## 2 Aero-Thermal Modeling of Rotating Detonation Combustors
### Unsteady Reynolds-Averaged Navier-Stokes Evaluation
Since all species diffuse with the same diffusivity, mass is conserved and all mass fractions add up to 1 (\(\mathrm{\sigma H_{2}}+\mathrm{\sigma O_{2}}+\mathrm{\sigma H_{2}O}+\mathrm{ \sigma N_{2}}=1\)). The thermodynamic properties were based on a real gas model with a temperature-dependent coefficient polynomial for each individual species with the coefficients found in [PERSON] et al. (2016). The Sutherland law was used to model the viscosity for each individual species, and the constants are depicted in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(T_{\text{ref}}\) & \(\sigma_{viscosity,ref}\) & \(\boldsymbol{C^{te}}_{viscosity}\) & \(\boldsymbol{k_{\text{ref}}}\) & \(\boldsymbol{C^{te}}_{thermal}\) _conductivity_ \\ \hline \(\mathrm{H_{2}}\) & 273.16 & \(8.45\times 10^{-6}\) & 66.8 & 0.17290 & 175.3 \\ \(\mathrm{O_{2}}\) & 273.16 & \(1.92\times 10^{-5}\) & 125.0 & 0.02449 & 268.8 \\ \(\mathrm{N_{2}}\) & 273.16 & \(1.656\times 10^{-5}\) & 104.7 & 0.02407 & 178.1 \\ \(\mathrm{H_{2}O}\) & 273.16 & \(1.006\times 10^{-5}\) & 523.3 & 0.0161 & 1276.0 \\ \hline \end{tabular}
\end{table} 
Table 1: Sutherland law coefficients for viscosity and thermal conductivity. For this study, a one-step Arrhenius reaction with three reactive species (H\({}_{2}\)O, O\({}_{2}\) and H\({}_{2}\)) [11] was used. To solve the turbulent flow, the k-epsilon model was used for the 2D simulations, while for the three-dimensional simulations, the k-\(\omega\) SST model was chosen for its better accuracy in the near-wall region and in free-shear layer flows. Before examining the combustor's flow field, the solver was verified with the Sod shock tube problem [27] to characterize the dynamic capabilities of the solver. The propagation of a detonation front was analyzed in a one-dimensional domain for a stoichiometric mixture of H\({}_{2}\)-air at 0.1 MPa and 273 K. The detonation was initiated through a narrow rectangle at high pressure and temperature located at the inlet of the tube (\(x\) = 0). The length of the tube was 0.1 m and the numerical cell width was 0.5 mm (Figure 1). The time step was 1 \(\times\) 10\({}^{-8}\) s, similar to the simulations of [PERSON] et al. [28]. The calculated detonation speed was around 2000 ms\({}^{-1}\) and with a relative error below 1% compared to the theoretical Chapman-Jouguet velocity (1986 ms\({}^{-1}\)[11]). Figure 1a depicts the pressure along the channel at two different time intervals (\(t\) = 2.0 \(\times\) 10\({}^{-5}\) s and \(t\) = 3.0 \(\times\) 10\({}^{-5}\) s, respectively). The pressure jump behind the detonation front and following pressure decrease related to the expansion fan are observed at positions \(x\) = 0.05 m and \(x\) = 0.07 m. Figure 1b represents iso-Mach contours where the detonation front is identified by a narrow and large Mach gradient, behind which the flow accelerates to M = 1.05. The temperature behind the shock attained 3000 K, comparable to the equilibrium value behind the detonation front. Figure 2a shows the static temperature flow-field of an unfolded two-dimensional RDC with a perimeter of 0.3 m and an axial length of 0.15 m. The numerical grid size was 0.5 mm in both \(x\) and \(y\) directions and the timestep was 1.0 \(\times\) 10\({}^{-8}\) s. The outlet boundary condition was set to supersonic flow conditions, and at the inlet, a total pressure of 0.5 MPa and a total temperature of 330 K were applied. Periodicity was imposed at both sides of the RDC.