# Effects of Diverging Nozzle Downstream on Flow Field Parameters of Rotating Detonation Combuster
## 3 Numerical Results and Discussion
### Effect of Different Diverging Nozzles on Temperature Parameters
It is obvious that the static temperature of the RDC outlet decreases significantly with the increasing of the diverging angle. Figure 19: The averaged static temperature in different axial cross-sections of the RDC at \(t\) = 3980 \(\upmu\)s.  
Figure 20: The outlet static temperature contour of the RDC with different \(\theta\) at \(t\) = 3980 \(\upmu\)s.  
Figure 21 presents the circumferential static temperature distribution along the radial direction of the three typical heights. Obviously, a larger \(\theta\) leads to a greater degree temperature reduction at all selected heights. The result shows that with \(\theta\) changing from 0\({}^{\circ}\) to 45\({}^{\circ}\), the maximum outlet temperature decreases from 3126.77 K to 2181.53 K in the 1% height, from 3067.37 K to 2025.03 K in the 50% height, and from 3126.77 K to 2089.01 K in the 99% height. The analysis shows that 50% height results in a higher temperature reduction. The reasonable explanation is that the diverging nozzle establishes a wider space for the mixture to expand, mix, and exchange heat with the surrounding airflow, causing its temperature to drop more than that at the other locations near the wall. In addition, it can be observed that when \(\theta\) is increased from 30\({}^{\circ}\) to 45\({}^{\circ}\), the difference between the two curves becomes inconspicuous, and when \(l\) = 1% height, the two temperature curves almost coincide. It indicates that the static temperature near the inner wall is no longer affected by the increasing of \(\theta\) significantly when the diverging angle increases to a certain degree. As a reflection of the change in the axial thermodynamic parameters of the RDC, Figure 22 demonstrates the variations of Mach number in different axial cross-sections of RDC at \(t\) = 3980 \(\upmu\)s. For the RDC with \(\theta\) = 0\({}^{\circ}\), the Mach number at the inlet is 0.62, and then it continuously increases to 1.19 at the outlet, which directly indicates that there is a transonic process inside the combustor. For the RDC with \(\theta\) = 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\), the Mach number is always higher than that of \(\theta\) = 0\({}^{\circ}\). Especially in the diverging section, the Mach number increases significantly rapidly, and the maximum value at the outlet ranges from 2.01 to 2.39 with \(\theta\) changing from 15\({}^{\circ}\) to 45\({}^{\circ}\). This is because the supersonic flow is further accelerated in the diverging section, while the static temperature decreases, resulting in the  
Figure 21: The circumferential outlet static temperature distribution of RDC at \(t\) = 3980 \(\upmu\)s.  
increasing of the Mach number. Since the outlet of RDC with \(\theta=0^{\circ}\) has flowed at supersonic speed, the supersonic flow of the outlet due to the diverging nozzle is reasonable and acceptable. Figure 23 shows the circumferential distribution of the Mach number in the three typical heights at the outlet of the RDC. Combined with the outlet Mach number displayed in Figure 22, there is an interesting finding that although the average Mach number at the outlet is greater than 1 in the RDC with \(\theta=0^{\circ}\), there exist some regions where the Mach number is less than 1 (as shown in Figure 23). This kind of local supersonic flow poses a serious challenge for controlling the outlet stability and design of RDC-based gas turbines. However, once the diverging nozzle is adopted, the airflows of the outlet reach a coincident supersonic flow state at all outlet positions, and the outlet flow of the RDC becomes relatively uniform and controllable. It is worth noting that, in combination with the previous Figures 15 and 21, the parameters the near inner wall of the RDC outlet are not significantly affected by the diverging nozzle when \(\theta\) is bigger than \(30^{\circ}\). This is because the detonation wave propagates rapidly in the RDC. When the outer wall expands to a certain extent, the oblique shock wave is late to expand and transfer heat in the large space.