## 7 Conclusions  
This paper focuses on pump-jet-vectored propulsion submersibles, aiming to investigate their hydrodynamic characteristics during oblique and yaw maneuvers, particularly the effects of drift angle, mixed-flow pump rotational speed, and vector nozzle angle on the submersible's lateral force, lateral moment, and flow field distribution. Through numerical simulations and experimental validation, the following main conclusions are drawn:  
1. Influence of drift angle on the flow field: The drift angle has a minimal impact on the pressure distribution around the submersible. However, changes in the curvature of the mixed-flow pump housing can cause alterations in the surrounding fluid pressure field, notably creating a relative adverse pressure region in the middle of the pump housing. The resistance of the hull to fluid flow leads to the formation of a local stagnation point high-pressure region at the bow. This phenomenon is particularly evident during yaw maneuvers, where a negative pressure region appears on the starboard side of the bow, and a local high-pressure region appears on the port side. As the rotational speed and vector nozzle angle increase, the high-pressure range and intensity of the stagnation point at the bow are significantly enhanced. 2. Influence of rotational speed and vector nozzle angle on the flow field: During oblique and yaw maneuvers, an increase in rotational speed significantly improves the performance of the pump-jet-vectored thruster, with the intensity and reach of the jet wake also increasing. In yaw maneuvers, an increase in the vector nozzle angle results in a decrease in the submersible's forward speed and an increase in the lateral moment, thereby reducing the radius of the yaw maneuver trajectory as the vector nozzle angle increases. 3. Mathematical model of hydrodynamic coefficients: Using the least squares method for fitting, this paper establishes a mathematical model of the submersible's speed coefficient, angular velocity coefficient, and mixed-flow pump rotational speed. This model can effectively predict the variations in lateral force, lateral moment, and angular velocity of the submersible under different rotational speeds, providing a theoretical basis for optimizing the submersible's maneuverability and dynamic performance. Through numerical simulations and experimental validation, this study systematically analyzes the hydrodynamic characteristics of pump-jet-vectored propulsion submersibles during oblique and yaw maneuvers. It reveals the influence patterns of drift angle, rotational speed, and vector nozzle angle on the submersible's flow field and mechanical performance. The research results provide important theoretical guidance for submersible design and optimization, especially in enhancing maneuverability and propulsion efficiency. Future research can further explore the dynamic behavior of pump-jet-vectored propulsion submersibles in complex marine environments, particularly the flow field characteristics and maneuverability under multi-degree-of-freedom movements. Additionally, the structural design of submersibles can be optimized by combining experiments and numerical simulations to improve stability and propulsion efficiency under complex conditions. The application of novel propulsion technologies, such as magnetohydrodynamic drive, also deserves further investigation to explore more possibilities for submersible propulsion systems. T.X. and W.H. : funding acquisition, writing--original manuscript; R.L. and S.Z. : proofreading, supervision, simulation; R.C. : experiment; W.L. : data collation; S.Z. and R.C. : analysis. All authors have read and agreed to the published version of the manuscript. This study was supported by the National Natural Science Foundation of China [52179086, 52269022], central government guides local science and technology development fund projects [23 ZYQA0320], and excellent postgraduate innovation star project in Gansu Province [2025 CXZX-519, 2025 CXZX-591, 2025 CXZX-596].