## 4 Analysis
### Impact of the Angle of Attack on the Vertical Plane  
The NEU-based model was employed to assess the changes in the vertical angle of attack \(\beta\) in the range of 0-90\({}^{\circ}\), with an interval of 5\({}^{\circ}\). The effects of the vertical angle of attack on the lateral force, rolling moment, and yawing moment of the submarine were discussed. In order to simplify calculations and avoid an inconvenience caused by coordinate transformation, the submarine was kept fixed, and only the incident flow direction was changed. Figure 19 presents a schematic representation of the angle of attack (\(\beta\)) for ease of calculations. In the performed calculations, the incident flow velocity was maintained at \(V\) = 1 m/s. According to the changes in the incident flow direction, the boundary conditions of the computational domain were changed, as presented in Figure 20. The upstream and upper boundaries of the submarine were set as inflow velocity inlets, whereas the downstream and lower boundaries were defined as outflow pressure outlets. The lateral surfaces of the submarine were considered solid walls. The sizes of the computational domain and grid settings were the same as those given in the previous section. The force conditions and the corresponding values of \(\tau\), \(\chi\), and \(\sigma\) at various angles of attack (\(\beta\)) are illustrated in Figure 21. The plotted results in Figure 21a,d reveal that, with the increase of \(\beta\), the average magnitudes of \(F_{z}\) and \(F_{y}\) gradually rise. After \(\beta\) exceeds 55\({}^{\circ}\), the rate of increase in the mean value of \(F_{z}\) slows down, while the magnitude of \(F_{y}\) remains relatively small and increases slowly before \(\beta\) reaches 25\({}^{\circ}\). After reaching its peak  
Figure 19: Plots of the angle of attack (\(\beta\)). at \(\beta=55^{\circ}\), the magnitude of \(F_{y}\) starts to decrease gradually. As a result, the corresponding value of \(\tau\) accelerates its increase when \(\beta\) exceeds \(25^{\circ}\), and \(\tau\) starts to exceed \(10\%\). At an angle of attack \(\beta=55^{\circ}\), \(\tau\) reaches its maximum value of \(36.26\%\). Subsequently, as \(\beta\) exceeds \(55^{\circ}\), \(\tau\) starts to decrease and then gradually stabilizes. From Figure 21b,e, it can be seen that, when \(\beta\) is greater than \(25^{\circ}\), the \(\chi\) values are all greater than 2. When \(\beta\) is equal to \(50^{\circ}\), \(\chi\) reaches its first peak value of 5.81. Subsequently, \(\chi\) starts to decrease and reaches its valley value at \(\beta=70^{\circ}\), where \(\chi\) is equal to 4.12. At \(\beta=80^{\circ}\), \(\chi\) reaches its second peak value of 6.71. After \(\beta\) exceeds \(80^{\circ}\), \(\chi\) undergoes slow changes. Figure 21c,f show that, when \(\beta\) is less than \(55^{\circ}\), the magnitude of \(M_{z}\) continues to increase. It gradually decreases between \(55\) and \(75^{\circ}\), but it starts to increase again after \(75^{\circ}\). As for the pitching moment \(M_{y}\), there is a zero-crossing point between \(80^{\circ}\) and \(85^{\circ}\), which leads to a significant increase in the value of \(\sigma\). However, in general, the plot of \(\sigma\)  
Figure 21: Force statistics for various angles of attack (\(\beta\)). (**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 \(\sigma\). Figure 20: Boundary conditions of the computational domain. exhibits a slight increase when \(\beta\) is less than \(25^{\circ}\), but after crossing \(25^{\circ}\), it gradually exceeds \(40\%\). When considering the roll moment values from Figure 21b, it can still be seen that the overall trend of \(M_{z}\) magnitude is consistent with that of the \(M_{x}\) magnitude.