Chagelishvili [Phys. Rev. Lett. 79, 3178 (1997)] discovered a linear mechanism of acoustic wave emergence from vorticity fluctuations in shear flows. This paper illustrates how this "nonresonant" phenomenon is related to the non-normality of the operator governing the linear dynamics of disturbances in shear flows. The non-self-adjoint nature of the governing operator causes the emergent acoustic wave to interact strongly with the vorticity disturbance. Analytical expressions are obtained for the nondivergent vorticity perturbation. A discontinuity in the x component of the velocity field corresponding to the vorticity disturbance was originally identified to be the cause of acoustic wave emergence. However, a different mechanism is proposed in this paper. The correct "acoustic source" is identified and the reason for the abrupt nature of wave emergence is explained. The impact of viscous damping is also discussed. © 2009 The American Physical Society.

R. I. Sujith

Department of Aerospace Engineering

Joseph George

ACOUSTIC SOURCES

Analytical expressions

Different mechanisms

Linear dynamics

NON-NORMALITY

NON-SELF-ADJOINT

Nonresonant

Velocity field

Viscous damping

VORTICITY FLUCTUATIONS

Acoustic waves

Shear flow

Vorticity

acoustics

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IIT Madras

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

A combustion chamber has a hydrodynamic field that convects the incoming fuel and oxidizer into the chamber, thereby causing the mixture to react and produce heat energy. This heat energy can, in turn, modify the hydrodynamic and acoustic fields by acting as a source and thereby, establish a positive feedback loop. Subsequent growth in the amplitude of the acoustic field variables and their eventual saturation to a limit cycle is generally known as thermo-acoustic instability. Mathematical representation of these phenomena, by a set of equations, is the subject of this paper. In contrast to the ad hoc models, an explanation of the flame-acoustic-hydrodynamic coupling, based on fundamental laws of conservation of mass, momentum, and energy, is presented in this paper. In this paper, we use a convection reaction diffusion equation, which, in turn, is derived from the fundamental laws of conservation to explain the flame-acoustic coupling. The advantage of this approach is that the physical variables such as hydrodynamic velocity and heat release rate are coupled based on the conservation of energy and not based on an ad hoc model. Our approach shows that the acoustic-hydrodynamic interaction arises from the convection of acoustic velocity fluctuations by the hydrodynamic field and vice versa. This is a linear mechanism, mathematically represented as a convection operator. This mechanism resembles the non-normal mechanism studied in hydrodynamic theory. We propose that this mechanism could relate the instability mechanisms of hydrodynamic and thermo-acoustic systems. Furthermore, the acoustic-hydrodynamic interaction is shown to be responsible for the convection of entropy disturbances from the inlet of the chamber. The theory proposed in this paper also unifies the observations in the fields of low Mach number flows and zero Mach number flows. In contrast to the previous findings, where compressibility is shown to be causing different physics for zero and low Mach number flows, we show that the heat release rate can also introduce the distinction between the zero and low Mach number flows. Therefore, during the thermo-acoustic interaction, we should consider the coupling of convection modes that arise from the interaction of acoustic and hydrodynamic fields with the entropy modes.

Fulltext

Physics of Fluids

Acoustic-hydrodynamic-flame coupling-A new perspective for zero and low mach number flows

A linear mechanism for the emergence of acoustic waves from aperiodic vorticity disturbances is presented for a simple stagnation point flow. The non-modal approach, which made possible the discovery of a similar mechanism in unbounded shear flows, is employed. The phenomenon is shown to be closely related to the non-normal nature of the linearized operator governing the evolution of small perturbations about the mean flow. The vorticity disturbance acts as an acoustic source by interacting with the mean velocity gradient and drives the acoustic waves in a non-resonant manner. The physical mechanism by which the acoustic and vortical components of the disturbance exchange energy with the mean flow is presented. The proposed mechanism is relevant for numerous applications involving mass addition or removal, such as aeroacoustic instabilities in a solid rocket motor. Similar phenomena are expected to occur in accelerated flows, like the flow in a nozzle. Numerical simulations using a high order, dispersion relation preserving scheme is used to demonstrate how a localized vorticity disturbance of small amplitude generates sound waves when strained by the gradients in the base flow. The simulations reveal that the vorticity disturbance acts as a quadrupole source of sound, whose origin is explained using physical arguments. Linear simulations using a spectral code verify the results of the nonlinear simulations performed at small amplitude.

