## 4 Results
A similar trend can be found in the hydrodynamic force along the \(z\)-axis and the moment about the \(y\)-axis of the stabilizers, as shown in Figure 8. The force along the \(x\)-axis and the moment about the \(y\)-axis of the stabilizers in the X-rudder plane configuration were significantly larger than the force and moment of stabilizers in the Cross-rudder plane or Y-rudder plane configurations. On the other hand, the hydrodynamic force along the \(z\)-axis and the moment about the \(y\)-axis in the Cross-rudder plane configuration were slightly smaller than the corresponding force and moment of the Y-rudder plane configuration when the stabilizers were turned down, while a discrepancy between the hydrodynamic forces of these two rudder plane configurations was found in the positive deflection angle range. Figure 7: Hydrodynamic forces and moments of the rudders in horizontal plane. (**a**) Rudder force along \(x\)-axis, (**b**) Rudder force along \(y\)-axis, (**c**) Rudder moment about \(x\)-axis, (**d**) Rudder moment about \(z\)-axis. The variation of hydrodynamic forces and moments under the stern rudder plane configurations were then fitted using the least squares method, in order to estimate the hydrodynamic coefficients of the rudders and stabilizers. Table 2 depicts the hydrodynamic coefficients under the three stern-rudder plane configurations. The first-order derivative of the force along the \(y\)-axis of the X-rudder plane configuration was about 1.36 times and 1.45 times greater than that of the corresponding force acting on the Cross-rudder plane and Y-rudder plane configurations, respectively. The first-order derivative of the force along the \(z\)-axis of the X-rudder plane configuration was about 1.18 and 1.41 times greater than those of the Cross-rudder plane and Y-rudder plane, respectively. Furthermore, the second-order derivative of the rudder's force along the \(x\)-axis of the X-rudder plane configuration was about 1.67 times larger than that of the Cross-rudder plane configuration, but was approximately the same as that along the \(x\)-axis of the Y-rudder plane configuration. On the other hand, the first-order coefficients of the roll moment of the X-rudder plane were very small in comparison with the roll moment of the left rudder plane configurations. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Cross-Rudder Plane** & **X-Rudder Plane** & **Y-Rudder Plane** \\ \hline \(X_{\delta,\delta_{i}}^{\prime}\) & \(-5.27\times 10^{-3}\) & \(-8.81\times 10^{-3}\) & \(-9.06\times 10^{-3}\) \\ \(X_{\delta,\delta_{i}}^{\prime}\) & \(-4.91\times 10^{-3}\) & \(-9.66\times 10^{-3}\) & \(-7.\) 40\(\times 10^{-3}\) \\ \(Y_{\delta_{i}}^{\prime}\) & \(8.62\times 10^{-3}\) & \(1.17\times 10^{-2}\) & \(8.07\times 10^{-3}\) \\ \(Y_{\delta_{i},\delta_{i}}^{\prime}\) & \(-2.57\times 10^{-3}\) & \(-2.74\times 10^{-3}\) & \(-1.60\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hydrodynamic coefficients under the Cross-rudder plane, X-rudder plane, and Y-rudder plane configurations. Figure 8: Hydrodynamic forces and moments of the stabilizers in vertical plane. (a) X-force of the stabilizers, (b) Z-force of the stabilizers, (c) Stabilizer moment about \(y\)-axis. Local views of the pressure distribution on the rudder plane configurations are shown in Figures 9-11 for the static rudder test and static stabilizer test at a deflection angle of 15 degrees. It can be observed that the static pressure distributions were asymmetric on the deflected blades under the Cross-rudder plane configuration. A stagnation region surrounding the leading edge of the left blades of the rudder plane configuration can be seen. However, the asymmetric pressure field on all blades in the X-rudder plane and the two inclined blades in the Y-rudder plane configurations for any deflection angle can be observed.