# Research on a Strategy for Prediction Methods of Submarine Self-Propulsion Hydrodynamic Performance
## 3 Validation and Convergence Analysis of Numerical Methods
### Validation and Convergence Analysis of Numerical Methods for the E1619 Propeller  
Based on the geometric model of the E1619 propeller, the computational domain boundaries were set, and the mesh was divided. The propeller was placed in a rectangular flow field, with the inflow direction set at the velocity inlet, the outflow set as a pressure outlet, and the remaining boundaries set as wall surfaces. The velocity inlet was positioned at a distance of 3.0 times the propeller diameter from the propeller disk, the outlet was set at 6.0 times the propeller diameter, and the radial boundary was set at 3.0 times the propeller diameter. The entire computational domain was divided into an outward flow domain and a rotation domain. The boundary condition settings and mesh division for the computational domain are shown in Figures 7 and 8. Additionally, three sets of grids were used to carry out grid independence and convergence validation, with grid counts of 3.41 million, 6.10 million, and 11.05 million,  
Figure 8: E1619 propeller grid division. Figure 6: Submarine model resistance experiment and numerical simulation comparison. Figure 7: E1619 propeller model calculation domain setting. respectively. The boundary layer was set to 10 layers, with a \(Y+\) value of 30. A comparative analysis was conducted with an advance coefficient \(J=0.3\) (velocity \(V=3.37\) m/s) as the research subject. The analysis results are shown in Table 3. As shown in Table 3, the sensitivity of the propeller thrust coefficient and torque coefficient to grid density is low. The convergence analysis results are presented in Table 4. According to Table 4, the grid convergence rates \(R_{G}\) for the propeller thrust and torque calculations are both less than 1, indicating that the grid exhibits monotonic convergence. The grid uncertainties are 0.0029 and 0.0570, respectively, which are relatively small, suggesting that the numerical simulation uncertainty for the propeller model is within a reasonable range. Therefore, considering both computational efficiency and accuracy, the coarse grid was selected for further research. Additionally, using an advance coefficient \(J=0.2\)-\(1\) as an example, the numerical simulation results for the propeller's hydrodynamic performance were compared with the open-water experimental values [14; 24], as shown in Figure 9. In Figure 9, \(K_{T}\), \(K_{Q}\), and \(\eta\), respectively, represent the thrust coefficient, torque coefficient, and efficiency of the propeller. Figure 9 shows that the errors in the propeller thrust coefficient and torque coefficient are both below 4.2%. Overall, the numerical simulation results align well with the experimental values, with the calculation errors within an acceptable range. This verifies the reliability and applicability of the numerical calculation method used in this study for predicting propeller hydrodynamic performance. Validation of Numerical Methods for the SUBOFF Submarine with E1619 Propeller in Self-Propulsion State  
Building on the previous numerical simulation and method validation of the SUBOFF submarine and E1619 propeller, and considering the impact of the submarine hull wake effect on the propeller's hydrodynamic performance [34], the geometric model of the \"SUB-OFF Submarine + E1619 Propeller\" combination is shown in Figure 2. The computational domain settings used for the submarine resistance simulation were maintained, with the propeller's rotating domain and refinement zones integrated into the entire computational domain. The settings for other parts of the computational domain and refinement zones remain consistent with their original configurations. After establishing the computational domain, the mesh was divided for the SUBOFF submarine + E1619 propeller model and its flow field, with the total number of grids being 9.97 million.