# Numerical Study on the Route of Flame-Induced Thermoacoustic Instability in a Rijke Burner
## 2 Numerical Model
### Mathematical Model  
The governing equations of this model include conservation equations of mass, kinetics, species, and energy, combined with turbulence equations and radiation equations. The conservation equations are expressed by  
\[\frac{\partial(\rho\Phi)}{\partial t}+div(\rho u\Phi)=div(\Gamma grad\Phi)+S_{\Phi} \tag{1}\]  
where \(\rho\), \(t\), and \(u\) are the density, time, and velocity, respectively; \(\Phi\) represents the dependent transport variables [27]; \(\Gamma\) represents the diffusion coefficient; and \(S_{\Phi}\) is the source term of the variable \(\Phi\).  
The governing equations are solved with the finite volume method. Although at the steady state of the combustion field the average velocity is approximately 0.026 m/s, the corresponding Reynolds number is approximately 120, and the Mach number is 1.5 \(\times\) 10\({}^{-4}\) in this model, indicating that this is a laminar flow for the non-reacting condition. However, influenced by the highly nonlinear chemical reaction, it is inappropriate to ignore the turbulence effect in this model. The k-\(\epsilon\) model [28] is the most common model used in CFD to simulate mean flow characteristics for turbulent flow conditions. Here, the k-\(\epsilon\) model with enhanced wall treatment is used to simulate the reacting flow.  
The species is determined by the species transport model, while the finite-rate/eddy dissipation model is applied to compute the interaction between turbulence and chemistry. Besides, the material properties used are listed in Table 1 of Appendix A.  
The methane/air reaction is complicated and involves dozens of reactions and hundreds of elementary reactions. Several reaction mechanisms have been proposed to accurately describe the combustion process. The detailed reaction mechanisms are highly precise [29], while the computational costs are increased. In a numerical comparison between the different methane/air combustion mechanisms, [PERSON] [30] pointed out that the one-step mechanism agrees with experimental results quantitatively for fuel-lean conditions. [PERSON] et al. [31] concluded that moving from a one-step global mechanism to a two-step mechanism does not significantly increase the accuracy of predictions. According to the mentioned points, the one-step reaction mechanism is used for the reaction of methane and air, and is expressed as  
\[CH_{4}+2(O_{2}+3.76N_{2})\overset{K_{G}}{\rightarrow}CO_{2}+2H_{2}O+7.52N_{2} \tag{2}\]  
The rate of fuel consumption is given by the global Arrhenius equation:  
\[\frac{\mathrm{d}[CH_{4}]}{\mathrm{d}t}=-Aexp\left(\frac{-E_{a}}{R_{u}T}\right) [CH_{4}]^{m}[O_{2}]^{n} \tag{3}\]  
where the parameters \(A\), \(E_{a}/R_{u}\), \(m\) and \(n\) are given in Table 2 of Appendix A [32; 33].  
Due to the high temperatures of the flame, radiative heat transfer becomes one of the dominant mechanisms  
are compared with the previous numerical modeling of [PERSON] [23], and the results are presented in Figure 3. The convergence limit is assumed when the residual values are less than \(10^{-6}\).  
As shown in Figure 3, the temperature distributions before the flame stabilizer almost converged to the same value, while in the flame stabilizer area and the flame zone, the temperature increased sharply because of the heat released by the flame. The result of the 67,056-node scheme is closer to the result presented by [PERSON] [23] than the other two grid schemes.  
The peak temperatures of the computational grids with 26,312, 67,056, and 195,233 nodes were approximately 1804 K, 1847 K, and 1908 K, respectively, and the outlet temperatures were 408 K, 331 K, and 354 K, while the peak temperature and outlet temperature in [PERSON]'s simulation were 1927 K and 320 K, respectively. In addition, for the outlet temperature, the result of the 67,056-node scheme is much closer to the result of [PERSON]. Consequently, the mesh with 67,056 nodes was selected as the best computation grid for computational modeling.