## 5 The Shape Selection of Mechanical VGs  
The VGs were symmetrically mounted on both sides of the model according to the central line, as shown in Figure 6. We used the \(\lambda\) to denote the angle of the mechanical VGs to the flow direction. The height of the mechanical VGs was 0.1 \(H\), where, \(H\) was the height of the sail hull. The dimension of the mechanical VGs was higher than the thickness of the boundary layer on the surface of the submarine body. The length of the VGs was 0.1c, where c was the chord length of the sail hull. Considering the results of flow control in the air, we have investigated three shapes of mechanical VGs, semi-circular, triangular, and trapezoidal. The three shapes of mechanical VGs have the identical length of 0.1c, and the identical height of 0.1 \(H\), where c is the chord length of the sail hull, and \(H\) is the height of sail hull. At the leading edge, we have placed a pair of different shapes of mechanical VGs with an angle of 30\({}^{\circ}\) to the flow direction. Figure 7 shows the placement of mechanical VGs of different shapes. Figure 8 shows the comparison curves of radiated sound power from the original model and the models with semi-circular VGs, trapezoidal VGs, and triangular VGs. The models with trapezoidal VGs and triangular VGs had lower levels of radiated sound power than that of the original model. However, the radiated sound power from the model with semi-circular VGs was even higher than that of the original model. The corners of semi-circular VGs are not too sharp to produce enough vortices and cannot significantly destroy the formation of the horseshoe vortex. Therefore, the difference of the curves between the original model and the model with semi-circular VGs was minimal, especially  
Figure 5: The diagram of the horseshoe vortex nearby the sail hull. Figure 6: The sketch of the mechanical VGs on the model. (**a**) Top view; (**b**) side view. in the low-frequency range. The addition of semi-circular VGs enhanced the level of radiated sound power in the high-frequency range, due to the VGs also vibrating. We may conclude that semi-circular VGs cannot be utilized to suppress the flow-induced noise. Since the mechanical VGs with triangular and trapezoidal shapes have sharp corners and can produce enough vortices to change the horseshoe vortex's formation, the radiated sound power from the models with these shapes is lower than that of the original model. There are two corners in trapezoidal VGs; the vortices of two adjacent groups are produced, while only one group of vortices are produced from triangular VGs. When the fluid flows around the two corners of the trapezoidal VGs, the vortices of the two adjacent groups rotate at a certain phase. Then, these vortices are weakened by each other. The control effect of the horseshoe vortex by trapezoidal VGs is less than that by triangular VGs. That is why triangular VGs have a better flow-induced noise suppression, compared to the other two shapes. In Table 1, the total level of radiated sound power from the models with triangular and trapezoidal VGs is less than that of the original model in the frequency range from 10 Hz to 2000 Hz. The best noise reduction effect from mechanical VGs of the three shapes was the triangular. The conclusion is that if the flow-induced noise is suppressed by mechanical VGs, the VGs must have corners. If there is only one sharp corner in the VG, a better effect can be achieved. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Model** & **Total Level of Radiated Sound Power/dB\({}^{1}\)** & **Noise Reduction/dB \({}^{1}\)** \\ \hline Original & 113.51 & 0 \\ With triangular VGs & 108.23 & 5.28 \\ With trapezoidal VGs & 110.88 & 2.63 \\ With semi-circular VGs & 126.46 & \(-12.95\) \\ \hline \hline \end{tabular} \({}^{1}\) dB The reference is \(0.67\times 10^{-18}\).  
\end{table} 
Table 1: The level of radiated sound power from the original model and the models with triangular VGs (vortex generators), trapezoidal VGs, and semi-circular VGs.