## 8 The Optimized Distance between the VGs and the Sail Hull  
Since the placement of triangular VGs is related to the generation of the lateral vortices, which can negatively influence the intensity at the origin and the running path of the horseshoe vortex, the distance from the triangular VGs to the leading edge of the sail hull must also have an optimized value. Similar to the VGs moving towards the mixture line in the air, we put the triangular VGs at distances of 0.1c, 0.15c, and 0.2c in front of the leading edge, where c is the chord length as shown in Figure 17. The flow field and the sound field in these cases were calculated. The triangular VGs were at the angle of 30\({}^{\circ}\) to the direction of the flow. In the three cases, the pressure gradient change of the leading edge is shown in Figure 18.  
Compared to the original model, the triangular VGs stirred the turbulence. The pressure gradient stopped growing when the points were close to the leading edge. In other models, there exists turbulent disorders nearby the triangular VGs. The two new cores revealed the change in the pressure gradient,  
Figure 16: The diagram of the horseshoe vortex of the model with triangular VGs. Figure 15: The local streamline of Plane C between the original model and the model with triangular VGs. **(a)** The original model; **(b)** the model with triangular VGs. that is, one is adverse and the other is positive. The first core was the heads of triangular VGs, and the second core was the tails of triangular VGs. Compared to Figure 11, the movement of the triangular VGs significantly changed the turbulence. We observed that the vortices from the VGs reduced the intensity of adverse pressure gradients at the leading edge of the sail hull. With the distance increase, the intensity changed and the turbulent disturbance became more obvious. The origin of the horseshoe vortex was intensively destroyed and the intensity of the horseshoe vortex was weakened. This phenomenon showed that the flow-induced noise by the excitation of the horseshoe vortex would be further reduced if the triangular VGs are moved far from the leading edge. Figure 19 shows the streamline comparison between the original model and the model with triangular VGs at different distances from the leading edge. Compared to the original model, we observed that the origin of the horseshoe vortex was moved by the triangular VGs, and the dimension of the vortex core became small. The pressure gradient became mild, so that the formation intensity of the horseshoe vortex was decreased. In addition, the boundary layer separation was also delayed. Through the comparison, the vortex core changed by the triangular VGs was different, that is, the better effect was at 0.1c, where the vortex area had decreased by 53%, and the worse effect was at 0.15c, where the vortex area had increased by 94%. This phenomenon indicated that the position of the triangular VGs had an optimum value. If the distance from the triangular VGs to the leading edge of the sail hull is too close, the formation of the horseshoe vortex is mostly influenced by the physical properties of the VGs. The flow velocity near the leading edge approaches zero, and no vortices can be generated from the corners. The function of the VGs is to shorten the connection between the leading edge and the submarine body. Therefore, in this situation, the shape of the mechanical VGs should be progressively ascending. However, if the distance from the mechanical VGs to the leading edge of the sail hull is beyond 0.2c, the vortices from the corners ascend with the flow and do not energize the boundary layer. The pressure gradient at the  
Figure 17: The diagram of the movement of triangular VGs. (**a**) Side view; (**b**) top view. Figure 18: The contour map at the leading edge of Plane B of the model with triangular VGs at different distances. (**a**) 0.1c; (**b**) 0.15c; (**c**) 0.2c. origin of the horseshoe vortex is minimally destroyed by the vortices.