Measurement and Identification of Flame Describing Function (FDF) Based on Parallel Subsystem Model

Abstract

Because of the need for low pollutant emissions, industrial gas turbines typically use partially premixed gases for combustion. However, the nonlinear dynamic characteristics of partially premixed flames have not been studied sufficiently. Therefore, this study focuses on the dynamics of a partially premixed flame generated by a swirler with fuel holes on its surface and designs a flame describing function (FDF) identification method based on the parallel subsystem model. This method can separate the flame dynamic characteristics into a parallel connection of the nonlinear and linear models. The nonlinear model is related to the disturbance frequency and velocity perturbation amplitude, whereas the linear model depends only on the disturbance frequency. This method is verified using a simulation. Finally, experimental research on partially premixed flames is conducted. Based on the experimental data, the identification method successfully separates the FDF into a nonlinear model with saturation characteristics and a linear model with Gaussian distribution characteristics. The flame model obtained by the identification method is the foundation for the analysis of combustion thermoacoustic stability and active/passive control strategy.

1. Introduction

Owing to the increasingly strict requirements for NOx emissions from combustion products, lean premixed combustion has become a major combustion technology, particularly for efficient and clean power generation in heavy-duty gas turbines. However, under lean combustion conditions, severe combustion oscillations or thermoacoustic instability can occur in the combustion chamber [1]. This phenomenon not only generates huge noise and reduces combustion efficiency but also threatens the safe and stable operation of combustion equipment. Therefore, to suppress combustion oscillation, it is necessary to study the mechanism of combustion instability by obtaining a mathematical model of the combustion system and evaluating the flame stability [2]. In previous studies, the flame transfer function (FTF) or flame describing function (FDF) [3] was used to describe the flame dynamics, which is generally defined as the ratio of the overall heat release perturbation to the disturbance velocity perturbation at the reference plane. When using FTF to describe this process, the variation in the flame heat release rate perturbation with velocity perturbation is linear, and its characteristics are only related to the excitation frequency. A general FTF can be defined using Equation (1):

$$\begin{array}{c}\hfill \mathrm{FTF}\left(\omega \right)={\displaystyle \frac{\widehat{q}\left(\omega \right)/\overline{q}}{\widehat{u}\left(\omega \right)/\overline{u}}}\end{array}$$

(1)

where $\overline{q}$ and $\overline{u}$ represent the average values of the heat release rate and velocity at the reference plane, respectively. $\widehat{q}$ and $\widehat{u}$ denote the Fourier transform values of the heat release rate and velocity corresponding to the disturbance frequency, respectively. $\omega$ denotes the angular frequency of the disturbance.

For phenomena in thermoacoustic instability problems, linear analysis can provide an approximate estimate of system stability; however, it cannot capture more complex phenomena such as triggering instability, main frequency shift during oscillation initiation, mode switching, and hysteresis loops. It is also difficult to estimate the amplitude of the limit cycle [4]. To explain these phenomena and obtain more effective stability analysis results, it is necessary to determine the nonlinear characteristics of the flame. Considering the actual complex combustion process, the flame dynamic characteristics are often related to various parameters, such as the disturbance frequency, velocity perturbation, equivalence ratio, and combustion chamber structure. Generally, the FDF can be defined using Equation (2)

$$\begin{array}{c}\hfill \mathrm{FDF}(\omega ,|\widehat{u}/\overline{u}|,\varphi ,A,\dots )=G(\omega ,|\widehat{u}/\overline{u}|,\varphi ,A,\dots ){\mathrm{e}}^{\mathrm{i}\phi (\omega ,|\widehat{u}/\overline{u}|,\varphi ,A,\dots )}\end{array}$$

(2)

where $\varphi$ is the equivalence ratio, A is the combustion chamber structure, G is a function related to the frequency domain gain, and $\phi$ is a function related to the phase slope.

To date, flame dynamics and the response of flames to acoustic modulation have been extensively investigated. From the perspective of theoretical research, Crocco et al. [5] proposed a classical model for the atomization combustion process of propellant droplets to analyze the flame dynamics of liquid rocket engine combustion chambers. Park et al. [6] used a well-stirred reactor (WSR) model and a one-step kinetic mechanism to describe the combustion process for fully premixed gases and obtained a model similar to a first-order filter. Dowling [7] analyzed the interaction between unstable combustion and acoustic waves based on linear theory when studying the self-excited oscillation of a confined flame after a bluff body flame stabilizer and proposed a linear second-order delay saturation model. Li et al. [8] extended the model by introducing nonlinearity into flame gain and delay variation, which can reproduce important features in nonlinear dynamics such as single tone instability, “triggering”, “mode switching”, and frequency shift during the limit cycle.

Owing to the complex dynamic characteristics and strong coupling of acoustic, turbulent, and other mechanisms in actual combustion chambers, it is difficult to derive theoretical models. Therefore, researchers often use numerical simulations and experiments to identify systems based on certain design criteria and obtain approximate flame models and parameters. Zhu et al. [9] studied the low-frequency oscillation problem in atomization combustion. A low-order dynamic model conforming to the delay law was obtained through CFD simulations of the combustion process under changes in the atomizer inlet pressure. Bloxsidege et al. [10] developed a first-order filter model with time delay for a premixed flame in an afterburner by perturbing it upstream using a movable center body. Polifke et al. [11] found that axial and tangential flows have different effects on the flame response of fully premixed swirl stable combustion and proposed a Gaussian distribution model. Candel et al. [4] found that the flame has an amplifier effect in the low-frequency range and that its effect depends on the amplitude level; they proposed a premixed swirl flame model related to velocity perturbation. Laera et al. [12] proposed a high-order polynomial form of the nonlinear correction term when studying the thermal–acoustic instability of longitudinal and annular combustion chambers.

