Gas turbine combustors are susceptible to thermoacoustic instability, which manifests as large amplitude periodic oscillations in acoustic pressure and heat release rate. The transition from a stable operation characterized by combustion noise to thermoacoustic instability in turbulent combustors has been described as an emergence of order (periodicity) from chaos in the temporal dynamics. This emergence of order in the acoustic pressure oscillations corresponds to a loss of multifractality in the pressure signal. In this study, we investigate the spatiotemporal dynamics of a turbulent flame in a bluff-body stabilized combustor during the transition from combustion noise to thermoacoustic instability. During the occurrence of combustion noise, the flame wrinkles due to the presence of small-scale vortices in the turbulent flow. On the other hand, during thermoacoustic instability, large-scale coherent structures emerge periodically. These large-scale coherent structures roll up the wrinkled flame surface further and introduce additional complexity in the flame topology. We perform multifractal analysis on the flame contours detected from high-speed planar Mie scattering images of the reactive flow seeded with non-reactive tracer particles. We find that multifractality exists in the flame topology for all the dynamical states during the transition to thermoacoustic instability. We discuss the variation of multifractal parameters for the different states. We find that the multifractal spectrum oscillates periodically during the occurrence of thermoacoustic instability at the time scale of the acoustic pressure oscillations. The loss of multifractality in the temporal dynamics and the oscillation of the multifractal spectrum of the spatial dynamics go hand in hand.

We identify mechanisms through which open-loop control of thermoacoustic instability is achieved in a laminar combustor and characterize them using synchronization theory. The thermoacoustic system comprises two nonlinearly coupled damped harmonic oscillators – acoustic and unsteady heat release rate (HRR) field – each possessing different eigenfrequencies. The frequency of the preferred mode of HRR oscillations is less than the third acoustic eigenfrequency where thermoacoustic instability develops. We systematically subject the limit-cycle oscillations to an external harmonic forcing at different frequencies and amplitudes. We observe that forcing at a frequency near the preferred mode of the HRR oscillator leads to a greater than 90 % decrease in the amplitude of the limit-cycle oscillations through the phenomenon of asynchronous quenching. Concurrently, there is a resonant amplification in the amplitude of HRR oscillations. Further, we show that the flame dynamics plays a key role in controlling the frequency at which quenching is observed. Most importantly, we show that forcing can cause asynchronous quenching either by imposing out-of-phase relation between pressure and HRR oscillations or by inducing period-2 dynamics in pressure oscillations while period-1 in HRR oscillations, thereby causing phase drifting between the two subsystems. In each of the two cases, acoustic driving is very low and hence thermoacoustic instability is suppressed. We show that the characteristics of forced synchronization of the pressure and HRR oscillations are significantly different. Thus, we find that the simultaneous characterization of the two subsystems is necessary to quantify completely the nonlinear response of the forced thermoacoustic system.

In turbulent combustors, the transition from stable combustion (i.e. combustion noise) to thermoacoustic instability occurs via intermittency. During stable combustion, the acoustic power production happens in a spatially incoherent manner. In contrast, during thermoacoustic instability, the acoustic power production happens in a spatially coherent manner. In the present study, we investigate the spatiotemporal dynamics of acoustic power sources during the intermittency route to thermoacoustic instability using complex network theory. To that end, we perform simultaneous acoustic pressure measurement, high-speed chemiluminescence imaging and particle image velocimetry in a backward-facing step combustor with a bluff body stabilized flame at different equivalence ratios. We examine the spatiotemporal dynamics of acoustic power sources by constructing time-varying spatial networks during the different dynamical states of combustor operation. We show that as the turbulent combustor transits from combustion noise to thermoacoustic instability via intermittency, small fragments of acoustic power sources, observed during combustion noise, nucleate, coalesce and grow in size to form large clusters at the onset of thermoacoustic instability. This nucleation, coalescence and growth of small clusters of acoustic power sources occurs during the growth of pressure oscillations during intermittency. In contrast, during the decay of pressure oscillations during intermittency, these large clusters of acoustic power sources disintegrate into small ones. We use network measures such as the link density, the number of components and the size of the largest component to quantify the spatiotemporal dynamics of acoustic power sources as the turbulent combustor transits from combustion noise to thermoacoustic instability via intermittency.

