Synchronization based model for turbulent thermoacoustic systems

In most power generation devices, including gas turbines, rocket engines, and industrial burners, the combustion process usually occurs in an enclosed space. In these devices, the heat release rate fluctuations may interact with the acoustic field in the combustor, and vice versa. As a result, feedback may arise, and when this feedback is positive, there will be growth in acoustic fluctuations, which eventually saturates to limit cycle oscillations due to nonlinearities in the system. This phenomenon is known as thermoacoustic instability. The violent oscillations are usually harmful to combustion devices, they can cause the extinction of the flame, or damage the hardware [1, 2, 3, 4]. In the past few decades, lean-premixed-prevaporized (LPP) combustors are widely applied in the industry for reducing emissions. The flame is inherently more sensitive to perturbation when operating in fuel-lean conditions. Thus, the LPP combustors are more susceptible to thermoacoustic instability. Because of the wide application of LPP combustors, thermoacoustic instability has become a major concern in combustor design. Numerous studies in the past have focused on prediction of onset, evasion or suppression of thermoacoustic instabilities.

Rayleigh first explained the emergence of thermoacoustic instability in 1878. He showed that when the acoustic oscillation is in phase with the fluctuation of unsteady heat release rate, energy is added to the acoustic field, hence the amplitude of acoustic pressure will grow [5]. The amplitude would keep increasing until limited by the nonlinear effects. To identify the regime where thermoacoustic instability occurs, investigating the feedback mechanism is of crucial importance [6]. In many cases, the propagation of acoustic waves is considered to be a linear phenomenon since the amplitude of pressure oscillations corresponding to the perturbations is relatively low when compared to the mean pressure in the combustor [7]. The flame, however, is more complex due to the inherent nonlinearities associated with the chemical reactions and transport processes of heat and momentum [8]. The response of the flame to the acoustic perturbation is also highly nonlinear, especially in a turbulent environment [7].

Traditionally, modeling of thermoacoustic instability aims to determine the response of the flame to acoustic perturbation. In the past few decades, several low-order models based on quasi-linear approach were introduced. Among them, the most commonly used models are flame transfer function (FTF) [6, 9, 10, 11, 12, 13] and flame describing function (FDF) [6, 14, 15, 16]. FTF estimates the linear response of the flame to acoustic perturbations by either experiment or CFD. External velocity perturbation ($u^{\prime}$) is introduced to the combustor by using actuators (speakers or linear actuators). Then, the response in form of the fluctuation in the heat release rate ($\dot{q}^{\prime}$) is measured and evaluated as a function of the forcing frequencies ($\omega$). The flame transfer function, $F$, is, then, defined as shown in Eq. (1), where $\bar{\dot{q}}$ and $\bar{u}$ are the mean heat release rate and the mean velocity at a given condition.

Once the FTF is estimated, the global dynamical behavior of a thermoacoustic system can be evaluated by incorporating the acoustic driving due to heat release rate fluctuations (obtained from FTF) into the acoustic field equations. However, established on classic linear stability analysis, FTF cannot capture the nonlinear effect of the thermoacoustic system. Thus the amplitude of limit cycle oscillation, or triggering phenomenon in a subcritical Hopf bifurcation, cannot be estimated using FTF [17]. Subsequently, flame describing function (FDF) was introduced. Originated from theories of control systems [18] and introduced by Dowling et al. [6, 10], FDF extended the framework of FTF to the nonlinear regime. It defines the complex response of the flame as a function of both input frequency ($\omega$) and amplitude ($|u^{\prime}|$), as shown in Eq. (2).

In the FDF approach, the flame is considered as a nonlinear module coupled to the linear acoustic system. Compared to FTF, FDF can predict the amplitude of limit cycle oscillation. However, FDF is based on quasi-linear assumption, i.e. for a particular forcing, the flame only responds at the same frequency. In other words, the response of flame at the higher harmonic frequencies resulting from the forcing is ignored [19]. Also, FDF may not be suitable at modeling complex behaviors that include non-harmonic responses, e. g. chaos and intermittency. Moreover, in FDF, the flame is forced by acoustic perturbations at different frequencies and amplitude, which similar to FTF, does not include many aspects of inherent physics of the flame dynamics [20]. For a long time, the conditions of a thermoacoustic system were considered either operational stable or unstable (limit cycle thermoacoustic oscillation). Correspondingly, FTF/FDF approaches mainly focus on finding out the unstable regime and the amplitude of periodic oscillation. However, experimental observations have shown that thermoacoustic systems can exhibit complex dynamical behaviors other than limit cycle oscillations [21, 22, 23]. Many of these behaviors can be identified from laminar thermoacoustic systems. Beyond limit cycle oscillation, Kabiraj et al.  [24] observed a route to chaos through quasi-periodic oscillation in a laminar Rijke tube, which conforms Ruelle-Takens scenario. Various bifurcations in the same system were subsequently investigated [25], and complex behaviors including strange non-chaos were captured [26]. These finding were also supported by other groups. For example, in a separate study, Guan et al. [27] also identified an intermittency route to chaos in a laminar Rijke tube. Collectively, these studies show that even simple thermoacoustic system can exhibit rich dynamics.