To obtain more realistic nonlinear effects, the FDF must be identified from the experimental data. One of the key points in identifying the FDF is the selection of nonlinear model structures. In previous studies on nonlinear system identification, there was no mature or perfect theoretical system because of the diversity of nonlinear system forms. Correspondingly, there are no universal model parameter identification methods. Compared to the identification of linear systems, it is more difficult to identify nonlinear systems [13]. To adapt to the diversity of nonlinear characteristics and satisfy different identification purposes, various model structures have been developed, such as NARX, neural networks, and nonlinear models based on subsystem blocks (Wiener, Hammerstein, etc.). The NARX model is characterized by its ability to describe common nonlinearity and good function approximation ability, and is widely used in industrial scenarios [14]. However, this requires the early selection of model terms based on experience. Improper selection can lead to application difficulties, the curse of dimensionality, and poor robustness [15,16]. Neural network models can approximate nonlinear functions with arbitrary accuracy and have been applied to industrial parameter time-varying systems [17]. However, they are prone to overfitting and have poor readability. Nonlinear systems based on dynamic linear and static nonlinear subsystem blocks can be used to describe systems that can separate dynamic linear and static nonlinear systems. These models have the characteristics of easy identification, low computational complexity, and a good ability to reflect nonlinearity, which are widely used in biological and chemical processes [18,19].

In recent years, models based on linear and nonlinear subsystem blocks have been increasingly applied to thermoacoustics problems. Ghirardo et al. [20] described the feature of self-excited thermoacoustic oscillations with the sum of two Hammerstein models. Zhang et al. [21] used a Hammerstein–Wiener model to investigate the linear and nonlinear responses of a conical premixed laminar flame to oncoming acoustic disturbances. Gopinathan et al. [22] represented flame dynamics as the superposition of multiple Gaussian time delay distributions, where the nonlinear effect has a dependence on the disturbance amplitude.

The FTF and FDF of fully premixed flames have been studied to some extent. However, owing to the low pollutant emissions, industrial gas turbines typically use partially premixed combustion. Compared to the fully premixed flame, the characteristics of partially premixed flame require further investigations [23,24]. Therefore, in this study, the FDF of a partially premixed flame generated by a swirler with fuel holes on the surface is measured, and the influences of different input disturbance amplitudes on the flame are analyzed in combination with flame images. Based on existing flame model research results, this study attempts to separate the flame characteristics into a nonlinear model related to the velocity perturbation and excitation frequency, and a linear model related only to the excitation frequency. In order to avoid the unmeasurable variables between subsystems, which exist in the cascade subsystem models (e.g., Hammerstein and Wiener), the parallel block model is used to describe the measured flame frequency domain data. This method is more simple and intuitive, and can effectively show the saturation characteristics of flame in the experiment. Meanwhile, the characteristics at the lowest disturbance amplitude are not simply regarded as a linear transfer function to ensure flexibility. The particle swarm optimization (PSO) algorithm [25] is used to obtain the corresponding parameters. This identification method can analyze the influence of velocity perturbations and obtain a more accurate flame model, which serves as the basis for thermoacoustic oscillation monitoring or control tasks requiring parametric models [8].

The remainder of this paper is organized as follows: Section 2 introduces the parallel subsystem block model and corresponding identification algorithm. The effectiveness of the identification algorithm is verified using a simulation case. Section 3 presents the combustion experimental rig, measurement results, and an analysis of the flame frequency domain characteristics. Subsequently, the flame model is identified using the proposed algorithm, and an FDF model is obtained. Finally, Section 4 summarizes the study.

2. Identification Method

2.1. Identification of Parallel Subsystem Type Nonlinear Flame Model

When the combustion conditions remain unchanged, parameters such as the equivalence ratio and combustion chamber structure do not change significantly, and the amplitude of the velocity perturbation becomes the dominant factor affecting the dynamic response characteristics of the flame. In this case, the FTF can be extended to a descriptive function related only to the velocity perturbation amplitude and disturbance frequency as shown in Equation (3):

$$\begin{array}{c}\hfill \mathrm{FDF}(\omega ,|\widehat{u}/\overline{u}\left|\right)=G(\omega ,|\widehat{u}/\overline{u}\left|\right){\mathrm{e}}^{\mathrm{i}\phi (\omega ,|\widehat{u}/\overline{u}\left|\right)},\end{array}$$

(3)

where G represents the nonlinear gain caused by the velocity perturbation amplitude and disturbance frequency and $\phi$ represents the nonlinear phase change caused by the velocity perturbation amplitude and disturbance frequency.

The nonlinear system illustrated in Figure 1 is the parallel subsystem type model, which is characterized by a parallel connection of a nonlinear subsystem NL and a linear subsystem FTF.

According to the nonlinear characteristics of limit cycle oscillation [8], the velocity perturbation amplitudes affect both the flame gain and phase. Therefore, it can be assumed that the flame characteristics can be obtained by a parallel connection of a static nonlinear model and a dynamic linear model. The static nonlinear model is related to the velocity perturbation amplitude and disturbance frequency, whereas the dynamic linear model is only related to the disturbance frequency. The nonlinear model is given by Equation (4), and the FDF can be represented by Equation (5):

$$\begin{array}{cc}\hfill \mathrm{NL}(\omega ,|\widehat{u}/\overline{u}\left|\right)& =G(\omega ,|\widehat{u}/\overline{u}\left|\right){\mathrm{e}}^{\mathrm{i}\omega \phi (\omega ,|\widehat{u}/\overline{u}\left|\right)},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \mathrm{FDF}(\omega ,|\widehat{u}/\overline{u}\left|\right)& =\mathrm{NL}(\omega ,|\widehat{u}/\overline{u}\left|\right)\mathrm{FTF}\left(\omega \right),\hfill \end{array}$$

(5)

where G and $\phi$ are the calculated functions related to the velocity perturbation, which may have different forms at different frequencies.

Based on the above model assumptions, the key point of identification is to separate the nonlinear and linear models. Therefore, the identification method can be designed as follows. The identification flow chart is shown in Figure 2.

