## 3 Results and Discussions
### The Validation Accuracy  
In order to verify the validity and accuracy of the numerical method, a nonpropelled emergency rising simulation without any automatic steering devices is carried out first to reproduce the test results, as accurately as possible. The first working condition (1) in Table 1 is selected as the representative case of the numerical simulation due to its excessive yaw and roll simultaneously. For clarity, the initial conditions for simulation A are repeated as the following items:  
* The blown ballast mass is 44.886 kg, and locates its mass centroid with 362 mm forward from midship and 15.34 mm upward from central axis of model. * The model is launched at speed of 1.15 m/s from the depth of 8 m, while the sternplane angle is fixed as 0\({}^{\circ}\). * The opening states of flooding holes remain unchanged for all simulations, as does the metacentric height of 0.02 m.  
Comparing the simulation results with the experimental data to obtain the contrast curves as shown in Figure 7a-f. Firstly, it should be pointed out that the data of model tests and numerical simulation are aligned to the fifth second. As can be noticed from Figure 7f, at the beginning of the test, the towing sled is gradually accelerated from 0 m/s to about 1.15 m/s. The time histories of ship speed are converged to the target value in about 5 s. Due to the influence of the speed control accuracy of the towing sled, the actual speed of the model fluctuates slightly around the target speed of 1.15 m/s during the uniform motion phase. As shown in the figure, after the model is released, the axial velocity curves of the numerical simulation coincide with the model test. It can be seen that the effect of the slight velocity deviation around releasing is negligible. From the pitch curve of Figure 7a, it can be seen that the pitch curve of the model test has similar trends with that of numerical simulation. On one hand, both of them are difficult to maintain constant pitch state after reaching the maximum pitch angle. This is because the pitching moment provided by the buoyancy load is relatively small when the longitudinal position of blown ballast is close to the center of gravity of the ship, i.e., 362 mm. With the increase of roll angle, as shown in Figure 7c, the longitudinal hydrodynamic moment that trims bow down applied on the sail increases gradually, with further coupling the restoring moment provided by metacentric height, so that the pitch angle decreases after reaching its peak value. On the other hand, the deviation of the predicted peak value of pitch is lower than experimental results. This may be ascribed to the numerical dissipation of the turbulence model in solving the flow around the hull, especially the numerical solution during the rapid bow-lifting phase. The same phenomenon also exists in the prediction of incidence angle as illustrated in Figure 7b. the roll angle curves of numerical simulations and model tests also show a tendency of coincidence. Moreover, the peak-to-peak periods of rolling after surfacing are nearly identical, which indicates that both of them have the same metacentric height. In other words, it verifies the equivalence of static properties between the simulation model and the test model. To sum up, when the submarine blows off the ballast tank near its midship, it is prone to occurring excessive roll and threatening the safety of submarine and its crew. Furthermore, according to the transverse and vertical comparison of the drift angle, roll angle and drift angle, the maximum underwater yaw angle and the drift angle corresponding to the twice roll instability states both are larger than \(5^{\circ}\). At the same time, the peak values of the three curves show an obvious monotonous relationship, with the variation trend of parameters coinciding with the model test results. That is to say, the coupling effect between yaw and roll motion is verified by numerical simulation again.