## 2 Materials and Methods
### Low-Order Model of Thermoacoustic Oscillation  
The self-excited frequency (264 Hz) was below the cut-off frequency (1501 Hz, 1.2 m); thus, the combustor can be treated as one-dimensional. Figure 1a presents the low-order thermoacoustic oscillation model of the lean-premixed gas turbine combustor. The low-order thermoacoustic oscillation model of the combustor was modeled with MATLAB Simulink software using the method proposed by [PERSON] [34] and [PERSON] [35], and the governing conservation equations of the wave equation are as follows:  
\[p(x,t)=\overline{p}+\left[f\left(t-\frac{x}{\overline{c}+\overline{u}}\right) -g\left(t+\frac{x}{\overline{c}-\overline{u}}\right)\right] \tag{1}\]  
\[u(x,t)=\overline{u}+\frac{1}{\overline{\rho}\overline{c}}\left[f\left(t-\frac{ x}{\overline{c}+\overline{u}}\right)-g\left(t-\frac{x}{\overline{c}- \overline{u}}\right)\right] \tag{2}\]  
In Equations (1) and (2), \(p(x,t)\) and \(u(x,t)\) denote the total pressure and velocity fluctuation, \(f(t)\) and \(g(t)\) denote traveling waves in the combustion chamber, \(\overline{p}\) and \(\overline{u}\) denote the average pressure and velocity value, \(\overline{c}\) denotes the acoustic velocity in the combustion chamber and \(x\) denotes the relative position of traveling waves in the combustion chamber. The links between mean flow variables \(u\) and acoustic variables \(p\) can be described as in our previous research [33]:  
\[\frac{p^{\prime}}{p}\propto O\left(\gamma\overline{M}\right)\frac{u^{\prime}} {u} \tag{3}\]  
where gamma (\(\gamma\)) is the adiabatic exponent of gas and \(\overline{M}\) stands for the mean Mach number. In Figure 1b, \(R_{u}\) is the reflection coefficient of the upstream part of the combustor, and \(R_{d}\) is the reflection coefficient of the downstream part of the combustor. \(P(t)\) is the sound pressure signal in the combustion chamber. The lean-premixed flame heat release transfer function was modeled as the \(n\)-\(\tau\) equation, where n is the interaction index or combustion efficiency and \(\tau\) is a time delay; the travel times of the acoustic waves \(f\) and \(g\) are denoted as time delays \(\tau_{i}=x_{i}/c\) (\(i\)=1, 2, \(\ldots\),6); and \(x_{i}\) denotes the travelling distance of the acoustic waves. These considerations lead to the following transfer function:  
\[q^{\prime}(s)=\frac{k_{f}}{1+T_{f}s}e^{-s\tau_{f}}u_{1}^{\prime}(s) \tag{4}\]  
In Equation (4), \(k_{f}\) is the static gain, \(T_{f}\) is the time constant of the low pass element, \(\tau_{f}\) is the travel dead time and u\({}_{1}\) is the acoustic velocity upstream of the flame. In this paper, parameters for modeling are summarized in Table 2. The equivalence ratio of the propane flame was maintained at 1.0, and air for combustion was 0.105 MPa with a relative humidity of 50%. The lower heat value of propane is 46.56 kJ/kg, and the heat capacity of air is 1.009 kJ/(kg K). As displayed in Figure 2, the low-order model was implemented in MATLAB. A linear flame transfer function was used, and the \(n\)-\(\tau\) model was filtered by a first-order filter to capture the flame response [36; 37]. \[H(s)=\frac{\omega_{c}}{\omega_{c}+s}afe^{-\tau_{f}s} \tag{5}\] \begin{table}
\begin{tabular}{l l} \hline \multicolumn{2}{c}{**Geometries of the Combustor**} \\ \hline Burner inlet length (mm) & 550 \\ \hline Diameter (mm) & 114 \\ \hline Burner total length (mm) & 1200, 2400, 3600, 4800, 6000 \\ \hline Swirl section length (mm) & 150 \\ \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of the lean-premixed combustor. where \(a_{f}\) is the gain and \(\tau_{f}\) indicates the time delay. \(\omega_{c}=2\pi f_{c}\) denotes the cut-off frequency of the filter. The reflection coefficient \(R_{u}\) for the chocked boundary condition is defined as 0.95. The reflection coefficients of the open-ended downstream are sensitive to geometric dimensions and sound speed.