## 4 Numerical Results  
### Description of Test Cases  
In this paper, numerical simulations of liquid sloshing are performed on a 400,000 DWT Very Large Ore Carrier (VLOC) \"PACIFIC VISION\", which is designed by the Shanghai Merchant Ship Design and Research Institute (SDARI). Since liquid sloshing is mostly taking place in water ballast tanks, a ballast water tank of this VLOC, namely \"R2.03 AP\", is selected to carry out the numerical simulation of liquid sloshing in ship tanks under different rolling motions. The principal particulars of \"PACIFIC VISION\" and the parameters of the water ballast tank \"R2.03 AP\" are shown in Table 1 and Table 2, respectively [61]. The 3D design models of the hull and this water ballast tank are illustrated in Figure 5. \begin{table}
\begin{tabular}{l c} \hline \hline
**Description** & **Value** \\ \hline Length overall & 361.90 m \\ Length between perpendiculars (L.B.P.) & 355.00 m \\ Breadth moulded & 65.00 m \\ Depth moulded & 30.40 m \\ Scantling draft moulded & 23.00 m \\ Deadweight at scantling draft & 398,411.1 t \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Principal particulars of ‘PACIFIC VISION’, from [61]. Figure 4: Comparisons of numerical simulation results of the pressure impact at the probe \(P_{0}\). (**a**) 2D results given by [52,60]; (**b**) 3D results given by [52,60]; (**c**) 2D numerical simulation results of this paper; (**d**) 3D numerical simulation results of this paper. It is worth noting that a scaled model of \"R2.03 AP\" is applied to carry out the simulation due to the limitation of computing ability. The scale ratio is set to 1:50. Meanwhile, five particular points on the port sidewall are selected to analyze the sloshing-induced pressure, as illustrated in Figure 6. It can be seen from the lateral view that, when the liquid level is low, the arc structure on the side away from the \(y\)-axis can effectively reduce the liquid impact. When the liquid level is high, the free surface area will decrease due to the slope of the bulkhead near the \(y\)-axis. The numerical simulation in the ship tanks can reflect the sloshing phenomena more realistically than the rectangular and other simplified tanks. The parameters of the scaled model and the particular points are shown in Table 3. \begin{table}
\begin{tabular}{l c} \hline \hline
**Description** & **Value [mm]** \\ \hline Size of scale model in the \(x\)-direction & 322.80 \\ Size of scale model in the \(y\)-direction & 403.60 \\ Size of scale model in the \(z\)-direction & 619.48 \\ Height of Probe \(P_{1}\) (above tank bottom) & 20.0 \\ Height of Probe \(P_{2}\) (above tank bottom) & 80.0 \\ Height of Probe \(P_{3}\) (above tank bottom) & 140.0 \\ Height of Probe \(P_{4}\) (above tank bottom) & 200.0 \\ Height of Probe \(P_{5}\) (above tank bottom) & 260.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the scale model of “R2.03 AP” and the particular points. Figure 5: Illustration of the 3D design models of “PACIFIC VISION”. Figure 6: Illustration of the particular points on “R2.03 AP”. (**a**) custom view; (**b**) lateral view. One characteristic ballast load condition, LOADA02, is used to carry out numerical simulation. Parameters of \"PACIFIC VISION\" under this load condition are shown in Table 4, in which the terms \"T.C.G.\" and \"V.C.G.\" indicate the center of gravity of the ship of the breadth from the centerline and the height above the base line, respectively, and \"GM\" is ship's metacentric height. The undamped natural period of the ship \(T_{r}\) can be determined by Equation [62], as follows:  
\[T_{r}=\frac{2 cB}{\sqrt{GM}} \tag{12}\]  
where \(B\) is ship's breadth moulded; and \(c\) is the coefficient describing ship's transverse gyration radius, and is determined by:  
\[c=0.373+0.023\frac{B}{T}-0.043\frac{L}{100} \tag{13}\]  
where \(L\) and \(T\) is ship's length between perpendiculars and mean ship draft, respectively.