## 3 Results
### KP505 Propeller behind KCS Hull  
Table 9 shows the results of the propeller thrust coefficient \(KT\) for the experiments and CFD using the average and the rotating blade body force model for the behind-hull condition. The propeller rotational speed is 16.3 RPS and was obtained using a self-propulsion test in a towing tank at Osaka University for a 3.2 m long ship model at Froude number Fr = 0.26 [41]. Both models under-predicted the \(KT\) by around 1%. The rotating blade result was obtained by restarting from the average model result to save computational time. The time step is required to be very small for the rotating blade model to obtain a converged and stable result. In the presented work, the time step for the average model was around 0.0055 s, while it was 0.00022 s for the blade rotating model (around 1.3 degrees of propeller rotation). Figure 17 presents the flow field plotted by the axial velocity contour (\(u/U\)) and vector field for cross flow (\(v/U\), \(w/U\)) after the rudder, i.e., \(x/L=1.025\). \(U\) is the ship model speed. \(x/L=0.0\) is at the ship FP (front perpendicular). Figure 17a is the PIV (particle image velocimetry) measurement. Figure 17b is the result of the CFD using the average model. Figure 17c is the result of the CFD using the rotating blade model. The three results show a similar flow pattern. Because of the right-rotating propeller, high axial velocity is located mainly in the starboard. The highest \(u/U\) values in (a) and (c) are in a more similar position near \(z/L=-0.04\), where is the lower part of the starboard side flow field. The highest \(u/U\) in Figure 17b is a longer band region in the middle of the starboard side flow field. The clockwise-rotating flow is interfered by the rudder in the middle, so the portside cross flow was twisted to form an upward jet and it is pointing downward in the starboard. The hub vortex is separated, becoming two cores as well: the portside core is in the lower position and starboard core is in the higher position. A vortex was induced by outer upward flow  
Figure 16: Vector field solved by the rotating blade propeller model for E1619 propeller. The vectors are colored by the axial velocity. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Method** & \(KT\) & \(E\%D\) \\ \hline \(D\) & Experiment & 0.2522 & – \\ \(S\) & CFD (average model) & 0.2497 & 0.97\% \\ \(S\) & CFD (rotating blade model) & 0.2487 & 1.39\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: \(KT\) result for behind hull condition. and starboard downward cross flow around (\(y/L\), \(z/L\)) = (0.01, \(-\)0.02). This is the corner of the high \(u/U\) contour (in yellow color, i.e., \(u/U=1.2\)-1.3). In Figure 17a,c, the corner shape is sharper than the one in Figure 17b. Basically, the flow field away from the propeller is nearly steady even though the propeller rotational flow is unsteady. This can also be seen in Figure 18 for the flow field between the propeller and rudder, i.e., \(x/L=0.99\). The propeller plane is at \(x/L=0.9825\) (the ship AP is at \(x/L=1.0\) and the ship FP is at \(x/L=0\)). In Figure 18, both models show a similar flow field pattern, but the average model has a larger high-axial-velocity area in the starboard side. In conclusion, for the downstream steady flow field behind the propeller, the above comparison indicates that the rotating blade model result is closer to the experimental model, because it considers a more realistic geometry. Comparing it with the average  
Figure 17: Propeller wake field after rudder (\(x/L=1.025\)): (**a**) PIV measurement [41]; (**b**) CFD average model [41]; (**c**) CFD rotating blade model. model, the body forces of the rotating blade model are distributed only on the blade with lower thrust prediction (Table 9) such that the distribution of high velocity downstream is more scattered and has lower values. When the wake field is near the propeller, the rotating blade model reveals an unsteady phenomena. The \(x/L=0.985\) section is right behind the propeller plane (\(x/L=0.9825\)).