}\), \(\hat{F}\) and \(\hat{G}\) are the viscid flux terms; \(\hat{E}_{v}\), \(\hat{F}_{v}\) and \(\hat{G}_{v}\) are the viscous terms; \(\hat{S}\) is the source term. These vectors are given by  
\[\hat{Q}=\frac{1}{\hat{I}}\left[ \begin{array}{c}Y_{l}\rho_{m}\\ \rho_{m}u\\ \rho_{m}v\\ \rho_{m}w\\ Y_{v}\rho_{m}\\ Y_{ng}\rho_{m}\end{array} \right];\ \hat{E}=\frac{1}{\hat{I}}\left[ \begin{array}{c}Y_{l} \rho_{m}U\\ \rho_{m}Uu+\xi_{x}p\\ \rho_{m}Uv+\xi_{y}p\\ \rho_{m}Uw+\xi_{z}p\\ \rho_{m}h_{0}U\\ Y_{v}\rho U\\ Y_{ng}\rho U\end{array} \right];\ \hat{F}=\frac{1}{\hat{I}}\left[ \begin{array}{c}Y_{l} \rho_{m}V\\ \rho_{m}Vu+\eta_{x}p\\ \rho_{m}Vv+\eta_{y}p\\ \rho_{m}Vw+\eta_{z}p\\ \rho_{m}h_{0}V\\ Y_{v}\rho V\\ Y_{ng}\rho V\end{array} \right];\ \hat{G}=\frac{1}{\hat{I}}\left[ \begin{array}{c}Y_{l} \rho_{m}W\\ \rho_{m}Wu+\zeta_{x}p\\ \rho_{m}Wv+\zeta_{y}p\\ \rho_{m}Wu+\zeta_{z}p\\ \rho_{m}h_{0}W\\ Y_{v}\rho W\\ Y_{ng}\rho W\end{array} \right] \tag{2}\]  
\[\hat{E}^{V}=\frac{1}{\hat{I}}\left[ \begin{array}{c}0\\ \xi_{x}\tau_{xx}+\xi_{y}\tau_{xy}+\xi_{z}\tau_{xz}\\ \xi_{x}\tau_{yx}+\xi_{y}\tau_{yy}+\xi_{z}\tau_{yz}\\ \xi_{x}\tau_{xx}+\xi_{y}\tau_{zy}+\xi_{z}\tau_{zz}\\ \xi_{x}\tau_{xx}+\xi_{y}\tau_{zy}+\xi_{z}\tau_{zz}\\ \xi_{x}b_{x}+\xi_{y}b_{y}+\xi_{z}b_{z}\\ 0\\ 0\end{array} \right];\ \hat{V}=\frac{1}{\hat{I}}\left[ \begin{array}{c}0\\ \eta_{x}\tau_{xx}+\eta_{y}\tau_{xy}+\eta_{z}\tau_{xz}\\ \eta_{x}\tau_{yx}+\eta_{y}\tau_{yy}+\eta_{z}\tau_{yz}\\ \eta_{x}\tau_{zx}+\eta_{y}\tau_{zy}+\eta_{z}\tau_{zz}\\ \eta_{x}b_{x}+\eta_{y}b_{y}+\eta_{z}b_{z}\\ 0\end{array} \right];\ \hat{G}^{V}=\frac{1}{\hat{I}}\left[ \begin{array}{c}0\\ \zeta_{x}\tau_{xx}+\zeta_{y}\tau_{xy}+\zeta_{z}\tau_{xz}\\ \zeta_{x}\tau_{yx}+\zeta_{y}\tau_{yy}+\zeta_{z}\tau_{yz}\\ \zeta_{x}\tau_{zx}+\zeta_{y}\tau_{zy}+\zeta_{z}\tau_{zz}\\ \zeta_{x}b_{x}+\zeta_{y}b_{y}+\zeta_{z}b_{z}\\ 0\end{array} \right] \tag{3}\]\[\hat{S}=\frac{1}{J}\left[ \begin{array}{c}\dot{m}^{+}+\dot{m}^{-}\\ \rho_{m}g_{x}\\ \rho_{m}g_{y}\\ \rho_{m}g_{z}\\ 0\\ -\dot{m}^{+}-\dot{m}^{-}\\ 0\end{array} \right] \tag{4}\]  
where,  
\[b_{x_{i}}=u_{j}\tau_{x_{i}x_{j}}-q_{x_{i}},\ h_{0}=\left[h+\frac{1}{2}(u^{2}+v^ {2}+w^{2})\right], \tag{5}\]  
\(Y\) is the mass fraction, \(\alpha\) is the volume fraction, \(p\) is the pressure, \(\rho_{m}\) is the mixture density, \(u\), \(v\), and \(w\) are the cartesian velocity components, and \(h\) is the mixture enthalpy. Here, \(\xi_{x}\), \(\xi_{y}\), \(\xi_{z}\), \(\eta_{x}\), \(\eta_{y}\), \(\eta_{z}\), \(\zeta_{x}\), \(\zeta_{y}\), \(\zeta_{z}\), and \(J\) are the metrics and the Jacobian of the transformation, respectively. \(U\), \(V\), and \(W\) are the contravariant velocity components in the \(\xi\), \(\eta\), and \(\zeta\) directions, respectively.