## 3 Results and Discussions
### The Coupling Effect of Yaw and Roll
The postures statistics of the test scenarios with the maximum incidence angle of about 20\({}^{\circ}\) among the model tests are listed in Table 6. It is remarkable that under the premise of incidence angle approximately equal, the maximum roll angle \(\phi_{\text{max}}\) of the test conditions (1) and (2) is larger than 50\({}^{\circ}\), with occurring the excessive roll, while there is no excessive roll occurs for the test conditions (3)-(5) because \(\phi_{\text{max}}<30^{\circ}\). According to the theory of fluid hydrodynamics, it has been validated that the separated vortices emitted from leeward side of the main hull and sail are symmetrical when incidence angle ranges from 5\({}^{\circ}\) to 30\({}^{\circ}\). Thus, it may be known, the incidence angle is not the determinant factor of leading to excessive roll when the emergency rising is conducted at a certain speed with obtaining medium magnitude incidence angle. In contrast, before the model broaches the water surface, the maximum yaw deviation as well as maximum drift angle shows a significant positive \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Test Condition** & \(\mathbf{\Theta_{\text{max}}}\) & \(\mathbf{\theta_{\text{max}}}\) & \(\mathbf{\psi_{\text{max}}}\) & \(\mathbf{t_{\text{out}}}\) & \(\mathbf{\psi_{\text{out}}}\) & \(\mathbf{\beta_{\text{out}}}\) & \(\mathbf{\theta_{\text{out}}}\) & \(\mathbf{\phi_{\text{out}}}\) & \(\mathbf{V_{\text{out}}}\) & \(\mathbf{T_{\Phi\text{-}\Phi}}\) \\  & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** & **(\({}^{\circ}\))** \\ \hline (1)-8-362-0-1.15 & 19.24 & 12.98 & 53.49 & 16.92 & 14.11 & 15.53 & 5.52 & 40.23 & 0.66 & 5.12 \\ (2)-8-362-3-1.15 & 19.53 & 13.54 & 50.87 & 16.13 & 13.61 & 13.54 & 8.93 & 29.43 & 0.87 & 5.13 \\ (3)-8-362-9-0.95 & 18.98 & 20.10 & 15.42 & 15.21 & 7.43 & 6.36 & 19.89 & 9.19 & 1.02 & 5.68 \\ (4)-8-568-3-0.75 & 20.37 & 26.13 & 24.16 & 14.33 & 8.94 & 8.90 & 25.19 & 17.66 & 1.20 & 5.63 \\ (5)-8-568-3-0.75 & 19.68 & 26.46 & 9.57 & 14.07 & 3.32 & 2.51 & 26.09 & 4.49 & 1.22 & 5.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The working conditions of numerical simulation present in this work. Figure 6: **(a)** Typical curves of attitudes in condition of no rudder steering or propeller. The vertical dotted line marks the emergency time when completing the blowing process and beginning to ascend. **(b)** Comparison curves of maximum roll angle for working conditions listed in Table 6.  
correlation trend with the maximum roll angle as shown in Figure 6b. In other words, the strong coupling effect between roll and yaw motion in submarine emergency rising is verified from the perspective of model test. Based on the above analysis of test results, it is further deduced that under the condition of moderate incidence angle (\(\Theta_{\text{max}}<30\)), the larger yaw deviation is the root cause of the excessive roll for the emergency rising at an initial speed. In order to validate this inference in detail, a numerical simulation that is equivalent to model test conditions is carried out and discussed in Section 3.2. Once verifying the validity of numerical simulation, another two emergency rising simulations under the condition of course keeping would be further conducted in Sections 3.3 and 3.4 respectively, with making up for the deficiency of the model test.