## 5 Analysis of Results
### Dynamic Performance Analysis of Oblique Navigation of Submarine  
By conducting numerical simulations of sideslip motion under different drift angles and rotational speed conditions, the curves depicting the lateral velocity \(v\), lateral force \(Y\), and moment \(N\) of the submarine with respect to the drift angle \(\beta\) are obtained, as shown in Figure 10. In the context of the \(\beta\)-\(Y\) curve, the negative sign indicates that the lateral force acting on the submarine is opposite to the positive direction of the coordinate system. Since all the dynamics during sideslip motion stem from the spiral mixed-flow pump, higher pump speeds lead to enhanced performance in the submarine's dynamics. Operating the submarine at a certain angle (with constant drift angle) with high-performance propulsion significantly boosts the operational speed of the submarine. During sideslip motion, the projections of the resultant velocity and resultant force in the vertical direction (\(y\)-direction) correlated positively with the sine values corresponding to the drift angle. Larger drift angles result in higher sine values; thus, with a constant rotation speed of the spiral mixed-flow pump, both lateral force and lateral velocity increase. The \(\beta\)-\(N\) curve illustrates the relationship between drift angle and the lateral moment. At higher speeds, the lateral moment exhibits a faster growth rate (i.e., the rate of change in lateral moment with respect to drift angle) and higher values. For instance, at a drift angle  
Figure 9: Effect of drift angle on pressure and velocity around ship. of 9\({}^{\circ}\), the lateral moments for speeds of 1500, 2000, 2500, and 3000 rpm are 0.31, 0.6, 0.89, and 1.28 N\(\cdot\)m, respectively. During the transition from 7\({}^{\circ}\) to 9\({}^{\circ}\) of drift angle, the growth rates of lateral moment for speed levels of 1500, 2000, 2500, and 3000 rpm are 4%, 10%, 13%, and 17%, respectively. Based on the numerical results, the relationship between \(Y\), \(N\), and \(v\) was studied, and the relevant data can be found in Table 3. It has been proven that there are strong linear relationships between \(Y\)-\(v\) and \(N\)-\(v\), indicating that the lateral force as well as the lateral force moment curves are proportional functions of lateral velocity. By incorporating the hydrodynamic coefficients from Table 3 into Equations (7) and (8) for regression analysis, non-dimensionalization is performed using Equations (11) to (14) to obtain the non-dimensional velocity coefficients for the submarine, as shown in Table 4. Employing a polynomial fitting method, polynomial fitting is conducted on the velocity coefficients under different rotational speed conditions in Table 4 to establish a mathematical model depicting the relationship between the velocity coefficient of the pump-jet vector thrust submarine and the rotation speed \(n\) of the spiral mixed-flow pump, as expressed in Equation (15). \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{n}\) & \(\mathbf{Y_{\mathrm{p}}}\) & \(\mathbf{N_{\mathrm{p}}}\) & \(\mathbf{Y_{\mathrm{p/p}}}\) & \(\mathbf{N_{\mathrm{p/p}}}\) \\ \hline
1500 rpm & \(-2.1\times 10^{-3}\) & \(4.93\times 10^{-4}\) & \(7.02\times 10^{-4}\) & \(-2.2\times 10^{-4}\) \\
2000 rpm & \(-9.6\times 10^{-4}\) & \(2.65\times 10^{-4}\) & \(-1.01\times 10^{-3}\) & \(1.6\times 10^{-4}\) \\
2500 rpm & \(-1.53\times 10^{-3}\) & \(3.42\times 10^{-4}\) & \(-6.31\times 10^{-4}\) & \(6.75\times 10^{-5}\) \\
3000 rpm & \(-1.22\times 10^{-3}\) & \(3.27\times 10^{-4}\) & \(-3.31\times 10^{-4}\) & \(1.12\times 10^{-5}\) \\ \hline \end{tabular}
\end{table} 
Table 4: Speed coefficients of pump-jet vector propulsion submersibles at different speeds. Figure 10: Kinetic parameters of submersible with drift angle \(\beta\) ((**A**): \(v\)-\(\beta\); (**B**): \(Y\)-\(\beta\); (**C**): \(N\)-\(\beta\)).