## 2 Numerical Modeling
### Analysis Cases and Mesh Sensitivity  
The basic cases named Scheme A1 and Scheme B1 are introduced in Section 2.1.2. To find more improved solutions, several retrofitting options for the leading edge illustrated in Figure 11 were also studied. Among them, Scheme B1-M extends the core channel by 7.5 mm towards the leading-edge segment based on Scheme B1, while Scheme A2 reduces the channel outer radius (R3) and channel inner radius (R4) based on Scheme A1. Additionally, Scheme B2 decreases the channel outer radius (R3) from Scheme B1 but keeps the channel inner radius (R4) unchanged. Lastly, for Scheme B2-M, a length of 10.5 mm is added to the core channel towards its leading-edge segment. Figure 11: Leading-edge details in six different actively cooled configurations (mm). For a high-speed vehicle, an efficient cooling channel design must balance optimizing the cooling performance with minimizing the coolant flow rate. To evaluate potential cooling channel schemes for two all-movable rudders operating at an altitude of 30 km and Mach 8, two total coolant supply rates of the rudder branches--200 g/s and 100 g/s--were analyzed for both the series and parallel pipeline configurations. In the series configuration, at a total coolant flow rate of 200 g/s, both rudders received 200 g/s of coolant, with the upstream rudder having an inlet temperature of 300 K and the downstream rudder reaching 400 K. When the flow rate was reduced to 100 g/s, both rudders received 100 g/s, with the upstream rudder at 300 K and the downstream rudder at 400 K. In the parallel configuration, at a 200 g/s total coolant flow, each rudder received 100 g/s at an inlet temperature of 300 K. At the reduced total flow rate of 100 g/s, each rudder was supplied with 50 g/s, maintaining an inlet temperature of 300 K.  
The flow rate for each individual cooling channel was determined by dividing the total flow rate to a rudder by 100, given that 100 cooling channels are arranged within a 300 mm span. Figure 12 outlines the competitive selection process for cooling channels from the perspective of a single channel, with an 800 K temperature limit serving as the core criterion. For the series pipeline configuration, six cooling channel schemes were initially evaluated at a flow rate of 2 g/s and an inlet temperature of 400 K. Those that met the temperature standard proceeded to the second round, where the flow rate was reduced to 1 g/s and the inlet temperature was maintained at 400 K. In the parallel pipeline configuration, the systems were first tested under a flow rate of 1 g/s with an inlet temperature of 300 K. Systems that met the criteria advanced to the second round, with conditions adjusted to a 0.5 g/s flow rate and a 300 K inlet temperature. The operating pressure was maintained at 3 MPa, and variations in the thermo-physical properties due to pressure changes were neglected, with gravity considered along the \(z\)-direction. Finally, the selected preferred solution was investigated for its thermal-mechanical performance to determine the structural integrity. Please refer to Table 2 for details regarding all cases. The significance of high-quality mesh in numerical computations cannot be overstated. In this paper, the pre-processing software Fluent Meshing was utilized to generate three-dimensional hybrid meshes for our computational model. Both the solid and fluid domains were discretized using poly-hexcore, ensuring the filling state of the hexahedral grids in the core region and the polyhedral grids in the transition region. To maintain a non-dimensional distance (\(y^{+}<1\)), the boundary layer grids were produced in the near-wall region, with a first-layer height of 0.001 mm and a transition ratio of 0.272. Figure 13 illustrates the mesh details for the medium mesh at the leading edge of Scheme A1. The local size is set  
Figure 12: Cases studies for competitive selection of the optimal channel scheme. to 0.075 mm.