1. Introduction
Thermoacoustic instability is a complex phenomenon arising from the nonlinear coupling between unsteady heat release and acoustic pressure oscillations in continuous combustion systems. It poses a significant challenge in the design and operation of modern propulsion and power-generation devices, including gas turbines, rocket engines and industrial burners. When the rate of heat release is in phase with the pressure fluctuations, energy is fed into the acoustic field, potentially leading to self-sustained oscillations of large amplitude. These instabilities can degrade performance, reduce efficiency, and cause severe structural damage due to fatigue or resonance (Lieuwen & Yang Reference Lieuwen and Yang2005). In practical combustion systems such as gas turbines and rockets, thermoacoustic instabilities can arise from the complex interaction between heat release oscillations from multiple flames within the acoustic pressure field, highlighting the importance of analysing the nonlinear dynamical characteristics of flame–flame interactions.
In multi-flame systems, the interplay between flames can induce a rich variety of nonlinear oscillatory phenomena, often involving thermoacoustic interactions (Sujith & Unni Reference Sujith and Unni2021). More specifically, coupling between spatially distributed flames through both the acoustic field and hydrodynamic interactions can lead to coherent or incoherent flame responses, giving rise to in-phase and out-of-phase oscillations, spatially localised synchronisation, amplitude death and bifurcations. For instance, Kitahata et al. (Reference Kitahata, Taguchi, Nagayama, Sakurai, Ikura, Osa, Sumino, Tanaka, Yokoyama and Miike2009) observed limit-cycle oscillations and synchronisation of multiple candle flames, reporting in-phase and out-of-phase coupled oscillations that depended on the flame distance. Manoj, Pawar & Sujith (Reference Manoj, Pawar and Sujith2018) experimentally demonstrated the coexistence of amplitude death and phase-flip bifurcation in coupled flames oscillating in self-sustained manner, revealing a transition between in-phase and anti-phase states as coupling strength varies. Later, Dange et al. (Reference Dange, Pawar, Manoj and Sujith2019b ) investigated the physical mechanisms underlying mode transitions in these coupled flames, analysing how buoyancy-driven vortex interactions give rise to distinct synchronisation states, including in-phase, anti-phase, amplitude death and desynchronisation. For turbulent flames, Takagi et al. (Reference Takagi, Gotoda, Miyano, Murayama and Tokuda2018) numerically investigated the scale-free nature of buoyancy-induced turbulent flames, and demonstrated synchronisation between two coupled flames driven by transverse vortex interactions. Recently, in a practical system, Guan et al. (Reference Guan, Choi, Liu and Kim2024) employed inter-system complex network analysis to investigate flame–flame interaction in an axially staged sequential combustor. Their results suggested that a positive feedback loop can be established via coupling between the primary stage’s acoustic field and the secondary flame.
The collective dynamical behaviour of multiple oscillators within combustion systems is often represented using coupled oscillator equations, particularly in the thermoacoustic context. These low-order models provide a mathematical platform for characterising, for example: dissipative and time-delayed coupling between two prototypical thermoacoustic oscillators (Biwa, Tozuka & Yazaki Reference Biwa, Tozuka and Yazaki2015; Thomas et al. Reference Thomas, Mondal, Pawar and Sujith2018a,Reference Thomas, Mondal, Pawar and Sujithb); mutual coupling via a cross-talk section linking two model gas-turbine combustors (Moon et al. Reference Moon, Guan, Li and Kim2020; Guan et al. Reference Guan, Moon, Kim and Li2021); and interactions between multiple global eigenmodes that govern the dynamics of the entire combustor (Bonciolini et al. Reference Bonciolini, Faure-Beaulieu, Bourquard and Noiray2021; Orchini & Moeck Reference Orchini and Moeck2024). While most of these models capture the coupled thermoacoustic dynamics phenomenologically, some studies adopt models derived from first principles (Thomas et al. Reference Thomas, Mondal, Pawar and Sujith2018a,Reference Thomas, Mondal, Pawar and Sujithb; Orchini & Moeck Reference Orchini and Moeck2024), further supporting the relevance of coupled oscillator equations for modelling the nonlinear dynamics of multiple thermoacoustic systems.