International Journal of Aeroacoustics

Sound production by vorticity fluctuations in a stagnation point flow

The question of which norm should be used to characterize the fluctuating disturbance energy in studies on thermoacoustic instabilities remains a topic of continued debate. In this paper we formulate a strategy for developing mathematically consistent norms that do not support spurious growth of disturbance energy in the absence of 'physical' sources of energy. A critical analysis of various disturbance energy norms existing in literature is conducted and important conclusions regarding positive definitiveness and susceptibility to exhibiting unphysical growth are drawn. It is shown that the energy norm proposed by Cantrell and Hart [Interaction between sound and flow in acoustic cavities: mass, momentum and energy considerations, Journal of Acoustical Society of America 36 (1964) 697706] and Myers first order disturbance energy norm [Transport of energy by disturbances in arbitrary steady flow, Journal of Fluid Mechanics 226 (1991) 383400] are positive definite if M0&lt;1 and M0&lt;1/γ, respectively, where M 0 is the magnitude of the local mean flow Mach number, γ is the ratio of specific heats at constant pressure and volume and the stipulated conditions should be met at every point in the flow domain. Our analysis shows that the disturbance energy norm proposed by Chu [On the energy transfer to small disturbances in fluid flow (part I), Acta Mechanica 1 (1965) 215234] does not exhibit unphysical growth. It is also shown that this property is not unique to Chus disturbance energy norm and there exists a family of norms which satisfy this requirement. The analysis also leads to an interesting interpretation of the acoustic energy conservation principle of Cantrell and Hart. The potential held by various disturbance energy norms for exhibiting fictitious growth is quantified using tools from nonmodal stability theory. It is concluded that if the mean flow Mach number is small, Myers norm is a suitable measure of the disturbance energy. © 2011 Elsevier Ltd. All rights reserved.

Journal of Sound and Vibration

Disturbance energy norms: A critical analysis

Chu [On the energy transfer to small disturbances in fluid flow (part I), Acta Mechanica 1 (1965) 215234] proposed a positive definite energy norm for characterizing the level of fluctuation in a disturbance. In the absence of heat transfer at the boundaries, work done by boundary or body forces, heat and material sources of energy, this norm is a monotone, non-increasing function of time. In this paper, we show that Chus disturbance energy defines an inner product, with respect to which the conservation equations of fluid motion linearized about a uniform base flow are self-adjoint. This ensures that the eigenvectors of the linearized operator are orthogonal to each other, and the property that the energy norm is a non-increasing function of time in the absence of physical sources of energy follows as an immediate consequence. Examples from numerical simulations of Euler equations are presented to highlight the importance of choosing an energy norm that is consistent with the underlying physics. We demonstrate that the disturbance energy as measured by Chus norm does not exhibit spurious transient growth in the absence of physical sources of energy and hence is suitable for analyzing thermoacoustic instability. © 2011 Elsevier Ltd.

On Chus disturbance energy

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.

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

On the existence of two-dimensional nonlinear steady states in plane Couette flow

Journal of Applied Mechanics

Transient Growth Before Coupled-Mode Flutter

Journal of the Physical Society of Japan

Remarks on Kelvin's Solutions of Shear-flow Systems – a Generalized Modal Approach –

Emergence of acoustic waves from vorticity fluctuations: Impact of non-normality

Journal	Physical Review E - Statistical, Nonlinear, and Soft Matter Physics
ISSN	15393755
Open Access	No