## 4 Analysis
### Impact of the Heel Angle  
The effect of the heel angle change on the lateral force (\(F_{y}\)), rolling moment (\(M_{x}\)), and yawing moment (\(M_{x}\)) of the NEU-based model was evaluated in the presence of the vertical flow velocity of \(V=1\) m/s. The heel angle (\(\theta\)) was varied from 0 to \(80^{\circ}\) in increments of \(5^{\circ}\). As in the previous section, the hull was kept fixed, and the direction of the incoming flow was changed. Figure 23 illustrates the schematic diagram of the heel angle (\(\theta\)).  
Figure 23: Heel angle (\(\theta\)).  
The presence of the heel angle (\(\theta\)) is equivalent to a hull experiencing lateral flow, resulting in non-zero mean values for \(F_{y}\), \(M_{x}\), and \(M_{z}\). Figure 24 presents the average values of \(F_{y}\), \(M_{x}\), and \(M_{z}\) at different heel angles (\(\theta\)). Concerning the force \(F_{y}\), it increases steadily with increasing \(\theta\), but the rate of increase becomes noticeably slower between \(20^{\circ}\) and \(40^{\circ}\). Regarding the rolling moment \(M_{x}\), it increases rapidly with increasing \(\theta\) and peaks at around \(30^{\circ}\) before gradually decreasing to a steady state. In contrast, when \(\theta\) is less than \(25^{\circ}\), \(M_{z}\) increases continuously, but when \(\theta\) exceeds \(25^{\circ}\), it quickly reverses its direction and slowly rises after \(\theta\) reaches \(60^{\circ}\). Therefore, the presence of the heel angle intensifies the changes in the lateral force, rolling moment, and yawing moment.  
To further compare the magnitude of the lateral disturbance caused by the vertical velocity, Figure 25 shows the force distribution by subtracting the average values of \(F_{y}\), \(M_{x}\), and \(M_{z}\) as well as the values of \(\tau\), \(\chi\), and \(\sigma\) at different heel angles \(\theta\). By examining Figure 25a,d, it can be seen that both the average value of \(F_{z}\) and the amplitude of \(F_{y}\) gradually decrease with increasing \(\theta\). The value of \(\tau\) decreases rapidly and stabilizes after a heel angle of \(10^{\circ}\) and continues to decrease when \(\theta\) exceeds \(45^{\circ}\). As \(\theta\) exceeds \(55^{\circ}\), the value of \(\tau\) becomes less than \(10\%\).  
According to Figure 25b,e, it can be seen that the \(M_{x}\) amplitude shows a valley shape at about \(15^{\circ}\), followed by a peak at about \(\theta=30^{\circ}\), then gradually lessens before stabilizing. By considering Figure 25c,f, it is similarly evident that, as \(\theta\) increases, the overall trend of the \(M_{z}\) amplitude is similar to that of \(M_{x}\), but the \"peaks and valleys\" of \(M_{z}\) are not as distinct as those of \(M_{x}\). As a result of the change in the direction of the pitching moment \(M_{y}\) in terms of \(\theta\), a significant peak in the value of \(\sigma\) occurs at \(\theta=50^{\circ}\). Then, when \(\theta\) exceeds \(65^{\circ}\), the value of \(\sigma\) drops below \(40\%\).  
In summary, by performing calculations for the range of heel angles (\(\theta\)) from \(0\) to \(80^{\circ}\) under a free-stream velocity of \(v=1\) m/s, it can be concluded that the increase in \(\theta\) intensifies the magnitude of the lateral force \(F_{y}\). Furthermore, in practical scenarios, it is very unlikely that the heel angle will exceed \(50^{\circ}\). Therefore, the effects of the vertical velocity on the forces \(F_{y}\), \(M_{x}\), and \(M_{z}\) cannot be ignored.  
Figure 26 presents the vorticity and streamlines at a particular instant for different heel angles near the hull. It can be observed that, as the heel angle increases, the asymmetry in the vortex between the left and right sides of the hull becomes more obvious.  
Figure 25: Force statistics for various angles of heel (\(\theta\)). (**a**) \(F_{z}\) and amplitude of \(F_{y}\), (**b**) amplitude of \(M_{x}\), (**c**) \(M_{y}\) and amplitude of \(M_{z}\), (**d**) the values of \(\tau\), (**e**) the values \(\chi\), (**f**) the values of \(\sigma\).