## 4 Results  
To verify the simulation results, the force along the \(y\)-axis and moment about the \(z\)-axis under the X-rudder plane configuration obtained from the CFD-based simulation in Ansys Fluent were compared with the corresponding force and moment of a similar model (SWE submarine), which was previously estimated by [PERSON] et al. [14], as shown in Figure 6. It can be seen that the obtained results were consistent with the experimental results [9]. The difference between the hydrodynamic coefficients of the X-rudder plane configuration and the previous results may have resulted from the ignorance of strong turbulent flow induced by the rudders, stabilizers, and the submarine's body, as the fluid flow through the stern rudder plane configuration on the submarine is simulated in steady state. The comparison between the current and previous results is detailed in Table 1. Next, the numerical simulation setup was applied to estimate the hydrodynamic forces and moments of the rudder plane configuration. Figure 5: Local mesh in stern area and local mesh around the submarine. To predict the hydrodynamic forces and moments acting on the rudders and stabilizers, the submarine was forced to run steadily at a speed of 20 knots with different angles of the rudder and stabilizer. Both the rudder and stabilizer were rotated from 0 to 21 degrees about the \(z\)-axis and \(y\)-axis, respectively. The interval of the deflected angle for the simulation was 3 degrees for each of the cases. As the submarine has a symmetrical body, the rudders were rotated in one direction and the obtained results are presented in both rotating directions. However, it was necessary to rotate the stabilizers in both directions about the axis of rotation. The hydrodynamic forces and moments acting on each rudder-plane were estimated in the numerical simulation. Figure 7 presents the results of hydrodynamic forces and moments acting on the X-rudder plane, Y-rudder plane, and Cross-rudder plane configurations obtained from the CFD-based simulation. It can be seen that the forces \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Symbol**} & \multirow{2}{*}{**Current Study**} & \multicolumn{2}{c}{**Experiment**} \\  & & **([PERSON] et al. [14])** & **Difference (\%)** \\ \hline \(Y_{\delta_{s}}^{\prime}\) & \(1.17\times 10^{-2}\) & \(1.38\times 10^{-2}\) & 15.22 \\ \(Z_{\delta_{s}}^{\prime}\) & \(1.12\times 10^{-2}\) & \(1.41\times 10^{-2}\) & 20.57 \\ \(M_{\delta_{s}}^{\prime}\) & \(4.84\times 10^{-3}\) & \(6.10\times 10^{-3}\) & 20.66 \\ \(N_{\delta_{s}}^{\prime}\) & \(-5.05\times 10^{-3}\) & \(-5.70\times 10^{-3}\) & 11.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the linear coefficients of the X-rudder plane configuration between current study and the experimental study of [PERSON] [14]. Figure 6: Comparison between current results and the previous results for rudders and stabilizers on am SWE submarine in [PERSON] et al., 2022 [14]. (**a**) Y\(-\)force of rudders versus angle of attack of the rudder, (**b**) Y\(-\)force of rudders versus angle of attack of the rudders, (**c**) Y\(-\)force of stabilizers versus angle of attack of the stabilizers, (**d**) Y\(-\)force of stabilizers versus angle of attack of the stabilizers. and moments increased as the deflection angle of the rudder and stabilizer increased. The hydrodynamic forces along the \(x\)- and \(y\)-axes of the X-rudder plane were larger than the corresponding forces acting on the Cross-rudder plane and Y-rudder plane. The roll moment of the X-rudder plane was rather small, in comparison with the moment acting on the left rudder plane configurations. This can be explained by the symmetric arrangement of the X-rudder configuration, which induces the small value in hydrodynamic force along the \(z\)-axis as well as the roll moment.