## 2 Numerical Model and Calculation Scheme
### Thermo-Physical Property Calculation Model  
To accurately capture the variations in the thermophysical properties of n-decane under supercritical pressure, the density was calculated using the Peng-Robinson cubic equation of state (P-EoS) [25]. The P-R EoS has been extensively applied in research on the heat exchange of hydrocarbon fuel [8; 26; 27; 28] and has proven to be effective and accurate in calculating physical properties. This equation is expressed as Equation (12). \[p=\frac{\rho R_{u}T}{M_{w}-b\rho}-\frac{a\alpha(T)\rho^{2}}{M_{w}^{2}+uM_{w}b \rho+wb^{2}\rho^{2}} \tag{12}\]  
[PERSON]'s method was used to determine the dynamic viscosity and thermal conductivity of n-decane under high pressure [29]. The expressions for dynamic viscosity and thermal conductivity are expressed in Equations (13) and (14). \[\mu=\mu^{*}\frac{36.344(M_{w}T_{c})^{0.5}}{V_{c}^{2/3}} \tag{13}\]  
\[\lambda=\frac{31.2\mu^{0}\psi}{M_{w}}\left(G_{2}^{-1}+B_{6}y\right)+qB_{7}y^{ 2}T_{r}^{0.5}G_{2} \tag{14}\]  
The material for the cooling wall was high-temperature alloy steel GH3128 with a density of 8810 kg/m\({}^{3}\). The constant pressure-specific heat capacity and thermal conductivity of GH3128 were calculated using a polynomial piecewise linear interpolation with the data provided in Table 1.  
The measurement of the radial distribution of the fuel temperature in the cooling channel is challenging owing to the limitations of the measurement methods. Consequently, the convective heat transfer coefficient is typically used as a proxy for the overall heat transfer coefficient. However, the existing calculation methods for the overall heat transfer coefficient do not adequately consider the flow state of the hydrocarbon fuel and the presence of thermal stratification in the cooling channel. To address this issue, a modification of the overall heat transfer coefficient was proposed in our previous study [30]. The important equations for calculating the overall heat transfer coefficient are provided in Equations (15)-(21). \[k_{Z_{j}}=\frac{1}{\sum_{1}^{n-1}\Big{(}\frac{d_{out}}{2\lambda_{w}}ln\frac{d_ {g+1}}{d_{n}}\Big{)}+\sum_{1}^{i-1}\Big{(}\frac{d_{out}}{2\lambda_{eff,i}}ln \frac{d_{i+1}}{d_{i}}\Big{)}+\frac{d_{out}}{d_{in}}\frac{1}{h_{adj,z_{j}}}} \tag{15}\]  
\[\lambda_{eff}=\lambda_{l}+\lambda_{t} \tag{16}\]  
\[\lambda_{t}=\frac{\mu_{t}c_{p}}{Pr_{t}} \tag{17}\]  
\[\mu_{t}=\frac{\rho k}{\omega} \tag{18}\]  
\[h_{adj,z_{j}}=-\frac{\lambda_{l}}{T_{w,z_{j}}-T_{adj}}\frac{\partial T}{ \partial Y}\mid_{Y=d_{in}/2} \tag{19}\]  
\[\overline{T}_{w}=\frac{\int_{d_{in}/2}^{d_{out}/2}Y\overline{T}_{w,j}dY}{\int _{d_{in}/2}^{d_{out}/2}YdY} \tag{20}\]  
\[\mu=\mu^{*}\frac{36.344(M_{w}T_{c})^{0.5}}{V_{c}^{2/3}} \tag{13}\]0  
To evaluate the transient flow and dynamic heat transfer characteristics of the hydrocarbon fuel in the cooling channel, numerical studies under different dynamic working conditions were conducted. The three different dynamic conditions were the inlet mass flow, heat flux, and working pressure. The rates of change in the inlet mass flow and heat flux are defined in Equations (22) and (23). \[\mu=\mu^{*}\frac{36.344(M_{w}T_{c})^{0.5}}{V_{c}^{2/3}} \tag{13}\]1  
\[\mu=\mu^{*}\frac{36.344(M_{w}T_{c})^{0.5}}{V_{c}^{2/3}} \tag{13}\]2  
Stable conditions were considered as the initial boundary conditions of the study. The heat flux values corresponding to these stable conditions are listed in \[\mu=\mu^{*}\frac{36.344(M_{w}T_{c})^{0.5}}{V_{c}^{2/3}} \tag{13}\]3 These values were obtained for a mass flow rate of 1 g/s, two different back pressures of 5 MPa and 3 MPa, and four outlet fuel temperatures of 373.15 K, 473.15 K, 573.15 K, and 673.15 K. The heat flux values listed in Table 2 represent the heat transfer per unit area at the cooling channel boundary. These served as reference values for stable conditions and were used as the initial conditions for the numerical simulations in this study. By setting these initial boundary conditions, this study aims to analyze the transient flow and heat transfer