## 4 Results
### Effect of Stern Appendage  
The running attitude of a high-speed vessel is one of the major factors affecting the resistance performance. In case of excessive stern trim, attaching stern appendage to build lift can reduce the resistance by 3 to 10% by reducing the stern trim by \(1^{\circ}\) to \(3^{\circ}\)[17]. In the previous study by the authors [13], the stern flap was adopted as the stern appendage to improve the resistance performance of the initial hull. The design variables for the stern flap are shown in Figure 9, the ratio of chord length to \(L_{WL}\), and the flap angle \(\alpha\).  
Having carried out a series of model tests, [PERSON] et al. [13] found out the optimal parameter to be \(c/L_{WL}=0.8\%\) and \(\alpha=10^{\circ}\). This design of stern flap was employed for the hull forms considered in this study as well. In this study, the effect of the stern flap on the resistance and running attitude was predicted by CFD simulation.  
Figure 8: Comparison of wave pattern and pressure distribution of the initial and optimum hull form at design speed. (**a**) Initial hull (\(\Delta Y_{M}=0.000\)); (**b**) optimal hull (\(\Delta Y_{M}=-0.084\) ).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(KM_{\text{T}}\) **[m]** & \(KG_{0}\) **[m]** & \(G_{0}M\) **[m]** \\ \hline \(\Delta Y_{M}=-0.126\) & 3.189 & 1.056 & 2.133 \\ \(\Delta Y_{M}=-0.084\) & 3.475 & 1.056 & 2.419 \\ \(\Delta Y_{M}=-0.042\) & 3.796 & 1.056 & 2.740 \\ \(\Delta Y_{M}=0.000\) & 4.097 & 1.056 & 3.041 \\ \(\Delta Y_{M}=+0.042\) & 4.563 & 1.056 & 3.507 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The metacentric height of initial and varied hull forms.  
Figure 10 compares the resistance ratio, trim, and sinkage between hull forms with and without stern flap. It is clear that the stern flap leads to the reduction of the stern trim for all hull forms, as shown in Figure 10b. However, the stern flap affects the resistance and the sinkage differently, depending on hull forms. For instance, the stern flap did not lead to further resistance reduction for the optimal hull form (\(\Delta Y_{M}=-0.084\)).  
Wave patterns and pressure distributions of the optimal hull form with and without flap at the design speed are compared in Figure 11. For the optimal hull form, the attachment of the stern flap weakened the stern wave as a whole. However, this effect was counteracted by the amplified bow waves, thereby making little resistance changes overall. This is shown in the wave profile near the hull surface with the pressure distribution in Figure 11. The effects from hull-form optimization and the stern appendage optimization cannot necessarily be superposed because both are related to nonlinear flow physics. The stern flap is still advantageous for the sake of safety by reducing excessive stern trim angle down to 2.74\({}^{\circ}\).  
Figure 11: Comparison of wave pattern and pressure distribution of the optimal hull form with and without stern flap. (**a**) Optimal hull (\(\Delta Y_{M}=-0.084\)) without stern flap; (**b**) optimal hull (\(\Delta Y_{M}=-0.084\)) with stern flap.  
Figure 10: Variation of total resistance, trim, and sinkage depending on design parameter \(\Delta Y_{M}\) and stern flap. (**a**) Ratio of total resistance; (**b**) trim; (**c**) sinkage.  
Figure 9: Definition of the design parameters of the stern flap.