# Numerical Study on the Route of Flame-Induced Thermoacoustic Instability in a Rijke Burner
## 2 Numerical Model
### Unsteady Flow Field in the Limit Cycle State  
Figure 14 illustrates the axial velocity distribution and streamline at different times within a period of one limit cycle, where the yellow regions represent velocities above zero, the blue regions represent velocities below zero, and the gray regions represent the solid zone. Once the system is excited into the periodical oscillating state, the flow in the tube does not remain steady and uniform; rather, it begins oscillating back and forth. At the beginning of a period, the velocity was positive in the left 2/3 of the tube length and negative in the right 1/3 of the tube length, as shown in Figure 14a. At approximately 2/3 of the tube length, which is a node point of the velocity mode in the 1-D analysis, the axis velocity was zero. This result is in good agreement with the predicted velocity mode, and the velocity along the axis is in agreement with the velocity at time T in Figure 13. In the flame zone, due to the influence of the high-temperature flame, the flow was compressed and distorted, inducing the formation of a clockwise vortex after the flame area, as shown in Figure 14c. With the growth of this clockwise vortex, a counterclockwise vortex was formed at the inner wall of the left half of the tube in Figure 14d. These vortices were enlarged and changed the direction of flow in Figure 14e, making the flow negative on the left 2/3 of the tube length and positive on the right of the 1/3 tube length at time T+T/2, as shown in Figure 14f. After time T+T/2, the flow was compressed near the inlet face and was expanded in the flame zone. When the negative flow was heated by the flame, followed by the formation of new vortices, and the subsequent development and collapse of these new vortices as shown in Figure 14g-j, the direction of flow changed in the opposite direction. A new period began. The flow field variation in a limit cycle shown in Figure 14 demonstrates that vortices emerged and collapsed when thermoacoustic instability occurred. Thus, the computation of thermoacoustic instability should consider a turbulent flow, even in low Reynolds number cases. Figures 15 and 16 show the temperature distribution and kinetic rate of the reaction at T+T/2 and T+T/8T in a period, respectively. The distance between the flame and the flame stabilizer outlet was far in Figures 15a and 16a, but close in Figures 15b and 16b. In Figures 15a and 16a, as time proceeds, the positive flow velocity moved the flame downstream, as shown in Figure 14c, and the flow before the 2/3 tube length was compressed to the maximum limit. Then, the pair of vortices broke down, and the flow was reversed, as shown in Figure 14d, where a vortex formed and the flow velocity decreased with time,  
Figure 13: Axial velocity distribution along the centerline at different times of a limit cycle [m/s]. causing the flow to become negative. The negative flow pushed the flame in the upstream direction, reaching the nearest point shown in Figures 15b and 16b, at which time the flow in the left tube was compressed to the maximum limit, new separation points were generated, and new vortices changed the flow direction again, making the flame surface move downstream. Thus, it can be concluded that the distance between the flame and flame stabilizer varies with the change in the axial velocity in the limit cycle state; the flow between the flame stabilizer and the 2/3 tube length is compressed when the axial velocity is positive, while it is expanded when the axial velocity is negative. During this periodic compression and expansion, the streamline is stretched and distorted, leading to the periodic formation and disappearance of vortices in the tube. This periodic behavior influences both the shape and distance of the flame via feedback; as a result, the forward feedback between the heat release and flow perturbation results in thermoacoustic instability. Figure 14: _Cont._  
Figure 14: Cont. Figure 14: Axis velocity distribution in a cycle. Figure 15: Temperature distribution at different times.