## 1 Introduction  
Hypersonic vehicles (HVs) are next-generation aircrafts capable of long-duration cruising in the near-space atmosphere at speeds exceeding Mach 5, primarily designed for space transportation, surveillance, and military operations [1]. During high-speed flight, the outer surfaces of HVs experience significant compression, friction, and viscous dissipation with the surrounding atmosphere, resulting in intense aerodynamic heating [2]. The air rudder is a crucial component within the control system of HVs that protrudes from the streamlined fuselage surface. Under extreme heat conditions, it becomes vulnerable to reaching elevated temperatures, which can lead to material failure or actuation malfunction and pose substantial safety risks to HVs [3, 4]. To address these challenges, a thermal protection system (TPS) is essential. The TPS is an integrated set of technologies and components specifically designed to shield vehicle structures from overheating in harsh thermal environments. For the air rudder, the TPS functions to manage and mitigate thermal loads, thereby maintaining survivable temperatures, ensuring standard functionality, and preserving the structural integrity. According to the protection mechanism and operational characteristics, the TPS can be categorized into three types: passive, semi-passive, and actively cooled [5, 6]. Considering the structural and functional attributes of the air rudder, as well as the peak and distribution characteristics of the surface heat flux, not all TPSs are suitable for implementation. For instance, utilizing a passive heat-sink TPS in an air rudder would significantly increase its mass burden and fail to meet the lightweight requirements. Moreover, employing a semi-passive ablative TPS may alter the shape of the air rudder and impact its maneuvering function. Lastly, incorporating a passive multilayer insulating TPS for air ruddersbecomes redundant when an internal low-temperature environment isn't required. In this scenario, passive hot structures [7] become the most prevalent and extensively utilized type because of their exceptionally high-temperature resistance, light weight, ease of assembly, and absence of additional energy consumption. Common materials for hot structures include high-melting-point alloy, carbon fiber-reinforced carbon (C/C) matrixes, and carbon fiber-reinforced silicon carbide (C/SiC) matrixes. The Hopper vehicle [8] utilizes C/SiC composites for rudders, elevons, and body caps with a maximum temperature of 1700 K. The X-33 aircraft [9] employs C/C composites to protect its aerosurface leading edges from high heating. Taking the X-43 vehicle as an example [10], the control surface is made of hybrid materials consisting of Haynes 230 superalloy for the main framework and C/C composites for the leading edge. However, superalloys have a limited capacity for withstanding high temperatures, C/C composites are susceptible to oxidation at elevated temperatures (above 500 \({}^{\circ}\)C), and C/SiC composites are prone to fatigue microcracks due to their inherent brittleness. Consequently, the design of hot structures heavily relies on advancements in the thermal resistance of materials, which hinders product upgrades. Recently, there has been significant interest in the semi-passive heat-pipe-cooled TPS [11], as it enables the efficient transfer of substantial aerodynamic heat from the stagnation zone of high heat fluxes to the flat-tail region of low heat fluxes and radiates this heat into the surrounding atmosphere. [PERSON] et al. [12] suggested integrating flat heat pipes into the leading edges to enhance heat conduction and dissipation. Through an isothermal analytical approximation method, they demonstrated the feasibility of utilizing a composite material consisting of the niobium alloy Cb-752 in conjunction with a lithium working medium for cooling leading edges with a radius of 3 mm under flight conditions ranging from Mach 6 to 8. [PERSON] et al.