We perform an experimental and theoretical study to investigate the interaction between an external harmonic excitation and a self-excited oscillatory mode ($f_{n0}$) of a prototypical thermoacoustic system, a horizontal Rijke tube. Such an interaction can lead to forced synchronization through the routes of phase locking or suppression. We characterize the transition in the synchronization behaviour of the forcing and the response signals of the acoustic pressure while the forcing parameters, i.e. amplitude ($A_{f}$) and frequency ($f_{f}$) of forcing are independently varied. Further, suppression is categorized into synchronous quenching and asynchronous quenching depending upon the value of frequency detuning ($|\,f_{n0}-f_{f}|$). When the applied forcing frequency is close to the natural frequency of the system, the suppression in the amplitude of the self-excited oscillation is known as synchronous quenching. However, this suppression is associated with resonant amplification of the forcing signal, leading to an overall increase in the response amplitude of oscillations. On the other hand, an almost 80 % reduction in the root mean square value of the response oscillation is observed when the system is forced for a sufficiently large value of the frequency detuning (only for $f_{f}<f_{n0}$). Such a reduction in amplitude occurs due to asynchronous quenching where resonant amplification of the forcing signal does not occur, as the frequency detuning is significantly high. Further, the results from a reduced-order model developed for a horizontal Rijke tube show a qualitative agreement with the dynamics observed in experiments. The relative phase between the acoustic pressure ($p^{\prime }$) and the heat release rate ($\dot{q}^{\prime }$) oscillations in the model explains the occurrence of maximum reduction in the pressure amplitude due to asynchronous quenching. Such a reduction occurs when the positive coupling between $p^{\prime }$ and $\dot{q}^{\prime }$ is disrupted and their interaction results in overall acoustic damping, although both of them oscillate at the forcing frequency. Our study on the phenomenon of asynchronous quenching thus presents new possibilities to suppress self-sustained oscillations in fluid systems in general.

Gas turbine engines are prone to the phenomenon of thermoacoustic instability, which is highly detrimental to their components. Recently, in turbulent combustors, it was observed that the transition to thermoacoustic instability occurs through an intermediate state, known as intermittency, where the system exhibits epochs of ordered behaviour, randomly appearing amidst disordered dynamics. We investigate the onset of intermittency and the ensuing self-organization in the reactive flow field, which, under certain conditions, could result in the transition to thermoacoustic instability. We characterize this transition from a state of disordered and incoherent dynamics to a state of ordered and coherent dynamics as pattern formation in the turbulent combustor, utilizing high-speed flame images representing the distribution of the local heat release rate fluctuations, flow field measurements (two-dimensional particle image velocimetry), unsteady pressure and global heat release rate signals. Separately, through planar Mie scattering images using oil droplets, the collective behaviour of small scale vortices interacting and resulting in the emergence of large scale coherent structures is illustrated. We show the emergence of spatial patterns using statistical tools used to study transitions in other pattern forming systems. In this paper, we propose that the intertwined and highly intricate interactions between the wide spatio-temporal scales in the flame, the flow and the acoustics are through pattern formation.

Thermoacoustic instability is the result of a positive coupling between the acoustic field in the duct and the heat release rate fluctuations from the flame. Recently, in several turbulent combustors, it has been observed that the onset of thermoacoustic instability is preceded by intermittent oscillations, which consist of bursts of periodic oscillations amidst regions of aperiodic oscillations. Quantitative analysis of the intermittency route to thermoacoustic instability has been performed hitherto using the pressure oscillations alone. We perform experiments on a laboratory-scale bluff-body-stabilized turbulent combustor with a backward-facing step at the inlet to obtain simultaneous data of acoustic pressure and heat release rate fluctuations. With this, we show that the onset of thermoacoustic instability is a phenomenon of mutual synchronization between the acoustic pressure and the heat release rate signals, thus emphasizing the importance of the coupling between these non-identical oscillators. We demonstrate that the stable operation corresponds to desynchronized aperiodic oscillations, which, with an increase in the mean velocity of the flow, transition to synchronized periodic oscillations. In between these states, there exists a state of intermittent phase synchronized oscillations, wherein the two oscillators are synchronized during the periodic epochs and desynchronized during the aperiodic epochs of their oscillations. Furthermore, we discover two different types of limit cycle oscillations in our system. We notice a significant increase in the linear correlation between the acoustic pressure and the heat release rate oscillations during the transition from a lower-amplitude limit cycle to a higher-amplitude limit cycle. Further, we present a phenomenological model that qualitatively captures all of the dynamical states of synchronization observed in the experiment. Our analysis shows that the times at which vortices that are shed from the inlet step reach the bluff body play a dominant role in determining the behaviour of the limit cycle oscillations.

