There have been conflicting opinions in literature, about the consequences of ignoring the linear coupling terms in finite-dimensional thermoacoustic models. We try to clarify this issue in this paper. We study consequences of ignoring the linear coupling terms on the non-normal thermoacoustic interactions in a horizontal Rijke tube. In the Galerkin formulation, we use pseudospectra and Henrici index of non-normality to study the impact of linear coupling between the Galerkin modes on the non-normality and the ensuing transient growth. This linear coupling is important for the energy transfer between the eigenmodes of the system (which are not the same as the natural acoustic modes of the duct). Although ignoring this linear coupling results in negligible shift in the eigenfrequencies, it may have serious consequences on the non-normality of the thermoacoustic system and its dynamics. Nonlinear phenomena such as triggering, boot-strapping and limit cycle oscillations may be completely missed out when this linear coupling is ignored. Ignoring the linear coupling between the Galerkin modes is justified only when the non-normality and the nonlinear dynamics of the system are not altered significantly.

R. I. Sujith

Department of Aerospace Engineering

Dynamics

Energy transfer

Gait analysis

Galerkin methods

Acoustic modes

combustion

EIGEN-FREQUENCIES

EIGENMODES

Galerkin

GALERKIN FORMULATION

GALERKIN MODES

Limit cycle oscillations

Linear coupling

LINEAR COUPLINGS

Non-linear dynamics

NON-LINEAR PHENOMENON

NON-NORMALITY

PSEUDOSPECTRA

Rijke tube

RIJKE TUBES

Thermoacoustic instability

TRANSIENT GROWTH

Coupled circuits

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IIT Madras

Impact of Linear Coupling on Thermoacoustic Instabilities

Combustion Science and Technology

Non-normal transient growth of energy is a feature encountered in many physical systems. Its observation is intimately related to the norm used to describe the system dynamics. For a multi-physics problem such as thermoacoustics, where a heat source is in feedback with acoustic waves and a flow field, the appropriate metric is an ongoing matter of debate. Adopting a systemic perspective, it is argued in the present paper that an energy norm is, in principle, a matter of choice, but one that is critically tied to the dynamics described by the system model. To illustrate our arguments, it is shown that different norms exhibit the non-normal dynamics of thermoacoustic systems differently, but that this difference is fully explicable by the energy flux and source terms related to the formulation of the model. The non-normal dynamics as such is unaffected by the choice of norm, and transient growth merely results from a maximization of the flux and source terms governing the energy balance associated with the specific model formulation. Investigating transient growth for arbitrary energy norms requires the capability to handle semi-norm optimization problems. In the present study, we propose an approach to do so using the singular value decomposition. Non-normal transient growth around a stable fix point is then investigated for a low-order model of a simple thermoacoustic configuration of a premixed flame enclosed in a duct with non-zero mean temperature jump and bulk mean flow. The corresponding optimal mode shapes and pertinent parameters leading to transient growth are identified and discussed. For transient growth resulting from the interaction of the flame with the acoustic field, it is found that heat sources with a fast response lead to more transient growth than slow heat sources, because the system can bear a larger source term before becoming linearly unstable. Furthermore, the amount of transient energy growth does not increase monotonically with the amplitude of the initial perturbation of the flame. © 2016, © The Author(s) 2016.

Fulltext

International Journal of Spray and Combustion Dynamics

A systems perspective on non-normality in low-order thermoacoustic models: Full norms, semi-norms and transient growth

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. © The Author(s) 2016.

Non-normality and nonlinearity in thermoacoustic instabilities

A systematic approach for non-modal stability analysis of thermoacoustic systems with a localized heat source is proposed. The response of the heat source to flow perturbations is obtained from unsteady computational fluid dynamics combined with correlation-based linear system identification. A model for the complete thermoacoustic system is formulated with a Galerkin expansion technique, where the heat source is included as an acoustically compact element. The eigenvalues of the resulting system are obtained from discretization of the solution operator, the maximum growth factor is estimated from the pseudospectra using Kreiss' theorem. The approach is illustrated with a simple Rijke tube configuration. Results obtained with a simple "baseline" model for the heat source dynamics based on King's law - widely used in hot wire anemometry - are compared against the more advanced treatment developed here. Analysis of pseudospectra diagrams shows that the choice of the heat source model does influence the sensitivity of eigenvalues to perturbations and hence the non-normal behavior. The maximum growth factor for the system with the heat source model based on King's law is more sensitive to changes in the heat source location than the CFD-based heat source model. © 2010 Elsevier Inc. All rights reserved.

Applied Mathematics and Computation

Identification of heat transfer dynamics for non-modal analysis of thermoacoustic stability

The role of non-normality and nonlinearity in flame-acoustic interaction in a ducted diffusion flame is investigated in this paper. The infinite rate chemistry model is employed to study unsteady diffusion flames in a Burke-Schumann type geometry. It has been observed that even in this simplified case, the combustion response to perturbations of velocity is non-normal and nonlinear. This flame model is then coupled with a linear model of the duct acoustic field to study the temporal evolution of acoustic perturbations. The one-dimensional acoustic field is simulated in the time domain using the Galerkin technique, treating the fluctuating heat release from the combustion zone as a compact acoustic source. It is shown that the coupled combustion-acoustic system is non-normal and nonlinear. Further, calculations showed the occurrence of triggering; i.e. the thermoacoustic oscillations decay for some initial conditions whereas they grow for some other initial conditions. It is shown that triggering occurs because of the combined effect of non-normality and nonlinearity. For such a non-normal system, resonance or 'pseudoresonance' may occur at frequencies far from its natural frequencies. Non-normal systems can be studied using pseudospectra, as eigenvalues alone are not sufficient to predict the behaviour of the system. Further, both necessary and sufficient conditions for the stability of a thermoacoustic system are presented in this paper. © 2008 Cambridge University Press.

Journal of Fluid Mechanics

Non-normality and nonlinearity in combustion-acoustic interaction in diffusion flames

The role of non-normality and nonlinearity in thermoacoustic interaction in a Rijke tube is investigated in this paper. The heat release rate of the heating element is modeled by a modified form of King's law. This fluctuating heat release from the heating element is treated as a compact source in the one-dimensional linear model of the acoustic field. The temporal evolution of the acoustic perturbations is studied using the Galerkin technique. It is shown that any thermoacoustic system is non-normal. Non-normality can cause algebraic growth of oscillations for a short time even though the eigenvectors of the system could be decaying exponentially with time. This feature of non-normality combined with the effect of nonlinearity causes the occurrence of triggering, i.e., the thermoacoustic oscillations decay for some initial conditions whereas they grow for some other initial conditions. If a system is non-normal, then there can be large amplification of oscillations even if the excited frequency is far from the natural frequency of the system. The dependence of transient growth on time lag and heater positions is studied. Such amplifications (pseudoresonance) can be studied using pseudospectra, as eigenvalues alone are not sufficient to predict the behavior of the system. The geometry of pseudospectra can be used to obtain upper and lower bounds on the growth factor, which provide both necessary and sufficient conditions for the stability of a thermoacoustic system. © 2008 American Institute of Physics.

Journal	Combustion Science and Technology
ISSN	00102202
Open Access	No