Non-normality and nonlinearity in thermoacoustic instabilities

Analysis of thermoacoustic instabilities were dominated by modal (eigenvalue) analysis for many decades. Recent progress in nonmodal stability analysis allows us to study the problem from a different perspective, by quantitatively describing the short-term behavior of disturbances. The short-term evolution has a bearing on subcritical transition to instability, known popularly as triggering instability in thermoacoustic parlance. We provide a review of the recent developments in the context of triggering instability. A tutorial for nonmodal stability analysis is provided. The applicability of the tools from nonmodal stability analysis are demonstrated with the help of a simple model of a Rjike tube. The article closes with a brief description of how to characterize bifurcations in thermoacoustic systems

Experimental investigation of non-normality of thermoacoustic interaction in an electrically heated Rijke tube

An experimental investigation of the non-normal nature of thermoacoustic interactions in an electrically heated horizontal Rijke tube is performed. Since non-normality and the associated transient growth are linear phenomena, the experiments have to be confined to the linear regime. The bifurcation diagram for the subcritical Hopf bifurcation into a limit cycle behavior has been determined, after which the amplitude levels, for which the system acts linearly, have been identified for different power inputs to the heater. There are two main objectives for this experimental investigation. The first one deals with the extraction of the linear eigenmodes associated with the acoustic pressure from experimental data. This is accomplished by the Dynamic Mode Decomposition (DMD) technique applied in the linear regime. The non-orthogonality between the eigenmodes is determined for various values of heater power. The second objective is to identify evidence of transient perturbation growth in the system. The total acoustic energy in the duct has been monitored as the thermoacoustic system evolves from its initial condition. Transient growth, on the order of previous theoretical studies, has been found, and its parameteric dependence on amplitude ratio and phase angle of the initial eigenmode components has been determined. This study represents the first experimental confirmation of non-normality in thermoacoustic systems

Sensitivity and Nonlinearity of Thermoacoustic Oscillations

Nine decades of rocket engine and gas turbine development have shown that thermoacoustic oscillations are difficult to predict but can usually be eliminated with relatively small ad hoc design changes. These changes can, however, be ruinously expensive to devise. This review explains why linear and nonlinear thermoacoustic behaviour is so sensitive to parameters such as operating point, fuel composition, and injector geometry. It shows how non-periodic behaviour arises in experiments and simulations and discusses how fluctuations in thermoacoustic systems with turbulent reacting flow, which are usually filtered or averaged out as noise, can reveal useful information. Finally, it proposes tools to exploit this sensitivity in the future: adjoint-based sensitivity analysis to optimize passive control designs, and complex systems theory to warn of impending thermoacoustic oscillations and to identify the most sensitive elements of a thermoacoustic system

Precursors to self-sustained oscillations in aeroacoustic systems

In this study, we attempt to provide a repertoire of measures to forewarn the onset of impending flow-induced mechanical oscillations via online health monitoring. To illustrate the principles, the flow of air through a pipe terminated by a circular orifice plate is investigated at various flow velocities using a suitably placed pressure transducer. It is observed that the regimes corresponding to the production of a tone is presaged by operating conditions that display temporarily intermittent bursts of periodic pressure oscillations that emerge from a background of lower-amplitude aperiodic fluctuations. The various model-free measures prescribed in this paper serve as efficient precursors by characterizing these intermittent states, which can potentially arise in aeroacoustic systems when the flow is highly unsteady or turbulent

Sensitivity and Nonlinearity of Thermoacoustic Oscillations

Nine decades of rocket engine and gas turbine development have shown that thermoacoustic oscillations are difficult to predict but can usually be eliminated with relatively small ad hoc design changes. These changes can, however, be ruinously expensive to devise. This review explains why linear and nonlinear thermoacoustic behaviour is so sensitive to parameters such as operating point, fuel composition, and injector geometry. It shows how non-periodic behaviour arises in experiments and simulations and discusses how fluctuations in thermoacoustic systems with turbulent reacting flow, which are usually filtered or averaged out as noise, can reveal useful information. Finally, it proposes tools to exploit this sensitivity in the future: adjoint-based sensitivity analysis to optimize passive control designs, and complex systems theory to warn of impending thermoacoustic oscillations and to identify the most sensitive elements of a thermoacoustic system.We would like to thank the European Research Council (ALORS 2590620), the Department of Science and Technology (DST), and the Office of Naval Research Global (ONRG) for their financial support

