## 1 Introduction
[13] developed a conjugated-based, three-dimensional numerical method to investigate the heat transfer characteristics and limitations of the heat-pipe-cooled TPS. [PERSON] et al. [14] employed a three-dimensional numerical method to predict the thermal performance of a platelet heat-pipe-cooled TPS. The results demonstrated that the TPS design, incorporating IN718 as the solid material and sodium as the working fluid, successfully satisfies the cooling requirements for a leading edge with a radius of 15 mm under flight conditions of Mach 8 at an altitude of 34 km. In Ref. [15], utilizing the well-established numerical methods, the feasibility of a platelet heat-pipe-cooled leading edge with sodium and lithium as the working medium is discussed under nominal and failure conditions. Calculations of the temperature and stress were performed for different wall materials and thicknesses to study the structural robustness. Despite the numerous advantages of heat-pipe-cooled TPSs, their broader application in air rudders has been limited due to the restricted capacity for bearing heat loads and the complexities entailed in the design, fabrication, and operation. In contrast, the actively cooled TPS [16; 17] has emerged as a superior alternative for critical components operating in extreme thermal environments. This is particularly true for future HVs that will require faster speeds, longer cruising durations, and stronger maneuverability. The implementation of the actively cooled TPS can be achieved through various methods, including convective cooling [18; 19; 20], film cooling [21], and transpiration cooling [22]. Convective cooling offers a simpler and more feasible solution compared to the latter two cooling methods. [PERSON] et al. [23] present an integrated fluid-thermal-structural finite element approach to study the response of hydrogen-cooled leading edges subjected to intense aerodynamic heating. [PERSON] et al. [24] theoretically investigated the transient heat transfer by combined conduction, convection, and radiation in a semitransparent porous medium, considering gas injection through the layer. The results demonstrated that the actively cooled TPS alters the temperature distribution in the solid matrix and significantly reduces the time required to reach steady-state conditions. Additionally, it was found that the thickness of the porous layer and the radiation properties substantially impact the temperature distribution and insulation performance. [PERSON] and [PERSON] [25] conducted an experimental and numerical study on forced convection in highly porous metal foams,finding that foam-water combinations exhibit significantly higher thermal dispersion than foam-air combinations. The numerical model was validated and further utilized to investigate various combinations of solid materials and fluids. [PERSON] [PERSON] and [PERSON] and [PERSON] (2018) present a thermo-mechanical analysis of actively cooled metal foam sandwich panels using experimental and analytical approaches. By pumping compressed air through the metal foam core, this actively cooled system could effectively eliminate elevated temperatures and control the thermo-mechanical deformation. Using a finite difference method, [PERSON] et al. [PERSON] et al. (2018) modeled both traditional and actively cooled TPSs. The numerical predictions indicate that the conventional TPS fails to ensure that the internal titanium alloy structure remains within the operating temperature limit, whereas active cooling offers a viable solution. [PERSON] et al. [PERSON] et al. (2018) proposed an actively cooled TPS composed of a porous ceramic core and ceramic matrix composite material, which enhances heat dissipation by introducing flowing gas into the core layer. Simulation methods were utilized to identify porous materials with optimal mechanical and thermal flow dynamic properties, while additive manufacturing techniques were employed to fabricate leading-edge components. Finally, experimental verification was conducted in a plasma wind tunnel. [PERSON] et al.