## 4 Results and Discussions
### The Dynamics of Flame Response  
In this section, we processed a sequence of CH* chemiluminescence and schlieren images to reveal the interaction mechanism between flow and reaction. We analyzed the heat release response at four excitation frequencies: 10 Hz, 20 Hz, 30 Hz, and 100 Hz for \(u^{\prime}/\dot{u}=0.10\) and 0.50. The 10 Hz case corresponds to the maximum gain of the FDF, whereas the 20 Hz and 30 Hz cases correspond to the local gain valley and peak, respectively. We also analyzed the response under relatively high-frequency excitations of 100 Hz. The instantaneous CH* chemiluminescence images and the corresponding schlieren images under excitation amplitudes of 0.10 and 0.50 are shown in Figures 8-11. The phase angle of 0\({}^{\circ}\) corresponds to the maximum positive value of velocity fluctuation, while 180\({}^{\circ}\) corresponds to the minimum negative value of velocity fluctuations. Figure 7: The comparison of time delay and convection time under different excitation frequencies and amplitude: (**a**) The time delay calculated through the phase-lag. The red frame in the picture highlights the anomalous time delay associated with 20 Hz excitations. (**b**) The convection time calculated through the height of the COM and the constant convective velocity (\(u_{cv}\) = 3.52 m/s). Figure 8: The instantaneous CH* chemiluminescence images in an excitation cycle at different excitation frequencies with \(u^{\prime}/\bar{u}\) = 0.10. Figure 9: The instantaneous schlieren images in an excitation cycle at different excitation frequencies with \(u^{\prime}/\bar{u}\) = 0.10. Figure 11: The instantaneous schlieren images in an excitation cycle at different excitation frequencies with \(u^{\prime}/\tilde{u}=0.50\). Figure 10: The instantaneous CH\({}^{\star}\) chemiluminescence images in an excitation cycle at different excitation frequencies with \(u^{\prime}/a=0.50\). Under low-amplitude velocity excitation, flame height, and overall phase remain relatively constant within an excitation cycle, particularly at high frequencies. Because of the short high-frequency excitation period, the HRR distribution only causes local lift and curling at the nozzle exit, without affecting the downstream flame region. The flame dynamic is mainly characterized by the changes in the wrinkle of the flame sheet, which can be observed more prominently from a side view. The number of wrinkles increases with increasing excitation frequency. High excitation amplitudes result in significant changes to flame morphology, with distinct differences in change characteristics depending on excitation frequency. At a low-frequency excitation of 10 Hz, the flame undergoes multiple processes, including curling, bifurcation, overall extinguishment downstream, and gradual strengthening of the reaction upstream, as well as the formation of a vortex structure. In contrast, high-frequency excitation results in the formation of multiple distinct regions of heat release within a single cycle, with pronounced, vortex-like curling at their edges. Additionally, the main region of heat release increases in intensity from the nozzle and gradually decreases after reaching its maximum. The number of distinct regions of heat release increases as frequency increases. Specifically, a gradual transition between four layers of heat release regions was observed when the excited frequency was 100 Hz. A clearly observable propagation process of large vortex structures induced by flow perturbation can be seen in the instantaneous schlieren images. The convective velocity of the disturbance mentioned earlier can be analyzed through schlieren images. The minimum excitation velocity corresponds to the phase of 180 degrees, which is also the origin of vortex structure formation. The height at which perturbations propagate downstream over a single excitation period is defined as the convective wavelength \(\lambda_{f}\)[50]. The convective wavelengths at different excitation frequencies and a fixed excitation amplitude of 0.5 are identified in Figure 11 with red lines.