## 2 Methods: Experiment and Numerical Simulation
### Theory of Numerical Simulation Method
#### 2.2.3 Fluid Equations of Motion  
The three-dimensional viscous incompressible flow field around the emergency rising submarine is determined by solving the URANS equations. The derive process of fluid equations of motion based on the local velocities results in apparent body forces appears on the right-hand side (RHS) of the transport equations. Therefore, the conservation equations for mass and momentum based on local velocities are, respectively [28]:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\rho U_{j}}{\partial x_{j}}\ =\ 0 \tag{4}\]  
\[\frac{\partial\rho U_{i}}{\partial t}+\frac{\partial\rho U_{i}U_{j}}{\partial x _{j}}\ =\ -\frac{\partial P}{\partial x_{i}}+\frac{\partial}{\partial x_{j}} \left[\mu\left(\frac{\partial U_{i}}{\partial x_{j}}+\frac{\partial U_{j}}{ \partial x_{i}}\right)\right]-\frac{\partial}{\partial x_{j}}\rho\overline{u^ {\prime}_{i}u^{\prime}_{j}}+F_{B} \tag{5}\]  
\[F_{B}\ =\ -\rho\left(\frac{d^{2}R_{0}}{dt^{2}}+\frac{d\Omega}{dt}\times \mathbf{r}_{1}+\Omega\times(\Omega\times\mathbf{r}_{1})+2\Omega\times U_{i}\right) \tag{6}\]  
Among the above equations, the instantaneous fluid velocity is decomposed into the mean components \(U_{i},\,U_{j}\) and fluctuating components \(u_{i},\,u_{j}\), where the subscripts identify corresponding components along \(i\) and \(j\) directions respectively. \(x_{i}\) and \(x_{j}\) are position coordinates of fluid particles along \(i\) and \(j\) directions. Additionally, \(\rho\) is the density, \(\mu\) is viscosity, and \(P\) is the mean component of the instantaneous pressure. Furthermore, \(F_{B}\) is the apparent body force as a result of the translating-rotating frame of reference, and these terms in Equation (6), in order from left to right, account for apparent linear, angular, centripetal, and Coriolis accelerations sequentially. In this force, derived in [29], \(R_{0}\) is the position vector locating the submarine with respect to the inertial coordinate system, \(\Omega\) is the angular velocity vector of the body coordinate system, and \(\mathbf{r}_{1}\) is a vector that locates a fluid particle in body axes. Besides, the linear acceleration term can alternatively be written as follows when referenced to the body axes:  
\[\frac{d^{2}\mathbf{R}_{0}}{dt^{2}}\ =\ \overline{U^{\prime}}+\mathbf{\Omega}\times \overline{U} \tag{7}\]where \(\widetilde{U}=ue_{x}+ve_{y}+we_{z}\), \(\widetilde{U}^{\prime}=ie_{x}+\hat{v}e_{y}+\hat{w}e_{z}\) are the velocity and accelerate velocity vector of the submarine respectively. For turbulent simulations, the \(k\)-\(\omega\) based Shear Stress Transport (SST \(k\)-\(\omega\)) turbulence model has been selected to model the Reynolds stress term, \(\rho\overline{u}_{i}\overline{u}_{j}\), and equations are shown as following forms:  
\[ \begin{array}{c}\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{ \partial x_{i}}(\rho ku_{i})=\frac{\partial}{\partial x_{j}}\bigg{(}\Gamma_{k }\frac{\partial k}{\partial x_{j}}\bigg{)}+\widetilde{G}_{k}-Y_{k}+S_{k}\\ \frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial x_{i}}(\rho \omega u_{i})=\frac{\partial}{\partial x_{j}}\bigg{(}\Gamma_{\omega}\frac{ \partial\omega}{\partial x_{j}}\bigg{)}+\widetilde{G}_{\omega}-Y_{\omega}+S_{ \omega}+D_{\omega}\end{array}  \tag{8}\]  
In the above formula, \(k\) is the turbulent kinetic energy, \(\omega\) is the specific dissipation rate of turbulence, \(\Gamma_{k}\), \(\Gamma_{\omega}\) are the effective diffusions of \(k\) and \(\omega\), respectively. \(Y_{k}\), \(Y_{\omega}\) are the dissipation of \(k\) and \(\omega\); \(\widetilde{G}_{k}\), \(\widetilde{G}_{\omega}\) are the generator terms of \(k\) and \(\omega\); \(D_{\omega}\) is a cross-diffusion term; \(S_{k}\), \(S_{\omega}\) are the custom source terms. According to the previous numerical investigations, it has been proven that the maximum prediction deviation between several turbulence models, such as SST \(k\)-\(\omega\), Realizable \(k\)-\(\varepsilon\), does not exceed 5%, which has a negligible effect on the analyses of the emergency rising stability.