## 3 Model Validation and Simulation Details  
### Validation of the Numerical Simulation Method with Experimental Results  
#### 3.1.1 Experimental Setup  
The test article is an actively cooled thermal protection rudder surface model. Quartz lamps manufactured by China Guangzhou Langpu Optoelectronic Technology company (Guangzhou, China) are used to heat the metal skin on the surface, while internal cooling channels provide convective active thermal protection for the rudder shaft. The test article is entirely made of 304 stainless steel components and consists of the skin, structural frame, rudder shaft, and other components. During the test, the specimen is mounted vertically on a support frame and thermally insulated from the frame using thermal insulation tiles. Quartz lamp arrays are installed parallel to both sides of the specimen to provide overall heating of the rudder surface. The distance between the quartz lamp arrays and the rudder surface is 70 mm, with a heat flux loading of 80 kW/m\({}^{2}\). The heat flux is measured at the symmetrical central area of the quartz lamp arrays on both sides. Prior to computation, the model is simplified based on the provided data. This simplification includes reducing the complexity of the bolts and some structural components, as well as performing smoothing treatment, as shown in Figure 2a. The model is discretized, and the total number of cells is about 2.7 million, as shown in Figure 2b.  
The temperature of the rudder shaft under the condition of 80 kW/m\({}^{2}\) heating and waterless working conditions was calculated. Further, the fluid-solid coupled heat transfer steady-state calculation was carried out under the condition of 80 kW/m\({}^{2}\) heating and water supply at inlet flow rates of 1 and 0.2 m/s, and the results were compared and analyzed with the experimental results. Based on the existing test data and model settlement results, the evaluation of the cooling efficiency of the rudder shaft under small flow conditions was obtained.  
Figure 1: (**a**) Physical model of the air rudder shaft and (**b**) schematic diagram of the initial coolant channel.  
However, due to the complexity and high computational cost of creating a comprehensive numerical model that includes the quartz lamp arrays, the surrounding flow field, and the rudder shaft itself, directly applying this heat flux condition in simulation is relatively difficult. Therefore, in this work, under no cooling water conditions, different equivalent heat fluxes (60,000, 50,000, and 45,000 W/m\({}^{2}\)) are applied to the surface of the test article in the simulation to obtain transient temperatures, which are then compared with the experimental data. As shown in Figure 3, the result with a heat flux of 45,000 W/m\({}^{2}\) aligns with the experimental data. It is worth noting that under a heat flux density of 45,000 W/m\({}^{2}\), the temperature at intermediate times is slightly higher than the experimental results but gradually converges with the experimental results after 1600s, which is favorable for subsequent steady-state simulations for cases with inlet cooling water velocities of 0.2 m/s and 1 m/s.  
Figure 3: The fitting result of the heat flux density of the rudder surface.  
Figure 2: (**a**) Schematic diagram of model processing and (**b**) corresponding meshing diagram.