## 3 Results and Discussion
### Water Entry of 3D Torpedo
#### 3.2.2 Effect of Entry Angle  
In addition to the case of the entry angle of \(\beta=90^{\circ}\) under 10 m/s studied above, two more entry angles of \(\beta=75^{\circ}\) and \(60^{\circ}\) under the same entry velocity are also investigated to determine the effects of the entry angle on the water entry of the torpedo. Figures 12 and 13 summarize the evolution of the velocity fields within the middle plane of the calculation domain along the \(z\) direction and the development processes of the cavities during the water entry of the torpedo at different entry angles, respectively. Figure 11: Variations in (**a**) the impact load projecting onto the \(y\)-axis \(F_{y}\) and (**b**) the velocity \(u_{y}\) and the vertical displacement of the center of mass \(y-y_{0}\) of the torpedo with time when the torpedo vertically enters the water at different speeds of 5, 10, and 15 m/s. Figure 12: Evolution of velocity fields of the water domain induced by the torpedo entering the water at different entry angles of (**a**) \(\beta=75^{\circ}\) and (**b**) \(\beta=60^{\circ}\) under the same entry velocity of 10 m/s. Figure 13: Development of the cavity within the water domain induced by the torpedo entering the water at different entry angles of (**a**) \(\beta=75^{\circ}\) and (**b**) \(\beta=60^{\circ}\) under the same entry velocity of 10 m/s. As for the cavity development at a certain entry angle, the figures within the upper row show the cross-sectional view of the water phase within the middle plane of the computational domain along the \(z\) direction and those within the bottom row show the corresponding 3D views of the free surface of the water generated around the torpedo. Once the torpedo collides with water in a very short time, it causes the nearby fluid to splash radially outward with high velocities, as seen from the figures of \(t=55\) ms in Figure 12a and \(t=55\) ms in Figure 12b. However, in contrast to the symmetric splash curtain generated by the vertical water entry of the torpedo, the velocity along the \(x\)-positive component of the torpedo \(u_{x}\), occurring when the torpedo enters water at angles of \(\beta=75^{\circ}\) or \(60^{\circ}\), results in a larger impact force on the fluid in front of the torpedo. This results in an asymmetric splash curtain, as shown in Figures 12 and 13. Correspondingly, an asymmetric cavity is also simultaneously generated, which is larger at the front of the torpedo owing to the higher momentum transferred to the fluid there. As the torpedo moves downward, the fluid gradually flows back to contact with the torpedo from the bottom to the upper part of the torpedo. In addition, the cavity closes faster at the back of the torpedo than at the front of the torpedo. Furthermore, the cavity at the back of the torpedo closes faster for a smaller entry velocity. The variations in the impact loads, the velocities, and the displacements of the center of mass of the torpedo with time when the torpedo enters water at different entry angles of \(\beta=60^{\circ}\), \(75^{\circ}\), and \(90^{\circ}\) under the same entry velocity of \(10\) m/s are drawn in Figure 14. Compared to the impact load of the \(x\)-component of zero at the entry angle of \(\beta=90^{\circ}\) depicted in Figure 14a, entry angles of \(\beta=60^{\circ}\) and \(75^{\circ}\) lead to non-zero impact loads of the \(x\)-component. These two non-zero horizontal impact loads initially point to the negative direction of the \(x\)-axis, impeding the horizontal movement of the torpedo. In addition, a larger impact load of the \(x\)-component occurs at the smaller entry angle of \(\beta=60^{\circ}\) in the earlier stage of the torpedo entering the water. Correspondingly, the velocity of the \(x\)-component at the smallest entry angle of \(\beta=60^{\circ}\) is largest and it decreases the most sharply in the earlier stage, as shown in Figure 14c.