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 &amp; 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, Gmax, of the order of 105 to 107. However, recent investigations by L. Magri and M. P. Juniper have brought to light certain errors in that paper. When the errors are corrected, Gmax 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. © 2013 Cambridge University Press.

R. I. Sujith

Department of Aerospace Engineering

acoustics

Aircraft engines

Fighter aircraft

Rockets

ACOUSTIC DUCTS

ACOUSTIC INTERACTION

ACOUSTIC OSCILLATION

Diffusion Flame

FLAMES

INITIAL ENERGY

SELF-SUSTAINED OSCILLATIONS

TRANSIENT GROWTH

combustion

JUNIPERUS

Fulltext

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IIT Madras

Journal of Fluid Mechanics

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.

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

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

Proceedings of the Combustion Institute

Finding thermoacoustic limit cycles for a ducted Burke-Schumann flame

Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition

Spectra and Pseudospectra

Acta Mechanica

On the energy transfer to small disturbances in fluid flow (Part I)

Non-normality in combustion-acoustic interaction in diffusion flames: A critical revision

Journal	Data powered by TypesetJournal of Fluid Mechanics
Publisher	Data powered by TypesetCambridge University Press
ISSN	00221120
Open Access	No