## 3 Results and Discussions  
Cavity evolution of water entry for different entry angles, lengths and velocities of projectiles are shown in Figures 4 and 5. The initial time \(T=10\), and the time interval \(\Delta T\) between each cavity contour is 20. From Figure 4a, the entry angle of projectile is \(10^{\circ}\), during the water entry process,  
Figure 3: Between the cavity evolution of numerically result and predicted by the analytical cavity models. Figure 2: Comparison of numerical and analytical results of vertical water entry. at the early stage, the water surface is ascending quicker than the descending of cavity, and its interface appears relatively coarse, later, the projectile tends to touch and disturb the cavity surface, which in consequence interact with each other, the pitch angle of projectile changes remarkably, and the projectile will ricochet. Comparing the Figure 4a with Figure 4b, the projectile with the large length (8D) has better stability, and the variation of its pitch angle is decreased, even though it still vibrates. With the increase of its entry velocity (Figure 4c,d), the cavity wall during the whole water entering process keeps relatively smoother even the projectile still plans on the cavity surface; however, the shorter length projectile still has the tendency to ricochet (Figure 4c), while the long projectile moves stable along its entry angle. Figure 4: Evolution of water entry at initial angle 10\({}^{\circ}\): (**a**) 10_5D_V100, (**b**) 10_8D_V100, (**c**) 10_5D_V500, (**d**) 10_8D_V500. Figure 5: _Cont_. With the increase of its entry angle to 45\({}^{\circ}\), the projectile moves more stable after the entry. The deep seal of cavity appears for low speed projectile (Figure 5a,b), and the time of deep seal for short projectile (\(L\)/\(D\) = 5) is almost the same with the long one (\(L\)/\(D\) = 8). With the increase of velocity (Figure 5c,d), the appearance of deep seal is delayed. Meanwhile, smoothness of cavity surface can be maintained for longer. On the other hand, for a long projectile, its distance to the entry point is larger than that of the short one at the same time period, which shows that the decrease of velocity gradient of 8D projectile is smaller than that of projectile with short length (Figure 5D). Figure 6 shows the trajectories of water entry at different conditions. From Figure 6a, it is clear that, when the entry angle is small (10\({}^{\circ}\)), the projectile with \(L\)/\(D\) = 5 ricochets for both the entry velocity of 100 and 500 m/s, and with \(L\)/\(D\) = 8, even there is no appearance of ricochet, its deviation from the initial trajectory which is the line along the initial water entry angle is still large. With the increase of entry angle (Figure 6b,c), the trajectory deviations become small for both projectiles; therefore, the projectile length does not have much effect on its trajectory deviation at large entry angles. Figure 5: Evolution of water entry at initial angle 45\({}^{\circ}\): (**a**) 45_5D_V100, (**b**) 45_8D_V100, (**c**) 45_5D_V500, (**d**) 45_8D_V500. Figure 6: _Cont._  
Figure 7 shows the time histories of velocity magnitude during the water entry process. It is clear that the velocity decay with \(L=5D\) is faster than that of \(L=8D\), the velocity variations caused by the entry angles are small compared with that by its length modification. The pitch angle of projectile vibrates during the water entry process. At the initial stage of oblique water entry, the contact area at the head is asymmetric, the bottom side of the projectile head touches the water first, which generates the pitch moment and increases the pitch angle, then the attitude of projectile modifies, till its tail impacts the cavity surface and generates the opposite pitch moment, which make the pitch angle decrease and vibrate periodically. Figure 8: _Cont_. Figure 6: Trajectory curves of water entry for different initial inclination angles: (**a**) 10\({}^{\circ}\), (**b**) 45\({}^{\circ}\), (**c**) 80\({}^{\circ}\).