## 5 Conclusions  
The numerical simulations of the flow around a submarine vertical plane at high angles of attack were methodically carried out using the SST-DDES model in conjunction with AMR technology. A new omega vorticity identification approach was employed as the adaptive criterion. The numerical method was validated through flow calculations around a cylinder and water tank experiment of flow around the vertical plane of the submarine. The effects of different attack angles and heel angles on the submarine lateral force, rolling moment, and yawing moment were methodically analyzed. The following results can be obtained:  
(1) It is possible to simulate the unsteady flow around a submarine vertical plane via the SST-DDES model in conjunction with AMR technology. A comparison between the calculated average values of longitudinal force \(F_{x}\), vertical force \(F_{z}\), and pitching moment \(M_{y}\) and the experimental results shows an error range of 3.0% to 14.0%. In addition, comparing the time-history data of the lateral force \(F_{y}\), rolling moment \(M_{x}\), and yawing moment \(M_{z}\) with the experimental results reveals a significant consistency in the trends of \(F_{y}\), \(M_{x}\), and \(M_{z}\). The minimum errors in \(F_{y}\), \(M_{x}\), and \(M_{z}\) magnitudes are obtained as 13.9%, 7.4%, and 11.9%, while the minimum errors in their dominant frequencies are obtained as 0.6%, 0.9%, and 0.9%, respectively. (2) In calculating the flow around the vertical plane of the submarine, it was found that the vertical velocity of the hull has a special effect on the horizontal forces. Specifically, the magnitude of \(F_{y}\) accounts for approximately 20% of the average value of \(F_{z}\), and the ratio of the magnitude of \(M_{z}\) to the average value of \(M_{y}\) ranges from 71.77% to 161.97%. Therefore, when studying the motion of the vertical plane of the submarine, the effects of lateral force and yawing moment cannot be ignored. Similarly, the influence of the rolling moment cannot be neglected when the vertical velocity of the hull \(w\) exceeds 0.6 m/s; the effect of the rolling moment cannot be ignored. (3) When analyzing the influence of the angle of attack of the submarine vertical plane, it can be seen that, when the angle of attack exceeds 25\({}^{\circ}\), the roles of \(F_{y}\), \(M_{x}\), and \(M_{z}\) cannot be ignored. In particular, the influence of the yawing moment \(M_{z}\) is highlighted when its value approaches or exceeds the average value of the pitching moment \(M_{y}\). (4) The presence of the roll angle (\(\theta\)) during the motion of the vertical plane of the submarine intensifies the forces on \(F_{y}\), \(M_{x}\), and \(M_{z}\) induced by the vertical velocity. The forces on \(F_{y}\), \(M_{x}\), and \(M_{z}\) are caused by the vertical velocity. The effect of forces on \(F_{y}\), \(M_{x}\), and \(M_{z}\) due to the vertical velocity cannot be neglected. (5) During the motion of the vertical plane of the submarine, regardless of the angle of attack and the roll angle, it was found that the trend of changes in the magnitude of \(M_{z}\) closely follows that of \(M_{x}\). (6) In the case of the angle of attack above the vertical plane of the submarine, the lateral force, the rolling moment, and the yawing moment influence and affect each other. This is a complex nonlinear problem, and in future research, the effects of forces on these three parameters should be methodically examined during the dynamic process. We will focus on quantitatively describing these hydrodynamic forces using mathematical models, aiming to establish a mathematical model applicable to the up-surfacing motion of submarines at large angles of attack. Conceptualization, G.X. and Y.O. ; methodology, G.X. and J.C.; validation, H.W., W.W. and Y.O. ; writing--original draft preparation, G.X. ; writing--review and editing, Y.O. and H.W. ; supervision, W.W.; project administration, Y.O. ; funding acquisition, Y.O. All authors have read and agreed to the published version of the manuscript.