Detection and control of combustion instability based on the concept of dynamical system theory

Dynamic behavior and driving region of spray combustion instability in a backward-facing step combustor

We numerically study the dynamic behavior and driving region of spray combustion instability in a backward-facing step combustor using analytical methodologies based on dynamical systems theory, symbolic dynamics, complex networks, and machine learning. The global dynamic behavior of a heat release rate field represents low-dimensional chaotic oscillations with deterministically aperiodic intercycle dynamics. Spray combustion instability is driven in the formation and separation region of a large-scale organized vortex induced by the hydrodynamic shear layer instability at the edge of the backstep. This region corresponds fairly to that of the hub in an acoustic-energy-flux-based spatial network. The feature importance in a random forest is valid for clarifying the feedback coupling of spray combustion instability.

Identifying optimal location for control of thermoacoustic instability through statistical analysis of saddle point trajectories

We propose a framework of Lagrangian Coherent Structures (LCSs) to enable passive open-loop control of tonal sound generated during thermoacoustic instability. Experiments were performed in a laboratory-scale bluff-body stabilized turbulent combustor in the state of thermoacoustic instability. We use dynamic mode decomposition on the flow-field to identify dynamical regions where the acoustic frequency is dominant. We find that the separating shear layer from the backward-facing step of the combustor envelops a cylindrical vortex in the outer recirculation zone, which eventually impinges on the top wall of the combustor during thermoacoustic instability. We track the saddle points in this shear layer emerging from the backward-facing step over several acoustic cycles. A passive control strategy is then developed by injecting a steady stream of secondary air targeting the identified optimal location where the saddle points spend a majority of their time in a statistical sense. After implementing the control action, the resultant flow-field is also analyzed using LCS to understand the key differences in flow dynamics. We find that the shear layer emerging from the dump plane is deflected in a direction almost parallel to the axis of the combustor after the control action. This deflection, in turn, prevents the shear layer from enveloping the vortex and impinging on the combustor walls, resulting in a drastic reduction in the amplitude of the sound produced.

Fokker-Planck modeling of the stochastic dynamics of a Rijke tube

We derive and numerically validate a low-order oscillator model to capture the stochastic dynamics of a prototypical thermoacoustic system (a Rijke tube) undergoing a subcritical Hopf bifurcation in the presence of additive noise. We find that on the fixed-point branch before the bifurcation, the system is dominated by the first duct mode, and the Fokker-Planck solution for the first Galerkin mode can adequately predict the stochastic dynamics of the overall system. We also find that this analytical framework predicts well the dominant mode on the limit-cycle branch, but underperforms in the hysteretic bistable zone where the role of nonlinearities is more pronounced. Besides offering new insights into stochastic thermoacoustic behavior, this study shows that an analytical framework based on the Fokker-Planck equation can facilitate the early detection of thermoacoustic instabilities in a Rijke-tube model subjected to noise.

Interaction of acoustic pressure and heat release rate fluctuations in a model rocket engine combustor

We experimentally clarify the interaction of acoustic pressure and heat release rate fluctuations during a transition to high-frequency combustion instability in a model rocket engine combustor. The dynamical state of acoustic pressure fluctuations undergoes a transition from high-dimensional chaotic oscillations to strongly correlated limit cycle oscillations. The coherent structure in the heat release rate field emerges with the initiation of weakly correlated limit cycle oscillations. The effect of the heat release rate on acoustic pressure fluctuations predominates during high-dimensional chaotic oscillations. In contrast, the effect of acoustic pressure on the heat release rate fluctuations markedly increases during the correlated limit cycle oscillations. These are reasonably shown by an ordinal pattern-based analysis involving the concepts of information theory, synchronization, and complex networks.

Early warnings of tipping in a non-autonomous turbulent reactive flow system: Efficacy, reliability, and warning times

Real-world complex systems such as the earth's climate, ecosystems, stock markets, and combustion engines are prone to dynamical transitions from one state to another, with catastrophic consequences. State variables of such systems often exhibit aperiodic fluctuations, either chaotic or stochastic in nature. Often, the parameters describing a system vary with time, showing time dependency. Constrained by these effects, it becomes difficult to be warned of an impending critical transition, as such effects contaminate the precursory signals of the transition. Therefore, a need for efficient and reliable early-warning signals (EWSs) in such complex systems is in pressing demand. Motivated by this fact, in the present work, we analyze various EWSs in the context of a non-autonomous turbulent thermoacoustic system. In particular, we investigate the efficacy of different EWS in forecasting the onset of thermoacoustic instability (TAI) and their reliability with respect to the rate of change of the control parameter. This is the first experimental study of tipping points in a non-autonomous turbulent thermoacoustic system. We consider the Reynolds number (Re) as the control parameter, which is varied linearly with time at finite rates. The considered EWSs are derived from critical slowing down, spectral properties, and fractal characteristics of the system variables. The state of TAI is associated with large amplitude acoustic pressure oscillations that could lead thermoacoustic systems to break down. We consider acoustic pressure fluctuations as a potential system variable to perform the analysis. Our analysis shows that irrespective of the rate of variation of the control parameter, the Hurst exponent and variance of autocorrelation coefficients warn of an impending transition well in advance and are more reliable than other EWS measures. Additionally, we show the variation in the warning time to an impending TAI with rates of change of the control parameter. We also investigate the variation in amplitudes of the most significant modes of acoustic pressure oscillations with the Hurst exponent. Such variations lead to scaling laws that could be significant in prediction and devising control actions to mitigate TAI.

