## 3 Results and Discussion
### Effect of FMFRF on Blowoff Limits  
The influence of the external acoustic forcing on the blowoff limit was recorded at a mean fuel mass flow rate of 0.142 g/s with varying FMFRF amplitude, as shown in Figure 9. In this work, the blowoff limit is represented by the air velocity when blown off. The procedure involves fixing the fuel flow rate and then gradually increasing the air velocity in small steps of approximately 2% (0.258 m/s) every 30 s until the blowoff occurs. The average of five repeats determines the blowoff air velocity.  
It can be seen from Figure 9 that the blowoff limit increases until a 100% FMFRF amplitude and then reaches a plateau as the FMFRF amplitude increases. Previous work on the same burner demonstrated that flame stability worsens when the flow rate fluctuation is applied to the recirculating air [67]. However, it was found in our study that when acoustic forcing was applied to the fuel passage, the flame was still stable even when the FMFRF was beyond 100%. The extension of the blowoff limit at different fuel flow rates behaves very similarly, indicating that the periodic modulation of the jet fuel is a possible strategy to extend the blowoff limit of a swirling flame.  
The Damkohler number (\(Da\)) was introduced to qualitatively describe the impact of FMFRF. The \(Da\) number is defined as the ratio between the residence timescale, \(\tau_{\text{res}}\), and the chemical timescale, \(\tau_{\text{chem}}\). The residence timescale \(\tau_{\text{res}}\) is the ratio of the reaction zone length and the air flow velocity [68]. The chemical timescale can be estimated using the formula, \(\tau_{\text{chem}}=\
u/S_{L}^{2}\), where \(\
u\) is the kinematic viscosity and \(S_{L}\) is the laminar flame speed at stoichiometric conditions. This formulation has been adopted in a lot of studies to estimate the chemical timescales in premixed and non-premixed flames [5; 31; 69; 70]. According to the extinction criterion [24; 60], which was validated in a non-premixed swirling flame with a similar setup in Ref. [5], the blowoff events occur when the \(Da\) number is smaller than the critical value. Thus, a longer \(\tau_{\text{res}}\) or shorter \(\tau_{\text{chem}}\) is favorable for flame stabilization. As shown in Figures 6-8, the larger FMFRF pushes the fuel to a further  
Figure 9: The blowoff air velocity as a function of the varying FMFRF amplitude at a mean fuel mass flow rate of 0.142 g/s.  
downstream position and create a larger reaction zone, which is demonstrated by a broader OH* distribution. Since the air flow velocity was constant, the residence timescale \(\tau_{\text{res}}\) became longer consequently. Meanwhile, \(\
u\) and \(S_{L}\) both exhibit dependence on temperature in the relationship of \(\
u\propto T^{1.76}\)[71; 72] and \(S_{L}\propto T^{1.612}\)[73; 74; 75; 76] for gases. In the flame under FMFRF, the value of \(\
u\) and \(S_{L}\) is increased due to a higher temperature in the CRZ verified by the intensive SO\({}_{2}\)-PLIF. In contrast, the chemical timescale (\(\tau_{\text{chem}}\propto T^{-1.464}\)) decreased in the flame under FMFRF. These two beneficial impacts help to extend the blowoff limit. When FMFRF is over 100%, the flame height indicated by the OH*-CL remains constant, implying that \(\tau_{\text{res}}\) does not change. Hence, the blowoff limit arrives at a plateau.