# Research on the Hydrodynamic Performance of a Pentamaran in Calm Water and Regular Waves
## 4 Numerical Modelling
### Validation of the Numerical Method  
To evaluate the reliability of the numerical method, it is necessary to validate the numerical model. One of the best ways to do this is to compare simulation data with experimental data. However, there are no relevant test data applicable to the pentamaran or monohull. Another method is to carry out numerical simulations of the standard hull model, which can also prove the reliability of the numerical method. KCS is a modern container ship designed by KRISO (Korea Research Institute of Ship and Ocean Engineering) in Korea and was listed as one of the standard ship types for CFD assessment and verification in the Gothenburg-2000 conference [43]. The KCS ship model has a large bulbous bow and a complex tail surface, belonging to the hypertrophy ship category [49]. The main particulars and the geometry of the full-scale monohull are shown in Figure 8 and Table 7, respectively. The total mesh number was 3,000,000. Figure 9 shows the section of the mesh and the mesh at the bow and stern. For the computations in calm water, six conditions were selected in the range of low and medium speeds. The validation objects were total resistance coefficient, heave, and pitch. The total resistance coefficient was calculated as follows:  
\[C_{t}=\frac{R_{t}}{0.5\rho U^{2}S} \tag{13}\]  
where \(C_{t}\) is the total resistance coefficient, \(R_{t}\) is the total resistance, \(\rho\) is the density of water, \(U\) is the speed, and \(S\) is the wetted area of the hull. This is the same as \(C_{p}\) and \(C_{f}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Particulars** & **Symbols/Unit** & **Value** \\ \hline Scaling ratio & \(\lambda\) & 31.60 \\ Length between perpendiculars & \(L_{pp}\)/m & 7.278 \\ Breadth & \(R_{sat}\)/m & 1.01 \\ Molded depth & D/m & 0.60 \\ Draft & \(d\)/m & 0.341 \\ Displacement & \(\
abla\)/m\({}^{3}\) & 1.649 \\ Wetter area & \(S\)/m\({}^{2}\) & 9.512 \\ Waterline coefficient & \(C_{w}\) & 0.818 \\ Block coefficient & \(C_{b}\) & 0.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Main particulars of the monohull. Figure 8: The geometry of the KCS model. Figure 9: The KCS mesh. Figure 10 shows the simulation results of the total resistance coefficient of KCS at different speeds. The simulation results are in good agreement with the experimental data. Among the six calculation conditions, the largest error 2.29% occurred at \(F_{n}\) = 0.108, which was due to the difficulty of resistance convergence at low speed. When the speed was high, the simulation results were in good agreement with the experimental values. From the changing trend, the resistance coefficient first decreased with the increase in the speed, and then increased rapidly after reaching the condition of \(F_{n}\) = 0.195. Figure 11 shows the comparison of the hull motions of KCS between the simulation results and experimental data. The heave and pitch values were generally similar to the experimental data. From the changing trend, the heave value increased with the increase in the speeds, and the growth rate also increased. The pitch value first increased in the direction of the negative value (stern inclination), reached the maximum value in the negative direction at \(F_{n}\) = 0.26, and then changed in the direction of the positive value. The global view of the wave pattern of the KCS at \(F_{n}\) = 0.26 is shown in Figure 12. Compared with the experimental result, the calculated waveform was in good agreement with the experimental waveform in terms of wavelength and wave shape. The VOF method used in numerical simulation can accurately capture Kelvin waves for large ships. It should be noted that only the starboard side was measured and reflected on the port side for plotting. It was observed that a wave existed inside a certain angle known as the Kelvin angle from linear potential wave theory. The bow wave, having the maximum height on the hull surface, gradually disappeared, followed by a trough and crest lines of divergent wave  
Figure 11: Comparison of heave and pitch of KCS.