First, we select a certain frequency ${\omega }_i$ (the subscript i represents the determined frequency): because the dynamics of the linear model are independent of the velocity perturbation amplitude, the frequency domain characteristic obtained under a certain input $|\widehat{u}/\overline{u}{|}_{\mathrm{ref}}$ is taken as the reference value (the subscript ref represents the reference value). The frequency domain response obtained by different inputs is then divided by this reference value so that the influence of the linear model can be eliminated. The characteristics of nonlinear model ${\mathrm{NL}}_{\mathrm{id}}$ were obtained directly using the measurement data of the system (the subscript id represents the identification result) as shown in Equation (6). Based on the gain and phase characteristics, the structure of the nonlinear gain $G_{\mathrm{id}}$ and phase ${\phi }_{\mathrm{id}}$ can be assumed, and the parameters of the corresponding structure can be obtained using the optimization algorithm:

$$\begin{array}{cc}\hfill {\mathrm{NL}}_{\mathrm{id}}({\omega }_i,|\widehat{u}/\overline{u}\left|\right)& ={\displaystyle \frac{\mathrm{FDF}({\omega }_i,|\widehat{u}/\overline{u}\left|\right)}{\mathrm{FDF}({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}})}}={\displaystyle \frac{\mathrm{NL}({\omega }_i,|\widehat{u}/\overline{u}\left|\right)}{\mathrm{NL}({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}})}}\hfill \\ \hfill & ={\displaystyle \frac{G({\omega }_i,|\widehat{u}/\overline{u}\left|\right)}{G({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}})}}{\mathrm{e}}^{\mathrm{i}{\omega }_i\left[\phi \right({\omega }_i,|\widehat{u}/\overline{u}\left|\right)-\phi ({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}}\left)\right]}.\hfill \end{array}$$

(6)

The gain $G_{\mathrm{id}}$ and phase ${\phi }_{\mathrm{id}}$ of the nonlinear model to be identified can be expressed using Equation (7):

$$\left\{\begin{array}{c}{G}_{\mathrm{id}}({\omega }_i,|\widehat{u}/\overline{u}|)={\displaystyle \frac{G({\omega }_i,|\widehat{u}/\overline{u}\left|\right)}{G({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}})}},\hfill \\ {\phi }_{\mathrm{id}}({\omega }_i,|\widehat{u}/\overline{u}|)=\phi ({\omega }_i,|\widehat{u}/\overline{u}\left|\right)-\phi ({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}}).\hfill \end{array}\right.$$

(7)

According to Equations (6) and (7), the difference between the identified nonlinear model and the actual model is $\mathrm{NL}({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}})$. The identified gain is the actual gain divided by a frequency-dependent number $G({\omega }_i,|\widehat{u}/\overline{u}\left|\right)$, and the identified phase function is the actual phase function subtracted by a frequency-dependent number $\phi ({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}})$. Because of the multiplication of the nonlinear and linear models, the complex number $\mathrm{NL}({\omega }_i,|\widehat{u}/\overline{u}{|}_{\mathrm{ref}})$ can be regarded as the coefficient $\alpha \left({\omega }_i\right)$. In the case that Equation (8) is satisfied, although the identified parameters of the nonlinear model are $\alpha$ times different from the actual parameters, they can be compensated for by the identification of the linear model. The mapping relationship from the input to the output remains unchanged:

$$\left\{\begin{array}{c}{\mathrm{NL}}_{\mathrm{id}}({\omega }_i,|\widehat{u}/\overline{u}|)={\displaystyle \frac{1}{\alpha \left({\omega }_i\right)}}\mathrm{NL}({\omega }_i,|\widehat{u}/\overline{u}\left|\right),\hfill \\ {\mathrm{FTF}}_{\mathrm{id}}\left({\omega }_i\right)=\alpha \left({\omega }_i\right)\mathrm{FTF}\left({\omega }_i\right),\hfill \end{array}\right.$$

(8)

where ${\mathrm{FTF}}_{\mathrm{id}}$ denotes the identified linear model.

Subsequently, by traversing the frequency ${\omega }_i$, the nonlinear characteristics at different frequencies can be obtained. If the influence of the frequency is small, the nonlinear characteristics obtained at different frequencies should be approximately the same, and the error can be reduced by averaging to obtain nonlinear model parameters. When frequency has a significant impact on nonlinearity, the form of the nonlinear model can be set and identified based on the true frequency dependence of the characteristic curve. After identifying the nonlinear model characteristics ${\mathrm{NL}}_{\mathrm{id}}$, the effects of the different input amplitudes can be removed from the measurement output to obtain an equivalent linear system $\mathrm{FDF}/{\mathrm{NL}}_{\mathrm{id}}$. Using the optimization algorithm, the specific parameters of the equivalent linear model ${\mathrm{FTF}}_{\mathrm{id}}$ can be identified, and the nonlinear characteristics can be separated from the linear characteristics.

2.2. Simulation

To further illustrate the above method, one simulation case is presented to validate the identification process. The case uses polynomial-type nonlinearity which simultaneously affects the gain and phase. Simulation data are generated using MATLAB/Simulink (v. R2021a), and the model is constructed according to the structure shown in Figure 1.

Consider the linear flame model as a second-order delay model [26] and the nonlinear model as a polynomial model [8]. The structure of the linear model is given by Equation (9):

$$\begin{array}{c}\hfill \mathrm{FTF}\left(\omega \right)=k{\displaystyle \frac{{{\omega }_n}^2}{{\left(\mathrm{i}\omega \right)}^2+2\mathrm{i}\xi {\omega }_n\omega +{{\omega }_n}^2}}{\mathrm{e}}^{-\mathrm{i}\omega {\tau }_0},\end{array}$$

(9)

where k is the amplification coefficient, which represents the amplification effect of the flame heat release on the incoming flow. $\xi$ denotes the damping coefficient, which is related to the dissipative effect of acoustic waves. ${\omega }_n$ is the characteristic angular frequency, determined by the resonant frequency of the system. ${\tau }_0$ denotes the time delay.

