# Effects of Diverging Nozzle Downstream on Flow Field Parameters of Rotating Detonation Combuster
## 4 Conclusions  
Using methane-air as the reactive mixture, this paper numerically investigated the formation and propagation process of detonation wave and parameter changes of RDCs under different diverging nozzles in detail. The main conclusions can be drawn as follows.  
(1) The diverging nozzle in this study did not obviously affect the formation and propagation process of detonation waves in the RDC. However, the collisions time of flame front and pressure waves were delayed in the process of rotating detonation wave formation.  
(2) The pressure gain performance of the RDC was weakened by the diverging nozzle, which was reduced from 1.427 to 1.392, 1.270, and 1.129, respectively, but it still maintained a certain pressure gain performance. In addition, with the increasing of diverging angles, the high-pressure load in the RDC was alleviated, and the static pressure distribution of the RDC outlet became more uniform, which was beneficial to improving the aerodynamic load on downstream components.  
(3) Under the diverging nozzle, the static temperature distribution in the combustor was also improved. This is favorable to the stability control and component design of RDC-based engines. At the same time, the Mach number of the outlet was further enhanced as the diverging angle increased.  
(4) When the diverging angle increased to 30\({}^{\circ}\), further increases in the angle did not significantly affect the parameters near the inner wall of the RDC.  
[PERSON] was responsible for the results, analysis, and draft writing. [PERSON] was responsible for the numerical approach. [PERSON] and [PERSON] performed the numerical simulation of the RDC. [PERSON] and [PERSON] did the model validation.  
This research was funded by the Fundamental Research Funds for the Central Universities, grant number HEUCFP201719.  
The authors declare no conflict of interest.  
**Nomenclature**  
\(e\): internal energy (J/kmol)  
\(e_{t}\): total energy (J/kmol)  
\(E\): activation energy in the reaction (J/kmol)  
\(\overline{g}\): mass flow rate (kg/s)  
\(h\): height of detonation wave (mm)  
\(\overline{h}_{d}\): time-averaged height of detonation wave (mm)  
\(\Delta i\): grid size (mm)  
\(j\): the label of species  
\(l\): the ruler of the width between the inner wall and outer wall  
Figure 23: The circumferential outlet Mach number distribution of the RDC at \(t=3980\)\(\mu\)s.  
\begin{tabular}{l l} \(M\) & molecular weight (kg/kmol) \\ \(N_{n}\) & total number of the reactions \\ \(N_{s}\) & total number of the species \\ \(p\) & pressure (MPa) \\ \(\overline{p}\) & average pressure (MPa) \\ \(r\) & chemical reaction \\ \(R\) & universal gas constant (J/(kmol\(\cdot\)K)) \\ \(S\) & source vector \\ \(t\) & time (\(\upmu\)s) \\ \(t_{rot}\) & propagating period of detonation wave (\(\upmu\)s) \\ \(\Delta t\) & time step (\(\upmu\)s) \\ \(T\) & Temperature (K) \\ \(\overline{T}\) & average temperature (K) \\ \(u,v,w\) & velocities in the x, y, and z directions (m/s) \\ \(U,V,W\) & convective flux vectors \\
**Greek letters** \\ \(\beta\) & temperature exponent \\ \(\gamma\) & specific heat ratio \\ \(\eta_{c}\) & efficiency \\ \(\kappa\) & reaction rate constant \\ \(\xi\) & molar concentration (kmol/m\({}^{3}\)) \\ \(\pi_{c}\) & total pressure ratio \\ \(\rho\) & total density of the premixed gas (kg/m\({}^{3}\)) \\ \(\rho_{j}\) & density of the specie j (kg/m\({}^{3}\)) \\ \(v\) & stoichiometric coefficient \\ \(\Phi\) & physical variable vector \\ \(\omega\) & production rate (kmol/(m\({}^{3}\cdot\)s)) \\
**Superscripts** \\, & backward reaction \\, & forward reaction \\
**Subscripts** \\ \(cr\) & critical \\ \(d\) & detonation \\ \(w\) & injection \\ \(in\) & inlet of combustor \\ \(out\) & outlet of combustor \\ \(1\) & inner wall of combustor \\ \(2\) & outer wall of combustor \\ \end{tabular}