## 2 Methods: Experiment and Numerical Simulation
### Meshing Methodology  
In the present work, all simulations are accomplished in the commercial software Star-CCM+ with employing multi-level overset mesh technique, and the computational domain is decomposed into several overset regions and background region by region-based meshing methodology. This multiple overset region consists of a main hull, rudder and propeller three grid blocks as shown in Figure 4b, in which each moving component has its own grid to deal with complex motion problems. Specifically speaking, the hexahedral grids are generated using the trimmer mesher for the background region, while polyhedral grids are generated using the polyhedral mesher for the main hull subregion because polyhedral grids can achieve a reasonable level of accuracy with the relative smaller number of grids and shorter computation time [31]. Besides, it is worth noting that trimmer mesh also is applied to the sternplanes and propeller subregions to deal with some small gaps better. Furthermore, prism layer mesher is utilized to gridding boundary layers of the no-slip wall for the employment of wall function. Prism layer mesh are generated such that average near wall \(Y+\) values could be larger than 30, as required by the \(k\)-\(\omega\) turbulence model. \begin{table}
\begin{tabular}{c c c c} \hline \hline Parameters & Value & Parameters & Value \\ \hline \(L\)/m & 5.09 & \(I_{xx}\)/kg\(\cdot\)m\({}^{2}\) & 34.348 \\ \(D\)/m & 0.58 & \(I_{yy}\)/kg\(\cdot\)m\({}^{2}\) & 1001.61 \\ \(h\)/m & 0.02 & \(I_{zz}\)/kg\(\cdot\)m\({}^{2}\) & 1000.32 \\ \(m\)/kg & 862.80 & \(\Delta\downarrow\)/kg & 1107 \\ \(m_{blown}\)/kg & 44.886 & \((x_{blown},y_{blown},z_{blown})\)/m & (0.362, 0, 0.01534) \\ \(D_{\text{outer}}\)/m & 0.3365 & \((x_{G},y_{G},z_{G})\)/m & (0.14676, 0, \(-\)0.02824) \\ \(D_{\text{inner}}\)/m & 0.0664 & \((x_{B},y_{B},z_{B})\)/m & (0.16863, 0, \(-\)0.00823) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main computational parameters of buoyant model and self-designed propeller. In the study of naval ship computational fluid dynamics, the polyhedral meshes have been widely used in the computation of flow around objects with complex geometry due to its strong adaptability and high computational efficiency. Furthermore, the trimmed mesh of the hexahedral type has a relatively higher accuracy and is more suitable for the background mesh generation of multiphase flow with a free surface. In this paper, three meshing schemes, i.e., polyhedral mesh, trimmed mesh and hybrid mesh (polyhedral mesh for the overset area and trimmed mesh for background area), are analyzed for grid independence. Taking the pitch angle that is relatively stable in the emergency rising process as the comparison benchmark parameter, the grid number and calculation error of the three grid schemes are given in Table 3.  
It can be seen from Table 3, that the polyhedral mesh is not suitable for the calculation of multiphase flow in this paper because of its large deviation in predicting the pitch angle of the submarine after surfacing. To the contrary, both the trimmed mesh and the hybrid mesh scheme have higher prediction accuracy before and after surfacing, and the hybrid mesh has the minimum perdition error for submarine underwater pitch angle. Therefore, the third hybrid mesh scheme is adopted in this paper. Figure 4: (**a**) Longitudinal mesh section of computational domain for emergency rising simulation. (**b**) Local refined mesh for stabilizing fins and rudder gap. (**c**) Transverse mesh section of the overlapping region near the sailplane. (**d**) The schematic diagram of activating the prism layer shrinkage to morph the cells of the prism layers in small gaps. The polyhedral grid of propeller and rudder are illustrated in the subgraph.