The structure of the nonlinear model is expressed in Equations (10) and (11). This type of nonlinearity affects both the gain and phase. Parameters of the linear model and the nonlinear model are shown in

Table 1. Parameters of the linear second-order delay model and the nonlinear polynomial model.

k	${\mathit{\omega }}_{\mathit{n}}\left(\mathbf{rad}/\mathbf{s}\right)$	$\mathit{\xi }$	${\mathit{\tau }}_0\left(\mathbf{s}\right)$	${\mathit{a}}_0$	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{\tau }}_{\mathit{N}}\left(\mathbf{s}\right)$
$1.68$	$73.18\pi$	$0.41$	$0.006$	1	$0.07$	$-0.37$	$0.012$

. For simplicity, the frequency does not affect the form of the gain function in this case, and the influence of the frequency is reflected in the experimental results in Section 3:

$$G\left(\right|\widehat{u}/\overline{u}\left|\right)=a_2{|\widehat{u}/\overline{u}|}^2+a_1|\widehat{u}/\overline{u}|+a_0,$$

(10)

$$\phi \left(\right|\widehat{u}/\overline{u}\left|\right)={\tau }_N[1-(a_2{|\widehat{u}/\overline{u}|}^2+a_1|\widehat{u}/\overline{u}|+a_0\left)\right],$$

(11)

where ${\tau }_N$ represents the phase lag caused by the nonlinear effects. As the disturbance amplitude increases, the phase lag also increases.

Taking the frequency range $f\in [10,110]\mathrm{Hz}$ and the velocity perturbation range $|\widehat{u}/\overline{u}|\in [0.025,1.000]$, the partial frequency response corresponding to the FDF model is shown in Figure 3. The larger the velocity perturbation $|\widehat{u}/\overline{u}|$, the greater the deviation of the flame characteristics from the linear characteristics.

Because the nonlinear model affects both the gain and the phase, it is necessary to identify the characteristics of the gain model and phase model, respectively. First, a nonlinear gain model is identified. Regard the amplitude–frequency characteristics of FDF at $|\widehat{u}/\overline{u}|=1.0$ as a reference value. The relationship between the nonlinear gain characteristics and velocity perturbation is indicated by the red scatter points in Figure 4.

As the influence of the linear model is eliminated, the scatter points corresponding to the same velocity perturbations at different frequencies coincide. The scattered points indicated in Figure 4 have the characteristic of polynomial combination, so the nonlinear gain model to be identified is assumed to be described in the form of Equation (10).

Using the optimization algorithm, the nonlinear gain model parameters can be identified as shown in Equation (12), and this effect is represented by the blue curve in Figure 4:

$$\begin{array}{c}\hfill G_{\mathrm{id}}\left(\right|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=-0.54{|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|}^2+0.10\left|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\right|+1.43.\end{array}$$

(12)

For the nonlinear phase model, regard the phase–frequency characteristics of FDF at $|\widehat{u}/\overline{u}|=1.0$ as the reference value. Subtract the phase–frequency characteristics at this reference value from the characteristics at other velocity perturbation values and then divide the results by the corresponding frequency. The nonlinear phase characteristics ${\phi }_{\mathrm{id}}$ to be identified can be obtained. The relationship between the nonlinear phase characteristics and velocity perturbation is indicated by the red scatter points in Figure 5.

Because the influences of the linear model and frequency are eliminated, the scatter points corresponding to the same velocity perturbations at different frequencies coincide. The scattered points indicated in Figure 5 have the characteristic of polynomial combination, so the nonlinear phase model to be identified is assumed to be the form of polynomial combination.

Through parameter optimization based on the data at each frequency, the blue curve shown in Figure 5 can be obtained, and the parameters of the identified nonlinear phase model are shown in Equation (13):

$$\begin{array}{c}\hfill {\phi }_{id}\left(\right|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=(-4.47{|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|}^2+0.84|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|+3.63)\times {10}^{-3}.\end{array}$$

(13)

The frequency response of the equivalent linear model is indicated by the red scatter points in Figure 6. There is a resonant peak in the amplitude–frequency characteristics. The phase continued to decrease as the frequency is increased, and a slope change is observed. Therefore, the linear model to be identified is assumed to be a second-order delay model as shown in Equation (9):

Parameter optimization is performed to obtain the blue curve shown in Figure 6. The parameters of the identified linear model are expressed by Equation (14):

$$\begin{array}{c}\hfill {\mathrm{FTF}}_{\mathrm{id}}\left(\omega \right)=1.17{\displaystyle \frac{{229.91}^2}{{\left(\mathrm{i}\omega \right)}^2+2\mathrm{i}·0.41·229.91\omega +{229.91}^2}}{\mathrm{e}}^{-0.0096\mathrm{i}\omega }.\end{array}$$

(14)

As shown in Equation (8), a coefficient exists between the actual and identified models. According to Equation (14) and Table 1, the ratio of the identified linear gain model to the actual linear gain model is 0.7, which corresponds to the actual gain model G at the reference value $|\widehat{u}/\overline{u}|=1.0$. By multiplying this value by the identified nonlinear gain model represented by Equation (12), the parameters of the actual nonlinear gain model listed in Table 1 can be obtained.

For phase, as shown in Equation (13) and Table 1, the pure delay difference between the identified linear model and the actual linear models is 0.036, which corresponds to the actual nonlinear phase model $\phi$ at the reference value $|\widehat{u}/\overline{u}|=1.0$. By subtracting this value from the identified nonlinear phase model represented by Equation (13), the actual delay parameters of the nonlinear phase model in Table 1 can be obtained.

Thus, the amplitude–frequency comparison between the combination of the identified nonlinear and linear models and the simulation data is shown in Figure 7a, and the phase–frequency comparison is shown in Figure 7b, where the solid lines are the identification result, the scattered points are the simulation data, and the dashed line is the identified nonlinear model.

It can be seen from the comparison that the identification results under different velocity perturbations are consistent with the simulation data. Therefore, the identification of the FDF in this case is realized.

