## 2 Numerical Model  
This research focuses on a pump-jet vector thrust submarine model, which includes an axial symmetric structure with additional components like a conning tower and rudders. The main hull has a streamlined shape, with a total length of 1.52 m. The inflow section at the bow is 0.25 m long and the parallel amidship body is 0.83 m long. The submersible is powered by a screw centrifugal pump-jet propulsion system, which draws surrounding liquid into the pump chamber through high-speed rotation and expels it axially to generate reactive force. And the total length of the spiral mixed-flow jet propulsion system is 0.2 m, consisting of a diffuser shell, impeller, and spatial guide vanes (see Table 1 for relevant parameters). The maximum diameter of the streamlined body is 0.14 m, located at the parallel amidship body, with the conning tower positioned 0.25 m from the streamlined body, exhibiting an elliptical shape in cross-section. The rudders are arranged in a cross configuration, positioned at 1.27 m from the streamlined body. The forward direction along the horizontal length of the pump-jet vector thrust submarine is designated as the \(x\)-axis, with the positive \begin{table}
\begin{tabular}{c c} \hline \hline
**Design Parameter** & **Value** \\ \hline Design flow \(Q\) (m\({}^{3}\)/h) & 12.86 \\ Design head \(H\) (m) & 1.303 \\ Design speed \(n\) (rpm) & 1500 \\ Design specific speed \(n_{s}\) & 277.9 \\ Design efficiency \(\eta\) (\%) & 80 \\ Impeller inlet diameter \(D_{1}\) (mm) & 50 \\ Impeller outlet diameter \(D_{2}\) (mm) & 90 \\ Number of impeller blades \(Z_{1}\) & 2 \\ Guide vane inlet diameter \(D_{3}\) (mm) & 108 \\ Guide vane inlet diameter \(D_{4}\) (mm) & 80.7 \\ Guide vane outlet diameter \(D_{5}\) (mm) & 62 \\ Guide vane inner streamline outlet diameter \(D_{6}\) (mm) & 32 \\ Guide blade number \(Z_{2}\) & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters and structural parameters of spiral mixed-flow pump. Figure 1: Technical route. \(y\)-axis pointing to the starboard side of the submarine. The power system of the submarine is a spiral mixed-flow pump. Refer to Figure 2 for an illustration of the structural layout of the pump-jet vector thrust submarine. The operational range of the pump-jet vector thrust submarine is assumed to be within a rectangular water domain, with the volume of the water area set to be sufficiently large to minimize the influence of the water domain boundaries on the submarine's dynamic behavior. Specifically, the parameters are as follows: the total horizontal length is 12.5 L, the distance from the water domain inlet to the bow of the submarine is 8 L, and the width and height of the water domain are both 11 times the maximum diameter of the streamlined body of the submarine, meeting the requirements of the assumption. The model is geometrically discretized in the spatial domain using polyhedral mesh partitioning techniques. Due to the complexity of the flow field around the submarine, it is necessary to focus on densifying the fluid mesh around the hull and its appendage area. In the computational region far from the hull, the grid can be appropriately coarsened to reduce the overall computational workload without affecting convergence and stability. This research validates grid independence by monitoring the efficiency of the spiral mixed  
Figure 2: Structure size and partial grid diagram of the pump-jet vector propulsor. flow power system as a physical quantity. The results indicate that the calculation stabilizes after about 7.5 million grid cells, leading to a final determination of 7.8032 million grid cells for the mesh model. The dynamic mesh technique is an important method in CFD used to simulate the motion of objects in a fluid. It involves adjusting the grid within the computational domain to adapt to the object's motion, thereby capturing the interaction between the object and the fluid.