## 2 Numerical Modeling
### Physical Model Details
#### 2.1.3 Periodic Modeling of Cooling Channels
Additionally, adiabatic wall BCs are assigned to both the surfaces of the skin inner/outer wall within the entrance/exit zones and to the surfaces of the skin inner wall within the heating zone. In Scheme B1, for the sake of reducing the computation, the \(x\)-\(z\) mid-plane is defined as a symmetry plane and the representing cell is modeled symmetrically. The inlet/outlet, heating flux, periodic translational, and adiabatic BCs are similar to those defined in Scheme A1. Figure 8: The BCs of the representative units of the cooling channel in Schemes A1 and B1. Figure 7: The representative cell of the cooling channel in Scheme B1. The heat flux BCs are imposed on the outside wall of the leading-edge segment, inclined segment, and horizontal segment with varying magnitudes, as depicted in Figure 9. These heat fluxes are determined using engineering correlations (the altitude and speed are 30 km and 8 Mach, respectively) and subsequently refined through homogenization. For the leading-edge segment, it can be considered as an infinitely long cylinder. Then, the heat flux \(q_{\rm w,le}\) can be determined as follows:  
\[q_{\rm w,le}=q_{\rm sph}/\sqrt{2} \tag{1}\]  
\[q_{\rm sph}=\frac{2\theta{\rm sin}\theta\Big{[}\Big{(}1-1/\gamma_{\infty}Ma_{ \infty}^{2}\Big{)}{\rm cos}^{2}\theta+1/\gamma_{\infty}Ma_{\infty}^{2}\Big{]}} {[D(\theta)]^{0.5}}q_{s} \tag{2}\]  
where \(q_{\rm sph}\) refers to the heat flux within a sphere with the same radius as the cylinder; \(\theta\) is the angle between the radius vector and the flight direction (\(\gamma_{\infty}=1.4\)); \(Ma\) is the incoming stream Mach number; \(D(\theta)\) is a geometrically relevant parameter and can be found in Ref. [37]; \(q_{s}\) is the heat flux at the stagnation point from the modified Lee formula, as follows:  
\[q_{s}=\frac{0.5\sqrt{2}}{{\rm Pr}^{2/3}}\sqrt{\rho_{s}^{*}\mu_{s}^{*}\Big{(} \frac{du_{c}}{dx}\Big{)}_{s}}(h_{s}-h_{w}) \tag{3}\]  
where \({\rm Pr}=0.71\) is the Prandtl number; \(\rho_{s}^{*}\) and \(\mu_{s}^{*}\) denote the thermo-physical properties of the stationary air based on the reference enthalpy; \(\big{(}du_{c}/dx\big{)}_{s}\) refers to the velocity gradient at the stationary point; and \(h_{s}\) and \(h_{w}\) are the enthalpies of the stationary point and the wall, respectively. The heat flux upon the inclined segment (\(q_{\rm w,i}\)) can be similarly concerted by the heat flux on the surface of the cone, as follows:  
Figure 7.0  
Figure 7.1  
Figure 7.2 The heat flux imposed on the outside wall of the leading-edge segment, inclined segment, and horizontal segment. (**a**) 2D schematic for heat flux computation; (**b**) Heat flux calculated by empirical formulae and homogenization (channel interior details using Scheme A1 as an example). where \(\theta_{c}\) is the cone half angle, and \(A(\theta_{c})\), \(B(\theta_{c})\), and \(s/R_{0}\) are geometrically relevant parameters and can be found in Ref. [37]. The heat flux upon the horizontal segment (\(q_{\text{w,h}}\)) [38] can be calculated as follows:  
Figure 7.3  
where \(Re_{\text{local}}\) is the local Reynolds number.