Inspired by the observations, the underlying physics of these complex behaviors in thermoacoustic systems attracts wide research interest. Existing approaches based on one-way coupling and forced response cannot provide plausible explanations for these phenomena. However, an alternative viewpoint based on synchronization is getting attention recently. Pawar et al. [28] showed that the onset of thermoacoustic instability is a result of the mutual synchronization between the unsteady heat release rate and acoustic fluctuations. Depending on the dynamic states of the system, the two signals can be either synchronized or de-synchronized. Later studies by Mondal et al. [29] found various synchronization states including phase locking, intermittent phase locking, and phase drifting. Moon et al. [30] presented a thermoacoustic system that consists of two coupled turbulent combustors, and investigated their mutual synchronization. Premraj et al. [31] investigated the effect of system parameters on the synchronization behaviors of coupled oscillators.

These researches paved the way for building new models for thermoacoustic systems using the framework of synchronization. Godvarthi et al. [32] found that while varying the coupling strength, a system of locally coupled grid of chaotic Rossler oscillators coupled globally to a chaotic Van der Pol oscillator can become self-organized and transition to periodic oscillations. Guan et al. [33] introduced a low order model consisting of two Van der Pol oscillators. In our recent work, we presented a phenomenological model based on the synchronization framework for modeling laminar thermoacoustic systems [20]. We use a pair of damped linear oscillators to represent the unsteady heat release rate and acoustic fluctuation, the two oscillators are nonlinearly coupled. By adjusting the coupling strength, the model can replicate the Hopf bifurcation in a laminar thermoacoustic system, and capture the route to chaos, and also states including quasi-periodic oscillation, chaos, and strange non-chaos. In the present work, we attempt to extend the synchronization framework for thermoacoustic systems to the turbulent regime.

In turbulent combustors, turbulent reactive flow always generates low amplitude disordered acoustic fluctuations in the chamber. During stable operation, the sound is known as combustion noise. Traditionally, combustion noise is considered to be acoustic fluctuations and treated as background noise. However, recent studies  [34, 35, 36] showed that combustion noise possesses high-dimensional chaotic fluctuations and exhibits multifractal characteristics. These studies indicate that the combustion noise is dynamically complex, and should not be considered as noise. Meanwhile, an intermittency state was also discovered presaging the transition from combustion noise to limit cycle oscillation [37, 38]. When increasing the flow rate in the combustor, intermittent bursts of periodic oscillations with high amplitude can appear randomly among low amplitude aperiodic oscillations. Upon increasing the flow rate further, the system will finally enter a regime of periodic oscillation. The transition is found to be a process in which the system gradually self-organizes and loses its multifractality, as order emerges from chaos [36, 39]. Several universal power laws were identified [40, 41, 42]. These studies provide valuable early warning signals to evade combustion instability.

In this study, we attempt to extend our synchronization framework [20] for the thermoacoustic system further to the turbulent regime and capture the transition from combustion noise to limit cycle oscillation through the route of intermittency. The model is also expected to identify the self-organization and loss of multifractality in the transition. Since it is a phenomenological model, the primary goal is to qualitatively replicate the observations from experiments. In our previous work [20], we showed that a pair of coupled oscillators when coupled with proper coupling strength, can model the quasi-periodic route to chaos observed in laminar thermoacoustic systems. Based on that experience, in this work, we attempt to use a tuned pair of coupled oscillators to represent the simplified turbulent reactive flow. These two oscillators were then, further coupled with a third oscillator representing the acoustic field. We will show that, with proper coupling strength, this three oscillator model can reflect the combustion dynamics arising in a turbulent combustor.

The rest of the manuscript is organized as follows: in Sect. 2, we introduce the model based on the synchronization framework; In Sect. 4, we compare the dynamics observed in the model and the experiment. The results will show that the model can qualitatively replicate the transition from combustion noise to periodic oscillation in a turbulent thermoacoustic system. Power laws associated with self organization and loss of multifractality are also captured.

