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

Department of Aerospace Engineering

Journal of Sound and Vibration

Exact solutions for the longitudinal vibration of non-uniform rods: Author's reply

An analytical framework is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into account the non-orthogonality of the eigenmodes of the unsteady coupled system. The coupled system comprises the dynamics of the acoustic field and the propellant burn rate. In general, thermoacoustic systems are non-normal leading to non-orthogonality of the eigenmodes. For such systems, the classical linear stability predicted from the eigenvalue analysis is valid in the asymptotic (large time) limit. However, the short-term dynamics can be completely different and a generalized stability theory is needed to predict the linear stability for all times. Non-normal systems show an initial transient growth for suitable initial perturbations even when the system is stable according to the classical linear stability theory. The terms contributing to the non-normality in the acoustic field and unsteady burn rate equations are identified. These terms, which were neglected in the earlier analyses, are incorporated in this analysis. Furthermore, the short-term dynamics are analysed using a system of differential equations that couples the acoustic field and the burn rate, rather than using ad hoc response functions which were used in earlier analyses. In this paper, a solid rocket motor with homogeneous propellant grain has been analysed. Modelling the evolution of the unsteady burn rate using a differential equation increases the degrees of freedom of the thermoacoustic system. Hence, a new generalized disturbance energy is defined which measures the growth and decay of the oscillations. This disturbance energy includes both acoustic energy and unsteady energy in the propellant and is used to quantify the transient growth in the system. Nonlinearities in the system are incorporated by including second-order acoustics and a physics-based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to pulsed instability. Pulsed instability is shown to occur with pressure coupling for burn rate response. Transient growth is shown to play an important role in pulsed instability. © 2010 Cambridge University Press.

Journal of Fluid Mechanics

Thermoacoustic instability in a solid rocket motor: Non-normality and nonlinear instabilities

An analytical frame work is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into account the non-orthogonality of the eigenmodes of the unsteady coupled system that comprises of the acoustic field and the propellant burn rate dynamics. In general, thermoacoustic systems are non-normal leading to non-orthogonality of the eigenmodes. For such systems, classical linear stability predicted from the eigenvalue analysis is valid only in the asymptotic (large time) limit. However, the short term dynamics can be completely different and a generalized stability theory is needed to predict the linear stability for all times. Non-normal systems show initial transient growth for suitable initial perturbations even when the system is stable according to classical linear stability theory. The terms contributing to the non-normality in the acoustic field and unsteady burn rate equations are identified. These terms, which were neglected in the earlier analyses, are incorporated in the present analysis. Furthermore, the short term dynamics is analysed using a system of differential equations that couples the acoustic field and the burn rate, rather than using ad hoc response functions which were used in earlier analyses. In the present case, an SRM with homogeneous propellant grain is analysed. Nonlinearities in the system are incorporated by including second order acoustics and a physics based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to 'pulsed instability'. Pulsed instability is shown to occur with pressure coupling for burn rate response. Transient growth is shown to play an important role in pulsed instability.

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

Thermoacoustic instability in solid rocket motor: Nonnormality and nonlinear instabilities

We revisit a simple reduced order model (Matveev, K., and Culick, F. E. C, 2003, A Model for Combustion Instability Involving Vortex Shedding, Combust Sci. and Tech., 175, 1059) and re-examine its implications in light of the non-normal and nonlinear nature of combustion acoustic interactions. To this end, one-dimensional linear acoustic equations are used to model the acoustic field in an open-open duct. The Galerkin technique is then implemented to expand the acoustic pressure and velocity fluctuations in terms of the natural acoustic modes. The coupled thermoacoustic system is shown to be non-normal and nonlinear. This leads to complicated but interesting physics that were not examined in the earlier study. Examples showing transient growth leading to instability and bootstrapping in an initially decaying system are then presented. It is also shown that a vortex-based combustor reaches different limit cycle amplitudes for the same system parameters when subjected to different initial conditions. Further, the effect of damping on the non-normal behavior of the system is studied. A test case is presented that shows that for a lowly damped system, a slight increase in damping leads to high amplitude unstable oscillations. Finally, pseudospectral analysis is presented to study the non-normal behavior of such systems.

Combustion Science and Technology

Revisiting a model for combustion instability involving vortex shedding

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?

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.