Thermoacoustic systems with a turbulent reactive flow, prevalent in the fields of power and propulsion, are highly susceptible to oscillatory instabilities. Recent studies showed that such systems transition from combustion noise to thermoacoustic instability through a dynamical state known as intermittency, where bursts of large-amplitude periodic oscillations appear in a near-random fashion in between regions of low-amplitude aperiodic fluctuations. However, as these analyses were in the temporal domain, this transition remains still unexplored spatiotemporally. Here, we present the spatiotemporal dynamics during the transition from combustion noise to limit cycle oscillations in a turbulent bluff-body stabilized combustor. To that end, we acquire the pressure oscillations and the field of heat release rate oscillations through high-speed chemiluminescence ($CH^{\ast }$) images of the reaction zone. With a view to get an insight into this complex dynamics, we compute the instantaneous phases between acoustic pressure and local heat release rate oscillations. We observe that the aperiodic oscillations during combustion noise are phase asynchronous, while the large-amplitude periodic oscillations seen during thermoacoustic instability are phase synchronous. We find something interesting during intermittency: patches of synchronized periodic oscillations and desynchronized aperiodic oscillations coexist in the reaction zone. In other words, the emergence of order from disorder happens through a dynamical state wherein regions of order and disorder coexist, resembling a chimera state. Generally, mutually coupled chaotic oscillators synchronize but retain their dynamical nature; the same is true for coupled periodic oscillators. In contrast, during intermittency, we find that patches of desynchronized aperiodic oscillations turn into patches of synchronized periodic oscillations and vice versa. Therefore, the dynamics of local heat release rate oscillations change from aperiodic to periodic as they synchronize intermittently. The temporal variations in global synchrony, estimated through the Kuramoto order parameter, echoes the breathing nature of a chimera state.

In classical literature, blowout is described as loss of static stability of the combustion system whereas thermoacoustic instability is seen as loss of dynamic stability of the system. At blowout, the system transitions from a stable reacting state to a non-reacting state, indicating loss of static stability of the reaction. However, this simple description of stability margin is inadequate since recent studies have shown that combustors exhibit complex nonlinear behaviour prior to blowout. Recently, it was shown that combustion noise that characterizes the regime of stable operation is itself dynamically complex and exhibits multifractal characteristics. Researchers have already described the transition from combustion noise to combustion instability as a loss of multifractality. In this work, we provide a multifractal description for lean blowout in combustors with turbulent flow and thus introduce a unified framework within which both thermoacoustic instability and blowout can be described. Further, we introduce a method for predicting blowout based on the multifractal description of blowout.

We present the effect of noise on the hysteresis characteristics of a prototypical thermoacoustic system, a horizontal Rijke tube. As we increase the noise intensity, we find that the width of the hysteresis zone decreases. However, we find that the rate of decrease in hysteresis width is constant for all the mass flow rates considered in the present study. We also show that the subcritical transition observed in the absence of noise is no longer discernible once the intensity of noise is above a threshold value and the transition appears to be continuous. We compare our experimental observations with the results obtained from a numerical model perturbed with additive Gaussian white noise and we find a qualitative agreement between the experimental and the numerical results.

We investigate the scale invariance of combustion noise generated from turbulent reacting flows in a confined environment using complex networks. The time series data of unsteady pressure, which is the indicative of spatiotemporal changes happening in the combustor, is converted into complex networks using the visibility algorithm. We show that the complex networks obtained from the low-amplitude, aperiodic pressure fluctuations during combustion noise have scale-free structure. The power-law distributions of connections in the scale-free network are related to the scale invariance of combustion noise. We also show that the scale-free feature of combustion noise disappears and order emerges in the complex network topology during the transition from combustion noise to combustion instability. The use of complex networks enables us to formalize the identification of the pattern (i.e. scale-free to order) during the transition from combustion noise to thermoacoustic instability as a structural change in topology of the network.

