## 3 Results and Discussion
### Water Entry of 3D Torpedo
#### 3.2.1 Effect of Entry Velocity  
Three different entry velocities of 5, 10, and 15 m/s, at the same entry angle of \(\beta\) = 90\({}^{\circ}\), are applied to investigate the effects of the entry velocity on the water entry of a torpedo. A total of 1.52 \(\times\) 10\({}^{7}\) initialization particles are employed as the optimum strategy of particle discretization along with the adaptive refinement performed for the area around the torpedo. Figures 9 and 10 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 three different entry velocities, respectively. As evident, three stages of evolution are observed. The first stage is the impact stage. When a torpedo starts to collide with the water surface in a very short time, intensive compression waves are induced and dissipate rapidly to the surroundings. Meanwhile, the head of the torpedo is significantly impacted by the instantaneous impact load, which may result in considerable damage to the torpedo head. Typically, the impact process is accompanied by the exchange of momentum between the torpedo and the surrounding fluid near the impact surface. Consequently, the surrounding fluid, obtaining momentum from the torpedo, is squeezed away from its original location and splashes around at a high velocity, as shown in Figure 9a-c for \(t\) = 100, 60, and 40 ms, respectively. In addition, with a gradual increase in the entry velocity from 5 to 15 m/s, the surrounding fluid obtains growing momentum and thus splashes away at an increasing speed. Figure 9: Evolution of velocity fields of the water domain induced by the torpedo vertically entering water at different speeds of (**a**) 5 m/s, (**b**) 10 m/s, and (**c**) 15 m/s. The second stage is the cavity formation stage. After the impact stage, the torpedo continues to dive downward. Thus, more fluid around the head of the torpedo is squeezed out and gradually forms a growing V-shaped cavity, as shown in Figure 10. In addition, it is also found that a larger entry velocity results in a larger cavity formed at the same depth of the water domain. Figure 10: Development of the water cavity induced by the torpedo vertically entering the water at different speeds of (**a**) 5 m/s, (**b**) 10 m/s, and (**c**) 15 m/s. As for the cavity development at a certain entry velocity, 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. The third stage is the cavity closure stage. As the torpedo further enters into the water, the fluid, forming the free surface of the cavity around the torpedo, moves radially toward the torpedo under the hydrostatic force. Consequently, it gradually closes the space formed by the cavity and the torpedo surfaces as shown in Figure 10. For a low entry velocity, the cavity is small and thus is closed earlier. As for the case of a low entry velocity of 5 m/s (Figure 10a), the cavity starts to close from the head to the rear of the torpedo at \(t=100\) ms, and it is fully closed at \(t=370\) ms. Subsequently, the free surface of the fluid around the previous cavity continues to be squeezed upward, forming new splashing droplets (Figures 9a and 10a). As shown in Figure 10b, when the entry velocity increases to 10 m/s, the cavity requires approximately 170 ms to close, and the area where the cavity closes is from the middle part instead of the head of the torpedo. In addition, the induced droplets with a series of higher splashing heights are observed (Figures 9b and 10b). When the entry velocity continues to increase to 15 m/s (Figure 10c), the cavity does not close until 202 ms and the induced droplets with a series of highest splashing heights are captured (Figures 9c and 10c).