3. Experimental Results and Analysis

3.1. Experimental Setup

The experimental setup shown in Figure 8a is used to determine the FDF. The main component to realize gas premixing is the axial swirler, and the structure of the swirler is shown in Figure 9. The swirler consists of a cylindrical base and eight radially extending swirl blades. Each blade contains two holes with diameters of 1 mm for gas ejection. Air flows from the inlet below the rectifier cavity, and methane flows from the bottom center duct. Subsequently, methane sprays out from the small holes on the swirl blades (60 mm away from the outlet) and mixes with air. The swirl intensity is usually described by the swirl number $S_{\mathrm{N}}$, which is defined as the ratio of the tangential momentum flux to the axial momentum flux. After simplification, the swirl number can be expressed by Equation (15):

$$\begin{array}{c}\hfill S_{\mathrm{N}}={\displaystyle \frac{2}{3}}{\displaystyle \frac{1-z^3}{1-z^2}}\tan \theta ,\end{array}$$

(15)

where z is the hub ratio of swirl, which is the ratio of the inner radius $R_n$ to the outer radius $R_w$ of the swirler. $\theta$ is the swirl angle. For the swirler used in this experiment, the inner radius $R_n$ is 3.2 mm, and the outer radius $R_w$ is 8 mm. The corresponding hub ratio z is 0.4. The swirl angle $\theta$ is 40°, and the swirl number $S_{\mathrm{N}}$ is 0.62.

Two loudspeakers are arranged downstream of the rectifier cavity and used to apply a pressure perturbation to the flame. The speakers (YD176-8C, Southern Whale, Nanjing, China) have a rated power of 100 W and a commonly used frequency range of 20–6000 Hz. In the experiment, a power amplifier amplifies the signals from the signal generator and then send them to the loudspeakers. The power amplifier (PX3, YAMAHA, Shizuoka, Japan) has a frequency response range of 60–15 kHz. The pressure perturbation intensity in the duct can be controlled by adjusting the input voltage amplitude and gain level of the power amplifier.

Four pressure measurement sensors are arranged at 61 mm, 93 mm, 125 mm, and 157 mm from the burner outlet. The pressure sensors (112A22, PCB, New York, NY, USA) have a sensitivity of 14.5 mV/kPa and a measuring range of 0.5–250 kHz. The pressure signals from these sensors are measured and recorded, and the velocity perturbation of the reference plane is determined using the two-microphone method.

To obtain the dynamic characteristics of a flame, it is necessary to measure the heat release rate. In this study, an optical measurement method is used to separate the spectral line intensity corresponding to the OH* radicals in the flame light using a spectrometer (Omni-$\lambda$300, Zolix, Beijing, China). The electrical signal represents the heat release rate of the flame. It is pointed out that the heat release rate of partially premixed flame is not only related to the chemiluminescence intensity but also related to the equivalence ratio. However, since the equivalence ratio cannot be directly measured with an experimental setup, the chemiluminescence intensity is mainly used to characterize the heat release rate in experiments [27]. To record the flame dynamic characteristics, a color high-speed camera (Phantom-V210, Vision Research, NJ, USA) is used to capture flame images directly. The camera has a maximum resolution of 1024 × 800 pixels and an exposure rate of 2000 frames/s under the maximum pixel.

The airflow rate is set at 60 SLM, the equivalence ratio is 0.75, the outlet flow cross-sectional area is 156.52 ${\mathrm{mm}}^2$, and the average flow velocity of the gas mixture near the outlet in the duct is 6.89 m/s. A flame image captured using a high-speed camera under steady-state flow-injection conditions is shown in Figure 8b. Harmonic signals are used for the excitation. The excitation frequency range is 60–300 Hz. The reference plane is selected as the center of the swirler (60 mm away from the outlet). The reason for choosing this position instead of the flame is to investigate the effect of velocity perturbation at the swirler on the flame, and to minimize the effect of the swirler on the velocity perturbation calculations when using the two-microphone method. The FTF used in this paper can be more accurately described as the characteristic of the effect of velocity perturbation at the swirler on the heat release rate, rather than the velocity perturbation close to the flame (the effect of the acoustic properties were included in the transfer function measurements). Since the FTFs obtained from different reference planes can be reconstructed from each other with known information [28], the location of the reference plane does not affect the implementation of the identification method described in this paper. The power amplifier is used to ensure that velocity perturbation at the reference plane can reach 0.05–0.50 (due to the power limitation, the maximum velocity perturbation at 300 Hz is 0.40).

3.2. Measurement and Result Analysis

The experimentally measured flame dynamic characteristics are shown in Figure 10. As shown in Figure 10, when the excitation frequency range is $f\in [60,100]$ $\mathrm{Hz}$, the gain curve of FDF under the velocity perturbation range $|\widehat{u}/\overline{u}| \in [0.05,0.25]$ almost overlaps, similar to the linear characteristics of the FTF. Under other velocity perturbations and excitation frequencies, the gain curves are clearly different, indicating that the characteristics of nonlinear gain are related not only to the velocity perturbation but also to the frequency. For the phase characteristics, the results measured under different velocity perturbations are almost the same; therefore, the nonlinear influence on the phase is not considered.

In order to check the reliability of the experimental data, the frequency response with error bars for $|\widehat{u}/\overline{u}|=0.05$ and $f\in [60,300]\mathrm{Hz}$ is plotted as shown in Figure 11 (similar for the rest of the input amplitudes). It can be seen from Figure 11 that the uncertainty of multiple experimental calculations is small, which can ensure the reliability of the experiments and calculation results.

To analyze the measured dynamic characteristics of the flame, the corresponding flame images at one peak frequencies (60 Hz) and one valley frequency (110 Hz) of the frequency response gain are selected. The flame intensity is integrated in the radial direction and divided by the maximum value. The starting point is the moment when the pressure phase calculated at the reference plane (arranged 60 mm from the burner outlet) is 0. The changes in the flame intensity in the axial direction over time can be determined. The intensity of the color bar is shown in Figure 12a. These changes are shown in the upper parts of Figure 12b–e and Figure 13a–d. Meanwhile, a white-arrow dashed line with a slope corresponding to the value of the velocity perturbation is drawn, which represents the movement of the mass flow in the axial direction.

The overall flame intensity at each moment is obtained by performing axial integration on the radial integration results, and the variation trend is consistent with the heat release rate signal measured by the spectrometer. The overall flame intensity is then divided by the average flame intensity under the corresponding working conditions, and the change in the overall flame intensity over time is drawn as shown in the lower part of Figure 12b–e and Figure 13a–d.