Suppression of thermo-acoustic instabilities in horizontal Rijke tube using pulsating radial jets

Thermo-acoustic instability has been observed in gas turbines, rocket engines, and aero-engines. Acoustic perturbations grow and change the characteristics of the flow due to instability. The present work describes the use of pulsating air jets to suppress the thermo-acoustic instabilities. In present study pulsatile micro-jets are placed downstream of the burner radially which breaks the coupling between acoustic waves and unsteady heat release. A microphone connected to LIFA (LabVIEW Interface for Arduino) was used to detect the sound pressure levels. By controlling the airflow rate of the pulsatile jets, the sound pressure levels were suppressed down to the background noise level using minimum energy and time. A closed-loop control system is developed for this purpose, which works on the feedback signal acquired from microphone. To simulate the one dimensional combustion phenomenon, an experimental setup called Rijke tube was used. The suppression was most effective for the pulsatile jets of 27-33 Hz pulsation frequency range and at a flow rate of 6.8 LPM. This control strategy effectively controlled the combustion instability of around 35-42 dB.•The closed loop control method is built on DAQ and Arduino using the LabVIEW interface for Arduino (LIFA).•Developed closed loop active control method was observed to be effective for suppression of thermo-acoustic instability.•Optimum position of the radial planes of micro-jets with respect to the burner was decided to improve the efficacy of the pulsatile jets towards suppression of thermo-acoustic instability.

Imprints of log-periodicity in thermoacoustic systems close to lean blowout

In the context of statistical physics, critical phenomena are accompanied by power laws having a singularity at the critical point where a sudden change in the state of the system occurs. In this work we show that lean blowout (LBO) in a turbulent thermoacoustic system is accompanied by a power law leading to finite-time singularity. As a crucial discovery of the system dynamics approaching LBO, we unravel the existence of the discrete scale invariance (DSI). In this context, we identify the presence of log-periodic oscillations in the temporal evolution of the amplitude of the dominant mode of low-frequency oscillations (A_{f}) existing in pressure fluctuations preceding LBO. The presence of DSI indicates the recursive development of blowout. Additionally, we find that A_{f} shows a faster-than-exponential growth and becomes singular when blowout occurs. We then present a model that depicts the evolution of A_{f} based on log-periodic corrections to the power law associated with its growth. Using the model, we find that blowouts can be predicted even several seconds earlier. The predicted time of LBO is in good agreement with the actual time of occurrence of LBO obtained from the experiment.

Gravitational effect on the nonlinear dynamics of a buoyant turbulent flame

This study numerically examines the gravitational effect on the nonlinear dynamics of a buoyant turbulent flame utilizing analytical methods based on complex networks and dynamical systems. A dense (sparse) network structure is formed in the near (far) field in low gravity, as shown by the degree and cluster coefficient in the spatial network. The global dynamics of the vertical flow velocity fluctuations in the intermittent luminous zone is synchronous with that of the temperature fluctuations in low gravity. The synchronized state disappears as the gravity level is increased, leading to a desynchronized state. These behaviors are clearly identified by the symbolic recurrence plots.

Quenching and amplification of thermoacoustic oscillations in two nonidentical Rijke tubes interacting via time-delay and dissipative coupling

We numerically explore the quenching and amplification of self-excited thermoacoustic oscillations in two nonidentical Rijke tubes interacting via time-delay and dissipative coupling. On applying either type of coupling separately, we find that the presence of nonidentical heater powers can shrink the regions of amplitude death in both oscillators, while producing new regions of amplitude amplification in the weaker oscillator. We find that the magnitude of amplitude amplification grows with the heater power mismatch and with the total power input. These effects are also present when both types of coupling are applied simultaneously. This study highlights the critical role that nonidentical thermal loads can play in determining the amplitude response of coupled thermoacoustic systems, facilitating the design of control strategies for coupled oscillatorlike devices such as gas turbines.