Furthermore, several studies on multiple thermoacoustic oscillators have incorporated stochastic frameworks to account for the inherent randomness and external perturbations present in combustion processes. In an early study, Culick et al. (Reference Culick, Paparizos, Sterling and Burnley1992) modelled two longitudinal modes of a combustor using coupled stochastic differential equations subject to both additive and parametric noise. As for two physically distinct thermoacoustic oscillators, Thomas et al. (Reference Thomas, Mondal, Pawar and Sujith2018a) investigated the effect of external noise on a pair of Rijke tubes by introducing a Gaussian white noise term into a coupled oscillator model. In a more recent study, Lee, Gupta & Li (Reference Lee, Gupta and Li2024) numerically analysed the stochastic dynamics of a pair of mutually coupled van der Pol oscillators in a noisy environment. The authors systematically examined the Fokker–Planck dynamics – which represents the probability density function of the oscillation amplitude under stochasticity – across different coupling types, coupling strengths and noise levels. Given that Fokker–Planck models play a central role in the system identification of noisy thermoacoustic systems (Noiray & Schuermans Reference Noiray and Schuermans2013; Lee et al. Reference Lee, Guan, Gupta and Li2020; Son & Lee Reference Son and Lee2024), it is expected that coupled flame dynamics can be effectively diagnosed and predicted using a system identification framework, provided that a suitably chosen oscillator model is employed.
In this study, we build upon these previous works to experimentally and mathematically analyse the nonlinear dynamics arising from flame–flame interaction between two flames in an annular combustor. We focus in particular on the lock-in phenomenon of these flames, whereby self-excited oscillations synchronise with an external forcing signal (Kashinath, Li & Juniper Reference Kashinath, Li and Juniper2018), aiming to elucidate the role of mutual coupling and synchronisation in a stochastic environment. To this end, we integrate the coupled oscillator model, noise-induced dynamics and system identification frameworks to model and interpret the behaviour of two thermoacoustic oscillators under combined sinusoidal and stochastic forcing. This approach enables a mathematical explanation of the observed localised lock-in phenomenon, supported by both experimental and numerical investigations.
We begin by introducing the experimental set-up and data in § 2. This is followed by a description of the coupled stochastic oscillator model and the system identification framework in § 3. The results are then presented in § 4, with a particular focus on the effects of coupling and synchronisation, before concluding with a summary of key findings and limitations in § 5.
2. Experimental set-up and data
(a) Schematic of the experimental set-up. (b) Direct imaging of the two flames. (c) Photograph of the combustor. (d) Transient mode shape at 780 Hz identified using a three-dimensional Helmholtz solver.
We employ the experimental set-up illustrated in figure 1(a), which comprises an annular combustor (figure 1c), and an acoustic forcing device. The combustor has inner diameter 121 mm, outer diameter 216.5 mm, and height 292 mm. The outer wall is constructed from SUS304 stainless steel, while the inner wall consists of 3 mm thick quartz to allow optical access, as shown in figure 1(b). The top of the combustor features twelve circular outlets, each with diameter 32 mm. Only two inlets are installed at the base of the combustor, allowing flames to be stabilised at two of the twelve outlet positions, separated by 30
$^\circ$
(see figure 1
a). This configuration is specifically designed to isolate and investigate flame–flame interaction under well-controlled conditions, without the influence of additional flames or geometric symmetry constraints. Each inlet is equipped with an identical swirl nozzle with diameter 12.7 mm. Each nozzle incorporates six swirler vanes set at vane angle 60
$^{\circ }$
, corresponding to a geometrically estimated swirl number 1.22, using a swirler design identical to that in Worth & Dawson (Reference Worth and Dawson2013a
,
Reference Worth and Dawsonb
).
