## 3 Results and Discussions
### The Course Keeping Simulation Based on S Surface Controller
#### 3.3.2 The Course Keeping Simulation  
Based on the simulation A in Section 3.3.2, only a S surface controller is added into the simulation B through user defined function (UDF) to further simulate the emergency rising without propeller. The simulation results are compared with those of 3.2, as shown in Figure 9.  
Figure 9: Contrast curves of attitudes between CFD and MT in the condition of rudder steering but no propeller. The experimental data in Figure 7 are repeated here for distinct contrasting. Firstly, in order to verify the effectiveness of the S-plane control method, the manipulation curves of the rudder are described in Figure 9f. It can be seen from the enlarged local curves that the rudder manipulation is performed only when the deviation larger than 0.5\({}^{\circ}\), which is more in line with the actual steering mechanism, and further achieves the tracking control of the desired rudder angle. The yaw angle curve, as shown in Figure 9e, also indicates the effectiveness of the rudder steering. A local view of the rudder in the animation of Simulation B (see the \"Electronic supplementary material\", Video S2) reveals the rotation of the rudder, where both roll and yaw angles are significantly reduced by manipulating rudder. As shown in Figure 9a, the parametric comparison curves between two numerical simulation results (steering and no steering) and the experimental data (no steering) are presented. Firstly, compare the numerical results under two different conditions according to whether manipulating the rudder or not. As shown in Figure 9a-b, during the period of 5-15 s, the two predicted pitch curves as well as incidence curves substantially coincide, which shows that the numerical calculation has good computational stability. After t = 15 s, the pitch angle deviation between two numerical simulations increases gradually, which exactly reflects the effect of steering rudder on the trim. In other words, the manipulation of the rudder keeps the trim of the model before surfacing is relatively stable, which improves the longitudinal stability of the model to some extent. Obviously, this further leads to a nearly increase one time in the minimum pitch angle after the model surfacing, which is also understandable. In summary, the numerical simulation has high reliability for solving rudder steering process. From the yaw angle curve of Figure 9c, it can be seen that the underwater yaw angle before surfacing is limited to 0.2\({}^{\circ}\), which indicates that the model has excellent course keeping ability, and this further proves the effectiveness of the S surface controller. To be more precise, after t = 15 s, the yaw angle deviation of the model decreases gradually under the state of steering rudder, while the ones increases further when the rudder is not manipulated. Corresponding to this, according to curves of roll angle, the maximum roll angle in the rudder steering state is only 2\({}^{\circ}\). In contrast, when the rudder is not manipulated, the maximum roll angle of the model approaches to 50\({}^{\circ}\) after surfacing. Undoubtedly, there occurs an obvious excessive roll during the underwater process of the emergency rising. In conclusion, for two numerical simulations with identical computational parameters and meshing, merely changing the steering state of the rudder, the roll stability of the model is significantly enhanced with reducing the maximum roll angle by 96%, which further validates the effectiveness of numerical simulation for the rudder manipulation. Furthermore, by comparing the yaw angle and drift angle curves, it can be seen that under the steering rudder state, the underwater drift angle before surfacing is lower than 5\({}^{\circ}\), and the variation trend of drift angle also present obvious linear correspondence with the yaw angle curve. Thereby it is demonstrated that the increase of drift angle is caused by the variation of yaw angle.