## 2 Numerical Modeling  
### Physical Model Details  
#### 2.1.1 Description of Active Cooling System and Simplified All-Movable Rudder  
The active cooling system for HVs proposed in this paper, as illustrated in Figure 1, is an extension of the conventional active regenerative cooling technology. In this system, a portion of the low-temperature fuel is directly pumped into the cooling channels of the combustor wall to absorb the heat generated by combustion. Simultaneously, another portion of the low-temperature fuel is circulated through a network of cooling channels to dissipate the excess heat from the aerodynamically heated components, such as the wings, cones, air rudders, and elevators. Subsequently, these elevated-temperature fuels are injected into the combustion chamber where they undergo chemical reactions to generate thrust as required. As a thermal protection method of HVs, the active cooling system can not only reduce the wall temperature of the engine and various high-temperature components but can also increase the fuel temperature so as to reduce the ignition delay time and improve the combustion efficiency.  
The all-movable rudder, with its fully-rotational characteristic, can provide greater control force and higher rudder effectiveness during high-speed flight. For the effective management of the aerodynamic heat from the two rudders located at the tail of the vehicle, it is crucial to appropriately distribute the coolant supply pipeline layout throughout the fuselage in order to improve the performance of the active cooling system. The coolant supply pipeline for the two rudders is determined by the thermal management strategy and typically involves two configurations, as shown in Figure 2: series and parallel arrangements. In the series arrangement, both rudders are assumed to have identical inlet mass flow rates and are equal to the total mass flow rate of the rudder branch. However, due to heat transfer from the rudder surface and heat exchange at the rudder shaftgaps, the downstream rudder experiences a higher inlet temperature compared to that of the upstream one. Conversely, in the parallel arrangement, both rudders are assumed to receive equal mass flow rates--each half of the total flow rate--while maintaining the same inlet temperature for both rudders.  
Figure 3 provides an overview of the geometry and cross-section dimensions of the all-movable rudder investigated in this study. The key dimensions include a spread length of 300 mm (measured from the rudder root to the tip), a chord length of 450 mm at the rudder root with a thickness of 26 mm, and a chord length at the rudder tip of 225 mm  
Figure 1: An active cooling system concept in hypersonic vehicles.  
with a thickness of 13 mm. Moreover, the leading and trailing edges are chamfered with a radius of 2.25 mm to balance the aerodynamic heat and drag.  
Figure 3: Geometry model and dimensions. (**a**) Geometry and cross-section dimensions (mm); (**b**) Dimensions of the mid-section at 150 mm (axial) from the rudder root (mm); (**c**) All-movable rudder; (**d**) Simplified rudder.  
Figure 2: Arrangements of coolant supply pipelines for two all-movable rudders: (**a**) Series arrangement; (**b**) Parallel arrangement.  
However, the intricate geometric features of the all-movable rudder, including forward and backward sweeps as well as variable cross sections, made the actual design process exceedingly challenging. Consequently, to simplify the design process and reduce the computational cost, a preliminary design was implemented, using a simplified one with no forward or backward sweeps, while maintaining a consistent cross section. This approach aimed to satisfy the periodic boundary conditions required for the subsequent single-channel analysis. Specifically, the cross-sectional dimensions at 150 mm (axially) from the rudder root were used.