A direct numerical simulation method based on Reynolds Average Navier–Stokes (RANS) equations is used to carry out numerical prediction studies of submarine emergency rising in this paper. Firstly, a numerical simulation of the nonpropelled model without rudder manipulating is accomplished as the basis of this study. The numerical prediction results are in good agreement with the experimental data, which proves the feasibility and accuracy of the direct numerical simulation method. Meanwhile, both model tests and numerical simulation results reveal the strong coupling effect between roll and yaw motions during the underwater ascending process. Based on the above observation and analysis, another two numerical simulations, whose grids are identical with the non-manipulation simulation, are conducted respectively under the condition of rudder steering, i.e., course keeping simulation and self-propulsion simulation. An optimized S surface controller based on conditional determination is designed to manipulate the rudders. As a result, the yaw angle of the latter two simulations is limited within the range of 0.2° and 0.8° respectively, proving the effectiveness of the S surface controller. Correspondingly, the maximum roll angle is reduced by 96% and 70% respectively, which demonstrates that the roll stability is significantly enhanced by improving the course keeping ability of the model. Moreover, it is also proven from the perspective of reverse verification that, the excessive yaw deviation is the root cause of emergency rising roll instability for the situation of incidence angle lower than 30°.

Emergent buoyantly rising submarines encounter excess roll problems, partially owing to waves that significantly affect their behavior. This study predicts the behavior of a submarine, including when it rises in static water, beam sea, head wave, following wave, ${30}^{\circ }$ bow wave, ${60}^{\circ }$ bow wave, ${30}^{\circ }$ quartering wave, and ${60}^{\circ }$ quartering wave, using the computational fluid dynamics method. The beam sea has a slight effect on pitch prior to the submarine rising to the water surface, but the maximum roll angle in the beam sea is 4.43 times that in static water. After a submarine submerges in water, the pitching oscillation does not decay quickly owing to the yaw angle. The head wave and the following wave have a continuous significant effect on the pitch; the submarine sail remains under the water surface after it submerges from the highest position. The head wave and the following wave have a slight effect on the roll and yaw before the submarine rises to the water surface; however, the roll angle suddenly increases after the submarine submerges from the highest position. As the initial angle between the submarine centerline and wave direction increases, the effect of waves on the longitudinal motion decreases. The amplitude of the pitching oscillation decreases with an increase in the initial angle between the submarine centerline and wave direction, and the waterline when the submarine oscillates on the water surface decreases. The difference in the maximum roll angle between when a submarine rises in an oblique wave and when it rises in beam sea is below ${6.3}^{\circ }$. Submarines should try to avoid rising in a head wave and the following wave.

A mathematical model for predicting the maneuvering motion of a ship is constructed, using a planing craft with dual waterjet propulsion as the object of study. This model is based on the standard approach of the MMG (Maneuvering Modeling Group) and uses the Runge–Kutta algorithm to solve the differential equations. For the simulation of the turning and Z-shape maneuvering motion, the RANS equation is first solved using the program STAR-CCM + and then the PMM motion of the hull is simulated using the overlapping grid approach to derive the hydrodynamic derivative. The established method for predicting the ship’s maneuverability is feasible, as shown by the calculated results, which agree well with those obtained using data from the sea trials. This method was used to simulate the rudder rotation and Z-shape motion of the planing craft at medium and high speeds to predict the maneuverability index.