## 1 Introduction  
Scramjets are highly promising for hypersonic flight and offer benefits such as exceptional speed, stealth, and maneuverability. Scramjets are extensively employed as they provide higher specific thrust than any other air-breathing propulsion system [1]. The availability of scramjets was verified by the success of X-43A and X-51 [2]. Owing to the combination of high-speed, high-intensity combustion, and friction, there is a large heat load in the combuster in scramjets. To prevent potential damage, regenerative active cooling is adopted [3], where fuel is pumped into the cooling channel inside the combuster wall of the scramjet to dissipate heat before entering the combuster. Both liquid hydrogen and liquid hydrocarbon fuels are frequently utilized for the regenerative active cooling of scramjets. Liquid hydrocarbon fuel is often preferred owing to its convenient storage [4]. Furthermore, liquid hydrocarbon fuels can absorb a significant amount of heat through endothermic cracking and pyrolysis, which effectively reduces the engine wall temperature [5]. The structure of the regenerative cooling process is shown in Figure 1[6]. Additionally, a successful flight test of the X-51 confirmed the viability of regenerative cooling using hydrocarbon fuel as a coolant [7]. Numerous studies have been conducted on the steady flow and heat transfer characteristics of hydrocarbon fuel within a cooling channel, including investigations on the cross-sectional geometry of the cooling channel [8], working pressure [9, 10], heating mode [11, 12], and convective heat transfer correlation [13]. Owing to different off-design working conditions, sudden changes in the heat flux, inlet fuel mass flow, and working pressure in the cooling channel of the scramjet may occur. However, studies on transient flow and heat transfer in these scenarios are often limited to the instability phenomena within experimental flows. For example, [PERSON] et al. [14] identified the transition instability phenomenon in flow and heat transfer experiments of n-decane under supercritical pressure. These studies did not specifically analyze the influence of these factors on the transient flow and heat transfer characteristics of n-decane from the perspective of variable conditions. Consequently, a detailed analysis of the underlying mechanisms involved in transient flow and heat transfer processes is required. The cooling channels serve as a heat exchanger, with them being curved inside the combustor wall, and fuel flows through them as a coolant [15]. Owing to the significance of the convective heat transfer coefficient, extensive research has been conducted in this area. [PERSON] et al. [16] determined the convective heat transfer coefficient by calculating the average fuel temperature across a cross-section. However, this method does not consider the impact of the thermophysical properties of hydrocarbon fuels near the pseudocritical temperature. [PERSON] et al. [17] focused on the impact of dynamic viscosity on the convective heat transfer coefficient; however, their findings were limited to turbulent flows. Subsequent studies conducted by [PERSON] et al. [18], [PERSON] et al. [19], and [PERSON] et al. [20] expanded on the previous work by calculating the convective heat transfer coefficient in segments, considering the ratio of the average temperature across a cross-section within the critical temperature region. Furthermore, the thermophysical properties of hydrocarbon fuels in convective heat transfer calculations were explored. However, when piecewise convective heat transfer correlations are used, there is an inherent discontinuity in the overall heat transfer coefficient derived from the experimental data. [PERSON] et al. [21] addressed this issue by fitting three separate convective heat transfer correlations based on experimental data for the laminar, transition, and turbulent flow regions.