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. © 2011 Cambridge University Press.

R. I. Sujith

Department of Aerospace Engineering

Rahul R. Kulkarni

Active control

Eigenvalues

Instability

Linear controllers

Linear stability

Low-amplitude

NON-NORMALITY

Nonlinear effect

Nonlinear instability

Rijke tube

Thermoacoustic instability

THERMOACOUSTIC SYSTEMS

TRANSIENT GROWTH

Aeroacoustics

Eigenvalues and eigenfunctions

Linear control systems

Linear stability analysis

Quality assurance

Thermoacoustics

Controllers

acoustics

AMPLITUDE

Eigenvalue

FLUID MECHANICS

Nonlinearity

perturbation

temperature effect

Wave modeling

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Non-normality and its consequences in active control of thermoacoustic instabilities

Journal of Fluid Mechanics

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.

Fulltext

International Journal of Spray and Combustion Dynamics

Non-normality and nonlinearity in thermoacoustic instabilities

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. © 2013 Copyright Taylor and Francis Group, LLC.

Combustion Science and Technology

Thermoacoustic instabilities in a ducted premixed flame: Reduced-Order models and control

This study proposes a coefficient of Rayleigh Index (CRI) obtained by normalizing the acquired data which can circumvent the problem of arbitrary scale and unit while quantifying the Rayleigh Index. CRI is estimated from the simultaneous measurements of luminescence of a flame as an indicator of unsteady heat release and pressure fluctuation near the burner head. Experiments were carried out for six different mixture equivalence ratios ranging from 0.75 to 1.0 and a burner positioned at x/L = 0.2. The control technique of micro jets is found to completely suppress the thermo-acoustics by decoupling the pressure waves and unsteady heat release waves as indicated by increasing phase difference between them. The coefficient of Rayleigh Index is found to reduce to nearly zero with the use of radial micro jets for suppressing the thermo-acoustic instability. In such case, the flame luminescence was observed to significantly reduce. The proposed coefficient of Rayleigh index is very useful in quantifying the effectiveness of control technique for the thermo-acoustic instability. © 2019 Elsevier Ltd

Measurement: Journal of the International Measurement Confederation

Coefficient of Rayleigh Index based performance evaluation of radial micro jet injection technique for thermo-acoustic instability control

The present work describes the occurrence of thermo-acoustic instability inside a horizontal Rijke tube and its suppression using an open loop active control technique. The Rijke tube is provided with a co-axial pre-mixed gas burner as the source of heat, which could be placed at any desired position. Radial injection of air (less than 3% of the total mass flow) through micro-jets into the flame is used as a control technique to suppress the thermo-acoustic instability. The rise in heat content inside the Rijke tube, estimated from the temperature mapping, clearly shows reduction in the heat loss as a result of complete suppression of the thermo-acoustic instability. However, the stability achieved passively by means of a slight shift in the burner position does not result in any change in the heat content. There is a visible change in the appearance of the burner flame when the above two methods are used to suppress the thermo-acoustic instability. The flame is seen to significantly shrink in length and spread radially when the control technique was applied. The flame dynamics is believed to determine the heat loss and hence the heat content inside the Rijke tube. © 2016, © The Author(s) 2016.

Experiments on heat content inside a Rijke tube with suppression of thermo-acoustics instability

Non-normality in thermoacoustic systems has received attention recently. It has been shown that in a non-normal but classically linearly stable system, there can be significant transient energy growth of small perturbations before their eventual decay. This growth occurs in the absence of non-linear effects. This phenomenon can be explained by the non-normality of the governing linear operator or the non-orthogonality of the eigenvectors of the system. In this paper, we study various aspects of this transient energy growth for a general combustor, with localized heat release approximated by the popular n-r model. Galerkin technique is used to simplify the governing acoustic equations in a duct in the presence of a localized heat source with appropriate boundary conditions. Singularvalue decomposition (SVD) is used to compute the transient energy growth. SVD is used as a tool to obtain the maximum possible energy amplification and the optimal initial conditions required for this amplification. The necessary and sufficient conditions for no energy growth are discussed. A parametric analysis is performed to highlight the effect of system parameters on the maximum transient growth rate and to obtain regions of stability of the thermoacoustic system. © 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Proceedings of the Combustion Institute

Characterizing energy growth during combustion instabilities: Singularvalues or eigenvalues?

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.

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.

Physics of Fluids

Thermoacoustic instability in a Rijke tube: Non-normality and nonlinearity

Acta Acustica united with Acustica

The Rijke Tube: A Green's Function Approach in the Frequency Domain

About the Zero Mach Number Assumption in the Calculation of Thermoacoustic Instabilities

Journal	Journal of Fluid Mechanics
Publisher	Cambridge University Press
ISSN	00221120
Open Access	No