The dynamic transition from combustion noise to combustion instability was investigated experimentally in two laboratory-scale turbulent combustors (namely, swirl-stabilized and bluff-body-stabilized backward-facing-step combustors) by systematically varying the flow Reynolds number. We observe that the onset of combustion-driven oscillations is always presaged by intermittent bursts of high-amplitude periodic oscillations that appear in a near-random fashion amidst regions of aperiodic low-amplitude fluctuations. These excursions to periodic oscillations last longer in time as operating conditions approach instability and finally the system transitions completely into periodic oscillations. A continuous measure to quantify this bifurcation in dynamics can be obtained by defining an order parameter as the probability of the signal amplitude exceeding a predefined threshold. A hysteresis zone was observed in the bluff-body-stabilized configuration that was absent in the swirl-stabilized configuration. The recurrence properties of the dynamics of intermittent burst oscillations were quantified using recurrence plots and the distribution of the aperiodic phases was examined. From the statistics of these aperiodic phases, robust early-warning signals of an impending combustion instability may be obtained.

The transition in dynamics from low-amplitude, aperiodic, combustion noise to high-amplitude, periodic, combustion instability in confined, combustion environments was studied experimentally in a laboratory-scale combustor with two different flameholding devices in a turbulent flow field. We show that the low-amplitude, irregular pressure fluctuations acquired during stable regimes, termed ‘combustion noise’, display scale invariance and have a multifractal signature that disappears at the onset of combustion instability. Traditional analysis often treats combustion noise and combustion instability as acoustic problems wherein the irregular fluctuations observed in experiments are often considered as a stochastic background to the dynamics. We demonstrate that the irregular fluctuations contain useful information of prognostic value by defining representative measures such as Hurst exponents that can act as early warning signals to impending instability in fielded combustors.

Perturbations in a non-normal system can grow transiently even if the system is linearly stable. If this transient growth is sufficiently large, it can trigger self-sustained oscillations from small initial disturbances. This has important practical consequences for combustion–acoustic oscillations, which are a persistent problem in rocket and aircraft engines. Balasubramanian & Sujith (J. Fluid Mech., vol. 594, 2008, pp. 29–57) modelled an infinite-rate chemistry diffusion flame in an acoustic duct and found that the transient growth in this system can amplify the initial energy by a factor, ${G}_{max} $, of the order of $1{0}^{5} $ to $1{0}^{7} $. However, recent investigations by L. Magri and M. P. Juniper have brought to light certain errors in that paper. When the errors are corrected, ${G}_{max} $ is found to be of the order of 1 to 10, revealing that non-normality is not as influential as it was thought to be.

Recent investigations by L. Magri and M. P. Juniper have brought to light certain errors in the paper by Balasubramanian and Sujith (Journal of Fluid Mechanics, vol. 594, 2008, pp. 29–57). The revision is available in a separate note (Journal of Fluid Mechanics, vol. 733, 2013, pp. 681–683), which shows that these ducted diffusion flames exhibit a transient growth of order 1 to 10. These results are in contrast to those presented by Balasubramanian & Sujith (2008), who found the transient growth to be of order $1{0}^{5} $ to $1{0}^{7} $.

This paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.

Nonlinear self-excited thermoacoustic oscillations appear in systems involving confined combustion in the form of coupled acoustic pressure oscillations and unsteady heat release rate. In this paper, we investigate the nonlinear transition undergone by thermoacoustic oscillations to flame blowout via intermittency, in response to variation in the location of the combustion source with respect to the acoustic field of the confinement. A ducted laminar premixed conical flame, stabilized on a circular jet exit with a fully developed exit velocity profile, was investigated. Transition to limit cycle oscillations from a non-oscillatory state was observed to occur via a subcritical Hopf bifurcation. Limit cycle oscillations underwent a further bifurcation to quasi-periodic oscillations characterized by the repeated formation of elongated necks in the flame that pinch off as pockets of unburned fuel–air mixture. The quasi-periodic state loses stability, resulting in an intermittent state identified as type II through recurrence analysis of phase space trajectories reconstructed from the acoustic pressure time trace. In this state, the flame undergoes repeated lift-off and reattachment. Instantaneous flame images suggest that the intermittent flame behaviour is influenced by jet flow dynamics.

