## 4 Conclusions  
This work systematically investigates the complex water entry process of the 3D torpedo by applying an improved adaptive SPH algorithm. The initial flow field is generated by an algorithm of 3D meshless point generation. Based on adaptive particle refinement (APR) technology, the dynamic refinement/coarsening algorithm of particles is improved, avoiding the flow field noise in the dynamic refinement process and maintaining the mass conservation of the flow field. The guard area adopted in APR technology facilitates the realization of the coupling of two regions of different resolutions, avoiding the numerical error caused by the interaction between particles of different smooth lengths. In addition, the enhanced shifting algorithm is used to ensure the uniform distribution of particles during calculations, effectively preventing non-physical oscillations of the flow field and improving the accuracy of local SPH numerical simulations. Based on the improved SPH method, a parametric study on the water entry of the torpedo is conducted from two aspects: entry velocity and entry angle. When the torpedo vertically enters water at different velocities, all the induced impact loads acting on the head of the torpedo fluctuate significantly with two peak values in the initial stage and thereafter stabilize in the later stage. With the growth in the entry velocity, the impact load acting on the torpedo, the entry depth of the torpedo, the splash height of the droplets, and the size of the cavity generated around the torpedo increase. When the torpedo enters the water at different entry angles under the same initial entry velocity, both the vertical and the horizontal movement of the torpedo are observed, leading to increased and more complex variations in parameters along both the \(x\)- and the \(y\)-axes. First, the torpedo entering water at angles except \(\beta\) = 90\({}^{\circ}\) results in one side of the cavity being larger than another side, forming an asymmetric cavity. Second, with a decrement in the entry angle, the fluctuation in the horizontal impact load becomes more significant, the horizontal velocity increases, and the corresponding horizontal displacement of the center of mass of the torpedo increases, throughout the entire water entry process of the torpedo. On the contrary, opposite conclusions about variations in the fluctuation in the vertical impact load, the absolute value of the vertical velocity, and the absolute value of the vertical displacement of the center of mass of the torpedo with decreasing entry angle are observed. Third, the moment, the angular velocity, and the variation in the attitude angle of the torpedo obliquely entering water initially decrease to negative values, then increases to positive values, and finally decreases again. In addition, the decreasing entry  
Figure 15: Evolutions of the moment about the \(z\)-axis \(m_{z}\), the angular velocity \(\omega_{z}\), and the variation in the attitude angle \(\theta-\theta_{0}\) of the torpedo when the torpedo enters the water at different entry angles of \(\beta\) = 60\({}^{\circ}\), 75\({}^{\circ}\), and 90\({}^{\circ}\) under the same entry velocity of 10 m/s. angle increases fluctuations in the moment, the angular velocity, and the attitude angle of the torpedo. This work systematically investigates the complex nonlinear fluid-structure coupling performance of the water entry of the torpedo, which provides valuable guidance for the future design of the entry trajectory and the structural bearing capacity of the torpedo. However, there are three challenges for future works. Firstly, when a torpedo enters the water, the head will be subjected to a transient peak load during the impact stage, which may damage the structure. This aspect is not fully investigated along with the water entry process in this work. Therefore, it is necessary to develop an SPH-FEM coupling algorithm to solve the elastic deformation problem of the structure in the future, which will give more complicated guidance on the design of the torpedoes.