## 5 Analysis of Results
### Hydrodynamic Characteristics of Pump-Jet Vector Thrust Submarine Turning Motion  
The spiral mixed-flow water jet propulsor serves as the power source for propelling and steering the pump-jet vector thrust submarine, generating pivot-turning motion under the influence of the vector jet angle. By employing dynamic mesh technology to achieve pivot-turning motion under different vector jet angle conditions, a mathematical model is established using least squares fitting to depict the variation in angular velocity coefficients of the pump-jet vector thrust submarine with respect to the rotational speed of the spiral mixed-flow pump.  
Figure 13: Influence of vector jet angle of mixed-flow pump on flow field pressure and velocity in heading turning motion.  
Initially, monitoring of the pivot-turning motion under various vector jet angles and rotational speeds is conducted to derive the relationships between the vector jet angle \(\delta\) and the submarine hull's angular velocity \(r\), lateral force \(Y\), and lateral moment \(N\), as illustrated in Figure 14. The vector jet angle plays a crucial role in the submarine's steering motion, with larger jet angles yielding significant lateral force and lateral moment, resulting in pronounced rotational movement of the submarine and an increase in angular velocity. When fixing the vector jet angle, a linear relationship exists between the hull's angular velocity and the thrust generated by the pump-jet action. Higher rotational speeds of the spiral mixed-flow pump provide stronger propulsion for the hull, thus escalating the angular velocity of rotation.  
Therefore, in numerical investigations of the submarine's pivot-turning motion, a positive correlation between lateral force, the lateral moment, and the vector jet angle can be observed, indicating that increasing the vector jet angle can enhance the submarine's maneuverability during pivoting motions.  
Based on the numerical simulations, the hydrodynamic parameters of the pivot-turning motion hull, such as \(r\) (angular velocity), \(Y_{r}\) (normalized lateral force), and \(N_{r}\) (normalized lateral moment), are obtained. Utilizing Equations (16) and (17) for non-dimensionalization processing, different curves of hydrodynamic coefficients versus angular velocity at various rotational speeds are plotted, as shown in Figure 15.  
\[Y_{r}=\frac{Y^{\prime}{}_{r}}{0.5\rho L^{3}V} \tag{16}\]  
\[N_{r}=\frac{N^{\prime}{}_{r}}{0.5\rho L^{4}V} \tag{17}\]  
The positive correlation between hydrodynamic coefficients and angular velocity in the pivot-turning motion has been verified through data. However, this positive correlation does not exhibit linear conditions. An irregular linear positive correlation is utilized to describe the relationship between hydrodynamic coefficients and angular velocity, necessitating the establishment of complex mathematical functions through data fitting for predicting additional data. Table 5 presents the relevant data fitted using the least squares method. Employing a polynomial fitting approach, the angular velocity coefficients under different rotational speed conditions from Table 5 are fitted to establish a mathematical model depicting the variation in angular velocity coefficients of the pump-jet vector thrust submarine with respect to the rotational speed of the spiral mixed-flow pump, as shown in Equation (18).  
\[\left\{\begin{array}{ll}Y_{r}=0.069-8.19\times 10^{-5}n+2.72\times 10^{-8}n^{2}-2.7 6\times 10^{-12}n^{3}\\ N_{r}=-0.1+4.76\times 10^{-6}n-1.74\times 10^{-9}n^{2}+1.73\times 10^{-13}n^{3} \end{array}\right. \tag{18}\]