Our primary goal is to present a phenomenological, reduced-order model based on a synchronization framework for modeling the transition to thermoacoustic instability in a confined turbulent combustor. In our previous study [20], a synchronization-based model was proposed for a laminar combustor, where the acoustic field and the unsteady heat release rate were represented by a pair of nonlinearly coupled damped oscillators. By varying the coupling strength between the two oscillators, the system can reproduce the route to chaos observed in a laminar combustor [24]. Multiple bifurcations and different dynamical states from the experiment were captured, including limit cycle oscillation, quasi-periodic oscillation, strange-non chaos [26], and chaos. Similarly, in this work, the acoustic pressure and the unsteady heat release rate are considered as two sub-systems that are nonlinearly coupled. Unlike in a laminar combustor, at the decoupled states, the acoustic and heat release rate fluctuations are disordered due to underlying turbulent fluctuations. This effect needs to be included in the coupled oscillator model.

From our previous study, we realized that two second-order oscillators with proper coupling can be used to produce chaos [20], and can be used to model the underlying disorder from turbulence. Thus, a pair of coupled oscillators $y_{1}$ and $y_{2}$ are used to generate the chaotic oscillation that corresponds to combustion noise, when weakly coupled to oscillator $y_{3}$. Hence oscillators $y_{1}$ and $y_{2}$ form a sub-system representing the simplified turbulent reactive flow, where the time series of $y_{2}$ represents the unsteady heat release rate. Furthermore, $y_{3}$ can represent the pressure fluctuations in this three coupled oscillator model. The three oscillators have natural angular frequencies of $\omega_{1}$, $\omega_{2}$, and $\omega_{3}$. We will use the symbol $f$ to denote the frequency, which is related to angular frequency as $\omega~{}=~{}2\pi f$. The damping coefficients for these three oscillators are $\zeta_{1}$, $\zeta_{2}$, and $\zeta_{3}$, respectively. When decoupled (i.e. the coupling terms are zero), the three oscillators are all damped simple harmonic oscillators, as shown in Eqs. (3), (4), and (5).

where, the terms on the right hand side (NL) are the nonlinear coupling terms. These nonlinear terms used for this model are as follows:

Here, we define $C_{12},C_{21},C_{23},C_{32}$ as the coupling strength, where $C_{ij}$ denotes how strongly the $j^{th}$ oscillator affects the dynamics of the $i^{th}$ oscillator. Furthermore, the relative strength of the mutual coupling were defined using the ratio of $C_{ij}$s, $R_{1}~{}=~{}C_{21}/C_{12}$, $R_{2}~{}=~{}C_{23}/C_{32}$. Recent studies showed that the coupled interaction between the unsteady heat release rate and the acoustic field is asymmetric [43, 44]. Unsteady heat release rate exerts a stronger influence on the acoustic field than the converse, which implies $C_{32}>C_{23}$, or $R_{2}<1$. We select the natural frequencies of the three oscillators in a way that the output time series can match the trend of the dominant frequency in the experiment. To achieve this, we assign a fixed ratio between the natural frequencies of oscillator $y_{1}$ and $y_{2}$ ($\omega_{1}~{}=~{}1.125\omega_{2}$), resulting in intended aperiodic oscillation when weakly coupled to oscillator $y_{3}$. Meanwhile, to ensure that the dominant frequency of $\dot{q}^{\prime}$ increases during the transition as observed in the experiment, we set a linear relation between the natural frequency, $\omega_{2}$ and the coupling strength, $C_{32}$ ($\omega_{2}~{}=~{}2\pi C_{32}/20$). The natural frequency of the acoustics ($y_{3}$) is chosen to be constant ($\omega_{3}~{}=~{}1.3\times 2\pi$), since in the experiments the natural frequency of the combustor geometry is fixed. The other parameter values used in the Eq. (3-8) are selected to be: $\zeta_{1}~{}=~{}0.1$, $\zeta_{2}~{}=~{}0.05$, $\zeta_{3}~{}=~{}0.3$, $C_{12}~{}=~{}1.62$, $R_{1}~{}=~{}1.2$, and $R_{2}~{}=~{}0.54$. In the study, since $y_{1}$ and $y_{2}$ form a subsystem that represents unsteady heat release rate, $C_{12}$ is fixed. Meanwhile, $C_{32}$ is considered the control parameter which dictates the transitions between different dynamical states of the system. In an experimental turbulent thermoacoustic system, the control parameter is generally the Reynolds number of the flow, $Re$=$\bar{u}L/\nu$, where $L$ is the characteristic length scale of the burner and $\nu$ is the kinematic viscosity of the unburned mixture. With the current model and the value of parameters mentioned above, the transition between various states was observed for the range of $C_{32}$ from 14 to 18. The coupled equations (Eq: 3-5) are solved using the 4th order Runge-Kutta method in Matlab. Gaussian noise with variance $\sigma=5\%$ is then added to the time series. We note that for a fixed $C_{32}$, the model outputs normalized time series of heat release rate and pressure fluctuations.

