## 5 Conclusions  
In this study, numerical studies focused on the stern rudder plane configurations of a submarine submerged in deep water were performed using RANS-based simulation. Deflection angles ranging from \(-21\) degrees to \(+21\) degrees were considered, allowing for comparison of the hydrodynamic forces and moments of the rudders and stabilizers at different deflection angles.  
The variation in the hydrodynamic forces and moments about the \(x\)-, \(y\)-, and \(z\)-axes under the Cross-rudder plane, X-rudder plane, and Y-rudder plane configurations with the same span were predicted. The hydrodynamic coefficients of the rudders and stabilizers were then obtained through Taylor expansion of the hydrodynamic forces and moments. It was proved that the hydrodynamic forces of the X-rudder plane configuration are the largest, in comparison with the forces acting on the left cases (excluding the roll moment). The numerical analysis indicated that the first-order derivative of the Y-force for the X-rudder plane configuration exhibited a significantly higher magnitude, being approximately 1.36 and 1.45 times greater than those observed under the Cross-rudder and Y-rudder plane configurations, respectively. Furthermore, the first-order derivative of the Z-force associated with the X-rudder plane configuration also surpassed those of the Cross-rudder and Y-rudder plane configurations, being approximately 1.18 and 1.41 greater, respectively. The hydrodynamic force along the \(z\)-axis and the moment about the \(y\)-axis of the stabilizers in the Cross-rudder plane configuration were considerably larger than the corresponding force and moment under the Y-rudder plane configuration for positive deflection angles.  
The hydrodynamic forces and moments of the stern rudder plane configurations were assessed according to RANS-based simulation in steady state. In addition, fluid flow through the rudder and stabilizer at high deflection angles was not investigated. Therefore, CFD-based estimation of the hydrodynamic coefficients of the stern rudder plane configuration under unsteady fluid flow conditions should be assessed in further research.  
Figure 11: Static pressure contour on the Y-rudder plane configurations at an angle of deflection of 15 degrees: (**a**) deflection of rudders; (**b**) deflection of stabilizers.  
**Author Contributions:** Conceptualization and methodology, T.T.N.; resources, T.L.M.; software, T.L.P., T.T.N. and T.L.M.; data analysis and validation, T.L.P. and T.T.N.; writing-original draft preparation, T.L.P. and T.T.N.; writing-review and editing, T.L.P., T.T.N. and T.L.M.; Funding acquisition, T.T.N. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research is funded by Funds for Science and Technology Development of the University of Danang under project number B2020-DN02-85.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.  
**Conflicts of Interest:** The authors declare no conflicts of interest.