For the combustion experiments, premixed methane (99.95 % purity) and compressed air are used at equivalence ratio 0.82. The total mass flow rate of the fuel–air mixture is maintained at 1.5 g s−1. Prior to entering the premixing chamber, air is passed through a compressor (Kyungwon AL5N) and a dryer (Kyungwon T15K). The mass flow rates of both fuel and air are individually controlled using mass flow controllers (MKP TSC-145, MKP TSC-230). Dynamic pressure and temperature at the nozzle outlet are measured using pressure sensors (PCB Piezotronics Model 113B28) and thermocouples (KMQSS-125E-6), respectively. A loudspeaker (Sammi SU-150EF) is positioned aside the annular combustor to impose acoustic forcing. This configuration allows us to focus primarily on the transverse mode, consistent with our previous study (Guk, Seo & Lee Reference Guk, Seo and Lee2024). To investigate lock-in phenomena at the transverse thermoacoustic mode, a sinusoidal signal is generated using a signal generator (Keysight 33210A), and amplified through an amplifier (LTO MAC 2.2) to drive the loudspeaker. The forcing frequency is set at approximately 780 Hz, corresponding to the transverse mode frequency predicted by acoustic analysis (figure 1 d). This frequency is obtained by solving the three-dimensional Helmholtz equation using COMSOL Multiphysics v6.3 software. In this analysis, the inner and outer walls of the combustor are modelled as sound-hard boundaries, while the twelve upper outlets are treated as open boundaries.
Probability density functions of the pressure fluctuation amplitude
$P(a)$
at (a) nozzle 1 (
$P_1$
) and (b) nozzle 2 (
$P_2$
) under varying forcing frequencies (
$f_{\!f}$
). The inset plots in (a) and (b) show the time series signals of the mean-subtracted dynamic pressure measured at each nozzle. (c–f) Phase portraits of the pressure fluctuations (1 s) for reconstructed via time-delay embedding at forcing frequencies (c) 720 Hz, (d) 740 Hz, (e) 760 Hz and (f) 780 Hz. All pressure signals are band-pass filtered around the transverse acoustic mode, and mean-subtracted using a 0.008 s moving average window.
Figure 2(a,b) present the probability density functions of the pressure fluctuation amplitude under varying external forcing frequencies. The signals measured at both nozzles indicate a transition to high-amplitude oscillations under the acoustic excitation with frequency 780 Hz, which corresponds to the transverse acoustic mode identified in figure 1(d). One possible explanation for this observation, as will be detailed later using a phenomenological model, is that the two flames have individually synchronised with the sinusoidal forcing while also interacting through mutual coupling. Furthermore, figure 2 shows that the pressure oscillation amplitude during lock-in is greater at the nozzle closer to the loudspeaker (
$P_1$
) than at the one further away (
$P_2$
), indicating the localised lock-in of each flame within the annular combustor.
Figure 2(c–f) present the phase portraits at forcing frequencies (
$f_{\!f}$
) 720, 740, 760 and 780 Hz, respectively, obtained using Takens’ embedding theorem (Takens Reference Takens1981; Fraser & Swinney Reference Fraser and Swinney1986). These portraits reveal the emergence of high-amplitude oscillations – i.e. potential limit cycles – as
$f_{\!f}$
approaches 780 Hz, which corresponds to the transverse acoustic mode identified through COMSOL analysis. Again, it is worth noting that the oscillation amplitudes differ between the two flames at this frequency (see figure 2
f).
3. Mathematical framework
3.1. Coupled stochastic oscillator model with sinusoidal forcing
In this subsection, we introduce a low-order system model employed to diagnose flame–flame interaction within an annular combustor. The model comprises two coupled stochastic oscillators, each subject to a sinusoidal excitation term that represents the acoustic forcing responsible for inducing lock-in phenomena. The deterministic component of the model is based on the van der Pol oscillator equation, which effectively captures the limit-cycle and fixed-point dynamics of thermoacoustic systems (Noiray & Schuermans Reference Noiray and Schuermans2013; Lee et al. Reference Lee, Guan, Gupta and Li2020). Additionally, the model incorporates a stochastic noise term to represent the inherent combustion noise within the combustor, forming the Langevin-type equation. This stochastic forcing is modelled as additive white Gaussian noise, a common assumption in reduced-order thermoacoustic models. Such an approach simplifies the representation of stochastic dynamics and enables efficient numerical parameter identification with minimal complexity in the noise formulation.
