## 2 Numerical Model and Methods
### Numerical Method  
In this study, the numerical simulations were performed using commercial software, namely ANSYS Fluent (14.0, ANSYS, Pittsburgh, Commonwealth of Pennsylvania, US, 2011). Considering the complex chemistry reaction with detonation and the basic assumptions of compressible ideal gas, the density based Navier-Stokes solver was applied to solve the two-dimensional unstable Euler  
Figure 1: **(a)** Three-dimensional geometric model; and **(b)** Two-dimensional computational domain of CRD combustors. CJ, Chapman–Jouquet. Equation (1). Viscosity and thermal diffusion were ignored. Many past papers [15; 16; 17; 18; 28; 29] have verified the applicability of the Euler solution in numerical simulations of detonation waves. \[\frac{\partial\mathbf{\Phi}}{\partial t}+\frac{\partial\mathbf{U}}{\partial x}+\frac{ \partial\mathbf{V}}{\partial y}=\mathbf{\Omega} \tag{1}\]  
where the dependent variable vector \(\mathbf{\Phi}\), the convective flux vectors \(\mathbf{U}\) and \(\mathbf{V}\), and the source vector \(\mathbf{\Omega}\) are respectively defined as:  
\[\mathbf{\Phi}=\big{(}\rho,\rho u,\rho v,\rho e_{t},\rho_{j}\big{)}^{T} \tag{2}\]  
\[\mathbf{U}=\Big{(}\rho u,\rho u^{2}+P,\rho uv,(\rho e_{t}+P)u,\rho_{j}u\Big{)}^{T} \tag{3}\]  
\[\mathbf{V}=\Big{(}\rho v,\rho uv,\rho v^{2}+P,(\rho e_{t}+P)v,\rho_{j}v\Big{)}^{T} \tag{4}\]  
\[\mathbf{\Omega}=\big{(}0,0,0,0,\omega_{j}\big{)}^{T} \tag{5}\]  
where \(\rho\) is the density of the premixture; \(\rho_{j}\) is the density of the species \(j\); \(u\) and \(v\) is, respectively, velocity in the X and Y direction; \(P\) is the pressure that can be calculated by the thermal equation of state for a perfect gas; \(P=\rho RT/M\); \(M\) is molecular weight; and \(R\) is the gas constant. Moreover, the total internal energy \(e_{t}\) is defined as,  
\[e_{t}=e+\frac{1}{2}\Big{(}u^{2}+v^{2}\Big{)} \tag{6}\]  
where the caloric equation of state for the internal energy of the reacting mixture is, \(e=e\big{(}\rho_{j},T\big{)}\). Since viscosity was ignored in this study, a laminar finite rate model is acceptable for combustion with relatively small turbulence-chemistry interactions (such as detonation) according to the author's experience. Additionally, the numerical results of [PERSON] [30], [PERSON] et al. [31], and [PERSON] et al. [32] indicate that a laminar finite rate model can effectively describe many fundamental combustion and detonation phenomena. Therefore, a laminar finite rate model was selected as the chemical reaction model in this study. The production rate of each chemical species, \(\omega_{j}\), was calculated by combining the elementary chemical reactions in the kinetic model. \[\omega_{j}=M_{j}\sum_{\gamma=1}^{N_{n}}\Big{(}v^{\prime\prime}_{j,\gamma}-v^{ \prime}_{j,\gamma}\Big{)}\left(\kappa^{\prime\prime}_{\gamma}{\prod_{j=1}^{N_ {n}}\big{[}\xi_{j}\big{]}^{v^{\prime}_{j,\gamma}}}-\kappa^{\prime}_{\gamma}{ \prod_{j=1}^{N_{n}}\big{[}\xi_{j}\big{]}^{v^{\prime}_{j,\gamma}}}\right)(j=1,...,N_{s}) \tag{7}\]  
where \(v^{\prime\prime}_{j,\gamma}\) and \(v^{\prime}_{j,\gamma}\), respectively, represent the stoichiometric coefficient of species \(j\) in forward and backward reactions \(\gamma\); \(N_{n}\) represents the total number of the reactions; \(N_{s}\) represents the total numbers of the species; \(\xi_{j}\) represents the molar concentration of the species \(j\); \(\kappa^{\prime\prime}_{\gamma}\) and \(\kappa^{\prime}_{\gamma}\), respectively, represents the forward and backward reaction rate constant for reaction \(\gamma\); and the reaction rate constant is calculated by the following Arrhenius formula. \[k_{\gamma}=A_{\gamma}T^{\beta_{\gamma}}\exp\left(-\frac{E_{\gamma}}{RT}\right) \tag{8}\]  
where \(A_{\gamma}\), \(\beta_{\gamma}\) and \(E_{\gamma}\), respectively, represent the pre-exponential factor, temperature exponent, and activation energy in the reaction \(\gamma\). In addition, to effectively capture the shock and detonation waves, the flux term was dispersed by the advection upstream splitting method (AUSM) [33].