## 2 Aero-Thermal Modeling of Rotating Detonation Combustors
### Heat-Transfer Analysis: URANS vs. Reduced-Order Model with Heat-Flux Correlation  
The integral method established by [PERSON] and [PERSON] [35] was used to compute the heat transfer between the flow boundary layer and wall, based on the estimation of the boundary layer pressure gradient from the MOC code [20]. The calculation of the aerodynamic properties of the boundary layer involves the momentum integral Equation (6) that is simplified thanks to [PERSON]'s transformation for compressible flow:  
\[\frac{d\theta_{tr}}{dx}+\frac{\theta_{tr}}{M_{e}}\frac{dM_{e}}{dx}(2+H_{tr})=A \tag{6}\]  
\[A=0.123e^{-1.561H_{i}}\bigg{(}\frac{M_{e}a_{0}\theta_{tr}}{v_{0}}\bigg{)}^{-0. 268}\bigg{(}\frac{T_{e}}{T_{ref}}\bigg{)}^{0.732}\bigg{(}\frac{T_{e}}{T_{0}} \bigg{)}^{3.268}, \tag{7}\]  
where \(\theta_{tr}\) is the transformed momentum thickness defined as follows:  
\[\theta_{tr}=\int_{0}^{\Delta}\frac{U}{U_{e}}\bigg{(}1-\frac{U}{U_{e}}\bigg{)}dx. \tag{8}\]  
If a fully developed turbulence flow is considered, Equation (8) can be integrated to predict the value of the momentum thickness at a distance x from the leading edge. \[\theta_{tr}{}^{1.2155}=\frac{0.01173}{\Big{(}\frac{a_{0}}{v_{0}}\bigg{)}^{0. 2155}M_{e}^{B+0.2155}}\int_{0}^{x}M_{e}^{B}\frac{\Big{(}\frac{T_{e}}{T_{ref}} \Big{)}^{0.732}}{\Big{(}\frac{T_{0}}{T_{e}}\Big{)}^{3.268}}dx, \tag{9}\]  
where  
\[B=4.2+1.2155H_{i,fp}\bigg{(}\frac{h_{w}}{h_{0}}-1\bigg{)}. \tag{10}\]  
Figure 7: Total temperature contour for (**a**) unsteady Reynolds Averaged Navier Stokes (URANS) simulation and (**b**) reduced-order model; comparison between both methods for: (**c**) static inlet pressure and (**d**) total outlet pressure. Values with subscript \(e\) are defined at the edge of the boundary layer, while \(a_{0}\) and \(v_{0}\) are the speed of sound and kinematic viscosity based on the stagnation flow conditions. \(H_{i,fp}\) is the incompressible form factor for a flat plate, while Tref is evaluated at the reference enthalpy:  
\[\frac{h_{ref}}{h_{e}}=\frac{1}{2}\bigg{(}\frac{h_{w}}{h_{e}}+1\bigg{)}+0.22 \bigg{(}\frac{h_{aw}}{h_{e}}-1\bigg{)}. \tag{11}\]  
The adiabatic enthalpy (\(h_{aw}\)) is calculated as:  
\[\frac{h_{aw}}{h_{e}}=1+\frac{\gamma-1}{2}M_{e}^{2}(Pr)_{ref}^{2/3}. \tag{12}\]  
Finally, [PERSON] and [PERSON] defined the Stanton number:  
\[St=\frac{0.123e^{-1.151H_{i,fp}}\left(\frac{T_{e}}{T_{ref}}\right)^{0.732}}{ \left(\frac{T_{e}}{T_{ref}}\right)^{0.268}\left(\frac{M_{e}a_{0}a_{br}}{v_{0}} \right)0.268}Pr^{-2/3}. \tag{13}\]  
The heat flux (\(q_{wall}\)) between the fluid and the wall is computed in the following way:  
\[q_{wall}=St\rho_{e}u_{e}(h_{aw}-h_{w}). \tag{14}\]  
A simplified numerical test case was selected to verify the accuracy of the present correlation. Figure 8a depicts the test case that comprises a two-dimensional tube with a channel height of 5 mm with a premixed H\({}_{2}\)-air at stoichiometric conditions, with a high-pressure and high-temperature zone that ignited the combustible at the start of the simulation. The lower and upper walls were isothermal with a grid that was refined at the wall to guarantee that the viscous sublayer was well resolved. The Mach number and total conditions were measured 0.4 mm from the wall and represent the properties at the edge of the boundary layer. The wall and stagnation properties were then loaded into the model to predict the boundary layer properties to solve Equation (14) and retrieve the heat flux. Figure 8b details the heat flux on the lower wall, predicted by the URANS solver and by the correlation of [PERSON] and Tucker [35]. Through this test case, the value of \(H_{i,fp}\) = 1.7 was set as the optimal value to correctly reproduce the heat flux peak in flows that are facing shock and detonation transients.