The two oscillators are mutually coupled through both a linear dissipative coupling term (Bélair & Holmes Reference Bélair and Holmes1984) and a nonlinear coupling term (Bonciolini & Noiray Reference Bonciolini and Noiray2019), both of which can significantly influence the Fokker–Planck dynamics, as reported by Lee et al. (Reference Lee, Gupta and Li2024). In other words, by analysing the stochastic dynamics of pressure fluctuations (e.g. figure 2), the coupled flame–flame interaction can be systematically examined using the proposed system model. The low-order model is
\begin{align}& \ddot {x_1}-\big(\epsilon + \alpha {x_1}^2\big)\dot {x_1} + {\omega }^2 x_1 - k_\epsilon (\dot {x_1} - \dot {x_2}) - k_\alpha \big(x_1^2 \dot {x_1} - x_2^2 \dot {x_2}\big) = F_1 + \sqrt {2 {d}}\, \eta _1, \end{align}
\begin{align}& \ddot {x_2}-\big(\epsilon + \alpha {x_2}^2\big)\dot {x_2} + {\omega }^2 x_2 - k_\epsilon (\dot {x_2} - \dot {x_1} ) - k_\alpha \big(x_2^2 \dot {x_2} - x_1^2 \dot {x_1}\big) = F_2 + \sqrt {2 {d}}\, \eta _2, \end{align}
where
$x_1$
and
$x_2$
denote the system variables (normalised pressure fluctuations), and the overdot indicates their time derivatives. The parameters
$\epsilon$
and
$\alpha$
represent the linear growth rate and nonlinear saturation coefficient, respectively, while
$\omega$
is the angular oscillation frequency. The terms
$k_\epsilon$
and
$k_\alpha$
denote the linear dissipative and nonlinear coupling strengths, which phenomenologically represent the combined effects of physical mechanisms such as shared acoustic feedback, hydrodynamic interactions within the annular geometry, and potential radiative coupling. Also,
$\eta _1$
and
$\eta _2$
are unit white Gaussian noise terms, and
$d$
is the noise amplitude. The forcing terms
$F_1 = A_{\!f1} \sin (\omega t + \phi _{f1})$
and
$F_2 = A_{\!f2} \sin (\omega t + \phi _{f2})$
are sinusoidal excitations, with forcing frequencies identical to those of the respective oscillators. Note that although van der Pol type equations bear a mathematical analogy to first principles models in certain hydrodynamic systems (Lee et al. Reference Lee, Zhu, Li and Gupta2019), the system model employed here is purely phenomenological and describes only the pressure fluctuations associated with each flame. As such, heat release rate oscillations are not modelled explicitly but are assumed to be implicitly embedded within the system variables. Similarly, spatial variations in acoustic forcing – such as local mode coupling effects near each flame (O’Connor et al. Reference O’Connor, Acharya and Lieuwen2015) – are not resolved explicitly in this model. Instead, their influence is phenomenologically lumped in the forcing terms
$F_1$
and
$F_2$
, which represent the effective amplitude and phase of the acoustic excitation at each nozzle. This formulation enables the model to account for asymmetric acoustic driving across the flames.
By postulating the above system model, we make the following assumptions. First, we assume that the two oscillators share the same angular frequency
$\omega$
, which corresponds to the transverse acoustic mode. This allows us to focus on modelling the mode of interest. Second, since the two flames have identical geometry, fuel–air flow rates, inlet and outlet configurations, and boundary conditions, we assume that both oscillators possess the same linear (
$\epsilon$
) and nonlinear (
$\alpha$
) system parameters. Similarly, we assume that the level of inherent combustion noise (
$d$
) is identical for both flames. These assumptions enable us to isolate and investigate the effects of mutual coupling and local synchronisation.
