## 5 Analysis of Results
### Mechanical Performance Analysis of Turning Motion of Submarine  
During the pivot-turning motion of the submarine, where the vector jet angle of the spiral mixed-flow pump remains fixed, thrust in the axial direction provides forward propulsion while lateral force offers turning force, resulting in a composite motion known as pivot-turning. Due to this transition in motion form, the symmetry of the external flow field around the submarine is disrupted by the pivot-turning motion, further impacting the mechanical performance of the submarine itself. By employing dynamic mesh techniques, the pivot-turning motion of the submarine is achieved, establishing the relationship between the vector jet angle and the flow field during pivot-turning, as well as the hydrodynamic characteristic parameters of the submarine. Solving for the angular velocity coefficients is beneficial for gaining further insight into the mechanical principles governing the submarine. To delve deeper into understanding the impact of the vector jet angle on the trajectory of the pivot-turning motion, three-dimensional models of submarines with vector jet angles of 6\({}^{\circ}\), 9\({}^{\circ}\), 12\({}^{\circ}\), 15\({}^{\circ}\), and 18\({}^{\circ}\) are constructed. Numerical simulations of the pivot-turning motion under different vector jet angles at rotational speeds of 1500 rpm, 2000 rpm, 2500 rpm, and 3000 rpm are conducted for these various models. Figure 11 displays the three-dimensional models of the pivot-turning motion under different vector jet angles. Figure 12 shows the pressure and velocity contour maps of the submarine's pivot-turning motion at 0.4 s after operation under a vector nozzle angle of 6\({}^{\circ}\), with spiral mixed-flow pump speeds of 1500 rpm, 2000 rpm, 2500 rpm, and 3000 rpm. With a fixed vector jet angle, the pump-jet vector thrust increases with rotational speed, expanding the region of positive pressure stagnation points on the bow section of the submarine. The bow part of the pump-jet vector thrust submarine, the flow-facing surfaces of the cruciform tail fins, and the inflow surfaces of the spiral mixed-flow pump are areas of high pressure. Increased rotational speed enhances the hull rotation rate, amplifying the fluid resistance  
effect while intensifying interactions between the hull surface and surrounding liquid. Pressure increases significantly at the starboard side high-pressure area in the bow part as the rotational speed rises, enlarging the high-pressure region. Due to a lateral force acting from right to left on the submarine's hull, there is a noticeable increase in pressure on the starboard side of the stern, surpassing the port side, leading to a significant pressure imbalance across the hull sides as the rotational speed escalates. The velocity contour map of the submarine indicates that higher rotational speeds of the spiral mixed-flow pump provide greater thrust and movement speed for the pump-jet  
Figure 11: Spiral mixed-flow pump model with different jet angles. Figure 12: Influence of mixed-flow pump speed on flow field pressure and velocity in heading turning motion. vector thrust submarine, inducing phenomena such as increased jet velocity and trail length. This powerful jet flow may pose risks to nearby objects located within its influence range. Figure 13 depicts the pressure and velocity contour maps of the submarine at different vector jet angles and spiral mixed-flow pump design speeds after 4 s of operation. As the vector jet angle increases, the forward speed decreases while lateral moment and lateral force increase. The pivot-turning trajectory radius decreases as the vector jet angle increases, resulting in varied trajectories over the same time period. Under the influence of lateral moment and lateral force, the hull experiences enhanced forces from right to left, creating a stark pressure difference between the port (flow-facing) and starboard (flow-receiving) sides of the bow part of the submarine.