# Numerical Study on the Route of Flame-Induced Thermoacoustic Instability in a Rijke Burner
## 4 Conclusions  
A two-dimensional CFD simulation of a Rijke burner was conducted in this work. The self-excited thermoacoustic instability was numerically captured and analyzed, and the following conclusions are drawn.  
1. The onset and growth of the thermoacoustic instability were successfully captured. The simulated growth of the instability \(G\) was \(1.22\times 10^{-2}\) rad/s\({}^{2}\), which is less than the maximum possible growth rate.
2. The simulated oscillation frequency and mode were in good agreement with the 1-D analytical results. The simulated pressure oscillation frequency was 6% higher than the analytical results based on the inlet parameters, since the flow temperature and the sound speed were increased by the reaction heat. The normalized pressure and normalized velocity along the center line were in agreement with the prediction results of the second acoustic mode.
3. As the system was excited into a limit cycle oscillation state, the amplitude of the velocity perturbation was larger than the average steady-state flow; thus, in some local areas, the flow velocity may be below zero, resulting in local backward flow in the burner.
4. In the limit cycle state, the distance between the flame and honeycomb outlet varied with the periodic change in the axial velocity. The flow between the flame stabilizer and the 2/3 tube length was periodically compressed and expanded, while the streamline was periodically stretched and distorted, inducing the periodic formation and disappearance of vortices in the tube. This is a typical behavior of flame-induced vortices, which leads to the flow field transition from laminar flow to turbulent flow.
5. The periodic behavior of the flow field induced by the unsteady heat release influenced both the shape and distance of the flame via feedback; thus, the forward interaction between the heat release and flow perturbation resulted in thermoacoustic instability.  
Figure 16: Kinetic rate of reaction at different times.  
**Author Contributions:** N.D.: Conceptualization, methodology, validation, data curation, writing--original draft; software; J.Z.: project administration, funding acquisition, supervision, writing--review and editing; Y.D.: writing--review and editing, supervision. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research was funded by \"The National Natural Science Foundation of China (No.517/5437)\", \"The Key Research and Development Program of Shaanxi Province (No.2017 ZDCXL-GY-02-02)\", \"The Key Laboratory of Compressor of China (No.SKL-YSJ201802)\", \"The World-Class Universities (Disciplines) and \"The Characteristic Development Guidance Funds for the Central Universities (No.PY3A056)\".  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.  
**Conflicts of Interest:** The authors declare that they have no conflict of interest.  
**Appendix A. Supplementary Data for This Work Are Listed**