## 2 Aero-Thermal Modeling of Rotating Detonation Combustors
### Reduced-Order Model Based on the Method of Characteristics  
This section details the reduced model implementation, followed by a validation with a commercial two-dimensional URANS solver and finally a grid sensitivity of the proposed model. The reduced-order model that we used to retrieve the flow field consists of an iterative procedure to estimate injection velocity that certifies a stable operation of the RDC [20], in a similar way to the work of [PERSON] and [PERSON] [29]. The flow field is solved in the relative frame of reference of the rotating shock, and the numerical procedure relies on a one-dimensional chemical solver [30] to model the detonation front. In our approach, the complete flow field is computed with a two-dimensional method of characteristics solver [31]. Figure 3: (**a**) Instantaneous streamlines (in white) used to compute the two pathlines. (**b**) Pressure evolution for two selected pathlines. Figure 2: (**a**) Iso-temperature contour of the rotating detonation combustor (RDC). (**b**) Pressure-gain coefficient as a function of time. The initial inputs to the solver are inlet total pressure and inlet total temperature. This input (P\({}_{0,1}\) and T\({}_{0,1}\)) is represented in Figure 4, which shows the triangular injection zone, where the supersonic shock moves at the velocity U, towards the swirl-free fresh reactants with velocity V\({}_{\rm inj}\), a pressure P\({}_{1}\) and a temperature T\({}_{1}\). According to the velocity triangles, the relative incoming velocity (W\({}_{\rm Cl}\)), traveling at the Chapman-Jouguet velocity, defines the inclination of the refilling line (\(\beta\)) and of the detonation front. Immediately behind the detonation wave, the relative Mach number is sonic (M\({}_{\rm w}=1\)) and the detonation jump conditions are computed (P\({}_{2}\)/P\({}_{1}\), T\({}_{2}\)/T\({}_{1}\)) based on the ZND theory with the shock and detonation tool box [32]. In the present reduced-order model, both the edge of the refilling zone and the detonation front are considered to be straight lines. The edge of the refilling zone follows the same direction as the streamlines (Wcj), and the detonation wave is perpendicular to the incoming fresh mixture [20]. Therefore, the triangular zone is characterized by a 90 degree angle between the detonation wave and the edge of the refilling zone. The model solves an unfolded rotating detonation combustor, illustrated in Figure 5a, in which a single period is represented. To analytically solve the triple point, which intersects the oblique shock wave, slip line and detonation front, continuity of pressure and flow direction (\(\theta_{3}^{{}^{\prime}}=\theta_{3}\)) is guaranteed across the slip line, as introduced by [PERSON] and [PERSON] [33] for planar detonations. However, in rotating detonation combustors, the flow conditions just in front of the oblique shock wave are not known beforehand and need to be updated each time the solver iterates over the injection velocity. The oblique shock equations are used to retrieve the correct oblique shock-wave angle and the slip line, which ensures pressure equality. This then provides the needed boundary conditions to initiate the method of characteristics [20]. At each iteration, the pressure at the inlet wall (P\({}_{\rm wall}\)) is evaluated, and convergence is obtained if the pressure at the start of injection of the fresh mixture matches the injection pressure. At the end of the iterative procedure, absolute flow properties are computed for all the points. Figure 5b shows how the reduced model solves the flow field. The blue lines depict the characteristic lines from the method of characteristics, while the green lines show the oblique shock streamlines downstream of the oblique shock (in red). The reduced model needs a CPU time of 200 s on a laptop, while the 2D URANS required 40 h on 4 cores of an Intel Xeon -E5-2660 v3 processor @ 2.6 Ghz (64 GB RAM) on the RICE pundue cluster (built through a partnership with HP and Intel in April 2015 [34]).