## 3 Results and Discussion  
### Numerical Model Validation  
A classic study on water entry of a 3D horizontally placed cylinder [44] is conducted to validate the mathematical model and the SPH numerical method developed in this work. As shown in Figure 5, the 3D computational domain contains a large pool, filled with water, and a cylinder, horizontally located upon the pool. The pool is sufficiently large with dimensions of 1 m in length, 0.4 m in width, and 0.6 m in height to ensure that the calculation of the fluid flow is not affected by boundaries. The diameter and the length of the cylinder are 0.05 and 0.2 m, respectively. The densities of the water and the cylinder are 1000 and 1370 kg/m\({}^{3}\), respectively. The kinematic viscosity of water is \(10^{-6}\) m\({}^{2}\)/s and the initial velocity of the cylinder entering the water is set as 6.22 m/s. First, grid independence verification is performed by applying four different particle resolutions, including \(d_{\text{cld}}/\Delta d_{\text{p}}=5,10,15,\) and 20. The corresponding time evolutions of the penetration depth \(y-y_{0}\) and the vertical velocity \(u_{y}\) of the cylinder entering water achieved from this numerical work and the reported experimental studies [44; 45] are summarized in Figure 6. As evident, the numerical variation trends of both parameters gradually converge to the corresponding experimental ones with increasing particle resolution. Considering the numerical results achieved at a particle resolution of \(d_{\text{cld}}/\Delta d_{\text{p}}=20\) as an example (Table 1), the errors between the numerical and the reported experimental results at 0.12 s are only 2.11% and 4.15% for the penetration depth \(y-y_{0}\) and the vertical velocity \(u_{y}\), respectively. This comparison result validates that the mathematical model and the SPH numerical method developed in this work can effectively give rise to accurate convergence results. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(y-y_{0}/\text{m}\) & \(u_{y}/\text{m}\cdot\text{s}^{-1}\) \\ \hline Exp. [PERSON] and [PERSON] [44] & \(-0.285\) & \(\backslash\) \\ Sun [45] & \(\backslash\) & \(-1.421\) \\ SPH & \(-0.291\) & \(-1.362\) \\ Error (\%) & \(2.11\) & \(4.15\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Errors between the numerical results achieved at a particle resolution of \(d_{\text{cld}}/\Delta d_{\text{p}}=20\) and the reported experimental results. Figure 5: Three-dimensional computational domain of water entry of a horizontally placed cylinder. As calculated, the total number of particles is as high as \(1.7\times 10^{7}\) if the particle resolution of \(d_{\text{cld}}/\Delta d_{\text{p}}=20\) is applied for the whole 3D computational domain, which greatly reduces the computational efficiency. Thus, three adaptive refinement areas with three different particle resolutions of \(d_{\text{cld}}/\Delta d_{\text{p}}=20\), \(10\), and \(5\) are assigned outward from the cylinder to skillfully reduce the total number of particles and thus improve the computational efficiency. In particular, the distance from the boundary of the area of the highest particle resolution to the boundary of the cylinder is controlled at \(30\Delta d_{\text{p,max}}\) to capture the force of the fluid on the cylinder more accurately. Here, \(\Delta d_{\text{p,max}}\) is the initial distance between the two neighboring particles within the area of the highest particle resolution, Based on this adaptive technique, the total number of particles is reduced to \(1.4\times 10^{6}\), which is approximately 92% less than that generated under the condition of the uniformly distributed particle resolution of \(d_{\text{cld}}/\Delta d_{\text{p}}=20\). Therefore, the adaptive technique significantly reduces the usage of the computer memory and thus improves computational efficiency. Figure 7 shows the evolutions of the velocities within the middle plane along the \(z\) direction, as achieved from this work and the reported experimental work [44]. The results showed significant consistency.