## 4 Results and Discussion of the Initial Cooling Channel
### Transient Numerical Simulation Result Under Phase-Change Cooling  
To further analyze phase change processes and evaluate boiling heat transfer mechanisms, transient calculations were conducted alongside steady-state computations. The methods largely mirrored those used in steady-state calculations, utilizing the VOF two-phase model with liquid water as the primary phase and water vapor as the secondary phase. Simulations were performed with inlet velocities of 0.1 and 0.5 m/s, with a time step of 0.0005 s.  
Taking the initial structure as an example, the location, main method, morphology, and heat transfer mode of bubble generation are explored to provide a reference for subsequent analysis. As shown in Figure 11, at a 0.1 m/s inlet velocity, small bubbles first appear on the upper surface of the liquid collection ring at 440 s, with gradual growth observed in the 440-446 s period. At this stage, most of the flow remains below the saturation temperature, and boiling is just beginning, placing the flow in the surface boiling region. By 450 s, the bubbles have grown, forming a dispersed bubble film on the upper surface of the liquid collection ring, with the onset of bubble, slug, and annular flows marking the transition to the saturated nucleate boiling phase. At 490 s, bubbles coalesce, forming a continuous film on the upper surface of the liquid collection ring, indicating a fully developed annular flow and a transition to film boiling. Figure 11: Phase change results with an inlet velocity of 0.1 m/s. Figure 10: Schematic diagram of the location of the fluid average pressure cross section. Examining the flow state at \(t=475\) s, the fluid within the cooling channel undergoes phase change, as shown in Figure 12, with the gas phase reaching or exceeding the phase change temperature of water (\(\sim\)373 K), leading to intense boiling at the channel's top and slug flow formation. These transient results correspond with steady-state findings but reveal a more intense boiling mode, especially at the 0.1 m/s inlet velocity, where conditions are closer to the risk of heat transfer deterioration. The results thus confirm the rationale for selecting a safer 0.2 m/s inlet velocity in steady state calculation. At a 0.5 m/s inlet velocity, as shown in Figure 13, small vapor bubbles form around 825 s, with most of the flow remaining below saturation, marking the onset of surface boiling. Subsequently, over a prolonged period, the bubbles no longer grow and maintain a weak surface boiling state, which aligns with the steady-state calculation results. This phenomenon is attributed to the relatively high inlet velocity, where heat transfer is predominantly governed by forced convection of the single-phase liquid. As a result, the cooling water only progresses to a surface boiling state and cannot advance to more intense boiling regimes. For boiling heat transfer calculations, it is important to note that the saturation temperature for evaporation and condensation varies with the thermal environment. In this study, boundary conditions for wall heat flux also change over time, limiting steady-state calculations. Moreover, steady-state methods cannot capture dynamic changes in temperature and pressure over time, making it challenging to fully assess the cooling process. Therefore, transient simulations are essential for validating steady-state results and gaining a comprehensive understanding of real-time conditions for more accurate parameter analysis. Figure 12: At a water supply inlet velocity of 0.1 m/s and water supply time of 475 s, (**a**) gas volume fraction distribution on the upper surface of the rudder shaft liquid collection tank and (**b**) temperature distribution on the upper surface of the rudder shaft liquid collection tank. An inlet velocity of 0.5 m/s was applied, with a standard atmospheric pressure boundary condition at the outlet. As of the latest computation, transient calculations reached a maximum time of 1250 s, surpassing the period of most critical cooling conditions.