## 4 Numerical Results
### Snapshots of Flow Field  
The snapshots of flow field at several typical time steps under different filling ratio, maximum rolling angle, and rolling period are illustrated in Figures 7-9.  
Figure 7 gives the comparison of snapshots at several typical time steps under case 1 and case 2. Both of these cases have the same rolling period, 1.45 s, and maximum rolling angle, 10\({}^{\circ}\), but different filling heights. The filling height in case 1 is 100 mm, while 300 mm in case 2. From this figure, it can be seen obviously that the profiles of free surface under these two cases are different, especially at time 6.53 s. This phenomenon is reasonable since the sloshing periods of the liquid in these two cases are different due to different filling height.  
Figure 8 gives the comparison of snapshots at several typical time steps under case 1 and case 3. Both of these cases have the same rolling period, 1.45 s, and filling height is 100 mm, but different maximum rolling angles. The maximum rolling angle in case 1 is 10\({}^{\circ}\), while 20\({}^{\circ}\) in case 3. From this figure, it can be seen that the phenomenon of fluid accumulation, bending, and slamming on a free surface occurs in case 3, but not in case 1. Therefore, it can be concluded that a larger rolling angle would induce a more violent liquid sloshing in the ship tank.  
Figure 9 gives the comparison of snapshots at several typical dimensionless time steps under case 3 and case 4. Both of these cases have same maximum rolling angle, 20\({}^{\circ}\), and filling height, 100 mm, but different rolling periods. The rolling period in case 3 is 1.45 s, while 1.8 s in case 4. From this figure, it can be seen that, although the phenomenon of fluid accumulation, bending and slamming on free surface occurs in these two cases, the liquid sloshing in case 4 is more violent than that in case 3. This difference can be explained by the fact that the sloshing intensity of the liquid in sloshing tanks is not positively correlated with the ship's undamped natural rolling frequency.  
From Figures 7-9, it can also be seen that the movement of the free surface, like the phenomena of fluid accumulation, bending and slamming, occurs in the sloshing tank. This proves that the traditional quasi-static method, assuming the free surface is parallel to the sea level, cannot reflect sloshing phenomena realistically.  
\begin{table}
\begin{tabular}{l c} \hline
**Parameters** & **Value** \\ \hline Initial particle spacing, \(l_{0}\) & 0.015 m \\ Operating distance of Gradient model, \(r_{Grad}\) & 2.1 \(l_{0}\) \\ Operating distance of Laplacian model, \(r_{Lap}\) & 3.1 \(l_{0}\) \\ Time step size, \(\Delta t\) & 0.0005 s \\ Density of fluid, \(\rho\) & 1000 kg/m\({}^{3}\) \\ Free surface coefficient, \(\beta\) & 0.97 \\ Blending parameter, \(\gamma\) & 0.01 \\ \hline \end{tabular}
\end{table}
Table 6: Main calculation parameters used in the MPS method.  
\begin{table}
\begin{tabular}{c c c c} \hline
**ID** & **Number of** & **Number of** & **Performed Time** & **Total Computational** \\  & **Fluid Particle** & **Boundary Particle** & **Step** & **Time** \\ \hline
1 & 3696 & 22,013 & 15,000 & 4.2 h \\
2 & 10,290 & 23,289 & 15,000 & 13.3 h \\
3 & 3696 & 22,013 & 29,000 & 10.5 h \\
4 & 3696 & 22,013 & 36,000 & 13.0 h \\ \hline \end{tabular}
\end{table}
Table 7: The relationship among the number of particles, computation time, and step size in all cases.  
Figure 8: Snapshots of flow field at different rolling angles. (**a**) case 1; (**b**) case 3.  
Figure 7: Snapshots of flow field in different filling ratios. (**a**) case 1; (**b**) case 2.