GENERAL

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

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

This paper presents the effect of axial acoustic fields on swirl flows. Investigations were performed for pressure drop of 0.03 bar across the swirler, at the open end of a duct fitted with acoustic drivers at the closed end. The swirl flow field was characterized using particle image velocimetry (PIV). The swirl flow fields are altered with respect to the acoustic velocity oscillations in an acoustic cycle. The length of the CTRZ is shortened with increase in counter flowing acoustic velocity. The jets from the swirler vanes contract in co-flowing acoustic velocity and expand in counter-flowing acoustic velocity. The vorticity field of the swirl flow fluctuates in the presence of acoustic velocity oscillations. Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc.

47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition

An experimental investigation of swirl flow in axial acoustic fields

A systematic approach to study the nonmodal stability analysis of the thermoacoustic systems is proposed. The heat source is obtained using unsteady computational fluid dynamics in combination with correlation based linear system identification. The time domain simulation of the full system is obtained from a Galerkin expansion technique and the heat source is modeled as a compact element in the system. The eigenvalues of the resulting system are obtained from the discretization of the solution operator. The maximum growth factor is estimated from the pseudo spectra using Kreiss' theorem. The approach is illustrated in a simple Rijke tube configuration. For modeling the heat source, the King's law which is widely used in hot wire anemometry and the heat source model obtained from CFD and linear identification are used. The sensitivity of the eigenvalues near the zero of the real axis in a pseudospectra diagram for different perturbations for the same set of parameters show different behavior for different heat source models and hence different non-normal behaviors. The maximum growth factor for the system with King's law heat source model is more sensitive to the change in the heat source location than that of the CFD model heat source. © 2009 American Institute of Physics.

AIP Conference Proceedings

Identification of heat transfer dynamics for nonmodal stability analysis of thermoacoustic systems

The present study investigates the effect of acoustic oscillations on swirl flow characteristics, using 2C and stereo PIV. The experiments were performed in a sudden expansion square geometry. A high amplitude acoustic field was established in the duct using external acoustic drivers (woofers) that were mounted on the test section downstream of the swirler. A constant acoustic pressure of 600 Pa was maintained at the exit of the swirler. The experiments were performed for different Strouhal numbers. It was observed that the fluctuations in axial velocity in the recirculation zone are much higher than that of the acoustic velocity. However, no significant change in the tangential velocity was observed. It was seen that the maximum fluctuation amplitude increases with Strouhal number in both low and high acoustic velocity fluctuations regimes. It was found that the use of vane passage size as the characteristic dimension leads to Strouhal numbers in the range of 0.2 to 0.5, where the axial velocity fluctuations in the recirculation zone are high. Copyright © 2009 by E. Arunraj, R. I. Sujith and Chakravarthy.

39th AIAA Fluid Dynamics Conference

Investigation of the effect of acoustic oscillations on swirl flow using PIV

This paper describes an experimental investigation of the effect of transverse liquid injection (distilled water) on the acoustic field generated by confined supersonic flow past a cavity. The effect of cavity depth in the absence/ presence of liquid injection on the acoustic field is studied. In the presence of liquid injection, the effects of injection pressure and injection location on the acoustic field are investigated. The dependence of suppression/amplification of the acoustic amplitudes (compared with the absence of injection) on the injection condition and cavity depths is ascertained. In general, liquid injection in the presence of cavity resulted in higher amplitudes. Even square cavities that did not exhibit oscillations in the absence of injection generated significant acoustic amplitudes in the presence of liquid injection at high injection pressures. Phase-locked schlieren imaging was performed to sequence the fluiddynamic events. Schlieren images indicate the interaction of the shock wave generated by the flow past the liquid jet and the oblique shock generated at the cavity-leading edge. The schlieren images show that the amplitude increase observed for higher injection pressures for a range of cavity depths is accompanied by an unsteady normal shock ahead of the cavity-leading edge (which does not span the entire width of the test model, but is located locally in front of the liquid jet). Copyright © 2008 by T. K. Ganesh Anavaradham, R. I. Sujith, O. J. Shreenivasan, S. R. Chakravarthy, and S. Panneerselvam. Published by the American Institute of Aeronautics and Astronautics, Inc.

Journal of Propulsion and Power

Effect of liquid injection on acoustic field from supersonic flow past cavities

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.

