## 4 Free Surface RANS Solver and Validation
### CFD Validation with Respect to Running Attitude  
The computational time for the medium grid selected was five hours, which was not considered to be short enough to carry out iterative simulations for the SQP optimization. Consequently, further reduction in computational time without sacrificing the accuracy was necessary. In order to shorten the initial converging period before the running attitude is settled from the force and moment balance, the simulation started with the running  
Figure 4: Computational domain and boundary conditions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Grid** & \begin{tabular}{c} **Hull Surface** \\ **Mesh** \\ \end{tabular} & \begin{tabular}{c} **No of Grid** \\ **Points** \\ \end{tabular} & \(C_{P}\times\mathbf{10^{3}}\) & \(C_{TM}\times\mathbf{10^{3}}\) & \(C_{TM}\times\mathbf{10^{3}}\) & \begin{tabular}{c} **Sinkage** \\ **(a/LBP)** \\ \end{tabular} & \begin{tabular}{c} **Trim** \\ **(c)** \\ \end{tabular} & \begin{tabular}{c} **Computational** \\ **Time (h)** \\ \end{tabular} \\ \hline \begin{tabular}{c} Exp. \\ Coarse \\ \end{tabular} & - & - & - & - & 3.711 & \(-\)0.0019 & \(-\)0.169 & - \\ \begin{tabular}{c} Coarse \\ \end{tabular} & \(114\times 51\) & 1.62 M & 0.744 & 2.852 & \begin{tabular}{c} 3.596 \\ (96.9\%) \\ \end{tabular} & \(-\)0.0022 & \(-\)0.180 & 3 \\ \begin{tabular}{c} Medium \\ \end{tabular} & \(162\times 71\) & 2.70 M & 0.779 & 2.850 & \begin{tabular}{c} 3.629 \\ (97.8\%) \\ \end{tabular} & \(-\)0.0021 & \(-\)0.173 & 5 \\ \
\begin{tabular}{c} Fine \\ \end{tabular} & \(230\times 101\) & 4.61 M & 0.774 & 2.853 &
\begin{tabular}{c} 3.627 \\ (97.7\%) \\ \end{tabular} & \(-\)0.0022 & \(-\)0.206 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of grid convergence test for the original KCS hull (\(Fn=0.26\) ). attitude being fixed to the values of original hull. This assumes that the running attitude would be essentially independent on the hull form variation. Table 4 and Figure 5 compare the results obtained from various running attitude conditions for the original KCS hull. The first row denoted as \"Free\" represents the simulation with the running attitude being freely set by the force and moment balance. The second row (\"Fixed\") means that the running attitude is fixed at the static equilibrium, i.e., zero sinkage and trim throughout the simulation. For the third row (\"Fixed with static sinkage & trim\"), the sinkage and trim values are first fixed to the values of \"Free\" condition and then simulation is conducted all along. In Table 4, it is observed that the fixed with static sinkage & trim case yielded very close value of \(C_{TM}\), which is only 0.2% different from that of free condition in a considerably shorter computational time of two hours. In Figure 5, \"Free\" and \"Fixed with static sinkage & trim\" conditions give rise to almost identical wave profiles, which support the close match of the resistance coefficient. On the other hand, the \"Fixed\" condition fails to give a close match to the \"Free\" solution, which differs by 3.2%. This is because the running attitude was kept fixed to the incorrect values all along the simulation. The convergence history in Figure 5 indicates that the \"Fixed with static sinkage & trim\" case converged much faster at around 6000-7000 iterations. This is in support of the significant saving of computational time, approximately 60%, enabled by this treatment. Although the present running attitude treatment is shown to be effective for the original KCS hull, this may not necessarily be the case for other modified hulls. The comparison between the results using two different running attitude conditions for the optimal hull is listed in Table 5. As a matter of fact, these results are taken from the final result of optimal hull, which will be given in Section 6.2. Even though the underlying assumption of independence of running attitude on hull form variation may be seemingly unreasonable, the excellent agreement between the results in Table 5 is just remarkable.