## 1 Introduction  
The excellent performance of a pump-jet propulsor equipped with a tapered vectoring nozzle in the power system of a submarine has attracted attention [1]. The method of propelling by inducing internal low-pressure suction of a liquid medium through the high-speed rotation of the propulsion pump impeller, and utilizing the impeller's solid wall to interact with the liquid and expel it at high speed from the nozzle, shows potential value in delaying cavitation, reducing noise, and enhancing propulsion efficiency [2]. Understanding and mastering the dynamic behavior of submarines under the action of a pump-jet propulsor can further lead to proposing comprehensive and systematic structural optimization suggestions, thereby enabling more extensive exploration of marine resources and enhancing national defense capabilities [3, 4]. Submarines engaging in restricted water area navigation, hull parallelism, obstacle avoidance, and shore docking statuses generally involve yaw motion. Previous research has mainly focused on aspects such as jet propulsion device layout, submarine flow around, and internal flow field. Existing research results struggle to accurately predict the complex dynamic characteristics of submarines under multi-degree-of-freedom navigation, horizontal plane yaw motion, and other challenging operational conditions [5; 6]. Some studies indicate that physical quantities like longitudinal tilt coefficient, lateral force coefficient, and torque coefficient during submarine yaw have a significant impact on submarine maneuverability [7]. In some cases, essential conclusions may require reference to related studies on ship navigation and aircraft yaw. Therefore, current research faces difficulties in supporting submarine development due to a lack of critical findings to aid in its optimization. The innovative propulsion devices for submarines proposed based on interdisciplinary fusion ideas provide stronger power in yaw or heading movements. A novel propulsion device with additional nozzles at the duct tail developed by [PERSON] [8] and colleagues significantly enhances lateral dynamics of watercraft. The hydraulic-driven jet propulsion device designed by [PERSON] [9] and team facilitates forward-movement, reverse, oblique, pitching actions in watercraft. In recent years, the introduction of magnetohydrodynamic drivers has opened up a new path in submarine power system design [10; 11; 12], moving away from solid blade driving behavior to exhibit superior operational performance in suppressing cavitation and reducing noise; however, this research is still in its initial stage, with unresolved issues regarding magnetic fluid stability [13; 14; 15; 16; 17]. While these structurally optimized research outcomes offer potential value for improving submarine power behavior, they do not comprehensively cover the dynamic characteristics of submarines during yaw and heading motion. Based on current research progress, structural improvement technologies are taking the lead over theory, indicating a shift from theoretical guidance towards scientific research routes. With the rapid development of computer technology and image display techniques, computational fluid dynamics (CFD) has become a necessary tool for studying fluid behavior [18; 19; 20; 21; 22; 23], offering new methods for researching submarine power behaviors under complex conditions. [PERSON] [24] and fellow researchers numerically evaluated the yaw motion of submarines based on dynamic grid technology, indicating the dual-loop integral controller excels in controlling the movement process of submarines. [PERSON] [25], using block-grid technology, studied the influence of gap sizes at the impeller top on the performance of pump-jet vectoring nozzle injectors, suggesting that smaller gap sizes benefit enhancing injector efficiency.