Early detection of lean blowout using recurrence network for varying degrees of premixedness

Lean premixed combustors are highly susceptible to lean blowout flame instability, which can cause a fatal accident in aircrafts or expensive shutdown in stationary combustors. However, the lean blowout limit of a combustor may vary significantly depending on a number of variables that cannot be controlled in practical situations. Although a large literature exists on the lean blowout phenomena, a robust strategy for early lean blowout detection is still not available. To address this gap, we study a relatively unexplored route to lean blowout using a nonlinear dynamical tool, the recurrence network. Three recurrence network parameters: global efficiency, average degree centrality, and global clustering coefficient are chosen as metrics for an early prediction of the lean blowout. We observe that the characteristics of the time series near the lean blowout limit are highly dependent on the degree of premixedness in the combustor. Still, for different degrees of premixedness, each of the three recurrence network metrics increases during transition to lean blowout, indicating a shift toward periodicity. Thus, qualitatively, the recurrence network metrics show similar trends for different degrees of premixing showing their robustness. However, the sensitivities and absolute trends of the recurrence network metrics are found to be significantly different for highly premixed and partially premixed configurations. Thus, the results indicate that prior knowledge about (i) the degree of premixedness and (ii) the route to lean blowout may be required for accurate early prediction of the lean blowout. We show that the visible structural changes in the recurrence network can be linked to the changes in the recurrence network metrics, helping to better understand the dynamical transition to lean blowout. We observe the power law degree distribution of the recurrence network to break down close to the lean blowout limit due to the intermittent dynamics in the near-LBO regime.

Low-order modeling of the mutual synchronization between two turbulent thermoacoustic oscillators

We use a low-order oscillator model to investigate the mutual synchronization of a thermoacoustic system consisting of two turbulent lean-premixed combustors coupled via a cross-talk tube. The model consists of two Van der Pol (VDP) oscillators coupled via dissipative and time-delay terms. We show that, despite its simplicity, the model can reproduce many of the synchronization phenomena observed experimentally, such as amplitude death, desynchronization (quasiperiodicity), synchronization (phase locking), and nonlinear energy pumping from a limit-cycle mode to a damped mode. This study shows that the mutual synchronization dynamics of a turbulent thermoacoustic system can be reproduced with just a simple coupled VDP model. This suggests that such a model could be used to identify new strategies for quenching limit-cycle oscillations in turbulent thermoacoustic systems, such as gas turbines and rocket engines.

Convolutional neural networks to predict the onset of oscillatory instabilities in turbulent systems

Many fluid dynamic systems exhibit undesirable oscillatory instabilities due to positive feedback between fluctuations in their different subsystems. Thermoacoustic instability, aeroacoustic instability, and aeroelastic instability are some examples. When the fluid flow in the system is turbulent, the approach to such oscillatory instabilities occurs through a universal route characterized by a dynamical regime known as intermittency. In this paper, we extract the peculiar pattern of phase space attractors during the regime of intermittency by constructing recurrence networks corresponding to the phase space topology. We further train a convolutional neural network to classify the periodic and aperiodic structures in the recurrence networks and define a measure that indicates the proximity of the dynamical state to the onset of oscillatory instability. We show that this measure can predict the onset of oscillatory instabilities in three different fluid dynamic systems governed by different physical phenomena.

Recurrence network analysis exploring the routes to thermoacoustic instability in a Rijke tube with inverse diffusion flame

Inverse diffusion flame (IDF) is a reliable low NO_x technology that is suitable for various industrial applications including gas turbines. However, a confined IDF may exhibit thermoacoustic instability, a kind of dynamic instability, which is characterized by catastrophically large amplitude pressure oscillations. Transition to such instability for an inverse diffusion flame is less explored compared to other types of flame. In the present study, thermoacoustic instability in a Rijke tube with IDF is achieved by varying air flow rate and input power independently, and the onset of thermoacoustic instability is examined using the framework of recurrence network (RN). During the transition to thermoacoustic instability, we find new routes and two new intermediate states, here referred to as "amplitude varying aperiodic oscillations" and "low amplitude limit cycle-like oscillations." Furthermore, we show that recurrence network analysis can be used to identify the dynamical states during the transition to thermoacoustic instability. We observe an absence of a single characteristic scale, resulting in a non-regular network even during thermoacoustic instability. Furthermore, the degree distributions of RN during combustion noise do not obey a single power law. Thus, scale-free nature is not exhibited during combustion noise. In short, recurrence network analysis shows significant differences in the topological information during combustion noise and thermoacoustic instability for IDF with those for premixed flames, reported earlier.

Complex network analysis of the gravity effect on premixed flames propagating in a Hele-Shaw cell

We study the effect of gravity on spatiotemporal flame front dynamics in a Hele-Shaw cell from the viewpoint of complex networks. The randomness in flame front dynamics significantly increases with the gravitational level when the normalized Rayleigh number R_{a} is negative. This is clearly identified by two network entropies: the flame front network entropy and the transition network entropy. The irregular formation of large-scale wrinkles driven by the Rayleigh-Taylor instability plays an important role in the formation of high-dimensional deterministic chaos at R_{a}<0, resulting in the increase in the randomness of flame front dynamics.

