## 4 Conclusions  
This paper presents a direct numerical simulation method based on RANS equation for submarine emergency rising emulation. Numerical simulations for 3 various conditions, i.e., 2 nonpropelled model simulations based on whether rudder steering or not, and self-propulsion simulation with rudder steering, are performed using Star-CCM+ solver. Here are some conclusions that can be drawn from the above discussion:  
First, the predicted attitudes of the nonpropelled submarine without rudder steering agree well with the experimental results, which proves that the present CFD method coupled with overset grid technology is suitable for the direct numerical simulation of emergency rising. In addition, both numerical simulation and model tests indicate the strong coupling effect between roll and yaw motions, which serves as the theoretical foundation for the following studies. Second, during the rudder steering simulations, the rudder rotational rate is controlled by a S surface controller based on conditional determination mechanism to achieve excellent course keeping capability. Consequently, the heading angle is limited within the range of 0.2\({}^{\circ}\), while the maximum roll angle is reduced by 96%, which validates that the roll stability is significantly enhanced by improving the course keeping ability. More importantly, it is also proved that the excessive yaw deviation is the root cause of emergency rising roll instability at moderate incidence angles. Concisely, the model achieves optimal emergency rising stability under the condition of Simulation B. Third, in the self-propulsion simulations, the time histories of propeller revolutions and initial velocity are converged to the target value in about 5 s, and then release other 5 DOF motions to implement emergency rising. Due to the influence of propeller torque, the course keeping capability of the self-propelled model is slightly deteriorated. However, by manipulating the rudder, it is still possible to limit underwater yaw angle within 0.8\({}^{\circ}\), and further restrain the growth of drift angle, thereby significantly reducing the roll angle by 70%. Moreover, the coupling effect between roll and yaw is demonstrated through the submarine self-propulsion simulation again. Future work will focus on self-propulsion emergency rising simulation with stermplanes manipulating as well as the full-scale model simulation. There are some difficulties in acquiring the metacentric height curves of the model surfacing process. Furthermore, more work will be done to analyze the influence of scale effect on the roll stability of submarine emergency rising. The \"Electronic supplementary material\" mentioned in the main text is available online at [[http://www.mdpi.com/2076-3417/9/16/3285/s1](http://www.mdpi.com/2076-3417/9/16/3285/s1)]([http://www.mdpi.com/2076-3417/9/16/3285/s1](http://www.mdpi.com/2076-3417/9/16/3285/s1)), Video S1: Simulation A Rising process without rudder steering.avi. Video S2: Simulation B Rising process with rudder steering.avi. Video S3: Simulation C Rising process with rudder steering and propeller rotating.avi. Video S4: Simulation C Rising process without rudder steering but propeller rotating.avi. [PERSON] and [PERSON] conceived, designed and performed the numerical simulation; [PERSON] wrote and revised the paper; [PERSON] designed the model test and analyzed the data; [PERSON] provided the funding; [PERSON] performed the academic guidance. The research work is supported by the National High technology Research and Development Program (863 Program) of China (2011 AA09A106), the National Natural Science Foundation of China (51009040, 51879061) and Special Financial Grant from China Postdoctoral Science Foundation (2014T70312). The authors are grateful for the help of Director [PERSON], who provided technical support for the implementation of the model tests.