## 2 Numerical Modeling
### Governing Equations and Solution Strategy  
The steady flow and heat transfer of the fluid in cooling channels are governed by the conservation equations of mass, momentum, and energy, as follows:  
Mass conservation equation:  
\[\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{7}\]  
Momentum conservation equation:  
\[\frac{\partial\big{(}\rho u_{i}u_{j}\big{)}}{\partial x_{j}}=-\frac{\partial \overline{P}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{[}\mu\Bigg{(} \frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}} \Bigg{)}-\frac{2}{3}\mu\frac{\partial u_{k}}{\partial x_{k}}\delta_{ij}\Bigg{]} +\frac{\partial}{\partial x_{j}}\Big{(}-\rho u_{i}^{\prime}u_{j}^{\prime} \Big{)}+\rho a_{i} \tag{8}\]  
Energy conservation equation:  
\[\frac{\partial(\rho u_{i}H)}{\partial x_{i}}=\frac{\partial}{\partial x_{i}} \Bigg{[}\bigg{(}\frac{\lambda}{c_{p}}+\frac{\mu_{t}}{Pr_{t}}\bigg{)}\frac{ \partial H}{\partial x_{i}}\Bigg{]} \tag{9}\]  
where \(\rho\), \(\mu\), \(\lambda\), and \(c_{p}\) are the density, dynamic viscosity, thermal conductivity, and specific heat capacity of the fluid, respectively; \(u_{i}\) is the velocity component in the \(x\), \(y\), and \(z\) coordinates; \(\overline{P}\) is the average pressure; \(\delta_{ij}\) is the Kronecker delta function that equals unity if \(i=j\) and zero otherwise; \(u_{i}^{\prime}\) is the velocity component fluctuation and \(-\rho u_{i}^{\prime}u_{j}^{\prime}\) is therefore the turbulent stress; \(a_{i}\) is the acceleration component in the \(x\), \(y\), and \(z\) coordinates; \(H\) is the specific enthalpy; \(Pr_{t}=\left(\mu_{t}/\rho\right)/\left(\lambda_{t}/\rho c_{p}\right)\) is the turbulent Prandta number; \(\mu_{t}\) is the turbulent viscosity; \(\lambda_{t}\) is the turbulent thermal conductivity. It is important to note that the energy equation presented here is a simplified form: the source term is subtracted because the chemical reaction of the coolant is not considered, and the pressure and viscosity terms are neglected because of the variable property flow. The selection of an appropriate turbulence model is crucial for accurately simulating supercritical flow and heat transfer. Recently, several turbulence models have been comparatively studied in relevant numerical simulations [40]. It has been observed that the Shear Stress Transport k-\(\omega\) (SST k-\(\omega\)) turbulence model demonstrates a superior predictive performance. Consequently, in this work, the SST k-\(\omega\) turbulence model was employed to simulate the turbulence flow and heat transfer. Its governing equations are presented below:  
Turbulent kinetic energy equation:  
\[\frac{\partial(\rho u_{i}k)}{\partial x_{i}}=\frac{\partial}{\partial x_{j}} \Bigg{(}\Gamma_{k}\frac{\partial k}{\partial x_{j}}\Bigg{)}+G_{k}-\gamma_{k} \tag{10}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Thermal Properties** & **Thermal Conductivity** & **Specific Heat** & **Expansion Coefficient** \\  & **(W/m\(\cdot\)K)** & **(J/kg\(\cdot\)K)** & **(I/K)** \\ \hline Value & 91.74 & 460.6 & \(1.2\times 10^{-5}\) \\ \hline
**Mechanical Properties** & **Density** & **Young’s Modulus** & **Poisson’s Ratio** \\  & **(kg/m\({}^{3}\))** & **(Gpa)** & **Poisson’s Ratio** \\ \hline Value & 8900 & 190 & 0.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the nickel-based, high-temperature alloy.