The probabilistic solution to (3.1) can be derived using the stochastic averaging technique. We begin by applying the method of variation of parameters, substituting the instantaneous state of the system variable with its corresponding amplitude and phase. Specifically, the following transformations are applied to (3.1):
\begin{align} x_1 &= a_1 \cos \! {(\omega t + \phi _1)}, \end{align}
\begin{align} x_2 &= a_2 \cos \! {(\omega t + \phi _2)}, \end{align}
\begin{align} \dot {x_1} &= - a_1 \omega \sin \! {(\omega t + \phi _1)}, \end{align}
\begin{align} \dot {x_2} &= - a_2 \omega \sin \! {(\omega t + \phi _2)}, \end{align}
where
$a_1$
and
$a_2$
denote the amplitudes, and
$\phi _1$
and
$\phi _2$
represent the phases, of the first and second oscillators, respectively. Note that this transformation is frequently used in analyses of stochastic oscillators (Roberts & Spanos Reference Roberts and Spanos1986). Upon applying this transformation and utilising trigonometric identities, (3.1) can be reformulated into four explicit ordinary differential equations – two governing the amplitudes, and two governing the phases. By further assuming that the amplitude and phase of each oscillator vary much more slowly than the system variable itself, the method of stochastic averaging (Roberts & Spanos Reference Roberts and Spanos1986) can be applied. This yields equations describing the drift and diffusion terms for each oscillator:
\begin{align} \boldsymbol{m}_1(a) &= \frac {\epsilon a}{2} + \frac {k_\epsilon a}{2} + \frac {\alpha {a}^3}{8} + \frac {k_\alpha a^3}{8} + \frac {d}{2 \omega ^2 a} + \frac {A_{\!f1}}{2\omega }, \end{align}
\begin{align} \boldsymbol{m}_2(a) &= \frac {\epsilon a}{2} + \frac {k_\epsilon a}{2} + \frac {\alpha {a}^3}{8} + \frac {k_\alpha a^3}{8} + \frac {d}{2 \omega ^2 a} + \frac {A_{\!f2}}{2\omega }, \end{align}
\begin{align} \boldsymbol{\sigma}_1^2(a) &= \frac {d}{{\omega }^2}, \end{align}
\begin{align} \boldsymbol{\sigma}_2^2(a) &= \frac {d}{{\omega }^2}, \end{align}
where
$\boldsymbol{m}_1(a)$
and
$\boldsymbol{m}_2(a)$
are drift terms, and
$\boldsymbol{\sigma}_1(a)$
and
$\boldsymbol{\sigma}_2(a)$
are diffusion terms, for the first and second oscillators, respectively. Readers are referred to Lee et al. (Reference Lee, Gupta and Li2024) for a more detailed derivation of the procedure.
It should be noted that the final term in the drift expressions, involving the forcing amplitudes (
${A_{\!f1}}/{2\omega }$
and
${A_{\!f2}}/{2\omega }$
), appears if and only if the phase of the forcing signal matches that of the corresponding oscillator (i.e.
$\phi _1 = \phi _{f1}$
and
$\phi _2 = \phi _{f2}$
). Otherwise, the effect of the external forcing vanishes through the process of stochastic averaging, and no such term remains in the drift component. This implies that localised synchronisation – in which each oscillator locks into the phase of its respective excitation signal – is crucial for modulating the probability density function of the pressure fluctuation amplitude.
