# Effects of Diverging Nozzle Downstream on Flow Field Parameters of Rotating Detonation Combuster
## 2 Numerical Model and Method
### Numerical Method  
To simulate the complex chemical reaction with detonation and compressible flow inside the combustor, it was necessary to introduce some assumptions. The premixed gas was assumed to be compressible ideal gas, ignoring transmission properties such as viscosity, thermal conduction, and diffusion [37; 38; 39]. Thus, the three-dimensional Euler governing equation suitable for numerical simulation of detonation waves can be obtained as follow, which has been verified in many papers [40; 41; 42].  
\[\frac{\partial\mathbf{\Phi}}{\partial t}+\frac{\partial\mathbf{U}}{\partial x}+\frac{ \partial\mathbf{V}}{\partial y}+\frac{\partial\mathbf{W}}{\partial z}\ =\ \mathbf{S} \tag{1}\]  
where the conservative variable vector \(\Phi\), the convective flux vectors \(U\), \(V\), and \(W\), and the source vector \(S\) are respectively defined by column vectors as,  
\[\mathbf{\Phi}\ =\ (\rho,pu,\rho v,\rho w,\rho e_{t},\rho_{1},...,\rho_{N_{s}-1})^{T} \tag{2}\]  
\[\mathbf{U}\ =\ \left(\rho u,\rho u^{2}+P,\rho vu,\rho vu,(\rho e_{t}+P)u,\rho_{1}u,...,\rho_{N_{s}-1}u\right)^{T} \tag{3}\]  
Figure 1: (**a**) Three-dimensional geometry; (**b**) size of the premixed rotating detonation combustor (RDC).  
\[\mathbf{V}\,=\,\left(\rho v,\rho uv,\rho v^{2}+P,\rho wv,(\rho e_{t}+P)v,\rho_{1}v,..., \rho_{N_{s}-1}v\right)^{T} \tag{4}\]  
\[\mathbf{W}\,=\,\left(\rho w,\rho uw,\rho vw,\rho w^{2}+P,(\rho e_{t}+P)w,\rho_{1}w,...,\rho_{N_{s}-1}w\right)^{T} \tag{5}\]  
\[\mathbf{S}\,=\,\left(0,0,0,0,\omega_{1},\ldots,\omega_{N_{s}-1}\right)^{T} \tag{6}\]  
where \(\rho\) is the total density of premixed mixture; \(\rho_{j}\) is the density of the species \(j\); \(N_{s}\) is the total number of the species; \(u\), \(v\), and \(w\) are the velocities in the \(x\), \(y\), and \(z\) directions, respectively; \(p\) is the pressure that can be calculated by the ideal gas equation of state; and \(e_{t}\) is the total energy per unit mass.  
\[e_{t}\,=\,e+\frac{1}{2}\left(\mu^{2}+v^{2}+w^{2}\right) \tag{7}\]  
where \(e=e(\rho_{j},T)\) is the internal energy of the premixed mixture.  
Since the viscosity was ignored, the laminar finite rate model was selected to solve the detonation problem [43; 44]. Therefore, the net production rate of each chemical species (\(\omega_{j}\)) can be obtained by  
\[\omega_{j}\,=\,M_{j}\sum_{r\,=\,1}^{N_{n}}\left(v^{\prime\prime}_{jr}-v^{ \prime}_{j,r}\right)\left(\kappa^{\prime\prime}_{r}\prod_{j\,=\,1}^{N_{n}} \left[\xi_{j}\right]^{v^{\prime}_{jr}}-\kappa^{\prime}_{r}\prod_{j\,=\,1}^{N_{ n}}\left[\xi_{j}\right]^{v^{\prime\prime}_{jr}}\right)\quad(j\,=\,1,...,N_{S}) \tag{8}\]  
where \(v^{\prime\prime}_{jr}\), and \(v^{\prime}_{j,r}\) are the stoichiometric coefficient of the species \(j\) in forward and backward reactions \(r\), respectively. \(N_{n}\) is total number of the elementary reactions. \(\xi_{j}\) is the molar concentration of the species \(j\). \(\kappa^{\prime\prime}_{r}\) and \(\kappa^{\prime}_{r}\) are the forward and backward reaction rate constant for reaction \(r\). The reaction rate constant is expressed by the following Arrhenius formula:  
\[\kappa_{r}\,=\,A_{r}T^{\beta_{r}}\,\exp\left(-\frac{E_{r}}{RT}\right) \tag{9}\]  
where \(A_{r}\), \(\beta_{r}\), and \(E_{r}\) are the pre-exponential factor, temperature exponent, and activation energy in the reaction, respectively. \(R\) is the universal gas constant.  
In this study, the numerical simulations were performed by using the commercial software ANSYS FLUENT 14.0. Density based Navier-Stokes solver was utilized to solve the above-mentioned three-dimensional Euler equation. In order to capture the main propagation characteristics of shock waves and detonation waves effectively, the flux term was dispersed by the advection upstream splitting method (AUSM). Furthermore, the third-order upwind scheme and the four-step Runge-Kutta method with second-order accuracy were applied to discrete the convective term and time term, respectively.