# Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD
## 4 Results and Discussion
### Effect of Head Wave and Following Wave
The blown loads are in front of the submarine \(CG\), and the downward force is behind the \(CG\) (the same as when the submarine sails backward); thus, the submarine continuously maintains the nose-up position after 11.62 s.  
As shown in Figure 11(a), the roll angle of the submarine rises in the head wave and the following wave prior to 5 s is approximately the same as that of the submarine rising in static water. Prior to 10 s, the roll-developing rule of the submarine is the same for three situations, for which the maximum roll angle of the submarine in each situation is presented is below 10\({}^{\circ}\). As shown in Figure 11(b), prior to 10 s, the developing rule of the yaw angle keeps unchanged and the maximum yaw angle in all three situations are below 2.3\({}^{\circ}\). The CFD results indicate that the head and following waves have a slight effect on the transverse motion of the submarine before it falls in the water from the highest position. After falling to water, the yaw angle of the submarine rises in the head wave begins to oscillate. This is mainly caused by the pitching oscillation. The second-order wave force always points downstream, and the wave decays with the increase in depth, so the yawing moment caused by the second-order wave force when the nose is up is in opposite direction to that when the nose is down. Figure 11: Falling procedure of the submarine in the following wave: the wave spreads from left to right. Red water surface indicates wave peak, whereas blue water surface indicates wave valley. When the submarine rises in static water, the rolling oscillation gradually decays after 10 s, and the roll angle is always below 2\({}^{\circ}\) after 15 s; however, the rolling oscillation quickly develops after 10 s in situations where the submarine rises in the head or following waves. When the submarine rises in the head wave, the yaw angle increases rapidly after 10 s and reaches the maximum value (10.79\({}^{\circ}\)) at 14.22 s. The roll angle increases rapidly after the yaw angle starts to increase rapidly as well; the maximum roll angle (47.53\({}^{\circ}\)) occurs at 14.74 s. When the submarine rises in the head wave, the rapid increase in the roll angle is due to the following two reasons: first, the submarine experiences an oblique wave as a result of the yaw angle; the components of the first-order wave force in the transverse direction will result in a rolling moment. Second, the side force acts on the sail and main body as a result of the drift angle, and the side force on the sail leads to a large rolling moment because the vertical position of the sail is significantly higher than that of the submarine \(CG\). When the submarine rises in the following wave, the moment when the roll angle rapidly increases occurs sooner than that of the rapid increase in the yaw angle. The minimum roll angle \(-23.10^{\circ}\) occurs at 17.22 s, and the maximum yaw angle 8.17\({}^{\circ}\) occurs at 18.12 s. When a submarine rolls to the left, the axial force acts on the sail and main body; the axial force on the sail will lead to yawing to the right. The CFD results indicate that there is a coupling relationship between the roll and yaw motions for emergent buoyantly rising submarines. This is in agreement with an earlier study by [PERSON] et al. [5] and [PERSON] et al. [8].