## 3 Results  
### Navier-Stokes Solution vs. Reduced-Order Model  
To resolve the boundary layer within the three-dimensional rotating detonation combustor, we modelled the combustor as an unrolled annulus, as shown in Figure 10a. The annulus has a thickness of 5 mm, and inlet stagnation conditions equal to 330 K and 0.5 MPa with a turbulent intensity of  
Figure 9: **(a)** The log–log plot of the numerical extrapolated error as a function of the first-layer wall thickness; **(b)** log–log plot of the numerical error as a function of the timestep; **(c)** log–log plot of the extrapolated numerical error in terms of the characteristic mesh size H.  
5%. For this 3D simulation, turbulent equations were solved via the URANS k-omega SST model. The lateral walls were kept at a constant temperature of 450 K. The first grid point is 2.0 \(\times\) 10\({}^{-8}\) m away from the wall, to ensure a y+ below one and avoid the need for wall functions. This resulted in a three-dimensional mesh of 6.9 million cells. In order to ensure a converged result, we plot the mean heat-flux level at consecutive timesteps in Figure 10b and observe that the heat-flux fluctuation is below 0.4%. Total computational time was around 400 h on 16 cores of an Intel Xeon -E5-2660 v3 processor @ 2.6 Ghz (64 GB RAM). To speed up the computational time, the simulations were initially run on a two-dimensional mesh (described in Section 2.1 until full convergence (3.5 ms, as shown in Figure 2). Afterwards, the solution was interpolated into a 3D mesh and run for another 0.5 ms, which represented three passages of the detonation wave.  
Figure 11a illustrates the iso-heat-flux contour (in logarithmic scale) at a certain timestep. At the detonation front, the peak heat flux is around 200 MW/m\({}^{2}\), followed by a smooth decrease set by the subsequent flow expansion. Since the temperature of the fresh mixture is below the wall temperature, a negative heat flux is measured during refill time of the mixture. The unsteady sweeping of the oblique shock wave augments the total temperature and total pressure, and hence increases the heat in this area. Figure 11b compares the time-averaged heat flux from both models, at different streamwise positions. Our model predicts the same level of averaged heat flux in the region of the detonation front with a maximum value up to 6 MW/m\({}^{2}\) and a decrease towards the outlet, at 0.1 m, and the model under-predicts the 3D URANS by 16%. The bulk heat flux estimated by the URANS solution is 4.8 MW/m\({}^{2}\) and 5.1 MW/m\({}^{2}\) for the reduced-order model. Interestingly, the experimental data reported in [23] shows a similar magnitude of heat flux along the axis of the combustor. The maximum bulk heat-flux variation with the 2D MOC code was about 40% (0.4 MW/m\({}^{2}\)) between the injection location and the triple point, while our 3D simulations predicted a variation of 33%.  
Figure 11: (**a**) Wall heat-flux contour (3D URANS solution). (**b**) Comparison of the averaged heat flux computed with the reduced-order model and 3D URANS simulation.  
Figure 10: (**a**) Numerical domain with mesh refinement close to the boundary layer. (**b**) Mean time-averaged heat-flux evolution as a function of time.