# Numerical Study on the Route of Flame-Induced Thermoacoustic Instability in a Rijke Burner
## 2 Numerical Model
### Self-Excited Pressure Oscillation--The Exponential Growth of Thermoacoustic Instability
(2018, 2019), the analytic pressure modes and velocity models of the acoustic perturbation for the acoustically closed-open Rijke-type burner can be expressed by Equations (6) and (7):  
\[P_{n}=\cos((2n-1)\pi x/2\;\mathrm{L})n=1,2,3\cdots\infty \tag{6}\]  
\[V_{n}=\sin((2n-1)\pi x/2\;\mathrm{L})\;n=1,2,3\cdots\infty \tag{7}\]  
and the corresponding acoustic frequencies can be calculated by:  
\[f_{n}=\frac{2n-1}{4L}a\;n=1,2,3\cdots\infty. \tag{8}\]  
where \(f_{n}\) is the \(n\)-th acoustic frequency, and \(a\) is the sound speed. According to the Rayleigh criterion, oscillations are encouraged when heat fluctuates in phase with pressure perturbation ([PERSON] and [PERSON], 2019). It can be determined from Equation (6) that the second acoustic mode may be excited when the unsteady heat is located at the center of the tube length because the flame is near the antinode of the second acoustic mode, which has a three-quarter wave mode shape. The frequency of this mode is 168.8 Hz with the inlet parameter by solving Equation (8). The frequency component of the pressure oscillation from 1.4 s to 1.6 s is calculated and displayed in Figure 10. The excited frequency is 179 Hz, which is 6% higher than the predicted acoustic frequency of 168.8 Hz for the non-reacting flow. The small discrepancy is due to the flow in the tube being heated by the flame, which induced an increase in the sound speed. Furthermore, the increased sound speed resulted in an increase in the oscillation frequency. Thus, researchers should take the frequency increase effect into consideration when predicting the thermoacoustic instability frequency with the inlet parameters. To compare the simulated oscillating mode with the prediction, the dynamic pressure along the centerline at time 1.400445 s was normalized by the maximum pressure amplitude, meanwhile the velocity along the center line at time 1.40585 s was normalized by the maximum velocity amplitude. In Figure 11, the normalized pressure and axial velocity of this work are compared with the analytical second acoustic pressure mode and velocity mode, and the normalized pressure and axial velocity in reference ([PERSON] and [PERSON], 2019). The simulation pressure distribution along the centerline is in good agreement with the analytical results  
Figure 10: Fast Fourier transform (FFT) plot of the gauge pressure. and the results of [PERSON]. While for the velocity distribution, two areas appear to have significant differences between the numerical results and the analytic results. The first area is in the flame stabilizer, where the simulated velocity is two times the analytical velocity because the flame stabilizer occupied half of the total volume of the entire flame stabilizer zone. The other inconsistent area is on the outlet face, because it was set as a pressure outlet, and the mass flow in the tube must be conserved. To further study this problem, it will be better to impose constraints on the velocity derivative on the outlet boundary. Aside from these two areas, the simulated velocity distribution agrees with the prediction results. Figures 12 and 13 show the instantaneous axial pressure distribution and axial velocity along the centerline of the Rijke burner at 8 different times of a limit cycle: namely, at times T, T/8, T/4, 3T/8, T/2, 5T/8, 3T/4, and 7T/8. In Figure 12, it can be seen that the pressure node point was at 1/3 length, and the pressure antinode point occurred downstream of the flame stabilizer at approximately 2/3 length, which matches the theoretical pressure maximum location for a three-quarter wave shown in Figure 11. From Figure 13, one can see that the velocity node point occurred slightly before the 2/3 tube length, which is also the pressure summit point of the pressure distribution. Figure 11: Comparison of the simulated pressure and velocity along the centerline with the analytical prediction results. Figure 12: Pressure distribution along the centerline at different times of a limit cycle [Pa].