Bursting during intermittency route to thermoacoustic instability: Effects of slow-fast dynamics

Intermittency observed prior to thermoacoustic instability is characterized by the occurrence of bursts of high-amplitude periodic oscillations (active state) amidst epochs of low-amplitude aperiodic fluctuations (rest state). Several model-based studies conjectured that bursting arises due to the underlying turbulence in the system. However, such intermittent bursts occur even in laminar and low-turbulence combustors, which cannot be explained by models based on turbulence. We assert that bursting in such combustors may arise due to the existence of subsystems with varying timescales of oscillations, thus forming slow-fast systems. Experiments were performed on a horizontal Rijke tube and the effect of slow-fast oscillations was studied by externally introducing low-frequency sinusoidal modulations in the control parameter. The induced bursts display an abrupt transition between the rest and the active states. The growth and decay patterns of such bursts show asymmetry due to delayed bifurcation caused by slow oscillations of the control parameter about the Hopf bifurcation point. Further, we develop a phenomenological model for the interaction between different subsystems of a thermoacoustic system by either coupling the slow and fast subsystems or by introducing noise in the absence of slow oscillations of the control parameter. We show that interaction between subsystems with different timescales leads to regular amplitude modulated bursting, while the presence of noise induces irregular amplitude modulations in the bursts. Thus, we speculate that bursting in laminar and low-turbulence systems occurs predominantly due to the interdependence between slow and fast oscillations, while bursting in high-turbulence systems is predominantly influenced by the underlying turbulence.

Recurrence analysis of slow-fast systems

Many complex systems exhibit periodic oscillations comprising slow-fast timescales. In such slow-fast systems, the slow and fast timescales compete to determine the dynamics. In this study, we perform a recurrence analysis on simulated signals from paradigmatic model systems as well as signals obtained from experiments, each of which exhibit slow-fast oscillations. We find that slow-fast systems exhibit characteristic patterns along the diagonal lines in the corresponding recurrence plot (RP). We discern that the hairpin trajectories in the phase space lead to the formation of line segments perpendicular to the diagonal line in the RP for a periodic signal. Next, we compute the recurrence networks (RNs) of these slow-fast systems and uncover that they contain additional features such as clustering and protrusions on top of the closed-ring structure. We show that slow-fast systems and single timescale systems can be distinguished by computing the distance between consecutive state points on the phase space trajectory and the degree of the nodes in the RNs. Such a recurrence analysis substantially strengthens our understanding of slow-fast systems, which do not have any accepted functional forms.

Application of recurrence quantification analysis for early detection of lean blowout in a swirl-stabilized dump combustor

Lean blowout (LBO) is a serious issue in modern gas turbine engines that operate in a lean (premixed) mode to follow the stringent emission norms. When an engine operates with a lean fuel-air mixture, the flame becomes unstable and is at times carried out of the combustion chamber by the unburnt flow. Thus, the sudden loss of the flame, known as lean blowout, leads to fatal accidents in aircrafts and loss of production in power plants. Therefore, an in-depth analysis of lean blowout is necessary as the phenomenon involves complex interactions between flow dynamics and chemical kinetics. For understanding the complex dynamics of this phenomenon, recurrence analysis can be a very useful method. In the current study, we observe a transition to LBO as the global fuel-air ratio is reduced from stoichiometric condition and perform recurrence quantification analysis (RQA) with the CH^∗ chemiluminescence data obtained experimentally. The extent of fuel-air mixing is varied with an objective of developing some robust early predictors of LBO that would work over a wide range of premixing. We find some RQA measures, such as determinism, laminarity, and trapping time, which show distinctive signature toward LBO and thereby can be used as early predictors of LBO for both premixed and partially premixed flames. Our analysis shows that the computational time for laminarity and trapping time is relatively less. However, computational time for those measures depends upon the dynamics of the combustor, size of the data taken, and choice of recurrence threshold.

Dynamical systems approach to study thermoacoustic transitions in a liquid rocket combustor

Liquid rockets are prone to large amplitude oscillations, commonly referred to as thermoacoustic instability. This phenomenon causes unavoidable developmental setbacks and poses a stern challenge to accomplish the mission objectives. Thermoacoustic instability arises due to the nonlinear interaction between the acoustic and the reactive flow subsystems in the combustion chamber. In this paper, we adopt tools from dynamical systems and complex systems theory to understand the dynamical transitions from a state of stable operation to thermoacoustic instability in a self-excited model multielement liquid rocket combustor based on an oxidizer rich staged combustion cycle. We observe that this transition to thermoacoustic instability occurs through a sequence of bursts of large amplitude periodic oscillations. Furthermore, we show that the acoustic pressure oscillations in the combustor pertain to different dynamical states. In contrast to a simple limit cycle oscillation, we show that the system dynamics switches between period-3 and period-4 oscillations during the state of thermoacoustic instability. We show several measures based on recurrence quantification analysis and multifractal theory, which can diagnose the dynamical transitions occurring in the system. We find that these measures are more robust than the existing measures in distinguishing the dynamical state of a rocket engine. Furthermore, these measures can be used to validate models and computational fluid dynamics simulations, aiming to characterize the performance and stability of rockets.

