# Effects of Diverging Nozzle Downstream on Flow Field Parameters of Rotating Detonation Combuster
## 3 Numerical Results and Discussion  
### Effect of Diverging Nozzle on the Formation and Propagation Process of Detonation Wave  
Figures 6 and 7 respectively illustrate the typical temperature and pressure field evolutions of RDCs with \(\theta\) = 0\({}^{\circ}\) from ignition to stable propagation. The size of the ignition zone is 20 mm \(\times\) 20 mm \(\times\) 20 mm and it is located near the RDC inlet region, as shown in Figures 6a and 7a. It can be seen that when a clockwise CJ detonation kernel is initiated, the surrounding premixed stoichiometric methane-air mixture is ignited and combusted. After a short period of development, there are two kind of waves formed in the combustor (Figures 6b and 7b). One is the rotating detonation wave propagating clockwise, whose shock wave front is tightly coupled to the chemical reaction front, and another is a reverse wave composed of weak pressure wave and deflagration flame front. With the advancement of the detonation process, the reverse weak pressure wave and deflagration flame front begin to decouple. The detonation wave continues to propagate along the ignition direction and further develops, as shown in Figures 6c and 7c. After further development, the waves in RDC begin to collide (shown in Figures 6f and 7d). When \(t\) = 320 \(\upmu\)s, the forward shock wave collides with the reverse weak pressure wave. Soon afterwards, when \(t\) = 400 \(\upmu\)s, the detonation flame front collides  
Figure 4: Changing of static pressure with time at monitoring point (127.4 mm, 0 mm, 147 mm). Figure 5: The relationship between detonation cell size and initial pressure provided by Purdue [49]. with the reverse deflagration flame front. For the leading detonation wave and the uncoupled reverse distribution, the pressure wave propagates faster than the flame front, which causes that the collision of the pressure waves to occur before that of the flame front. After that, with the collision and reaction generated by detonation combustion, the flow field enters an unstable stage, described in Figures 6g and 7g. Subsequently, due to the presence of the low pressure region before the detonation wave near the inlet, the fresh premixture can enter into the combustor to maintain the forward propagation of the detonation wave, while part of original detonation wave near the outlet degenerates into the oblique shock due to the lack of energy supply after the previous collisions. Finally, after \(t=920\)\(\upmu\)s, the detonation wave basically develops to achieve stable self-sustaining propagation in the combustor. After analyzing the wave evolution process in RDC with \(\theta=0^{\circ}\), the comparative analysis of RDC with diverging nozzle is carried out in this section (taking \(\theta=30^{\circ}\) as an example). From the evolution of temperature and pressure in Figures 8 and 9, it can be seen that the process from ignition to the stable self-sustaining propagation of detonation wave is almost consistent for the RDC with and without diverging nozzles. However, the collisions time of pressure waves and flame fronts in the RDC with \(\theta=30^{\circ}\) is delayed compared to that of \(\theta=0^{\circ}\) (about 20 \(\upmu\)s and 60 \(\upmu\)s, respectively). The main cause of this phenomenon is the width between the inner wall and outer wall, which becomes bigger at the diverging section. The movement of the wall boundary causes flow in the reaction zone after the shock wave to expand [50; 51], which leads to bending of the detonation wave front and the decrease of the propagation velocity. Then the collisions of pressure waves and flame fronts are delayed. Figure 8: Temperature contour in the RDC with \(\theta=30^{\circ}\). Figure 7: Pressure contour in the RDC with \(\theta=0^{\circ}\). It can be speculated from the above analysis that although the diverging nozzle causes collision time to slightly lag, the formation and propagation process of detonation waves in RDCs follow basically the same laws.