# Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD
## 3 CFD Implementation
### Meshing  
An overset mesh can simulate complex relative motions between multiple objects, and it is widely used in CFD simulations of ships and airplanes [21; 22]. The mesh constructed for this simulation included two regions: the overset region and the numerical water tank region. The overset region moved along with the submarine, whereas the water tank region was static; the exchange of data between the two regions was achieved by interpolation.  
The overset region was defined as a cylinder region out of the submarine; a polyhedral mesh was used in this region. Polyhedral meshes can adapt to complex geometric shapes, and only a small number of cells are sufficient to ensure calculation accuracy [23]. The mesh had to be refined in the regions where the curvature of submarine geometry changes quickly, such as the region near the holes and appendages; there was only a slight deformation in these regions (as shown in Figure 3). The mesh gradually becomes coarse from the submarine model to the cylinder boundary. Note that when the rising speed of the submarine is 0.01 m/s, the average of the wall y\({}^{+}\) values is approximately 15; it becomes approximately 120 when the submarine is moving at maximum speed (i.e., \(\sim\)1 m/s). The mesh in the boundary layer was sufficiently refined to ensure simulation accuracy, given that the average of wall y\({}^{+}\) should be 20\(\sim\)100 [24; 25].  
As shown in Figure 4, a trim mesh was used in the numerical water tank to obtain a good simulation accuracy. The mesh near the water surface should be refined to capture the water surface sufficiently and predict the behavior of the submarine correctly when it moves near the water surface. There were more than 20 cells per wave height and more than 80 cells per wave length, and the height of the refinement zone near the water surface was 1.5 times the wave height. The cells in the region around the submarine rising path were the same size as the cells over the cylindrical surface in the overset region, ensuring a good accuracy of the interpolation between the two regions.  
To analyze the uncertainty of the simulation, three different grid sizes were used to predict the behavior of the submarine when it rose in static water. The coarse grid included 1.25 million cells, the medium grid included 3.53 million cells, and the fine grid included 10.37 million cells. The predicted results using the medium grid was almost the same as those using the fine grid (as shown in Table 2) ; therefore, a medium grid was used for all the simulations.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Submarine Property** & **Value** \\ \hline Length & 5045.2 mm \\ Width & 812.8 mm \\ Height & 958.6 mm \\ Maximum Diameter & 812.4 mm \\ \(m_{\
abla F}\) & 1147.2 Kg \\ \(x_{\
abla F}\) & 223.8 mm \\ \(y_{\
abla F}\) & 0 mm \\ \(z_{\
abla F}\) & \(-9.4\) mm \\ \(\overline{BG}\) & 13.65 mm \\ \(I_{\
abla F(x)}\) & 57.2 Kg \(\cdot\) m\({}^{2}\) \\ \(I_{\
abla F(y)}\) & 2813.2 Kg \(\cdot\) m\({}^{2}\) \\ \(I_{\
abla F(z)}\) & \(\text{Kg}\cdot\text{m}^{2}\) \\ \(I_{\
abla F(xz)}\) & 0.77 Kg \(\cdot\) m\({}^{2}\) \\ \hline \(m_{casting}\) & 244.4 Kg \\ \(x_{casing}\) & 155.4 mm \\ \(y_{casing}\) & 0 mm \\ \(z_{casing}\) & \(-127.3\) mm \\ \(I_{casing}(x)\) & 17.5 Kg \(\cdot\) m\({}^{2}\) \\ \(I_{casing}(y)\) & 665.2 Kg \(\cdot\) m\({}^{2}\) \\ \(I_{casing}(z)\) & 650.8 Kg \(\cdot\) m\({}^{2}\) \\ \hline \(m_{blown}\) & 44.89 Kg \\ \(x_{blown}\) & 400 mm \\ \(y_{blown}\) & 0 mm \\ \(z_{blown}\) & \(-13.3\) mm \\ \(I_{blown}(x)\) & 2.63 Kg \(\cdot\) m\({}^{2}\) \\ \(I_{blown}(y)\) & 245.2 Kg \(\cdot\) m\({}^{2}\) \\ \(I_{blown}(z)\) & 248.6 Kg \(\cdot\) m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical Characteristics of Submarine.