Effect of disorderness in two-dimensional discrete adiabatic combustion waves in a heterogeneous solid medium under stable regime

The effect of disorderness for the propagation of stable combustion waves in a two-dimensional discrete adiabatic system are studied by considering two systems, namely, (i) periodic and (ii) disorder. The discrete periodic system is modeled by regular arrangement of burnt and unburnt point heat sources. In contrast, the discrete disordered system is modeled by the concatenation of regular burnt and randomly distributed unburnt heat sources. Four different kinds of discrete disordered systems (S1, S2, S3, and S4) are modeled by varying the joint frequency count of unburnt heat sources without changing the domain size of the system. Results show that the disordered structure of the discrete system plays a crucial role in the combustion dynamics, shape of the combustion wave, and it affects the propagation of stable combustion waves with the manifestation of sparse and chaotic finger patterns. Quantitative analysis performed for the width and roughness of combustion front show that the disordered structure and ignition temperature (ε) of the discrete system plays a crucial role. The burn rates of discrete system are sensitive to change in structure of the system and ε. It is observed that the burn rates obtained for the discrete disordered system is always less than that of burn rates of the discrete periodic system. The burn rates calculated for the present model are compared with the available experimental data on combustion of thermite systems.

Bursting and mixed mode oscillations during the transition to limit cycle oscillations in a matrix burner

We investigate the route to self-excited thermoacoustic instability in a laminar flow multiple flame matrix burner. With an increase in the equivalence ratio, the thermoacoustic system that is initially quiet (stable operation) transitions to limit cycle oscillations through two distinct dynamical states, namely, bursting oscillations and mixed mode oscillations. The acoustic pressure oscillations transition from quiescence to large amplitudes during bursting oscillations. Such high amplitude bursting oscillations that occur well ahead of the onset of limit cycle oscillations can potentially cause structural damage. The thermoacoustic system exhibits hysteresis. The transition to limit cycle oscillations is replicated in a phenomenological model containing slow-fast time scales.

Spatiotemporal dynamics and early detection of thermoacoustic combustion instability in a model rocket combustor

We numerically study the spatiotemporal dynamics and early detection of thermoacoustic combustion instability in a model rocket combustor using the theories of complex networks and synchronization. The turbulence network, which consists of nodes and vertexes in weighted networks between vortices, can characterize the complex spatiotemporal structure of a flow field during thermoacoustic combustion instability. The transfer entropy allows us to identify the driving region of thermoacoustic combustion instability. In addition to the order parameter, a phase parameter newly proposed in this study is useful for capturing the precursor of thermoacoustic combustion instability.

Coupled interaction between unsteady flame dynamics and acoustic field in a turbulent combustor

Thermoacoustic instability is a result of the positive feedback between the acoustic pressure and the unsteady heat release rate fluctuations in a combustor. We apply the framework of the synchronization theory to study the coupled behavior of these oscillations during the transition to thermoacoustic instability in a turbulent bluff-body stabilized gas-fired combustor. Furthermore, we characterize this complex behavior using recurrence plots and recurrence networks. We mainly found that the correlation of probability of recurrence ( C P R ), the joint probability of recurrence ( J P R ), the determinism ( D E T ), and the recurrence rate ( R R ) of the joint recurrence matrix aid in detecting the synchronization transitions in this thermoacoustic system. We noticed that C P R and D E T can uncover the occurrence of phase synchronization state, whereas J P R and R R can be used as indices to identify the occurrence of generalized synchronization (GS) state in the system. We applied measures derived from joint and cross recurrence networks and observed that the joint recurrence network measures, transitivity ratio, and joint transitivity are useful to detect GS. Furthermore, we use the directional property of the network measure, namely, cross transitivity to analyze the type of coupling existing between the acoustic field ( p ' ) and the heat release rate ( q ˙ ' ) fluctuations. We discover a possible asymmetric bidirectional coupling between q ˙ ' and p ' , wherein q ˙ ' is observed to exert a stronger influence on p ' than vice versa.

