## 4 Results and Discussion
### Parameter Analysis of Flow Path in CRD Combustors  
Due to the coupled interaction between the chemistry reaction, detonation wave, shear wave, and oblique shock wave, amongst other factors, understanding the detailed variation and distribution of various thermodynamic parameters in CRD combustors is very complex, which hinders our understand of the pressure gain mechanism of CRD. Figure 6: Pressure ratios using different combustors. Figure 7: (**a**) \(\Delta S\) and (**b**) \(\Delta G\) using different combustors. Figure 8 shows the variation in pressure (average total pressure \(\overline{P}_{t}\) and average static pressure \(\overline{P}_{s}\)) and temperature (average total temperature \(\overline{T}_{t}\) and average static temperature \(\overline{T}_{s}\)) of the CRD combustor cross-section in the axial direction at \(t\) = 6920 \(\upmu\)s, which corresponds to the temperature field shadowgraph in Figure 4f. Figure 8 shows that due to the existence of chemical reactions between the range of 0-100 mm, both \(\overline{T}_{t}\) and \(\overline{T}_{s}\) increased continuously, to a maximum point of 2419.3 K and 2019.7 K, respectively. Then, the above two parameters remained relatively stable when the combustion ended and the oblique shock appeared. For \(\overline{P}_{t}\) and \(\overline{P}_{s}\), there was an initial increase due to the detonation wave. As the detonation combustion ended, the maximum of \(\overline{P}_{t}\) was obtained at the axial distance of about 80 mm. Then, in the transition region (combining deflagration, detonation, and shock) between the detonation wave and the oblique shock, \(\overline{P}_{t}\) began to drop due to the influence of several different states. Since combustion remained in the transition region, there was a distance difference between the maximum pressure and maximum temperature. Combined, the above results and the temperature field shadowgraphs in Figure 4f demonstrate that the height of the detonation wave and oblique shock wave was, respectively, about 80 mm and 100 mm. The transition region between the detonation wave and the oblique shock was about 20 mm. Moreover, Figure 9 shows the variations in \(\Delta S\) and \(\Delta G\) in the axial direction. The inlet parameters were used as baselines in the calculation of \(\Delta S\) and \(\Delta G\). The results showed that \(\Delta S\) increased somewhat linearly during the distance of 0-100 mm, which meant that \(\Delta G\) dropped rapidly. However, after combustion progress, both variables were continuously produced at a low rate due to the influence of the oblique shock wave. Figure 8: Pressure and temperature of CRD combustor cross-section in the axial direction. Figure 9: (**a**) \(\Delta S\) and (**b**) \(\Delta G\) of the CRD combustor cross-section in the axial direction. To further study the variation and distribution of various thermodynamic parameters in CRD combustors, Figure 10 presents temperature and entropy shadowgraphs containing typical streamlines. The streamlines were gained in a rotating coordinate system. When the flow field was stable in the rotating coordinate system, the streamlines coincided with the path lines, so the parameters along the path line can be represented by those along the streamline in the rotating coordinate system. This approach was applied and explained in more detail in [15; 16]. As the left and right sides of computational domain were periodic boundaries, one streamline could be divided into two parts with the same streamline number, as shown in Figure 10. In Figure 10a, all the streamlines passed through the detonation wave. Streamline 1 reached the outlet when it was about to pass through the oblique shock. Streamline 2 passed through the oblique shock wave beside the detonation wave. Streamline 3 passed through the lowest portion of the detonation wave and Streamline 4 passed through the highest portion of the detonation wave in the axial direction.