Fulltext

Physics of Fluids

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

We revisit a simple reduced order model (Matveev, K. and Culick, F. E. C, (2003) "A Model for Combustion Instability Involving Vortex Shedding", Combust Sci. and Tech., 175, 1059-1083) and re-examine its implications in the light of the non-normal and nonlinear nature of combustion acoustic interactions. One-dimensional linear acoustics is used to model the acoustic field in an open-open duct. The Galerkin technique is used to expand the acoustic pressure and velocity fluctuations in terms of the natural acoustic modes. The coupled thermoacoustic system is shown to be non-normal and nonlinear. This leads to complicated, but interesting physics which were not examined in the earlier study. Examples showing triggering and bootstrapping in an initially decaying system are then presented. It is also shown that a vortex based combustor reaches different limit cycle amplitudes for the same system parameters, when subjected to different initial conditions. Further, the effect of damping on the non-normal behavior of the system is studied. Pseudospectra analysis reveals that for a lightly damped system, an increase in damping might lead to higher nonnormal interaction of modes.

44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit

Revisiting a Model for Combustion Instability Involving Vortex Shedding

This article presents an investigation of the nonlinear response of a diffusion flame to unsteady velocity disturbances. The infinite rate chemistry flame model is employed to study unsteady two-dimensional co-flow non-premixed combustion. In this model, the flame geometry is given by the stoichiometric level surface, which is obtained by the equation for the Schvab-Zeldovich variable. In this article, this equation is solved using an integral equation approach. An approximate analytical expression is obtained for the flame length variation as a function of time. The heat release rate is obtained from thermodynamic calculations. The variation of the heat release rate with the velocity fluctuations is analyzed. It is seen that the flame response to a single frequency of excitation contains several frequencies even though the governing equation is linear. It is shown that the heat release rate has exponential dependence on the amplitude of velocity fluctuations. The nonlinear nature of flame response becomes significant at lower Fourier numbers (for mass transport). A nonlinear model for the transfer function to describe the response of the heat release rate to velocity perturbation has been constructed. Copyright © Taylor & Francis Group, LLC.

Nonlinear response of diffusion flames to uniform velocity disturbances

Detection of vortices from PIV data has been a challenging task driving efforts in experimentation with various algorithms and physical criteria. In the present study, three algorithms are employed and are arranged in a predictor-corrector fashion to reduce computation time while maintaining accuracy and robustness. Two predictor methods, geometric gridding and streamlines method have been employed to ensure that there is redundancy in finding the regions of interest at the predictor stage. A redundancy in the predictor method also ensures generality and robustness in detection. The corrector method is a pattern matching algorithm using the Hamel-Oseen model for velocity which acts on the reduced domain obtained through the predictor algorithms. The performance of the proposed scheme is demonstrated by applying them to detect vortices from two component PIV data obtained from swirl flow measurements. Copyright © 2008 by Arun Manohar, A. V. Varun and R. I. Sujith.

46th AIAA Aerospace Sciences Meeting and Exhibit

An automated predictor-corrector method for vortex detection

Design of a linear controller for a thermoacoustic system taking the consequences of nonnormality into account is presented in this paper. The analysis is performed in the framework of the classical n-τ model of Crocco. The controllers based on classical stability analysis focuses on the stability of the individual eigenmodes of the system. However, the nonnormal nature of a thermoacoustic system which implies non-orthogonality of the eigenvectors causes redistribution of the acoustic energy between the eigenmodes even in the linear regime. Transient growth in a linearized system, even when the individual eigenvectors are decaying is an important characteristic of a nonnormal system. This short term growth amplifies the small disturbances present in the system. Traditional linear controllers based on classical linear stability analysis do not take transient growth into account. High enough amplitudes of the transient growth can cause the system to enter into the region where nonlinear effects are significant and linear controller designed would fail in this case. It is shown that, controlling the dominant mode in a nonnormal thermoacoustic system alone can cause the instability to occur at another frequency which was initially unexcited. This manifests as secondary peaks in the FFT of the evolution of the acoustic pressure and velocity. Hence, a controller based on the pole placement technique which is applicable to the transients as well as the asymptotic stability of the system designed, and its effectiveness is demonstrated with an example of a horizontal Rijke tube model. Copyright © 2008 by Kulkarni R.R. , K. Balasubramanian and R.I. Sujith.