It can be observed from Figure 12 that at 60 Hz, as time increases, the position of the high luminous flame intensity in the axial direction swings up and down under different velocity perturbations $|\widehat{u}/\overline{u}|$, and the overall images show alternating light and dark changes. As the velocity perturbation is increased, the flame rising angle is increased, which is consistent with the slope of the mass streamline represented by the white dashed arrow. Meanwhile, as can be seen from the overall light intensity curve in the lower part of the subfigures, there is a large gap between the peak and valley values, which corresponds to a large heat release rate perturbation in the overall period. These features are responsible for the first peak of the measured FDF.

With an increase in the velocity perturbation, the overall light intensity perturbation also increases approximately linearly under a small velocity perturbation. However, when the amplitude of the velocity perturbation increases further, the growth rate of the overall light intensity perturbation decreases and eventually remains unchanged. According to the definition of the FDF, in the case of a large velocity perturbation, the increase rate of the heat release rate perturbation as a numerator cannot keep up with the increase rate of the velocity perturbation as a denominator, resulting in its ratio not being a fixed value but gradually decreasing, which explains the nonlinearity of the first peak.

Figure 13 shows that at 110 Hz, as time increases, although the position of the high luminous flame intensity in the axial direction also oscillates periodically under different speed perturbations $|\widehat{u}/\overline{u}|$, the duration of the transition section is very small, and the overall image shows no changes in light and dark. The overall light-intensity perturbation values at different times are relatively averaged, and the difference between the peak and valley values is very small. These features correspond to the overall smaller perturbations observed in the heat release rate, which explains the appearance of the first valley in the measured FDF.

These characteristics can be further analyzed in combination with flame images. Using Abel transformation, the flame surface structures under each phase at 60 and 110 Hz are obtained as shown in Figure 14a,b.

At a frequency of 60 Hz, as shown in Figure 14a, the structural characteristics of $|\widehat{u}/\overline{u}|=0.10$ and $|\widehat{u}/\overline{u}|=0.20$ do not change much under small perturbation. Only the brightness is increased with the perturbation amplitude, and the flame maintains its linear characteristics. When the amplitude of the perturbation is increased further, the flame is developed away from the axis and becomes elongated. The flame diameter decreases and then increases periodically, and the light–dark changes intensify, corresponding to a gain peak at 60 Hz.

At 110 Hz, as shown in Figure 14b, as the velocity perturbation increases, the shape of the flame surface also changes away from the axis, but the overall area is almost unchanged, and there is no alternation of light and dark, corresponding to the valley gain value at 110 Hz.

In addition, although the characteristics of the peak can be obtained, the FDF defined in this study is calculated according to the measured value of the overall heat release rate, which does not imply that the local flame forced response at the frequency corresponding to the valley value is small. As shown in Figure 13, with the addition of excitation, although the overall brightness at different times is relatively consistent, the flame still has a relatively obvious periodic swing in the axial direction. Only because the two transition sections at the beginning and end of the swing overlap does the overall integral not change significantly. The actual dynamic phenomena may be related to the superposition of multiple flame/flow processes [29]. Therefore, the response characteristics of the local flame require further study.

3.3. Identification of Flame Describing Function

As shown in Figure 10, the flame nonlinearity measured in this experiment is reflected in the gain characteristics and has no significant impact on the phase. Therefore, this subsection focuses on the identification of nonlinear gain and linear FTF models. According to the analysis in Section 3.2, when the velocity perturbation range is $|\widehat{u}/\overline{u}|\in [0.05,0.25]$ and the excitation frequency range is $f\in [60,100]\mathrm{Hz}$, the relationship between the flame gain and velocity perturbation is very small and exhibits linear characteristics. However, in other cases, nonlinear characteristics are significantly affected by the velocity perturbation. Because linearity can be considered a special form of nonlinearity, it presents a fixed amplification factor for different inputs. To maintain consistency of the equations, Equation (16) can be used to describe this characteristic segmentally, and the specific form of the nonlinear model is determined based on measurement data:

$$\mathrm{NL}(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=\left\{\begin{array}{ccc}{G}_1(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)\hfill & \hfill & \mathrm{for}|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|\in [0.05,0.25],\omega \in [120\pi ,200\pi ],\hfill \\ G_2(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)\hfill & \hfill & \mathrm{for}\mathrm{other}\mathrm{experimental}\mathrm{cases}.\hfill \end{array}\right.$$

(16)

First, nonlinear characteristics must be separated. In order to directly reflect the existing saturation characteristics, in this subsection, the amplitude–frequency characteristics of FDF are not directly analyzed, but the frequency response of the heat release rate signals is analyzed to obtain the nonlinear effects and related parameters of the velocity perturbation. The system characteristics are given by Equation (17):

$$\begin{array}{c}\hfill {\displaystyle \frac{\widehat{q}(\omega ,|\widehat{u}/\overline{u}\left|\right)}{\overline{q}}}=\mathrm{FTF}\left(\omega \right){\mathrm{NL}}^q(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right).\end{array}$$

(17)

Since Equation (17) is related to the calculation of the heat release rate, the NL function in the equation is superscripted with q. When calculating the FDF as shown in Equation (5), the amplitude response of NL should gradually approach 0 as the amplitude of the disturbance increases. In contrast, when performing calculations related to the heat release rate as shown in Equation (17), the ${\mathrm{NL}}^q$ function should gradually increase until it reaches the upper limit that represents the saturation characteristic.

According to the above analysis criteria, it is necessary to select data at the corresponding frequency when the heat release rate perturbation signals are large for analysis and identification. Therefore, experimental data measured near the peak frequencies $f\in [60,100]\cup [140,170]\mathrm{Hz}$ are selected. First, the gain characteristics are determined when the nonlinearity is not excited. The corresponding relationship between the heat release rate perturbation and velocity perturbation at $|\widehat{u}/\overline{u}|\in [0.05,0.25],f\in [60,100]\mathrm{Hz}$ is shown in Figure 15.

