## 3 Model Validation and Simulation Details
### Grid Independence Test  
In this study, computational grids with sizes of 2.33 million, 3.59 million, 5.47 million, 11.88 million, and 17.94 million cells were generated. Steady-state simulations were performed under identical operating conditions to assess the variation rates of the relevant physical parameters, such as the rudder shaft surface temperature, the gas fraction within the water supply pipes, and the temperature within the water supply pipes. The solid material used in the calculations was GH4099 steel, and the fluid materials were liquid water and water vapor. The calculations employed a pressure-based solver with the SST k-omega turbulence model [25; 26]. The phase change model used was the VOF model, and interphase mass transfer was modeled using the Lee model [24], with an initial mass transfer coefficient tentatively set to 100 and 0.1. The initial conditions were derived from the temperature distribution obtained through thermal boundary fitting at time 0 s.  
\begin{table}
\begin{tabular}{c c} \hline
**Boundary Name** & **Boundary Conditions** \\ \hline inlet & Speed entry 0.1 m/s, 300 K \\ outlet & Pressure outlet, 1 bar \\ Heat top & Heat flux density, 4 \(\times\) 104 W/m\({}^{2}\)\(\cdot\)K \\ Heat middle 1 & Heat flux density, 1 \(\times\) 104 W/m\({}^{2}\)\(\cdot\)K \\ Heat middle 2 & Heat flux density, 4 \(\times\) 104 W/m\({}^{2}\)\(\cdot\)K \\ Heat middle 3 & Heat flux density, 1 \(\times\) 104 W/m\({}^{2}\)\(\cdot\)K \\ Heat middle 4 & Heat flux density, 0 W/m\({}^{2}\)\(\cdot\)K \\ Heat middle 5 & Heat flux density, 0 W/m\({}^{2}\)\(\cdot\)K \\ \hline \end{tabular}
\end{table}
Table 4: Boundary Conditions of the numerical model.  
Figure 6: (**a**) Curve of heat flux density in each area of the rudder axis changing with time. (**b**) The transient thermal environment of the control surface.  
Figure 7a shows the phase fraction distribution of the fluid part for five different mesh sets. The results for the 5.47 million and 11.88 million mesh grids are similar, and distinct bubbles can be observed. The axial variation in the gas phase volume fraction was examined, and Figure 7c demonstrates that, under different mesh scales, there are clear gas-liquid interfaces, which approximately appear at the same axial position. Figure 7b,d,f present the computed axial temperature distributions for both the fluid and solid parts under different mesh scales. It can be observed that the axial temperature distribution patterns are consistent, indicating that the temperature calculation results are relatively insensitive to the mesh scale. Figure 7e compares the axial pressure variations in the fluid part for five different mesh sets. There are discrepancies between the results of the two smaller mesh sets and the three larger mesh sets, while the pressure results for the 5.47 million, 11.88 million, and 17.94 million mesh grids are similar.  
Figure 7: **(a)** Axial gas volume fraction distribution, (**b**) fluid axial temperature distribution, (**c**) gas phase volume fraction distribution in the fluid part, (**d**) temperature distribution on the rudder shaft surface, (**e**) axial temperature of the rudder shaft surface, and (**f**) fluid axial pressure comparison.  
The comparison results of gas fraction and temperature at different monitoring points were presented in Table 5. Considering the balance between computational accuracy and time cost, the mesh grid with 11.88 million cells was finally considered for computation in the subsequent numerical simulations.