Strange nonchaotic and chaotic attractors in a self-excited thermoacoustic oscillator subjected to external periodic forcing

We experimentally investigate the synchronization dynamics of a self-excited thermoacoustic system forced beyond its phase-locked state. The system consists of a laminar premixed flame in a tube combustor subjected to periodic acoustic forcing. On increasing the forcing amplitude above that required for phase locking, we find that the system can transition out of phase locking and into chaos, which is consistent with the Afraimovich-Shilnikov theorem for the breakdown of a phase-locked torus. However, we also find some unexpected behavior, most notably the emergence of a strange nonchaotic attractor (SNA) before the onset of chaos. We verify the existence of the SNA and chaotic attractor by examining the correlation dimension, the autocorrelation function, the power-law scaling in the Fourier amplitude spectrum, the permutation entropy in a pseudoperiodic surrogate test, and the permutation spectrum. In summary, this study explores the SNA and chaotic dynamics of a thermoacoustic system forced beyond its phase-locked state, opening up new pathways for the development of alternative strategies to control self-excited thermoacoustic oscillations in combustion devices such as gas turbines and rocket engines.

Simulation of knock probability in an internal combustion engine

In spark-ignition internal combustion engines, fluctuations of the in-cylinder pressure trace and the tendency of combustion knock are usually different from one cycle to another. These cycle-to-cycle variations are affected by the initial state at ignition time and the subsequent burning. The occurrence of the phenomena is unpredictable, and their stochastic nature offers challenges in the optimization of engine control strategies. In this paper, a simulator providing a series of cycle-to-cycle varied in-cylinder pressures is introduced. The Wiebe function and Livengood-Wu integration are used to describe the determinacy of combustion. Various means, including the Markov chain, are introduced to express the stochastic quantities during combustion. In addition, the combustion of a given knock probability is simulated.

On the emergence of critical regions at the onset of thermoacoustic instability in a turbulent combustor

We use complex network theory to investigate the dynamical transition from stable operation to thermoacoustic instability via intermittency in a turbulent combustor with a bluff body stabilized flame. A spatial network is constructed, representing each of these three dynamical regimes of combustor operation, based on the correlation between time series of local velocity obtained from particle image velocimetry. Network centrality measures enable us to identify critical regions of the flow field during combustion noise, intermittency, and thermoacoustic instability. We find that during combustion noise, the bluff body wake turns out to be the critical region that determines the dynamics of the combustor. As the turbulent combustor transitions to thermoacoustic instability, during intermittency, the wake of the bluff body loses its significance in determining the flow dynamics and the region on top of the bluff body emerges as the most critical region in determining the flow dynamics during thermoacoustic instability. The knowledge about this critical region of the reactive flow field can help us devise optimal control strategies to evade thermoacoustic instability.

Dynamic behavior of combustion instability in a cylindrical combustor with an off-center installed coaxial injector

We have intensively studied the dynamic behavior of combustion instability in a cylindrical combustor with an off-center installed coaxial injector. The most interesting discovery in this study is the appearance of a deterministic chaos in a transition from a dynamically stable state to well-developed high-frequency thermoacoustic combustion oscillations with increasing the volume flow rate of nitrogen with which oxygen is diluted. The presence of deterministic chaos is reasonably identified by considering an extended version of the Sugihara-May algorithm [G. Sugihara and R. May, Nature 344, 734 (1990)] as a local predictor and the multiscale complexity-entropy causality plane based on statistical complexity.

Characterization and detection of thermoacoustic combustion oscillations based on statistical complexity and complex-network theory

We present an experimental study on the characterization of dynamic behavior of flow velocity field during thermoacoustic combustion oscillations in a turbulent confined combustor from the viewpoints of statistical complexity and complex-network theory, involving detection of a precursor of thermoacoustic combustion oscillations. The multiscale complexity-entropy causality plane clearly shows the possible presence of two dynamics, noisy periodic oscillations and noisy chaos, in the shear layer regions (1) between the outer recirculation region in the dump plate and a recirculation flow in the wake of the centerbody and (2) between the outer recirculation region in the dump plate and a vortex breakdown bubble away from the centerbody. The vertex strength in the turbulence network and the community structure of the vorticity field can identify the vortical interactions during thermoacoustic combustion oscillations. Sequential horizontal visibility graph motifs are useful for capturing a precursor of themoacoustic combustion oscillations.

