# Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD
## 4 Results and Discussion
### Effect of Beam Sea  
As shown in Figure 6, the difference in pitch when the submarine rises in static water and beam sea is always below 0.1\({}^{\circ}\) prior to 5.3 s. When the submarine rises in both static water and beam sea, the submarine sail reaches the water surface at 6.30 s. The maximum pitch angle when the submarine rises in beam sea (19.16\({}^{\circ}\)) is only 0.47\({}^{\circ}\) larger than that when the submarine rises in static water (18.69\({}^{\circ}\)). When oscillating on the water surface, the deck of the submarine that rises in the beam sea is always above the water surface, similar to when the submarine rises in static water, as shown in Figure 7. The beam sea has a slight effect on pitch prior to the submarine sail reaching the water surface, which is consistent with the study of [PERSON] et al. [28]. When the submarine rises in static water, the pitching oscillation decays rapidly and the pitch angle is always below 1\({}^{\circ}\) after 12.3 s. When the submarine rises in the beam sea, the pitching oscillation on the water surface does not decay rapidly, and the pitch angle oscillates between \(-5^{\circ}\) and 5\({}^{\circ}\). The difference between the pitching oscillating period and the wave period is only 3%; thus, the pitching oscillation is assumed to be caused by the first-order wave force. There is a periodic pitching moment acting on the submarine, mainly due to the yaw changing after the submarine reaches the water surface (the yaw angle is already larger than 7\({}^{\circ}\) at 10 s); therefore, the submarine encounters oblique waves, and the first-order wave force will result in a pitching moment. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Submarine Station** & **Wave Station** & **Wave Angle** \\ \hline
1 & & static water \\
2 & baseline depth: 4 m & 0\({}^{\circ}\) \\
3 & metacentric height: 13.65 mm & 30\({}^{\circ}\) \\
4 & \(m_{blowm}\): 44.89 Kg & wave height: 0.4 m & 60\({}^{\circ}\) \\
5 & \(x_{blowm}\): 400 mm & wave length: 15 m & 90\({}^{\circ}\) \\
6 & \(x_{blowm}\): 400 mm & 120\({}^{\circ}\) \\
7 & & & 150\({}^{\circ}\) \\
8 & & & 180\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Initial conditions for all the simulations. Figure 6: Effect of beam sea on the pitch of an emergent buoyantly rising submarine. As shown in Figure 7(a),b, when the submarine rises in static water, the roll angle is always below \(0.1^{\circ}\) prior to \(3.7\) s, and slowly increases after \(3.7\) s; the maximum roll angle prior to the submarine emerging through the water surface is only \(2.75^{\circ}\). When the submarine rises in the beam sea, the roll angle is \(-0.1^{\circ}\) prior to \(0.27\) s, and the rolling oscillation occurs prior to the submarine rising to the water surface. During the transformation of the water surface above the submarine from peak to valley, the rolling angle velocity increases as a result of the first-order wave force, whereas during the transformation of the water surface above the submarine from valley to peak, the rolling angle velocity decreases owing to the first-order wave force. Figure 7: Effect of beam sea on the submarine posture on water surface: a wave spreads from right to left. Red water surface indicates wave peak, whereas blue water surface indicates wave valley. As shown in Figure 8c,d, the beam sea has a slight effect on the axial velocity prior to 5 s. The side velocity oscillates within a range of \(\pm\)0.1 m/s, which is mainly caused by the first-order wave force. After 5 s, the side velocity continues to oscillate periodically and has a velocity of approximately 0.1 m/s; however, the minimum side velocity is approximately \(-\)0.5 m/s. The velocity is significantly larger when the submarine translates to the left side than when the submarine translates to the right, which is mainly due to the effect of the second-order wave force.