# Numerical Study on the Route of Flame-Induced Thermoacoustic Instability in a Rijke Burner
## 1 Introduction  
In 1859, [PERSON] discovered strong acoustic oscillations when heat was added to the lower half of a vertical tube opened at both ends [1]. Several similar devices were proposed to study the heat-driven acoustic phenomenon in past decades, and these devices were grouped under the general heading \"Rijke tube\" [2]. Among these devices, one type in which the flow is combustible gas and the heat is produced by the flame is called a Rijke burner or a Rijke tube burner. Experimental research on the Rijke burner is intuitive and can yield highly reliable information about thermoacoustic instability, such as temperature, pressure, and velocity. Abundant nonlinear behaviors, such as limit cycle oscillation, quasiperiodic, frequency-locked, and chaotic behaviors, were discovered [3, 4, 5]. Based on experimental results, some classical flame models have been proposed [6, 7, 8, 9, 10]. However, experimental methods can collect data only at finite points. The parameters of the entire field cannot be obtained by experimental methods, and a comprehensive understanding of the flow field of thermoacoustic instability is still lacking. Analytical studies have been conducted to explain the mechanisms of thermoacoustic instability since the 1960s [11, 12]. [PERSON] established the linear theory of combustioninstabilities and first derived the nonlinear acoustic model in combustion chambers based on experimental research [13; 14; 15; 16]. [PERSON] and [PERSON] investigated the role of nonnormality and nonlinearity in flame-acoustic interactions and found that nonnormality and nonlinearity lead to triggering and redistribution of energy between eigenmodes [17]. [PERSON] et al. predicted the bootstrapping instability mechanism observed in rocket motors by approximated modal analysis [18; 19]. [PERSON] et al. observed the stability switch phenomenon with a one-dimensional (1-D) Rijke-type thermoacoustic system [20]. Through the analytical method, we have obtained a thoughtful understanding of the mechanism of thermoacoustic instability, especially for predicting the frequency, amplitude, and stability of self-excited thermoacoustic instability. With the development of computational technology, computational fluid dynamics (CFD) simulations have increasingly helped to describe the mechanisms of thermoacoustic instability and to provide guidance for experimental research, making up for the shortage of experimental work. Many numerical studies have been conducted on combustion instabilities. Because of the nonlinearity in the combustion system, the interactions between reaction mechanisms, hydrodynamics, and acoustics reveal different spatial and temporal scales and are sensitive to various parameters. [PERSON] investigated the occurrence of combustion instabilities in a Rijke tube-type combustor using a two-dimensional (2-D) finite volume method, captured the instability growth to a limit cycle of the pressure oscillation, and predicted the frequency and magnitude of the thermoacoustic instability [21]. [PERSON] et al. simulated a three-dimensional (3-D) premixed chamber with ANSYS FLUENT software and studied the dependence of a homogeneous mixture on various parameters such as the premixing length, airflow rate, and equivalence ratio to optimize the combustor [22]. [PERSON] also successfully simulated the lean premixed combustion instabilities in a 2-D Rijke-type burner by CFD simulation and found that the radiation and reaction mechanism play critical roles in predicting the occurrence of thermoacoustic instability [23]. [PERSON] et al. simulated a bluff-body flame holder, and the proper orthogonal decomposition modes of the simulated results and experiments showed quantitative agreement [24]. [PERSON] et al. used COMSOL software to investigate the driving mechanism of combustion instabilities in a small rectangular combustion duct and obtained the pressure waves in the longitudinal and transverse directions [25].