The standard Fokker–Planck equation describing the transient probability density function of the amplitude can be derived from (3.3):
\begin{align} \frac {\partial }{\partial t}P_1(a,t)&=-\frac {\partial }{\partial a} [\boldsymbol{m}_1(a) P_1(a,t) ] + \frac {\partial ^2}{\partial a^2}\bigg[\frac {\boldsymbol{\sigma}_1^2(a)}{2} P_1(a,t)\bigg], \end{align}
\begin{align} \frac {\partial }{\partial t}P_2(a,t)&=-\frac {\partial }{\partial a} [\boldsymbol{m}_2(a) P_2(a,t) ] +\frac {\partial ^2}{\partial a^2}\bigg[\frac {\boldsymbol{\sigma}_2^2(a)}{2} P_2(a,t)\bigg], \end{align}
where
$P_1(a,t)$
and
$P_2(a,t)$
denote the probability density functions of the pressure fluctuation amplitudes for the first and second oscillators, respectively – corresponding to the flames located closer to and further from the loudspeaker. If these functions remain constant over time, then (3.4) can be integrated with respect to time, yielding the stationary Fokker–Planck equation. Assuming local phase synchronisation between the forcing signal and each oscillator, the stationary Fokker–Planck equation takes the form
\begin{align} P_1(a) &= C_1 a \exp \left [\frac {(\epsilon + k_\epsilon ){\omega }^2}{2d} {a}^2 + \frac {(\alpha + k_\alpha ) {\omega }^2}{16d} {a}^4 + \frac {A_{\!f1}\omega }{d}a \right ]\!, \end{align}
\begin{align} P_2(a) &= C_2 a \exp \left [\frac {(\epsilon + k_\epsilon ){\omega }^2}{2d} {a}^2 + \frac {(\alpha + k_\alpha ) {\omega }^2}{16d} {a}^4 + \frac {A_{\!f2}\omega }{d}a \right ]\!, \end{align}
where
$P_1(a)$
and
$P_2(a)$
are the stationary probability density functions, and
$C_1$
and
$C_2$
are the normalisation constants, for the first and second oscillators, respectively. It is worth reiterating that the final term in the exponent appears only under the condition of local phase synchronisation.
3.2. Analysing flame–flame interaction through system identification
To identify the effects of mutual coupling and synchronisation in flame–flame interaction under external sinusoidal forcing, we adopt a system identification framework based on the low-order model described above. The overall identification procedure is as follows. First, using experimental data obtained under unforced conditions, we determine the linear and nonlinear system parameters, along with the background noise level, by fitting the system model with coupling strengths and forcing terms set to zero (
$k_\epsilon =k_\alpha =A_{\!f1}=A_{\!f2}=0$
). In this case, we assume that the intrinsic dissipative and nonlinear coupling effects are implicitly incorporated into the estimated linear and nonlinear system parameters, respectively, based on the additive nature of the coupling terms (see (3.5)). This assumption will be examined in greater detail using experimental results in the next section. Second, we repeat the system identification using data from the sinusoidally forced condition, focusing on the locked-in regime illustrated in figure 2. In this step, we estimate the dissipative and nonlinear coupling strengths, as well as the noise amplitude. Through this two-step process, we aim to elucidate the role of coupling and synchronisation in the experimentally observed lock-in behaviour.
The system identification process employed herein is based on noise-induced dynamics, a methodology widely used for analysing oscillatory systems subjected to stochastic forcing (Noiray & Schuermans Reference Noiray and Schuermans2013; Boujo & Noiray Reference Boujo and Noiray2017; Son & Lee Reference Son and Lee2024). Within this output-only identification framework, system parameters are extracted and optimised by minimising the discrepancy between experimentally observed and model-derived drift and diffusion terms, calculated via solutions of the adjoint Fokker–Planck equation (Boujo & Noiray Reference Boujo and Noiray2017). To enhance computational efficiency, we employ a derivative-free optimisation algorithm that modifies the classical Levenberg–Marquardt method (Marquardt Reference Marquardt1963) through the use of finite-difference approximations of the Jacobian. This improvement significantly reduces computational cost compared to the conventional Nelder–Mead method, while maintaining robust accuracy, thereby facilitating efficient diagnostics of noisy, nonlinear dynamical systems. For a comprehensive description of the system identification framework used in this study, readers are referred to Lee, Kim & Park (Reference Lee, Kim and Park2023).
4. Results and discussion
As shown in (3.4) and (3.5), the linear system parameter (
$\epsilon$
) and the linear dissipative coupling parameter (
$k_\epsilon$
) enter the formulation as a linear combination. Similarly, the nonlinear parameter (
$\alpha$
) and the nonlinear coupling parameter (
$k_\alpha$
) also appear in a linearly combined form. As a result, the system identification process based on the Fokker–Planck dynamics yields the combined parameters
$\epsilon + k_\epsilon$
and
$\alpha + k_\alpha$
, rather than identifying each constituent term individually.
