# Effects of Diverging Nozzle Downstream on Flow Field Parameters of Rotating Detonation Combuster
## 3 Numerical Results and Discussion
### Effect of Different Diverging Nozzles on Pressure Parameters  
Compared with the traditional isobaric combustor, the most attractive advantage of the RDC is the pressure gain performance. Therefore, it is an inevitable consideration to investigate the pressure distribution and pressure gain characteristic of RDCs with diverging nozzles. Figure 11 presents the variation of the RDC outlet total pressure with time. It can be clearly seen that the periodic fluctuations are consistent with different \(\theta\). However, with the adoption of the diverging nozzle, the total pressure exhibits a certain hysteresis compared to \(\theta\) = 0\({}^{\circ}\) in the time domain when the wave propagates stably. It is because the detonation wave collisions in RDCs with diverging nozzles occurs after that of \(\theta\) = 0\({}^{\circ}\), which is consistent with the analysis of the detonation wave formation process. It is also obvious that outlet total pressure decreases as the diverging angle \(\theta\) increases, and they are all smaller than the value of \(\theta\) = 0\({}^{\circ}\). To accurately define the pressure gain performance of the RDC, its pressure ratio can be obtained by:  
\[\pi_{C}\ =\ \overline{P}_{t,out}/\overline{P}_{t,in} \tag{12}\]  
where \(\overline{P}_{t,in}\) and \(\overline{P}_{t,out}\) are the time-averaged mass-weight total pressure at the inlet and outlet of RDC, respectively. Since the injection pressure of premixed mixture is a constant value of 1.209 MPa, the pressure ratio (\(\pi_{c}\)) of the RDC with \(\theta\) = 0\({}^{\circ}\), 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\) are 1.427, 1.392, 1.270, and 1.129, respectively. The results demonstrate that \(\pi_{c}\) decreases with the increasing of \(\theta\). Considering that the positive shock wave (indicated by \(h_{d}\)) is basically not affected by the diverging nozzle, it can be inferred that the effect of positive shock wave is not weakened obviously. Thus, it is necessary to investigate the distribution of pressure along the axial direction of the RDC. Figure 10: Changing of detonation wave height near (**a**) inner wall and (**b**) outer walls. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\theta\) & **0\({}^{\circ}\)** & **15\({}^{\circ}\)** & **30\({}^{\circ}\)** & **45\({}^{\circ}\)** \\ \hline \(h_{d}\)/mm & 106.68 & 106.54 & 105.23 & 105.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Time-averaged detonation wave height of RDC with different diverging nozzle. Figure 12 shows the variations of the averaged total pressure in different cross-sections of the RDC along the axis direction (here \(t=3980\)\(\upmu\)s). For the total pressure of mixture, its value gradually increases under the action of the detonation wave, reaching a maximum at about \(X=120\) mm. Obviously, in the tail region of RDC, the detonation wave degenerates into the oblique shock wave after an unstable stage, thereby reducing the ability of pressure gain, which causes the total pressure to decrease. The detailed mechanism causing pressure loss can be found in our previous studies [10; 11; 36]. It can be seen from the curves that the total pressure of RDC with \(\theta=15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\) is all lower than that of \(\theta=0^{\circ}\) from the initial stage, and their decline degree gradually increases with the axial distance and reaches the maximum at the outlet. Combined with the points discussed above, it can be concluded that although the diverging nozzle does not affect the time-averaged detonation wave height, it does reduce the overall total pressure of RDC from the inlet to the outlet. Figure 11: The time-varied outlet total pressure of the RDC with different \(\theta\). Figure 12: The averaged total pressure in different axial cross-sections of the RDC at \(t=3980\)\(\upmu\)s.  
Considering that the static pressure at the combustor outlet directly affects the aerodynamic load of downstream turbine blades, this is an important index during the design of RDC-based engines.