# 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  
To capture the dynamic characteristics of thermoacoustic insatiability, a transient simulation was conducted. Here, the steady-state reacting flow field was set as the initial condition of the dynamic flow. The time step was set to between \(1\times 10^{-5}\) s and \(1\times 10^{-4}\) s to obtain enough information in the time domain while meeting the convergent requirements in previous research [21; 23; 42]; here, it was set to \(1\times 10^{-5}\) s. The transient solution includes the growth of the self-excited instability, the subsequent limit cycle behavior of pressure and velocity, the oscillating frequencies and modes, and the periodic behavior of the flow field. The instability growth mechanism is characterized by increasing oscillations of pressure inside the combustor. Figure 7a shows the time evolutions of average pressure and average velocity 5 cm downstream of the flame stabilizer. Figure 7b shows the growth of total reaction heat released by the flame. The steady-state value of pressure, velocity, and heat release were 0 Pa, 0.1226 m/s, and 63.68 W, respectively. Approximately 0.05 s after the time integration was initialized, the pressure, velocity, and reaction heat started to increase exponentially. The oscillation grew until approximately 0.86 s and reached maximum amplitudes. After several periods of amplitude adjustment from 0.86 s to 1.1 s, the system reached a \"saturated\" state, which is referred to as the limit cycle. The pressure, velocity, and reaction heat maintained at amplitudes of approximately 208 Pa, 0.39 m/s, and 10.79 W, respectively. Figure 7: The time evolutions of pressure, velocity, and reaction heat release of the flame. (**a**) the time evolutions of pressure, velocity, (**b**) the time evolution of reaction heat release, (**c**) The phase relationship between velocity and pressure, and (**d**) the phase relationship between reaction heat release and pressure. Figure 7c illustrated the phase relationship between the average pressure and velocity 5 cm downstream of the flame stabilizer at the limit cycle oscillation state. It can be seen that the velocity had a 88.11\({}^{\circ}\) phase advance relative to the pressure. Figure 74 illustrated the phase relationship between the average pressure 5 cm downstream of the flame stabilizer and total heat release of the flame at the limit cycle oscillation state. The reaction heat release had a 24.3\({}^{\circ}\) phase delay relative to the pressure, which fulfills the [PERSON] criterion that oscillations are encouraged when heat fluctuates in phase with pressure perturbation [43]. It can be calculated that the reaction heat had a 114.34\({}^{\circ}\) phase delay relative to the velocity. This time delay between reaction heat and velocity has been proved to play a critical role in the stability of the Rijke-type thermoacoustic system [20]. Two important parameters to evaluate the growth of thermoacoustic instability are the growth rate of the acoustic disturbances \(\alpha\) and G. The definition and detail derivation of \(\alpha\) and G can be found in the reference [44]. The time evolution of the pressure and heat release of the flame from 0 s to 0.4 s are illustrated in Figures 8 and 9. It can be seen that both the pressure and heat release were growing exponentially after an adjustment of about 0.05 s. Two exponential functions are constructed to best fit the series of local maximum and minimum points  
The Equations (4) and (5) are solved by iteration, and the calculated growth rate \(\alpha_{1}\) is 13.77 s\({}^{-1}\), \(\alpha_{2}\) is 13.85 s\({}^{-1}\). The corresponding \(G\) is 1.22 \(\times\) 10\({}^{-2}\) rad/s\({}^{2}\), which is within the range of the maximum possible growth rate for this type of combustion-driven acoustic disturbance, \(G_{max}\) = 9.2 \(\times\) 10\({}^{-2}\) rad/s\({}^{2}\). From the analysis of [PERSON] et al.