## 3 Results and Discussion  
### Effect of FMRF on Flame  
Figure 5 presents the inverse Abel-transformed time-averaged OH*-CL images with and without FMRF varying up to 159%. Chemiluminescence imaging provides line-of  
Figure 4: The processes of laser intensity correction. (**a**) The images of the cuvette, (**b**) the profile of laser intensity, (**c**) uncorrected PLIF, and (**d**) corrected PLIF (CH\({}_{2}\)O-PLIF on the left and SO\({}_{2}\)-PLIF on the right). The sample image is from the flame forced by an 80% FMRF amplitude at a mean fuel mass flow rate of 0.142 g/s and air velocity of 17.1 m/s. sight integrated data. Even when performed in a high-speed mode, it is still challenging to interpret the instantaneous chemiluminescence structures due to their integrated nature. Thus, an Abel deconvolution of the averaged OH* images was carried out, allowing us to reconstruct a 2D flame structure and providing a more precise visualization of the flame's response to FMFRF. Since the Abel-transformed time-averaged OH*-CL images showed symmetric structures, only the left half of every image is shown. The flame anchored on the bluff body presents a spindle shape. An intense OH*-CL signal is observed in the CRZ and near the shear layer. The intensity of OH*-CL and the flame height both keep increasing with larger forcing amplitude until the FMFRF reaches 80% and then arrives at a constant value, and the flame height is up to 75 mm. It reflects a non-linear response of the heat release to the FMFRF, probably due to the reverse flow of the fuel near the nozzle when the FMFRF amplitude exceeds 100%, and the reversed hot products are premixed with the fuel inside the central tube, altering the OH*-CL emission rate [62], which is consistent with the observations in Ref. [63]. Figure 6 compares the time-averaged CH\({}_{2}\)O- and SO\({}_{2}\)-PLIF images of an unforced flame and a flame with a 159% forcing amplitude, under which condition no reverse of the fuel flow back to the injection tube is observed. Since the time-averaged images are symmetric, only half of the CH\({}_{2}\)O- and SO\({}_{2}\)-PLIF images for each condition are displayed. All the results are based on the left side of the images, but for the convenience of demonstration, the results for SO\({}_{2}\)-PLIF are flipped to the right side. Figure 5: The left half of the inverse Abel-transformed time-averaged OH*-CL images with varying FMFRFs (displayed at the top of every image) at the conditions with a mean fuel mass flow rate of 0.142 g/s and air velocity of 17.1 m/s. Figure 6: Comparison of time-averaged (**a**) CH\({}_{2}\)O- and (**b**) SO\({}_{2}\)-PLIF images at the unforced (left half) and forced cases (right half) with 80% FMFRF amplitude at a mean fuel mass flow rate of 0.142 g/s and air velocity of 17.1 m/s. When no external acoustic forcing is applied to the fuel flow, CH\({}_{2}\)O-PLIF presents a tooth-shaped structure visible immediately from the entrance of the combustion chamber. The hollow region of the CH\({}_{2}\)O root is filled with pure fuel, and the profile of the hollow region shows the boundary where the fuel begins to decompose. In the forced case, the tooth shape of CH\({}_{2}\)O-PLIF is not evident because the total CH\({}_{2}\)O intensity and distribution area are strongly enhanced. According to the earlier study in Ref. [61], the time-averaged axial jet velocity increases with a larger forcing strength, the fuel penetrates the downstream recirculation zone, and the CH\({}_{2}\)O-PLIF signal expands downstream. A similar phenomenon is observed in the spatial distribution of the SO\({}_{2}\)-PLIF signal, which exists between the CH\({}_{2}\)O-PLIF signal and the cold annular jet [50]. The intensity of the SO\({}_{2}\)-PLIF signal edge decreases due to temperature drop and SO\({}_{2}\) dilution in contact with cold air. Under external acoustic forcing, both the intensity and distribution area of the SO\({}_{2}\)-PLIF were enlarged greatly, indicating the temperature rise of the CRZ.