Synchronous behaviour of two interacting oscillatory systems undergoing quasiperiodic route to chaos

Thermoacoustic instability, caused by a positive feedback between the unsteady heat release and the acoustic field in a combustor, is a major challenge faced in most practical combustors such as those used in rockets and gas turbines. We employ the synchronization theory for understanding the coupling between the unsteady heat release and the acoustic field of a thermoacoustic system. Interactions between coupled subsystems exhibiting different collective dynamics such as periodic, quasiperiodic, and chaotic oscillations are addressed. Even though synchronization studies have focused on different dynamical states separately, synchronous behaviour of two coupled systems exhibiting a quasiperiodic route to chaos has not been studied. In this study, we report the first experimental observation of different synchronous behaviours between two subsystems of a thermoacoustic system exhibiting such a transition as reported in Kabiraj et al. [Chaos 22, 023129 (2012)]. A rich variety of synchronous behaviours such as phase locking, intermittent phase locking, and phase drifting are observed as the dynamics of such subsystem change. The observed synchronization behaviour is further characterized using phase locking value, correlation coefficient, and relative mean frequency. These measures clearly reveal the boundaries between different states of synchronization.

Recurrence networks to study dynamical transitions in a turbulent combustor

Thermoacoustic instability and lean blowout are the major challenges faced when a gas turbine combustor is operated under fuel lean conditions. The dynamics of thermoacoustic system is the result of complex nonlinear interactions between the subsystems-turbulent reactive flow and the acoustic field of the combustor. In order to study the transitions between the dynamical regimes in such a complex system, the time series corresponding to one of the dynamic variables is transformed to an ε-recurrence network. The topology of the recurrence network resembles the structure of the attractor representing the dynamics of the system. The transitions in the thermoacoustic system are then captured as the variation in the topological characteristics of the network. We show the presence of power law degree distribution in the recurrence networks constructed from time series acquired during the occurrence of combustion noise and during the low amplitude aperiodic oscillations prior to lean blowout. We also show the absence of power law degree distribution in the recurrence networks constructed from time series acquired during the occurrence of thermoacoustic instability and during the occurrence of intermittency. We demonstrate that the measures derived from recurrence network can be used as tools to capture the transitions in the turbulent combustor and also as early warning measures for predicting impending thermoacoustic instability and blowout.

Chaotic dynamics of a swirling flame front instability generated by a change in gravitational orientation

We have intensively examined the dynamic behavior of flame front instability in a lean swirling premixed flame generated by a change in gravitational orientation [H. Gotoda, T. Miyano, and I. G. Shepherd, Phys. Rev. E 81, 026211 (2010)PLEEE81539-375510.1103/PhysRevE.81.026211] from the viewpoints of complex networks, symbolic dynamics, and statistical complexity. Here, we considered the permutation entropy in combination with the surrogate data method, the permutation spectrum test, and the multiscale complexity-entropy causality plane incorporating a scale-dependent approach, none of which have been considered in the study of flame front instabilities. Our results clearly show the possible presence of chaos in flame front dynamics induced by the coupling of swirl-buoyancy interaction in inverted gravity. The flame front dynamics also possesses a scale-free structure, which is reasonably shown by the probability distribution of the degree in ε-recurrence networks.

Nonlinear dynamics and intermittency in a turbulent reacting wake with density ratio as bifurcation parameter

The flame or flow behavior of a turbulent reacting wake is known to be fundamentally different at high and low values of flame density ratio (ρ_{u}/ρ_{b}), as the flow transitions from globally stable to unstable. This paper analyzes the nonlinear dynamics present in a bluff-body stabilized flame, and identifies the transition characteristics in the wake as ρ_{u}/ρ_{b} is varied over a Reynolds number (based on the bluff-body lip velocity) range of 1000-3300. Recurrence quantification analysis (RQA) of the experimentally obtained time series of the flame edge fluctuations reveals that the time series is highly aperiodic at high values of ρ_{u}/ρ_{b} and transitions to increasingly correlated or nearly periodic behavior at low values. From the RQA of the transverse velocity time series, we observe that periodicity in the flame oscillations are related to periodicity in the flow. Therefore, we hypothesize that this transition from aperiodic to nearly periodic behavior in the flame edge time series is a manifestation of the transition in the flow from globally stable, convective instability to global instability as ρ_{u}/ρ_{b} decreases. The recurrence analysis further reveals that the transition in periodicity is not a sudden shift; rather it occurs through an intermittent regime present at low and intermediate ρ_{u}/ρ_{b}. During intermittency, the flow behavior switches between aperiodic oscillations, reminiscent of a globally stable, convective instability, and periodic oscillations, reminiscent of a global instability. Analysis of the distribution of the lengths of the periodic regions in the intermittent time series and the first return map indicate the presence of type-II intermittency.

Probing and exploiting the chaotic dynamics of a hydrodynamic photochemical oscillator to implement all the basic binary logic functions

The convective motions within a solution of a photochromic spiro-oxazine being irradiated by UV only on the bottom part of its volume, give rise to aperiodic spectrophotometric dynamics. In this paper, we study three nonlinear properties of the aperiodic time series: permutation entropy, short-term predictability and long-term unpredictability, and degree distribution of the visibility graph networks. After ascertaining the extracted chaotic features, we show how the aperiodic time series can be exploited to implement all the fundamental two-inputs binary logic functions (AND, OR, NAND, NOR, XOR, and XNOR) and some basic arithmetic operations (half-adder, full-adder, half-subtractor). This is possible due to the wide range of states a nonlinear system accesses in the course of its evolution. Therefore, the solution of the convective photochemical oscillator results in hardware for chaos-computing alternative to conventional complementary metal-oxide semiconductor-based integrated circuits.