Table 1 presents the identified system parameters of the low-order model before and after the lock-in of the two flames. Under sinusoidal forcing, the combined linear terms shift from negative to positive values, indicating the onset of a limit cycle associated with the lock-in condition. In addition, the combined nonlinear terms increase markedly, suggesting a loss of stabilising mechanisms within the system. Given that the key operating parameters – such as the mass flow rates of fuel and oxidiser, equivalence ratio and combustor geometry – remain unchanged, it is reasonable to assume that the intrinsic system parameters
$\epsilon$
and
$\alpha$
are also preserved. Consequently, the observed increase in the combined linear and nonlinear terms is likely attributable to an increase in the coupling components. Assuming negligible coupling in the unforced case, the linear dissipative and nonlinear coupling terms in the forced case are identified as
$k_\epsilon = 3.87 \times 10^1$
and
$k_\alpha = 4.16 \times 10^5$
, respectively, as shown in table 1. This finding implies that lock-in and the transition to a limit cycle under external forcing can be driven purely by flame–flame interaction. This aligns with established numerical and experimental studies on coupled thermoacoustic oscillators, which show that variations in coupling can induce significant dynamical transitions (Dange et al. Reference Dange, Manoj, Banerjee, Pawar, Mondal and Sujith2019a
; Lee et al. Reference Lee, Gupta and Li2024).
Identified system parameters, comprising the combined linear and linear dissipative coupling parameters (
$\epsilon + k_\epsilon$
), the combined nonlinear and nonlinear coupling parameters (
$\alpha + k_\alpha$
), the local forcing amplitudes of the first (
$A_{\!f1}$
) and second (
$A_{\!f2}$
) oscillators, and the noise amplitude (
$d$
).
Comparison between the experimentally obtained probability density functions of the pressure fluctuation amplitude (scatter points) and the corresponding mathematical model (solid lines) for (a) unforced and (b) forced cases. Blue and red markers represent data from the flames located closer to (
$P_1$
) and further from (
$P_2$
) the loudspeaker, respectively. In the forced case, the blue, red and grey lines indicate model predictions for the
$P_1$
,
$P_2$
and unsynchronised cases, respectively.
Next, we examine the effect of synchronisation under sinusoidal acoustic forcing. As discussed in § 3.1, the influence of external forcing is reflected in the system model only when local synchronisation occurs – namely, when the phase of each oscillator aligns with that of the acoustic forcing signal, resulting in a shift of the probability density function. This condition is critical: only under such phase alignment can the external forcing alter the drift dynamics and induce observable changes in system behaviour. Figure 3(a) compares the experimentally obtained probability density function with its numerical counterpart, reconstructed using the identified parameters under unforced conditions. The probability density functions measured from the two flames exhibit highly similar dynamics, supporting the assumption of identical system parameters. Furthermore, the close agreement between the experimental and numerical distributions confirms the validity of the system identification framework employed in this study. However, as shown in figure 3(b), when the two oscillators remain unsynchronised with the external forcing signal, the model fails to reproduce the dynamics of either flame. To address this, we identify the effective acoustic forcing amplitudes under synchronised conditions by minimising the discrepancy between the experimental and numerical probability density functions. This yields effective forcing amplitudes
$A_{\!f1} = 6.16 \times 10^3$
and
$A_{\!f2} = 5.02 \times 10^2$
. Incorporating these values into the system model results in good agreement between the experimental and numerical probability density functions, as also shown in figure 3(b). Notably, the identified forcing amplitude is significantly greater for the flame located closer to the loudspeaker (
$P_1$
) than for the one further away (
$P_2$
), suggesting that the transversely propagating excitation wave dissipates along the combustor. This difference leads to the emergence of dissimilar limit cycles under localised phase synchronisation.
(a) Probability density functions of the pressure oscillation amplitude for various forcing amplitudes in synchronised cases. The upper and lower dashed lines denote the experimentally identified excitation amplitudes at which local synchronisation occurs. (b) Pressure oscillation amplitude corresponding to the maximum probability, shown as a function of the dissipative and nonlinear coupling strengths. The cross marker indicates the experimentally identified coupling strengths.
