## 3 Description of the Model and the Accuracy Validation
### Parameters of Numerical Calculation  
The turbulence equations are solved numerically using the finite volume method, which is realized by the FLUENT codes. A rectangular flow field surrounding the model is built up to develop the turbulence completely. The distance between the model and the flow inlet, and that between the model and the flow outlet are 1L and 2L, respectively. The width of the flow field is six times the chord length, and the height is three times the chord length. The boundaries of the computational domain, including the inlet, the outlet, the plane of the model, and the outside, are set as the velocity inlet, pressure outlet, symmetrical boundary, and solid wall boundary, respectively. The velocity of the flow is 8.68 m/s. Considering time cost and computational power, we have performed the steady-state grid-independent verification. We have calculated the resistance value of the models with the grid numbers of 1 million, 1.5 million, 2 million, 2.5 million, 3 million, 3.5 million, and 4 million. Through the comparison in Table 1, we have found that the grid number of 3 million can obtain the appropriate resistance. Therefore, the grid number of the flow field calculation is 3 million. When we have performed NS calculation, a hexahedral mesh is used, and the boundary layer of the sail hull surface is encrypted. We have calculated the whole SUBOFF model by the LES. We have \begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Grid Numbers** & **Resistance Values** \\ \hline
1 & 1 million & 33.9004 \\
2 & 1.5 million & 34.1862 \\
3 & 2 million & 34.9821 \\
4 & 2.5 million & 35.6891 \\
5 & 3 million & 35.6893 \\
6 & 3.5 million & 35.6979 \\
7 & 4 million & 35.6986 \\ \hline \end{tabular}
\end{table} 
Table 1: Resistance values of the models with different grid numbers. Figure 1: Diagram of the model. The model is the sail hull with part of the submarine body, which is scaled from the SUBOFF model. The length (**L**), and the height (**h**) of the sail hull are 1.84 m, and 0.1 m, respectively. The chord length (**c**) is 0.184 m.  
obtained different \(y^{+}\) values by the change of first layer thickness. Through the comparison between the numerically calculated results and experimental results, the appropriate range of \(y^{+}\) value is required. Table 2 shows the comparison between the numerical calculation values and experimental values of SUBOFF resistance at different \(y^{+}\) ranges. For \(y^{+}<50\), the numerical calculation value is closest to the experimental value. Subsequently, the submarine tail flow has been carried out and is shown in Figure 2. \begin{table}
\begin{tabular}{c c c} \hline \hline \(y^{+}\) & **Numerical Simulation Value** & **Experimental Test Value** \\ \hline \(<\)50 & 102.0 & 102.3 \\ \(<\)100 & 109.3 & 102.3 \\ \(<\)250 & 120.2 & 102.3 \\ \(<\)300 & 98.1 & 102.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the resistance. Figure 2: Comparison of the tail flow from the SUBOFF between the numerical calculation and experimental test. Numerical calculation: (**a**) \(y^{+}<50\); (**b**) \(y^{+}<100\); (**c**) \(y^{+}<250\); and (**d**) \(y^{+}<300\). (**e**) The experimental test. For \(y^{+}<50\), the numerical simulation values are in good accordance with the experimental values. The method of wall function combined with the RNG model calculates the steady state of the flow field, where \(y^{+}=35\), and \(Re=1.6\times 10^{7}\). The mesh thickness of the first boundary layer, namely \(\Delta y_{\mathrm{P}}\), is 0.0001 m, and is calculated by Equation (21):  
\[\Delta y_{p}=\frac{Ly^{+}}{0.172\mathrm{Re}^{0.9}}, \tag{21}\]  
The transient simulation of the flow field is performed by the dynamic sub-lattice model in the LES. \[f_{max}=\frac{1}{2\Delta t},\ \Delta f=\frac{1}{n\Delta t} \tag{22}\]  
The convection term is discretized by second-order upwind scheme, whereas the diffusion term is discretized by a central difference scheme.