Subcritical bifurcation and bistability in thermoacoustic systems

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

Non-normality and nonlinearity in thermoacoustic instabilities

Analysis of thermoacoustic instabilities were dominated by modal (eigenvalue) analysis for many decades. Recent progress in nonmodal stability analysis allows us to study the problem from a different perspective, by quantitatively describing the short-term behavior of disturbances. The short-term evolution has a bearing on subcritical transition to instability, known popularly as triggering instability in thermoacoustic parlance. We provide a review of the recent developments in the context of triggering instability. A tutorial for nonmodal stability analysis is provided. The applicability of the tools from nonmodal stability analysis are demonstrated with the help of a simple model of a Rjike tube. The article closes with a brief description of how to characterize bifurcations in thermoacoustic systems

Flame blowout: Transition to an absorbing phase

The turbulent flame inside a gas turbine engine is susceptible to local extinction leading to global extinguishment or blowout at fuel lean conditions. Flame blowout is traditionally viewed as a loss of static stability of the combustor. However, flames often exhibit rich dynamics as blowout is approached suggesting that a more comprehensive description of the dynamics of flame blowout, which could lead to reduced order models, is necessary. A turbulent flame can be considered as a collection of a large number of flamelets. The population dynamics of these flamelets could be used to model the overall flame behavior as a contact process. In this context, flame blowout can be viewed as the population of flamelets approaching zero, in other words, extinction of flamelets. In this paper, we employ a cellular automata based model to study the emergent dynamics of the population of such flamelets. We show that the model is able to qualitatively capture interesting dynamics that a turbulent flame inside a combustor exhibits close to flame blowout. Furthermore, we show that flame blowout is similar to a threshold-like transition to an absorbing phase. Published by AIP Publishing

Novel perspectives on the dynamics of premixed flames.

The present study develops an alternative perspective on the response of premixed flames to flow perturbations. In particular, the linear response of laminar premixed flames to velocity perturbations is examined in the time domain, and the corresponding impulse response functions are determined analytically. Different flame types and shapes as well as different velocity perturbation models are considered. Two contributions to the flame response are identified: a convective displacement of the flame due to velocity perturbations, and a restoration mechanism, which is a consequence of the combined effects of flame propagation and flame anchoring. The impulse responses are used to identify the relevant time scales and to form non-dimensional frequencies. The link of the present results to previous studies formulated in the frequency domain is established. The time domain approach is found to facilitate analysis and interpretation of well-known properties of premixed flames such as excess gain, periodic cutoff and self-similar aspects of flame response. Characteristic time scales of response appear naturally and can be interpreted in a straightforward manner

Thermoacoustic instabilities in a ducted premixed flame: reduced-order models and control.

The problem of combustion instabilities arising from the thermoacoustic interactions in a ducted premixed flame model is considered. Contrary to the conventionally used low-order models to describe such systems, a high dimensional model governed by a set of the acoustic equations coupled to the equations for flame dynamics is employed. The flame front is discretized into finite flame elements to assign internal degrees of freedom to the front and track its evolution. Model reduction schemes, namely proper orthogonal decomposition (POD) and balanced truncation, are used to reduce the size of this model. Compared to POD modes, balanced modes show superior input–output characteristics, in agreement with the full model. POD modes on the other hand capture the transient growth in the model while the balanced modes do not. Performance of POD modes is highly dependent on the snapshots used for their computation. A linear quadratic Gaussian (LQG) framework using the reduced-order model based on 16 balanced modes is formulated to control thermoacoustic instabilities in the linear model. The controllers thus obtained are then used on the nonlinear model. They successfully curtail limit cycle oscillations in the nonlinear plant and also avoid subcritical transition to instability