Finally, we systematically examine the effects of synchronisation and coupling using the low-order model and the identified system parameters. Figure 4(a) presents the probability density function of the pressure oscillation amplitude for a flame under varying effective forcing amplitudes. The bottom line (
$A_f = 0$
) corresponds to the unsynchronised case, while the white dashed lines indicate the experimentally identified effective forcing amplitudes for the first and second flames. All other parameters are held constant, as specified in table 1. The results reveal that localised synchronisation with stronger forcing amplitudes causes a rightward shift in the probability density function, signifying the emergence of a limit cycle with larger amplitude. Similarly, the model-based evaluation of varying linear dissipative and nonlinear coupling parameters is shown in figure 4(b). These results indicate that increases in either parameter can lead to a transition from a fixed point to a limit cycle, as well as an increase in the amplitude of the oscillations. These modelling trends are qualitatively consistent with experimental observations, which show distinct limit-cycle amplitudes between the two flames despite the assumption of identical intrinsic properties. This suggests that coupling effects – rather than variations in local flame parameters – may be the primary contributor to the observed asymmetry. In conclusion, the low-order model reveals that: (i) local synchronisation is essential for achieving lock-in between two coupled flames, with the effective forcing amplitude modulating the overall probability density function; and (ii) dissipative and nonlinear coupling can drive bifurcations and alter the oscillation amplitude in the coupled flame system.
5. Conclusions
This study investigated the nonlinear dynamics of flame–flame interactions in an annular combustor subjected to external sinusoidal forcing and stochastic disturbances. By integrating experimental observations with a low-order stochastic oscillator model, we demonstrated that localised lock-in phenomena are governed by both individual synchronisation to the forcing signal and mutual coupling between flames. Specifically, findings from the stochastic system identification framework showed that the transition to self-sustained oscillations was driven not only by the external forcing but also by enhanced dissipative and nonlinear coupling that alters the system stability. The identified system parameters indicated that coupling alone – without any changes to the intrinsic flame properties – could induce bifurcations and markedly increase oscillation amplitudes. Furthermore, the analysis highlighted that local phase synchronisation was essential for the external forcing to influence the flame dynamics, as only synchronised oscillators exhibit shifts in probability density functions. Parametric studies further confirmed the distinct roles of synchronisation and coupling strengths in determining the dynamics of the coupled oscillators. To the best of our knowledge, this is the first time that a low-order model and a system identification framework have been applied to quantify the role of mutual coupling and local synchronisation in experimentally observed thermoacoustic lock-in phenomena. The proposed framework offers a systematic and predictive means of diagnosing complex thermoacoustic interactions in multi-flame systems, with potential applicability to practical configurations.
Nonetheless, several limitations should be acknowledged. First, although different forcing terms are applied, the low-order model proposed in this study assumes identical intrinsic properties for both flames, which may not fully capture the asymmetries present in the experimental set-up. In addition, the model simplifies the spatially varying interaction between each flame and the surrounding acoustic field. Future studies could address such spatial variations in the acoustic field by incorporating spatially dependent flame parameters or by developing models that explicitly account for the influence of coupled mode effects on local flame dynamics, such as the transverse–longitudinal interactions discussed by O’Connor et al. (Reference O’Connor, Acharya and Lieuwen2015). Furthermore, time-delay effects and higher-order acoustic modes could be incorporated in future work, as they may play a significant role in practical combustion systems.
Second, the assumption of white Gaussian noise may not accurately represent the coloured and correlated nature of combustion noise in real-world conditions. While this assumption enables tractable system identification and reduces model complexity, future work could incorporate more realistic noise characterisations to enhance model fidelity in thermoacoustic systems.
Finally, both the experiment and the analysis are limited to two interacting flames, modelled as mutually coupled self-sustained oscillators. This configuration was intentionally selected to isolate flame–flame interaction under well-controlled conditions, free from the influence of additional flames. Future studies could extend the framework to larger flame arrays, which may require increased modelling complexity, including nonlinear coupling mechanisms and the treatment of additional spatial modes.
Funding
This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry, and Energy (MOTIE) of the Republic of Korea (no. RS-2024-00455846). The author would like to thank S. Guk at Hanbat National University for his assistance with data acquisition.
Declaration of interests
The author declares no conflict of interest.