Chaos of radiative heat-loss-induced flame front instability

We are intensively studying the chaos via the period-doubling bifurcation cascade in radiative heat-loss-induced flame front instability by analytical methods based on dynamical systems theory and complex networks. Significant changes in flame front dynamics in the chaotic region, which cannot be seen in the bifurcation diagrams, were successfully extracted from recurrence quantification analysis and nonlinear forecasting and from the network entropy. The temporal dynamics of the fuel concentration in the well-developed chaotic region is much more complicated than that of the flame front temperature. It exhibits self-affinity as a result of the scale-free structure in the constructed visibility graph.

Characterizing the signature of flame flashback precursor through recurrence analysis

In this paper, it is shown that prior to flashback, small dynamical changes appear in the system. These changes appear as a drift in the recurrence plots and are found to be associated with a gradual increase in the determinism and recurrence rate. Thus, this study indicates that precursors to flame flashback exist and can be detected in the multidimensional phase space reconstructed from pressure measurements acquired during flashback. This observation could have broad academic as well as industrial implications.

Detecting deterministic nature of pressure measurements from a turbulent combustor

Identifying nonlinear structures in a time series, acquired from real-world systems, is essential to characterize the dynamics of the system under study. A single time series alone might be available in most experimental situations. In addition to this, conventional techniques such as power spectral analysis might not be sufficient to characterize a time series if it is acquired from a complex system such as a thermoacoustic system. In this study, we analyze the unsteady pressure signal acquired from a turbulent combustor with bluff-body and swirler as flame holding devices. The fractal features in the unsteady pressure signal are identified using the singularity spectrum. Further, we employ surrogate methods, with translational error and permutation entropy as discriminating statistics, to test for determinism visible in the observed time series. In addition to this, permutation spectrum test could prove to be a robust technique to characterize the dynamical nature of the pressure time series acquired from experiments. Further, measures such as correlation dimension and correlation entropy are adopted to qualitatively detect noise contamination in the pressure measurements acquired during the state of combustion noise. These ensemble of measures is necessary to identify the features of a time series acquired from a system as complex as a turbulent combustor. Using these measures, we show that the pressure fluctuations during combustion noise has the features of a high-dimensional chaotic data contaminated with white and colored noise.

Characterization of degeneration process in combustion instability based on dynamical systems theory

We present a detailed study on the characterization of the degeneration process in combustion instability based on dynamical systems theory. We deal with combustion instability in a lean premixed-type gas-turbine model combustor, one of the fundamentally and practically important combustion systems. The dynamic behavior of combustion instability in close proximity to lean blowout is dominated by a stochastic process and transits to periodic oscillations created by thermoacoustic combustion oscillations via chaos with increasing equivalence ratio [Chaos 21, 013124 (2011); Chaos 22, 043128 (2012)]. Thermoacoustic combustion oscillations degenerate with a further increase in the equivalence ratio, and the dynamic behavior leads to chaotic fluctuations via quasiperiodic oscillations. The concept of dynamical systems theory presented here allows us to clarify the nonlinear characteristics hidden in complex combustion dynamics.

Dynamics of self-excited thermoacoustic instability in a combustion system: Pseudo-periodic and high-dimensional nature

We have examined the dynamics of self-excited thermoacoustic instability in a fundamentally and practically important gas-turbine model combustion system on the basis of complex network approaches. We have incorporated sophisticated complex networks consisting of cycle networks and phase space networks, neither of which has been considered in the areas of combustion physics and science. Pseudo-periodicity and high-dimensionality exist in the dynamics of thermoacoustic instability, including the possible presence of a clear power-law distribution and small-world-like nature.

Analysis and prediction of aperiodic hydrodynamic oscillatory time series by feed-forward neural networks, fuzzy logic, and a local nonlinear predictor

Forecasting of aperiodic time series is a compelling challenge for science. In this work, we analyze aperiodic spectrophotometric data, proportional to the concentrations of two forms of a thermoreversible photochromic spiro-oxazine, that are generated when a cuvette containing a solution of the spiro-oxazine undergoes photoreaction and convection due to localized ultraviolet illumination. We construct the phase space for the system using Takens' theorem and we calculate the Lyapunov exponents and the correlation dimensions to ascertain the chaotic character of the time series. Finally, we predict the time series using three distinct methods: a feed-forward neural network, fuzzy logic, and a local nonlinear predictor. We compare the performances of these three methods.