# Effect of Waves on the Behavior of Emergent Buoyantly Rising Submarines Using CFD
## 4 Results and Discussion
### Effect of Head Wave and Following Wave  
As shown in Figure 9, prior to 1.43 s, the pitching angle velocity \(q\) of the submarine rising in the head wave is always below that of the submarine rising in static water, and the water surface above the submarine sail transforms from peak to valley during this period. During the period between 1.43 and 3.18 s, the pitching angle velocity \(q\) of the submarine rising in the head wave is always larger than that of the submarine rising in static water, and the water surface above the submarine sail transforms from valley to peak, or \(q\) is smaller when the water surface above the submarine sail transforms from peak to valley. In addition, prior to the submarine rising to the water surface, the pitching motion law for the following wave is the same as that for the head wave. Figure 9: Effect of head wave and following wave on the pitching motion. When the submarine rises in static water, the pitching oscillation decays rapidly after submerging in water from the highest position, and the absolute value of the pitch after 12.3 s is always below 1\({}^{\circ}\). When the submarine rises in the head wave, the water surface around the submarine transforms from peak to valley when the submarine submerges from the highest position. Thus, the first-order wave force aggravates the development of the pitch motion and the maximum downward angle of the nose (8.31\({}^{\circ}\)) is 23.3% larger than that in static water. Violent pitching oscillation occurs after the submarine falls into the water; the maximum upward angle of the nose prior to 19 s is 11.98\({}^{\circ}\) whereas the maximum downward angle of the nose prior to 19 s is 4.64\({}^{\circ}\); the posture of the submarine maintains its nose in the upward direction for only 2.22 s during the interval 10-17 s. The downward hydrodynamics due to the attack angle is significantly larger than the upward hydrodynamics; as a result, the submarine cannot rise back up to the water surface after falling (as shown in Figure 10). When the submarine rises in the following wave, the submarine slowly drops to the water surface from the highest position. As the wave peak spreads from the rear to the submarine and the water surface rises near the nose of the submarine, the nose of the submarine easily glides with the water (as shown in Figure 11). The wave valley spreads from the rear to the water surface near the submarine after the nose of the submarine falls on the water surface. The large bow-down moment acts on the submarine as a result of the first-order wave force; thus, the maximum bow-down angle of the submarine in the following wave (12.88\({}^{\circ}\)) is 1.91 times that of the angle in static water (6.74\({}^{\circ}\)). The submarine is always in the nose-up position after 11.62 s; however, the pitching oscillates violently, and the maximum pitch angle is 8.91\({}^{\circ}\). The submarine sail cannot rise to the water surface as it does in the head wave. When the submarine rises in the following wave, the wave and submarine sailing directions concide; however, the wave velocity is larger than that of the submarine, and downward hydrodynamics acts on the nose-up submarine as a result of the first-order wave force. The downward  
Figure 10: Position and posture of the submarine when experiencing the peak and valley of the head wave: the wave spreads from right to left. Red water surface indicates wave peak, whereas blue water surface indicates wave valley. force is larger than the sum of blown loads and the wave suction when the submarine is near the water surface; thus, the sail is always below the water surface. The downward force exponentially decays along the depth, and the submarine stops diving when the forces are balanced in the vertical direction. The wave suction acting on the submarine nose is larger than that on the stern.