## 1 Introduction  
In an emergency, submarines should take emergency-surfacing measures. During the plating process, but before breaking the water surface, excessive rolling, swaying, and yawing movements may occur. [PERSON] [1] documented the movements of floating circular cylinders rising vertically (no forward velocity), with and without simulated deck and appendage geometries. However, these experiments were performed at low Reynolds numbers. The findings revealed that the bare cylinders show unpredictable swaying and yawing motions due to unsteady vortex shedding in the wake. When vertical vanes were added on top of the cylinder, roll amplitudes ranging from 30\({}^{\circ}\) to 150\({}^{\circ}\) were observed. The free-running model experiments conducted by [PERSON] [2] showed a 60\({}^{\circ}\) roll angle during the ascent process. [PERSON] [3] conducted experiments on a full-scale diesel-electric submarine, which showed a lateral roll angle of 25\({}^{\circ}\) during emergency surfacing. However, it should be noted that this study did not consider the occurrence of very high flow, where the unsteady vortex shedding makes the maneuver inherently unpredictable. [PERSON] et al. [4] exploited a mathematical model based on quasi-steady state coefficients to examine the submarine surfacing problem. This study showed that the roll angle is very sensitive to the initial lateral inclination angle. When the initial bank angle was set as 2\({}^{\circ}\), the roll angle during ascent reached 9\({}^{\circ}\), whereas, in the absence of the initial roll angle, the roll angle during ascent was only 1.5\({}^{\circ}\). [PERSON] et al. [5] examined the effect of pitch angle, yaw angle, static stability height, and water flow holes on the translation roll angle of submarines through free surface model experiments. The obtained results indicated a positive correlation between the yawing moment and the roll angle and thereby concluded that limiting the yaw angle can reduce the roll angle. However, the intrinsic relationship between the yawing moment and the roll angle was not unanswered. During the emergency-surfacing process, submarines accelerate their vertical ascent from low speeds at high angles of attack, inducing unsteady flow and generating asymmetric vortex flows around the hull. The resulting lateral force (\(F_{y}\)), yawing moment (\(M_{z}\)), and rolling moment (\(M_{x}\)) may be the primary causes of excessive rolling and swaying during submarine surfacing. However, current research on the high-angle-of-attack motion of submarines in the vertical plane generally considers only the effect of high angles of attack in the vertical plane. While [PERSON] [6] considered high-angle-of-attack motion in developing quasi-steady state coefficients for the six-degrees-of-freedom equations of motion, the study overlooked the effect of submarine vertical plane motion on the horizontal plane forces and the rolling moment, only fitting the forces at various attack angles in a mean-value manner. Similarly, [PERSON] et al. [7] conducted water tank experiments with a planar motion mechanism (PMM) in a circulating water channel (CWC) to revise the equations of motion associated with the high-angle-of-attack motion of a submarine vertical plane; however, the effects of the vertical plane motion on the horizontal plane forces and the rolling moment were not addressed. In aerodynamics, research on lateral forces on symmetric bodies is more common at high angles of attack and is mainly applied to slender bodies. [PERSON] et al. [8] carried out a numerical simulation of the asymmetric phenomenon of slender bodies at high angles of attack via the separated vortex SA-DES approach and surface roughness application, which were suitably compared with the experimental results. [PERSON] et al. [9] examined the unsteady characteristics of asymmetric vortices through experiments on narrow elliptical cones at angles of attack from 0 to 90\({}^{\circ}\) and velocities of 10 to 40 m/s.