## 3 Results
### KP505 Propeller in Open Water Test
#### 3.1.1 Thrust and Torque Coefficient  
As shown in Figure 8, the propeller thrust and torque coefficient (\(KT\), \(KQ\)) predicted by the proposed rotating blade model were compared with the average model result and experimental data provided by NMRI (National Maritime Research Institute, Tokyo, Japan) \begin{table}
\begin{tabular}{c c c c} \hline \hline \(r\)/\(R\) & \(P\)/\(D_{P}\) & \(\theta_{s}\) (deg) & \(C\)/\(D_{P}\) \\ \hline
0.18 & 0.8347 & \(-\)4.72 & 0.2313 \\
0.25 & 0.8912 & \(-\)6.98 & 0.2618 \\
0.3 & 0.9269 & \(-\)7.82 & 0.2809 \\
0.4 & 0.9783 & \(-\)7.74 & 0.3138 \\
0.5 & 1.0079 & \(-\)5.56 & 0.3403 \\
0.6 & 1.013 & \(-\)1.5 & 0.3573 \\
0.7 & 0.9967 & 4.11 & 0.359 \\
0.8 & 0.9566 & 10.48 & 0.3376 \\
0.9 & 0.9006 & 17.17 & 0.2797 \\
0.95 & 0.8683 & 20.63 & 0.2225 \\
1 & 0.8331 & 24.18 & 0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Offset table of KP505 propeller. Figure 7: Technical drawing of KP505 propeller. Blue line is the mid-chord line made by Equations (3) and (4). Red and green lines are the leading and trailing edges, respectively, generated by Equations (5) and (6). for the T2015 workshop [1]. The result of both propeller models shows good agreement with the experiment, especially for the high \(J\) (advance coefficient) region. The \(KT\) values were from \(J\) = 0.3 to 1.0 in the experiment, and the rotating blade and average models are listed in Table 4. In the table, the error \(E\%D\) for the two models are included as well, and evaluated as follows:  
\[E\%D=\frac{D-S}{D}, \tag{30}\]  
wherein \(D\) represents the experimental data and \(S\) is the simulation value. For the rotating blade model, the \(KT\) in low \(J\) (\(J\leq\) 0.5) is under-predicted and the \(KT\) is over-predicted at high \(J\) (\(J>\) 0.5). The absolute error \(|E\%D|\) is less than or around 3%, except for at the two extreme values of \(J\), i.e., 0.3 and 1.0. The smallest error is less than 1%, at \(J\) = 0.5. Zero error is expected between \(J\) = 0.5 and 0.6. From low to high \(J\) excluding \(J\) = 1.0, the average model provides under-predicted errors increasing from around 3% to more than 10%. The \(|E\%D|\) of the average model is generally larger than that of the rotating blade model (except for at \(J\) = 0.7 and 1.0). For \(J\) = 0.1, \(KT\) is close to zero, leading to prediction difficulty. Furthermore, it is easy to obtain larger error value because the very small \(D\) value is the denominator of Equation (30). In the following discussion for \(KT\), \(J\) = 0.1 will not be included. As \(J\) decreases, the \(KT\) and \(KQ\) of both models predict lower values and deviate more clearly from the experimental data. Especially at low \(J\) value, the performance of both models is limited by the blade element theory which cannot consider foil section geometry. Equation (5) may not be suitable for high-angle attack conditions. Because \(C_{D}\) is assumed as a small constant, \(C_{L}\) is the dominated component which is computed in the current propeller computation. With the consideration of 2D blade geometry, more realistic body \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 & 0.387 & 0.365 & 5.80\% & 0.342 & 11.70\% \\
0.4 & 0.336 & 0.325 & 3.20\% & 0.298 & 11.17\% \\
0.5 & 0.285 & 0.283 & 0.65\% & 0.263 & 7.60\% \\
0.6 & 0.235 & 0.238 & \(-\)1.40\% & 0.224 & 4.71\% \\
0.7 & 0.185 & 0.190 & \(-\)2.95\% & 0.180 & 2.56\% \\
0.8 & 0.137 & 0.139 & \(-\)1.82\% & 0.132 & 3.41\% \\
0.9 & 0.083 & 0.085 & \(-\)2.82\% & 0.080 & 3.53\% \\
1.0 & 0.022 & 0.028 & \(-\)27.59\% & 0.023 & \(-\)6.74\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error analysis for thrust coefficient \(KT\). Figure 8: Open water curve for KP505 propeller. EFD(NMRI) = experiment, CFD(AVE) = average body force propeller model, CFD(ROT.B) = rotating blade body force body force.