## 4 Results and Discussions
### Axial Distribution of Heat Release Oscillation  
The axial distribution of heat release oscillations for various excitation frequencies is shown in Figure 12 at a velocity excitation amplitude of 0.50. Notably, distinct axial oscillation distributions are present for each different excitation frequency. At a low frequency of 10 Hz, the velocity excitation predominantly causes positive heat release oscillations upstream within the 0-180\({}^{\circ}\) phase range. Additionally, the positive oscillation region progressively diminishes and propagates downstream. After the 180-degree phase, when the excitation velocity reaches its minimum negative value, the entire middle and lower region begin to experience flameout. This results in a negative oscillation of heat release. Eventually, due to the increase in velocity fluctuation, the heat release in the upstream region increases, resulting in inducing positive oscillation and entering the next cycle. At 10 Hz excitation, positive or negative oscillations appear predominantly in the axial distribution of heat release, and the oscillation zones are relatively broad. Subsequently, the corresponding oscillation amplitude is also significant, and this behavior is associated with \(St_{f}<1\). This oscillation pattern reveals the primary reason for the large gain of the flame under low-frequency (10 Hz) excitation. Under 20 Hz excitation, unlike 10 Hz, both positive and negative oscillation zones coexist at various phases. Furthermore, the amplitudes of these two zones are equal, leading to a relatively small total heat release fluctuation value. As a result, this is the main reason for the appearance of the gain valley around 20 Hz. Under 30 Hz excitation, there are two  
Figure 12: The axial distribution of heat release oscillations with variation of phase angles at different excitation frequencies and \(u^{\prime}/\bar{u}=0.10\). The red region in the graph represents the positive value zone, while the blue region represents the negative value zone. opposing oscillation regions, but their relative relationship is significantly different from 20 Hz excitation frequency. Specifically, regarding the middle forward oscillation region, the corresponding region for 30 Hz excitation reaches its maximum value at \(\phi=120^{\circ}\) and then quickly decays, which causes single-region predominance for most phases. Similar to that seen under 10 Hz excitation, leads to the appearance of local gain peaks under 30 Hz excitation. With 100 Hz high-frequency excitation, multiple alternating oscillation regions appear. At different phase angles, the heat release oscillations have a wavenumber ranging from 2.0 to 2.5 within the flame length (\(L_{f}=87.6\) mm). This is consistent with the \(St_{f}\) (2.43) and the number of heat release layers (2-3) in Figure 10. Additionally, the oscillation amplitude of the downstream region decreases significantly relative to the upstream region. This multi-region alternating structure under high-frequency excitation means that positive and negative oscillations at the same moment have a counteractive effect, and thus the dispersed oscillation amplitude is relatively small, which leads to a relatively low gain under high-frequency excitation. This is the main reason for the low-pass filtering characteristic of the flame. The phase-lag varies significantly at different excitation amplitudes at 20 Hz, presenting another noteworthy phenomenon. Therefore, we analyzed the axial distribution of heat release oscillation under the condition of 20 Hz and 0.1 excitation amplitude. As shown in Figure 13, comparing the two, we found that, although the number of axial regions is consistent under two different amplitudes, the variation trends of the five oscillation regions differ significantly, especially in the phase range of \(\phi=0\)-\(180^{\circ}\).