## 1 Introduction  
Combustion instability has been an issue in combustion equipment with low NOx emissions. It is manifested as large amplitude fluctuations of pressure and heat release in the combustion chamber, which may lead to structural damage. This instability is attributed to the interaction of the unsteady heat release and the cavity acoustic modes of the combustion chamber [1]. Therefore, the response characteristic of heat release rate (HRR) to the flow perturbations is a key factor in better understanding combustion instability [2; 3; 4]. Previous studies have established related models to describe the response of heat release to flow disturbance. Among them, the n-\(\tau\) model and the flame transfer function (FTF) were based on the linearization assumption [5; 6]. The frequency and the corresponding growth rate of the thermoacoustic oscillations can be predicted through the combination of the linear flame response models and the thermoacoustic coupling model [7]. However, the linear FTF cannot interpret the nonlinear thermoacoustic features, such as the mode switching, nonlinear trigger, and frequency shifting [8], nor can it obtain the oscillation amplitude in limit cycle mode. Therefore, in order to determine the nonlinear response of the flame, the flame describing function (FDF) was proposed [9]. The FDF considers the effects of the forcing amplitude in addition to the frequency. [PERSON] et al. [10] used the FDF of a premixed flame to interpret the mode-switching phenomenon and predict the limit cycle amplitudes of oscillation. The theoretical results were in excellent agreement with the measurements. FTF and FDF can be obtained through CFD numerical simulation and theoretical analysis and experiments [5; 11; 12]. With the experiment, the HRR response can be measured using optical equipment, such as PMT, OH-PLIF, or ICCD camera integrated with CH* or OH* chemiluminescence filters [13; 14]. Much work has been reported related to the responses of the premixed flame with different configurations, such as laminar conical flames [15; 16], multiple conical flames [17], swirled flames [18; 19; 20], bluff body stabilization flames [21], and practical gas turbine burners [22; 23; 24]. Compared to the premixed flame, there are relatively few studies on the response characteristics of non-premixed combustion. Existing works mainly focused on the laminar jet diffusion flame. In order to understand the foundational mechanism, a lot of the literature has worked on the forced response of the Burk-Schumann diffusion flame with theoretical analysis. [PERSON] et al. [25] theoretically studied the heat release of the Burke-Schumann flame to the longitudinal velocity fluctuations with the assumption of the uniform flow field. It was found that the Da number plays an important role in determining the amplitude and phase of the heat release fluctuations with respect to the velocity fluctuations. Based on the same flame and assumptions, [PERSON] and [PERSON] [26] investigated the response of the flame to transverse velocity oscillations. The gain of heat release response was nearly independent of the amplitude at low frequencies. However, the gain decreased with the increase in the excitation amplitude. [PERSON] et al. [27; 28] compared the dynamics of the forced response of laminar non-premixed and premixed flames to the velocity fluctuations. Mass burning rate oscillations dominate the response of non-premixed jet flame; however, the perturbations of the flame speed and the burning area induced by the velocity fluctuations caused the oscillations of the HRR for the premixed flame [29]. Corresponding to the different response mechanisms, the gain and phase-lag of the FTF of the jet diffusion flame and premixed flame were also much different, especially under low flame Strouhal number (_Stf_). [PERSON] and [PERSON] [30] discussed the distributed FTF for the Burke-Schumann diffusion flame with Green's Function method. A dual-peak amplitude of the gain was observed in the fuel-rich zone and the fuel-lean zone, respectively.