# The Prediction of the Performance of a Twisted Rudder
## 4 Results and Discussions
### Hydrodynamic Performances of Twisted Rudders  
Three twisted rudders shown in Figure 13 were made respectively by rotating each cross-section of the flat rudder around the mid-chord point according to the initial, intermediate, and final corrected twist angles described in Figure 12. Afterward, the three twisted rudders were named TR-ini, TR-int, and TR-fin. To examine the cavitation performance of the tip area, the TR-ini and TR-int twist rudders were attached with an end plate and a rounded tip was tested for the TR-fin model. For the investigation of the hydrodynamic performance of the twisted rudders, the numerical simulations of cavitating flows were carried out at 30-knot ship speed and the same propeller rotation condition used for the flat rudder as shown in Table 2.  
The effectiveness of the twisted rudder TR-fin on the compensation of the onset flow angles was confirmed by recalculating the inflow angles from the distribution of the streamlines over each cross-section in the rudder span direction at \(\delta_{\text{R}}=0^{\circ}\) as shown in Figure 14. While the inflow angles are seen to be slightly overcompensated by the TR-ini rudder, nearly zero angles of attack and symmetrical pressure distributions are  
Figure 11: Prediction and correction points at which inflow angles were calculated by using the axial and transverse velocities: (a) z/s = 0.3; (b) z/s = 0.5; (c) z/s = 0.7; (d) z/s = 0.9. Figure 12: Distributions of twist angles along the span direction. shown for the TR-fin rudder. The intermediate version, TR-int rudder presented the next best improvement. Figure 15 shows the pressure distributions along the rudder span direction from z/s = 0.4-0.9 for the twisted rudders compared with those of the flat rudder. As seen in Figure 14, the pressure distributions for the TR-fin rudder are very close to each other on the inboard and outboard surfaces except for z/s = 0.9, where the three-dimensional  
Figure 14: Streamline and pressure distributions on the cross sections in the span direction of three twisted rudders: (**a**) TR-in at z/s = 0.5; (**b**) TR-int at z/s = 0.5; (**c**) TR-fin at z/s = 0.5; (**d**) TR-in at z/s = 0.7; (**e**) TR-in at z/s = 0.7; (**f**) TR-fin at z/s = 0.7. Figure 13: Designed twisted rudders: (**a**) TR-in; (**b**) TR-int; (**c**) TR-fin. flow effect near the tip region can be significant. The predicted pressure distributions for the flat rudder vary largely on each surface, especially at z/s = 0.7\(\sim\)0.9, at which the cavitation inception appeared on the outboard surface. These results indicate that the surface cavitation inception phenomenon on the twisted rudder will be significantly improved compared with the flat rudder. Figures 16-18 show the predicted and observed surface cavitation on the flat and three twisted rudders at the rudder angles of \(\delta_{\rm R}=-6^{\circ}\), \(-8^{\circ}\), and \(-16^{\circ}\), respectively. As shown in Figure 5, the surface cavitation inception was detected even at \(\delta_{\rm R}=0^{\circ}\) for the flat rudder. For the twisted rudders, no surface cavitation was observed before \(\delta_{\rm R}=-6^{\circ}\) in both the experiments and numerical simulations. While surface cavitation appeared on the outboard surface of the port flat rudder at \(\delta_{\rm R}=-6^{\circ}\), the same appeared on the inboard surfaces of each starboard twisted rudder. This was because each inflow angle distribution induced by the propellers on the starboard and port side was in opposite sign and both side rudders of a twin twisted rudder were made based on these twist angles. The starboard twisted rudder was unable to effectively compensate the inflow angles at negative rudder angles, and in the same way, the port twisted rudder was disadvantageous at positive rudder angles. As expected from the pressure distributions shown in Figure 15, it is seen that the cavitation occurrence is significantly decreased in the TR-fin rudder, followed by the TR-int and TR-ini rudders.