## 3 Results  
### Effects of the Combustor Length  
Figure 2(a)-e show the variation tendency of the sound pressure amplitude in a combustion chamber. When the combustor length ranges from 1.2 to 6.0 m, the upstream part of the combustor remains constant at 0.55 m. It can be inferred from Figure 3 that the sound pressure amplitude gradually declines as the length of the combustor reduces. The oscillation curve becomes sparse, which denotes that the oscillation frequency also experienced a gradually decreasing trend. The triggering time of thermoacoustic oscillations varies significantly while the combustor length changes. In Figure 2(a), at the length of 1.2 m, thermoacoustic oscillation emerges within 0.5 s; however, at the length of 6.0 m, Figure 2(e) shows that oscillation emerges after 1 s. Both sound pressure amplitude and oscillation frequency decline as the length grows, accompanied by the gradually increasing trigger time. Figure 3 also indicates that there is a significant difference in the climb rate during the excitation processes. The main reason for the sound pressure amplitude decreasing can be attributed to the variation in sound wavelengths. At the same time, there is a significant change in the length of the sound waves. In Figure 3, as the chamber length rises, the oscillation frequency gradually declines. This is because, as the length of the combustion chamber increases, the oscillation period of thermoacoustic oscillations is also extended. According to the equation for the difference between the oscillation period and frequency, \(f=1/T\), it \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Values** \\ \hline Fuel type & Propane, gas purity: 99.995\% \\ Thermal power (kW) & 0.013, 0.026, 0.039, 0.052, 0.065 \\ Lower heat value for propane (kJ/kg) & 46.56 \\ Equivalence ratio (\(\Phi\)) & 0.90 \\ Atmospheric pressure (MPa) & 0.105 \\ Gas mixture temperature (\({}^{\circ}\)C) & 0, 25, 50, 75, 100 \\ Burner lengths (mm) & 1.2, 24, 3.6, 4.8, 6.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter settings in this study. Figure 2: Low-order thermoacoustic oscillation model in MATLAB Simulink. can be seen that the increase in the oscillation period will lead to a decrease in oscillation frequency. In the sound wavelength calculation equation, \(\lambda=v/f\), v denotes the acoustic velocity and \(f\) denotes the oscillation frequency. The equation indicated that the sound wavelength \(\lambda\) increased as the oscillation frequency \(f\) declined. As the sound wavelength \(\lambda\) increased, the relative location where high amplitude oscillations are easily triggered also changed. Then, the intensity of thermoacoustic oscillation decreased. The detailed values of sound pressure amplitude and triggering time under different combustor lengths are displayed in Figure 4. As the combustor length increased from 1.2 m to 6.0 m, the thermoacoustic oscillation triggering time increased from 0.32 s to 0.91 s, while the sound pressure amplitude decreased from 3.2 \(\times\) 10\({}^{5}\) Pa to 2.1 \(\times\) 10\({}^{5}\) Pa. The sound pressure amplitude experienced a rapid descent, while the chamber length rose from 1.2 m to 2.4 m. However, the downward trend slowed down when the chamber length rose from 2.4 m to 6.0 m. The change trend of the triggering time curve was relatively consistent as the chamber length rose. The reason why the triggering time changed is closely related to the increase in the acoustic wave oscillation frequency; as \(f=1/T\), the oscillation period increased. Due to the extension of the oscillation period and chamber length, the coupling process between the sound waves and flame was extended. As a result, the triggering time of oscillation was prolonged. To accurately evaluate the impact of parameter settings on the triggering of thermoacoustic oscillations, we introduced the climb rate as an indicator of thermoacoustic oscillations. This method is similar to that in our previous research on calculating the decay time of thermoacoustic oscillations [32].