## 4 Results and Discussion of the Initial Cooling Channel  
### Steady-State Numerical Simulation Results with Phase-Change Cooling  
Based on grid independence verification, steady-state parametric calculations were performed using the 11.88 million mesh and corresponding settings for the initial cooling scheme. Simulations were conducted at inlet velocities of 0.05 m/s, 0.1 m/s, 0.2 m/s, and 0.5 m/s, respectively. Observing the temperature distribution on the surface of the rudder shaft under different inlet velocities shown in Figure 8, it was found that at higher heat flux densities, the temperature near the connections with the upper and lower flanges was higher, with the maximum temperature located at the top of the rudder shaft. Overall, the temperature of the rudder shaft decreases with an increasing cooling water flow rate. At an inlet velocity of 0.05 m/s, the temperature at the top of the cooled rudder shaft remains around 500 \({}^{\circ}\)C. At inlet velocities of 0.1 m/s and 0.2 m/s, the temperature gradually decreases, and when the inlet velocity increases to 0.5 m/s, the highest temperature of the cooled rudder shaft is slightly above 400 \({}^{\circ}\)C (see Figure 8e). Based on the steady-state calculation results, the boiling conditions in the tube under different inlet flow rate conditions are further analyzed and plotted in Figure 9. At an inlet velocity of 0.5 m/s as shown in Figure 9a, regions with higher heat flux density on the inner wall produce a nucleation center that generates bubbles. However, the central water flow has not reached saturation temperature, causing the bubbles to condense after detaching from the wall and entering the central water flow. The flow is in the surface boiling region. As shown in Figure 9b, when the inlet flow velocity is reduced to 0.2 m/s, the temperature of the central water flow inside the pipe reaches saturation temperature. The bubbles generated by the nucleation center gradually grow, with observable bubble flow and slug flow. The flow is in the saturated nucleate boiling region. When the inlet flow rate is further reduced to 0.1 m/s (see Figure 9c), the central water flow inside the pipe reaches saturation temperature, and in some areas, the fluid temperature is significantly higher than the saturation temperature. In addition to observable bubble flow and slug flow, some annular flow is present. Therefore, the flow is considered to be in the saturated nucleate boiling region. Some areas of the outlet pipe exhibit annular flow with droplet entrainment, indicating that this fluid is approaching or within the two-phase forced convective heat transfer region, which is relatively dangerous. When the inlet flow rate \begin{table}
\begin{tabular}{c c c c c} \hline Mesh Number (Million) & 233 & 359 & 547 & 1188 \\ \hline Maximum Axial Temperature (\({}^{\circ}\)C) & 479.846 & 479.281 & 479.146 & 478.584 \\ Gas Volume Fraction (Top) & 0.0659 & 0.0687 & 0.0892 & 0.0774 \\ Gas Volume Fraction (Upper & 0.0427126 & 0.0401731 & 0.0468843 & 0.0426627 \\ Through-Hole) & & & & \\ Gas Volume Fraction (Lower & 0.0007834 & 0.0003438 & 0.0003438 & 0.0017856 \\ Temperature at the Top of the & & & & \\ Shaft (\({}^{\circ}\)C) & 479.4 & 478.5 & 478.5 & 477.7 \\ Temperature in the Upper & & & & \\ Through-Hole (\({}^{\circ}\)C) & 178.355 & 175.288 & 173.607 & 172.766 \\ Temperature in the Lower & & & & \\ Through-Hole (\({}^{\circ}\)C) & 102.807 & 101.32 & 100.996 & 100.558 \\ \hline \end{tabular}
\end{table} 
Table 5: Comparison of temperature and gas volume fraction at monitoring points. is set to the minimum value of 0.05 m/s as shown in Figure 9d, the regions with higher heat flux density correspond to higher gas phase content inside the pipe. The fluid largely enters the annular flow with the droplet entrainment state and is in the two-phase forced convection heat transfer region. Some sections even approach the single-phase vapor flow.