## 3 Results and Discussions  
### The Coupling Effect of Yaw and Roll  
In view of the fact that this paper mainly uses CFD as a means to carry out emergency rising simulations that cannot be achieved by the model test at present, this section firstly describes the coupling effect between yaw and roll found in the model test by analyzing the experimental data in detail. By comparing the maximum roll angle of several different test conditions whose incidence angle are almost the same, it is explained that the incidence angle is not the key factor to determine whether the model excessive roll or not, at least not at moderate incidence angles. Based on the experimental data of test condition (1) given in Table 6, the typical curves of attitudes for non-propelled emergency rising without rudder steering are depicted in Figure 6a. The code in the first column of Table 6, taking 8-362-0-1.15 for example, indicates the test condition of \(d_{0}=8\) m, \(x_{blown}=362\) mm, \(\delta_{s}=0^{\circ}\) and \(u_{0}=1.15\) m/s. Apparently, the model occurs excessive roll and obvious yaw deviation simultaneously, while the pitch curve does not have a plateau stage. Meanwhile, there is an obvious positive correlation among roll, yaw and drift angles before surfacing. On one hand, as the model occurs roll, the v component of local vertical ascent velocity increases rapidly while the w component decreases correspondingly, such that the drift angle increases with the increase of the roll \begin{table}
\begin{tabular}{c c c} \hline \hline
**Items** & **Boundary Condition** & **Initial Condition** \\ \hline Submarine & No-slip wall & Specified for different simulations \\ Behind side & Pressure outlet & The hydrostatic pressure of flat VOF wave \\ Remaining sides & Velocity inlet & The velocity of flat VOF wave \\ Water & — & The volume fraction of heavy fluid of flat VOF wave \\ Air & — & The volume fraction of light fluid of flat VOF wave \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The boundary settings and initial conditions. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Simulation Condition** & **Rudder Steering** & **Propeller Rotating** & \(d_{0}\) **(m)** & \(u_{0}\) **(m/s)** & \(x_{blown}\) **(mm)** & \(\delta_{s}\) **(\({}^{\circ}\))** \\ \hline A & \(\times\) & \(\times\) & 8 & 1.15 & 362 & 0 \\ B & \(\surd\) & \(\times\) & 8 & 1.15 & 362 & 0 \\ C & \(\surd\) & \(\surd\) & 8 & 1.15 & 362 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The working conditions of numerical simulation present in this work. angle, and further results in an evident yaw deviation. On the other hand, the incipient increase of drift angle may be induced by the growth of yaw deviation, as shown at around t = 10 s, due to the lack of course keeping capability. The yaw deviation emerges at the earlier beginning and continues accumulating until the model rises to the surface. During the underwater rising process, the drift angle grows with the increases of yaw deviation, further causing excessive roll. Thus, it is hard to tell who induces whom exactly. In other words, there are complex coupling effects between yaw and roll motions, considering the drift angle as the medium. The comparison curves of maximum roll angle for working conditions listed in Table 6 are drawn in Figure 6b, where the x coordinate is labeled the test number 1-5 from left to right. It can be seen from this subgraph that there is an obvious linear correlation trend among the three curves of roll angle (magenta dashed line), yaw angle (red solid line) and drift angle (blue dot dash line) on the premise that the incidence angle (black dotted line) is approximately 20\({}^{\circ}\). Along with yaw deviation decreasing from 15\({}^{\circ}\) to smaller than 5\({}^{\circ}\), the maximum incidence angle gradually decreases from 60\({}^{\circ}\)to lower than 30\({}^{\circ}\). Meanwhile, the curves of yaw and drift angle approximately coincide, which demonstrates that it is yawing motion that causes the drift angle to change synchronously.