## 3 Results
### KP505 Propeller in Open Water Test
#### 3.1.2 Grid Independence Test  
Based on the Section 3.1.1 result, the condition \(J\) = 0.5, 0.7, and 0.8 are selected to check grid independency of the propeller model and CFD method of the presented work. \(J\) = 0.5 has the minimal \(KT\) error, which is under-predicted less and around 1% in Table 4. \(J\) = 0.7 has the minimal \(KQ\) error, which is over-predicted nearly only 0.2% in Table 5. The \(KT\) and \(KQ\) predictions are performed in balance at \(J\) = 0.8. with error about 1.7-1.8%. The averaged absolute error for \(J\) = 0.8 is 1.75% which is second lowest among all \(J\) values. It is only slightly higher than the error of \(J\) = 0.7 (1.57%), because the \(KQ\) error of \(J\) = 0.7 is really low. The analysis of grid independence and uncertainty here is based on the verification and validation theory suggested by ITTC 7.5-03-01-01 guideline [48]. A different grid size and density are required. The grid in the previous section and Table 2 is served as medium grid, and then the grid number is increased by (\(N_{i}\), \(N_{j}\), \(N_{k}\)) \(\times\sqrt{2}\) to become a fine grid. Reducing the grid number, the coarse grid is built by (\(N_{i}\), \(N_{j}\), \(N_{k}\))/\(\sqrt{2}\). Finally, the total grid number is 1.2 M (1,217,280) and 9.8 M (9,777,246) for the coarse and fine grid, respectively. The simulation values of fine, medium, and coarse grid are assigned in this section as \(S_{1}\), \(S_{2}\), and \(S_{3}\), respectively. The grid dependence test results for \(KT\) and \(KQ\) are presented in \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(J\)} & \multicolumn{2}{c}{**Experiment**} & \multicolumn{2}{c}{**Rotating Blade Model**} & \multicolumn{2}{c}{**Average Model**} \\ \cline{2-6}  & \(D\) & \(S\) & \(E\%D\) & \(S\) & \(E\%D\) \\ \hline
0.3 & 0.557 & 0.449 & 19.40\% & 0.462 & 17.09\% \\
0.4 & 0.497 & 0.429 & 13.64\% & 0.433 & 12.87\% \\
0.5 & 0.437 & 0.400 & 8.49\% & 0.404 & 7.54\% \\
0.6 & 0.376 & 0.361 & 4.02\% & 0.365 & 2.82\% \\
0.7 & 0.311 & 0.312 & \(-\)0.19\% & 0.316 & \(-\)1.73\% \\
0.8 & 0.247 & 0.251 & \(-\)1.68\% & 0.256 & \(-\)3.67\% \\
0.9 & 0.181 & 0.179 & 1.36\% & 0.184 & \(-\)1.39\% \\
1.0 & 0.096 & 0.093 & 3.30\% & 0.098 & \(-\)1.75\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Error analysis for propeller torque coefficient \(KQ\). Tables 6 and 7, respectively. All \(S_{2}\) values are the same as the \(S\) values in Section 3.1.1. The \(D\) value and \(E\%D\) calculation follows Section 3.1.1 as well. The grid independence is judged by the \(R_{G}\) value, which is calculated as below:  
\[R_{G}=\frac{S_{2}-S_{1}}{S_{3}-S_{2}}. \tag{31}\]  
Grid independence is accomplished for \(\mid R_{G}\mid<1\), meaning that the value difference between the medium and fine grid is smaller than the difference between medium and coarse grid. As shown in Tables 5 and 6, all CFD results achieve grid independence, i.e., as the grid number increases, the results converge. Monotonic convergence (M.C.) is reached for \(0<R_{G}<1\), and oscillating convergence (O.C.) is reached for \(-1<R_{G}<0\). Except for the result that the \(KQ\) of \(\mid\) = 0.5 is M.C., the other CFD results satisfy the O.C. condition. For M.C., the grid uncertainty (\(U_{G}\)) is estimated by the generalized Richardson extrapolation with factor of safety suggested by [48]. For O.C., \(U_{G}\) is simply evaluated from the minimal and maximal values among three different grid results [48]:  
\[U_{G}=|0.5(S_{max}-S_{min})|, \tag{32}\]  
which is presented as percentage of \(D\) in Tables 5 and 6. The result achieving a \(U_{G}>\mid E\%D\mid\) of \(S_{1}\) would be considered as valid. As a consequence, for a \(KT\) of \(J\) = 0.5 and 0.8 and a \(KQ\) of \(J\) = 0.7 and 0.8, validation is achieved. Once grid convergence is confirmed, it strongly depends on if \(S_{1}\) could produce small enough error, i.e., how accurate the simulation can reach as the grid grows finer. For \(J\) = 0.8, \(KT\) and \(KQ\) both are validated.