As shown in Figure 15, when the nonlinearity is not excited, as the velocity perturbation increases, the heat release rate perturbation also increases proportionally. The curves at different frequencies conform to a linear function relationship. Therefore, the heat release rate corresponding to the velocity perturbation $|\widehat{u}/\overline{u}|=0.25$ at each frequency is regarded as the reference value, and the amplitude–frequency characteristics at other perturbations are divided by this reference value to obtain the red scattered points in Figure 16.

As shown in Figure 16, under the same velocity perturbation, the scattered points representing the heat release rate perturbation after eliminating the linear gain influence approximately overlap, consistent with a linear function relationship. Through parameter optimization, the blue curve shown in Figure 16 can be obtained. The specific expressions and parameters are given in Equation (18), where the subscript $\left(1,\mathrm{id}\right)$ and the superscript q represent the identification result of $G_1$ when using the heat release rate:

$$\begin{array}{c}\hfill G_{1,\mathrm{id}}^q(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=4.11|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\mathrm{for}\right|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|\in [0.05,0.25],\omega \in [120\pi ,200\pi ].\end{array}$$

(18)

For the remaining data of $f\in [60,100]\cup [140,170]\mathrm{Hz}$, to determine the nonlinear form, the variation in the heat release rate perturbation with the velocity perturbation can be drawn as shown in Figure 17.

As shown in Figure 17, when the nonlinearity occurs, as the velocity perturbation increases, the heat release rate perturbation also increases, but the growth rate decreases. The curves at different frequencies are described using a polynomial relationship. Considering the proportion of velocity perturbation data at different frequencies, the amplitude–frequency characteristics of the heat release rate perturbation at $|\widehat{u}/\overline{u}|=0.35$ are regarded as the reference values, and the red scattered points in Figure 18 can be obtained.

As shown in Figure 18, under the same velocity perturbation, the scattered points of the heat release rate perturbation after scaling are relatively close, so a cubic polynomial relationship can be used to describe it. Using the PSO algorithm for parameter optimization, the blue curve in Figure 18 can be obtained. The specific expressions and parameters are given in Equation (19), where the subscript $\left(2,\mathrm{id}\right)$ and the superscript q represent the identification result of $G_2$ when using the heat release rate:

$$\begin{array}{c}\hfill G_{2,\mathrm{id}}^q(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=-1.92{|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|}^3-1.51{|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|}^2+3.61\left|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\right|\mathrm{for}\mathrm{other}\mathrm{cases}.\end{array}$$

(19)

Based on the image analysis results in Section 3.2, it can be seen that as the velocity perturbation increases, the growth rate of the maximum overall heat release rate slows down and eventually tends to remain unchanged. This is consistent with Dowling’s assumption of saturation of the heat release rate [7]. Therefore, based on Equation (19), the maximum value of 1.19 of the cubic function is selected as the saturation value, corresponding to $|\widehat{u}/\overline{u}|=0.56$. The saturation model is defined in Equation (20). When the velocity perturbation is greater than 0.56, the amplification factor of the saturation model remains at 1.19 at any frequency, and the effect is shown by the pink dashed line in Figure 18. However, it is undeniable that the nonlinear characteristics of the flame are not only saturated but also “saturate then increase again, decrease then increase then saturate, among others” [29]. Different cases need to determine the nonlinear model structure based on theoretical studies or actual measurements:

$$\begin{array}{c}\hfill G_s(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=1.19\mathrm{for}\mathrm{saturation}.\end{array}$$

(20)

Since the identification results of Equations (18) and (19) use different reference values, in order to achieve function continuity, an additional correction term can be multiplied on Equation (18), which is given in Equation (21):

$$\begin{array}{c}\hfill \beta \left(\omega \right)={\displaystyle \frac{{|\widehat{q}/\overline{q}(\omega ,|\widehat{u}/\overline{u}\left|\right)|}_{|\widehat{u}/\overline{u}|=0.25}}{{|\widehat{q}/\overline{q}(\omega ,|\widehat{u}/\overline{u}\left|\right)|}_{|\widehat{u}/\overline{u}|=0.35}}}.\end{array}$$

(21)

In summary, the nonlinear models ${\mathrm{NL}}^q$ in Equation (17) is given in Equation (22):

$${\mathrm{NL}}^q(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=\left\{\begin{array}{c}4.11\beta \left(\omega \right)|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\mathrm{for}\right|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|\in [0.05,0.25],\omega \in [120\pi ,200\pi ],\hfill \\ -2.22{|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|}^3-2.28{|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|}^2+4.68\left|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\right|\mathrm{for}\mathrm{other}\mathrm{cases},\hfill \\ 1.52\mathrm{for}\mathrm{saturation}.\hfill \end{array}\right.$$

(22)

Based on the above nonlinear modeling results, the characteristics of the heat release rate perturbation after removing the influence of the nonlinearity are obtained as shown in Figure 19. The amplitude–frequency response at a certain frequency is approximately constant, and the model characteristics are shown in Equation (23):

$$\begin{array}{c}\hfill {\displaystyle \frac{\widehat{q}(\omega ,|\widehat{u}/\overline{u}\left|\right)}{\overline{q}}}=K\left(\omega \right){\mathrm{NL}}^q(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right),\end{array}$$

(23)

where K is a function dependent on the frequency.

According to the FDF definition, by dividing $|\widehat{u}/\overline{u}|$ on both sides of Equation (23), Equation (24) can be obtained:

$$\begin{array}{c}\hfill \mathrm{FDF}(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=K\left(\omega \right){\mathrm{NL}}_{\mathrm{id}}(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right),\end{array}$$

(24)

where ${\mathrm{NL}}_{\mathrm{id}}$ is shown in Equation (25):

$${\mathrm{NL}}_{\mathrm{id}}(\omega ,|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\left|\right)=\left\{\begin{array}{c}4.11\beta \left(\omega \right)\mathrm{for}|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|\in [0.05,0.25],\omega \in [120\pi ,200\pi ],\hfill \\ -2.22{|{\displaystyle \frac{\widehat{u}}{\overline{u}}}|}^2-2.28\left|{\displaystyle \frac{\widehat{u}}{\overline{u}}}\right|+4.68\mathrm{for}\mathrm{other}\mathrm{cases},\hfill \\ {\displaystyle \frac{1.52}{|\widehat{u}/\overline{u}|}}\mathrm{for}\mathrm{saturation}.\hfill \end{array}\right.$$

(25)

According to Equation (24), after removing the nonlinear model ${\mathrm{NL}}_{\mathrm{id}}$, K is equivalent to the ratio of the heat rate perturbation to the velocity perturbation at a certain frequency, which is consistent with the characteristics of the linear model represented by Equation (1). The frequency response of the linear model to be identified under different velocity perturbations is indicated by the red scatter points in Figure 20. The scattered points correspond to different velocity perturbation amplitudes at the same frequency overlap, which satisfies the requirements of the linear model. Therefore, a linear model is identified based on these data.

