## 3 Model and CFD Methods
### CFD Methods  
Numerical simulation was conducted using the commercial CFD software, STAR-CCM+ 13.02, which employs the incompressible unsteady Reynolds-averaged Navier-Stokes (RANS) equations in combination with the k-\(\omega\) shear stress transport (SST) turbulence model. The selection of the k-\(\omega\) SST turbulence model was based on its effective prediction capability and reasonable computational cost. The simulation strategy was guided using the studies conducted by [PERSON] (2017), and [PERSON] (2018) as references to inform the design process, which is represented in a two-equation format:  
\[\left\{ \begin{aligned} \frac{\partial(\rho k)}{\partial t}+\frac{ \partial\left(\rho u_{i}k\right)}{\partial x_{j}}=P-\beta^{*}\rho\omega k+ \frac{\partial}{\partial x_{j}}\Big{[}(\mu+\sigma_{k}\mu_{t})\frac{\partial k} {\partial x_{j}}\Big{]}\\ \frac{\partial(\rho\omega)}{\partial t}+\frac{\partial\left(\rho u _{i}\rho\omega\right)}{\partial x_{j}}=\frac{\partial T}{\mu_{t}}P-\beta^{*} \rho\omega^{2}+\frac{\partial}{\partial x_{j}}\Big{[}(\mu+\sigma_{\omega}\mu_{ t})\frac{\partial\omega}{\partial x_{j}}\Big{]}+2(1-F_{1})\frac{\rho\sigma_{\omega 2 }}{\omega}\cdot\frac{\partial k}{\partial x_{j}}\cdot\frac{\partial\omega}{ \partial x_{j}}\end{aligned} \right. \tag{21}\]  
where \(\rho\) represents density, \(\mu_{t}\) denotes velocity vector, \(x_{j}\) indicates position vector, and \(k\) and \(\omega\) represent turbulent kinetic energy and turbulence dissipation rate, respectively. The coefficients \(\mu\) and \(\mu_{t}\) correspond to laminar and turbulent viscosity coefficients. The form of the production term \(P\) and function \(F_{1}\), as well as the values of coefficients \(\beta^{*}\), \(\gamma\), \(\sigma_{k}\), and \(\sigma_{\omega 2}\), can be inferred from the relevant literature for the SST model ([PERSON] et al., 2018). In order to validate the accuracy of the numerical computation method and facilitate comparison with experimental data, the SUBOFF model was simulated at a scale ratio of 1:1. This corresponds to a model length of 4.356 m and an incoming flow velocity of 5.144 m/s. The drag force values from the towing experiment were calculated for reference ([PERSON] et al., 2018). This computational grid is presented in Figure 3.  
To ensure the convergence of the computational grid, the mesh was encrypted according to the fineness ratio in the ITTC recommendation rules. Specifically, the grid was encrypted in all three directions using \(\mathrm{r_{G}}=\sqrt{2}\); the other parameters were unchanged ([PERSON] et al., 2018). Table 2 reports three sets of grids with varying refinement levels, including the corresponding drag values obtained from towing experiments. \begin{table}
\begin{tabular}{c c c c c} \hline
**Method** & **Grid Level** & **Cells (Millions)** & **Drag Force (N)** & **Relative Error (\%)** \\ \hline Experiment & - & - & 283.8 & - \\  & Coarse & 2.96 & 299.87 & 5.66\% \\ CFD & Medium & 7.25 & 291.76 & 2.81\% \\  & Fine & 16.78 & 290.57 & 2.39\% \\ \end{tabular}
\end{table} 
Table 2: Comparison between CFD methods and experimental value for the SUBOFF model (L = 4.356). Figure 3: The computational region grid for resistance. Symbol \(\Delta\) represents the difference in resistance between two types of grids:  
\[\Delta_{1}=R_{medium}-R_{fine} \tag{22}\]  
\[\Delta_{2}=R_{coarse}-R_{medium} \tag{23}\]  
The changes in \(\Delta\) were used to define the convergence ratio \(R_{ratio}\):  
\[R_{ratio}=\Delta_{1}/\Delta_{2} \tag{24}\]  
The computed convergence ratio, \(R_{ratio}=0.128\), falls within the \(0<R_{ratio}<1\) range. Therefore, the three sets of grids exhibit monotonic convergence, and the drag force calculated for the medium grid resulted in an error of 2.81% when compared to the experimental data. This error falls below the 5% threshold, indicating a satisfactory level of accuracy.