As mentioned earlier the primary goal of this work is to demonstrate that coupled oscillator models can reproduce combustion dynamics in a turbulent combustor. During construction of the model we intended to keep the nonlinear terms as simple (low-order) as possible while qualitatively reproducing the key dynamics observed in experiments. The current forms of nonlinear terms shown in Eq: 6-8 and the values of various constants were chosen after manual optimization and satisfy the requirements. We note that, in general, a large number of possible forms of nonlinear terms can be selected. A robust method for identifying and optimizing the nonlinear terms will be focus of our future work.

We primarily use the experimental data from Pawar et al. [28] as a benchmark to validate the model. The experimental study applies to a laboratory-scale turbulent combustor with a bluff body stabilized flame, where liquefied petroleum gas was burned with air. The incoming airflow first passes through a settling chamber. Then, it is premixed with the fuel and sent into the combustion section. The pressure fluctuation ($p^{\prime}$) in the combustor is estimated using pressure transducers. The heat release rate ($\dot{q}$) is acquired by measuring the chemiluminescence intensity of CH* using a photo-multiplier tube (PMT), from which its fluctuations ($\dot{q}^{\prime}$) is calculated. During the experiment, the fuel flow rate was kept constant at 25 slpm, while the flow rate of air varied from 400 slpm to 940 slpm, covering Reynolds number ($Re$) range from $1.09\times 10^{5}$ to $2.12\times 10^{5}$. We note that an increase in flow rate ($Re$) increases the mean heat release rate ($\bar{q}$) of an unconfined (de-coupled from acoustics) turbulent flame due to decreased residence time [45]. Thus, it is appropriate to use heat release rate intensity $\dot{q}^{\prime}_{rms}/\bar{\dot{q}}$ as the statistical measure of fluctuations. The acoustic characteristics of the chamber, however, remain unaffected by the $Re$, thus $p^{\prime}_{rms}$ is a good measure.

From the measurements, we confirm that the combustor exhibited stable operation at low $Re$. However, due to the inherent turbulence, the pressure and heat release rate keep fluctuating at a low amplitude. This regime refers to high dimensional chaos [46]. As $Re$ increases, the amplitude of pressure oscillation increases continuously. Meanwhile, among aperiodic oscillations, burst and periodic oscillations start emerging in a near-random fashion, which refers to the state of intermittency. When $Re$ further increases, both pressure and heat release rate oscillate periodically, and with significantly greater amplitude, corresponding to the onset of thermoacoustic instability.

We will use a second set of experimental data from Nair et al. [37] during our discussion on the power-law behaviors in the transition process (Sec: 4.5). This work used a swirl stabilized burner. The rest of the configuration and experimental technique remained mostly unchanged. The details of this setup can be found in [37].

In experiments, the flow in the combustor is fully developed (constant turbulence intensity, $u^{\prime}/\bar{u}$) and hence, $u^{\prime}$ grows linearly with $Re$ ($u^{\prime}\propto Re$). In these experiments, the measured $p^{\prime}$ and $\dot{q}^{\prime}$ which increase with $Re$ are composed of the contribution from turbulence and the thermoacoustic instability. The former arises from the fluctuations in flow velocity $u^{\prime}$, and hence can be assumed to be proportional to $Re$. These turbulence-induced effects are apparent when one measures the pressure in a turbulent flow inside the combustor without the flame or heat release rate in an unconfined flame, thus no thermoacoustic effects. It is to be noted that the model presented in Eq: 3 - 8, only accounts for the coupled dynamics between acoustics and heat release, thus discounts the increase in $p^{\prime}$ and $\dot{q}^{\prime}$ due to turbulence effects. Since, these discounted effects are proportional to $Re$, and in our model, we assumed $C_{32}\sim Re$, for comparison purposes, the experiment equivalent of pressure and heat release rate from our models can be obtained as $p^{\prime}=y_{3}C_{32}$, and $\dot{q}^{\prime}=y_{2}C_{32}$.

The presented synchronization based three-oscillator model is expected to replicate some of the key features of a turbulent combustor. The model should capture the transition from combustion noise to periodic oscillation, during which the amplitude of pressure and heat release rate oscillation increase monotonically. It should also predict different synchronous states during the transition, which mostly depends on the synchronization states of the pressure and heat release rate. Moreover, in the regime of combustion noise, the model is expected produce a multifractal behavior in pressure oscillations, which is the hallmark of a stable turbulent combustor. As the system transitions to thermoacoustic instability, self-organization is observed during experiments [40, 41] and the time series shows a scale-independent nature marked by power-law dynamics. We expect the model to be able to predict similar behavior.

In the following subsections, we will analyze the statistics and dynamics of the time series produced by the model, and qualitatively compare with those of the experimental data. We note that the primary expectation from the model with predefined model parameters is to qualitatively reproduce the experimental observations. The quantitative similarity is not expected, which requires tuning the model parameters based on the experiments.