As shown in Figure 20, the amplitude–frequency response of the FTF varies with frequency, and there are two peaks, which is consistent with the Gaussian distribution model characteristics of swirling premixed flame [30]. Therefore, this model is adopted for the linear model identification, and its impulse response expression is shown in Equation (26):

$$\begin{array}{c}\hfill h_k=k_1{\displaystyle \frac{1}{{\sigma }_1\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{t-{\tau }_1}{{\sigma }_1}}\right)}^2}+k_2[{\displaystyle \frac{1}{{\sigma }_2\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{t-{\tau }_2}{{\sigma }_2}}\right)}^2}-{\displaystyle \frac{1}{{\sigma }_3\sqrt{2\pi }}}{e}^{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{t-{\tau }_3}{{\sigma }_3}}\right)}^2}],\end{array}$$

(26)

where $h_k$ denotes the impulse response of the system. $k_1$, ${\tau }_1$ and ${\sigma }_1$ are the parameters related to acoustic wave propagation. $k_2$, ${\tau }_2$, ${\tau }_3$, ${\sigma }_2$, and ${\sigma }_3$ are the parameters related to circulation fluctuations.

Performing a Fourier transform on Equation (26) yields Equation (27):

$$\begin{array}{c}\hfill {\mathrm{FTF}}_{\mathrm{id}}\left(\omega \right)=k_1{e}^{-\mathrm{i}\omega {\tau }_1-{\displaystyle \frac{1}{2}}{\left(\omega {\sigma }_1\right)}^2}+k_2[e^{-\mathrm{i}\omega {\tau }_2-{\displaystyle \frac{1}{2}}{\left(\omega {\sigma }_2\right)}^2}-e^{-\mathrm{i}\omega {\tau }_3-{\displaystyle \frac{1}{2}}{\left(\omega {\sigma }_3\right)}^2}].\end{array}$$

(27)

The blue curve in Figure 20 can be obtained by parameter optimization, and the specific parameters are shown in

Table 2. Identified parameters of the linear Gaussian distribution model.

${\mathit{\tau }}_1\left(\mathbf{s}\right)$	${\mathit{\tau }}_2\left(\mathbf{s}\right)$	${\mathit{\tau }}_3\left(\mathbf{s}\right)$	${\mathit{\sigma }}_1$	${\mathit{\sigma }}_2$	${\mathit{\sigma }}_3$	${\mathit{k}}_1$	${\mathit{k}}_2$
$0.0056$	$0.0057$	$0.0015$	$0.0081$	$0.0016$	$0.0016$	$1.15$	$0.68$

. The scatter points of different input amplitudes at the same frequency are concentrated near the fitting results, satisfying the homogeneity requirements of the linear model. Meanwhile, because the nonlinear model in the experiments does not affect the phase calculation, the phase–frequency characteristics above the linear model are also the identification results for the phase–frequency characteristics of the FDF.

A gain comparison between the results after combining nonlinearity and linearity and the original measurement results is shown in Figure 21, where the solid line represents the identification results, the scatter points represent the experimental data, and the dashed line represents the identified nonlinear gain model.

As shown in Figure 21, the experimental measurement data represented by the scatter points under different velocity perturbations are basically consistent with the curve of the identified model. Therefore, the model obtained by combining Equations (25) and (27) better describes the characteristics of the swirling premixed flame obtained in this experiment. There are approximately two peaks in the frequency response, and the nonlinear influence caused by the velocity perturbation is reflected.

4. Conclusions

In this study, the characteristics of partially premixed flames were investigated. An FDF identification method based on a parallel subsystem block model was designed. This method can separate flame characteristics into a parallel connection of a nonlinear model and a linear model. The nonlinear model is related to the disturbance frequency and velocity perturbation amplitude, whereas the linear model depends only on the disturbance frequency. Subsequently, the method was verified for a simulation case. Finally, experiments were conducted on an actual swirl combustion experimental rig to determine the frequency response of the flame. The nonlinear and linear characteristics of the flame were successfully separated. Combined with the flame image analysis, it was found that the gain of this type of flame exhibited linear characteristics under small velocity perturbations in the low-frequency band. In the other cases, as the velocity perturbation amplitude increased, the gain gradually decreased and exhibited saturation characteristics. As for the phase, the changes in the velocity perturbation had little impact on the flame, and there was no nonlinear effect.

This study considered the ubiquitous nonlinear characteristics of a real flame under different velocity perturbations to design an identification method. According to the obtained linear model and nonlinear model, the different effects of flame characteristics and velocity perturbation nonlinearity can be specifically analyzed to obtain more accurate flame dynamic characteristics. This can not only help to analyze various nonlinear behaviors when combustion instability occurs but can also be combined with acoustic networks to predict thermoacoustic stability under different combustion chamber structures. Although not discussed in the paper, the proposed method provides a new modeling approach for thermoacoustic oscillation control. We believe that the subsystem modeling method has the potential to effectively reduce the difficulty of control and improve the quality of control compared to an overall nonlinear model. It should be emphasized that flame models under more complex combustion conditions are likely to be coupled linear/nonlinear, and the separation methods described in this paper may not necessarily yield the desired results. Therefore, in future work, we will try to study the separation principles of linearity/nonlinearity from theory or experiments, and generalize the method of this paper to higher-dimensional nonlinear problems associated with more variables.