An analysis of thermoacoustic instability is performed for a horizontal Rijke tube with an electrical resistance heater as the heat source. The governing equations for this fluid flow become stiff and are difficult to solve by the computational fluid dynamics (CFD) technique, as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore, an asymptotic analysis is performed in the limit of small Mach number and compact heat source to eliminate the above stiffness problem. The unknown variables are expanded in powers of Mach number. Two systems of governing equations are obtained: one for the acoustic field and the other for the unsteady flow field in the hydrodynamic zone around the heater. In this analysis, the coupling between the acoustic field and the unsteady heat release rate from the heater appears from the asymptotic analysis. Furthermore, a non-trivial additional term, referred to as the global-acceleration term, appears in the momentum equation of the hydrodynamic zone, which has serious consequences for the stability of the system. This term can be interpreted as a pressure gradient applied from the acoustic onto the hydrodynamic zone. The asymptotic stability of the system with the variation of system parameters is presented using the bifurcation diagram. Numerical simulations are performed using the Galerkin technique for the acoustic zone and CFD techniques for the hydrodynamic zone. The results confirm the importance of the global-acceleration term. Bifurcation diagrams obtained from the simulations with and without the above term are different. Acoustic streaming is shown to occur during the limit cycle and its effect on the unsteady heat release rate is discussed.

This paper investigates the non-normal nature of premixed flame–acoustic interaction. The thermoacoustic system is modelled using the acoustic equations for momentum and energy, together with the equation for the evolution of the flame front obtained from the kinematic G-equation. As the unsteady heat addition acts as a volumetric source, the flame front is modelled as a distribution of monopole sources. Evolutions of the system are characterized with a measure of energy due to fluctuations. In addition to the acoustic energy, the energy due to fluctuations considered in the present paper accounts for the energy of the monopole sources. The linearized operator for this thermoacoustic system is non-normal, leading to non-orthogonality of its eigenvectors. Non-orthogonal eigenvectors can cause transient growth even when all the eigenvectors are decaying. Therefore, classical linear stability theory cannot predict the finite-time transient growth observed in non-normal systems. In the present model, the state space variables include the monopole source strengths in addition to the acoustic variables. Inclusion of these variables in the state space is essential to account for the transient growth due to non-normality. A parametric study of the variation in transient growth due to change in parameters such as flame location and flame angle is performed. In addition to projections along the acoustic variables of velocity and pressure, the optimal initial condition for the self-evolving system has significant projections along the strength of the monopole distribution. Comparison of linear and corresponding nonlinear evolutions highlights the role of transient growth in subcritical transition to instability. The notion of phase between acoustic pressure and heat release rate as an indicator of stability is examined.

Non-normality can cause transient growth of perturbations in thermoacoustic systems with stable eigenvalues. This can cause low-amplitude perturbations to grow to amplitudes high enough to make nonlinear effects significant, and the system can become nonlinearly unstable, even though it is stable under classical linear stability. In this paper, we have demonstrated that this feature can lead to the failure of the traditional controllers that were designed on the basis of classical linear stability analysis. We have also shown in a simple model that it is possible to prevent ‘nonlinear driving’ by controlling transient growth, using linear controllers. The analysis is performed in the context of a horizontal Rijke tube.

An analytical framework is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into account the non-orthogonality of the eigenmodes of the unsteady coupled system. The coupled system comprises the dynamics of the acoustic field and the propellant burn rate. In general, thermoacoustic systems are non-normal leading to non-orthogonality of the eigenmodes. For such systems, the classical linear stability predicted from the eigenvalue analysis is valid in the asymptotic (large time) limit. However, the short-term dynamics can be completely different and a generalized stability theory is needed to predict the linear stability for all times. Non-normal systems show an initial transient growth for suitable initial perturbations even when the system is stable according to the classical linear stability theory. The terms contributing to the non-normality in the acoustic field and unsteady burn rate equations are identified. These terms, which were neglected in the earlier analyses, are incorporated in this analysis. Furthermore, the short-term dynamics are analysed using a system of differential equations that couples the acoustic field and the burn rate, rather than using ad hoc response functions which were used in earlier analyses. In this paper, a solid rocket motor with homogeneous propellant grain has been analysed. Modelling the evolution of the unsteady burn rate using a differential equation increases the degrees of freedom of the thermoacoustic system. Hence, a new generalized disturbance energy is defined which measures the growth and decay of the oscillations. This disturbance energy includes both acoustic energy and unsteady energy in the propellant and is used to quantify the transient growth in the system. Nonlinearities in the system are incorporated by including second-order acoustics and a physics-based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to ‘pulsed instability’. Pulsed instability is shown to occur with pressure coupling for burn rate response. Transient growth is shown to play an important role in pulsed instability.