The role of nonnormality in active control of combustion instability

A combustor with a divergent inlet passage and a coaxially movable bluff-body flame holder has been developed to enable self-excitation of high-amplitude acoustic oscillations, for the purpose of enhancing the rate of evaporation of water sprays. Commercially available liquid petroleum gas is used as fuel. The acoustic pressure levels and modal contents are controlled by varying the location of the bluff body along the axis of the combustor relative to its inlet. Locating the bluff body closer to the inlet yields higher acoustic pressure amplitudes over a wider range of fuel-air ratios. The excitation of high amplitudes is accompanied by a shift from the fundamental to the first harmonic at different fuel-air ratios for different air mass flow rates in the fuel-lean range. When high sound pressure levels are excited in the combustor, the heat transfer due to water cooling of the combustor walls and across the walls of the tailpipe increase considerably, up to 40% for an increase in sound pressure levels from 145 to 160 dB. Water-spray evaporation experiments are performed with low and high sound pressure levels of acoustic excitation for different water injection rates over a range of mass flow rates of fuel and air. For a given mass flow rate of air and fuel, it is found that the water evaporation rate increases significantly above a certain threshold acoustic pressure, up to 107 % for a sound pressure level of 156 dB, despite the considerable increase in heat loss due to the acoustic oscillations.

Characterization of an acoustically self-excited combustor for spray evaporation

Most gas turbine engines in service use a prefilming airblast atomizer in combination with strong swirling flow. An experimental investigation of the characteristics of a hollow cone prefilming airblast atomizer in a strong swirling air flow Is presented in this paper. The measured quantities include the spray cone angle by planar imaging, patternation using planar laser-Induced fluorescence, planar droplet velocity field using two-component particle image velocimetry, and droplet size and velocity distributions using a phase Doppler particle-size analyzer and laser Doppler velocimetry. Patternation of the spray field showed regions of high droplet concentration and volume flux fluctuations in various cross-sectional planes. The investigation revealed the presence of droplets of narrow size range up to 25 μ around the axis of the atomizer in the central toroidal recirculation zone of the swirler. The presence of vortical structures entraining the droplets is observed from the instantaneous velocity vector fields. The velocity data shows entralnment of the droplets in the central toroidal recirculation zone. Examining the size and the velocity data reveals that the centrifugal action of the swirling air flowfleld from the atomizer carries the bigger droplets outward, toward the periphery of the spray.

Characterization of a prefilming airblast atomizer in a strong swirl flowfield

This paper presents the effect of axial acoustic fields on non-evaporative sprays. Investigations were performed using distilled water as the spray liquid, at the open end of a duct fitted with acoustic drivers. The spray field was characterized using particle image velocimetry (PIV), phase Doppler particle size analyzer (PDPA) and three component laser Doppler velocimetry (3C-LDV) measurements. The spray velocity fields are altered with respect to the acoustic velocity oscillations in an acoustic cycle. Length of the zone having high rms velocity magnitudes is shortened with increase in counter flowing acoustic velocity. Variation in phase averaged droplet diameter and their distribution in presence of acoustic velocity oscillations in an acoustic cycle is noticed. The variation of droplet velocity, diameter and their distribution in presence of acoustic velocity oscillations increases with increase in acoustic pressure amplitude. A maximum variation of 25% in phase averaged droplet diameter is observed in an acoustic cycle. The atomization and diameter distribution of the spray is altered in an acoustic cycle. Smaller diameter droplets are entrained by the acoustic velocity oscillations. Copyright © 2008 by R. Kumara Gurubaran and R. I. Sujith.

An experimental investigation of non-evaporative sprays in axial acoustic fields

This paper presents the effect of axial acoustic fields on evaporative sprays. Investigations were performed using ethanol as the spray liquid, inside a duct fitted with acoustic drivers. The spray field was characterized using phase Doppler particle size analyzer (PDPA) and three component laser Doppler velocimetry (3C-LDV) measurements. The spray velocity fields are altered with respect to the acoustic velocity oscillations in an acoustic cycle. Variation in phase averaged droplet diameter and their distribution in presence of acoustic velocity oscillations in an acoustic cycle is noticed. The variation of droplet velocity, diameter and their distribution in presence of acoustic velocity oscillations increases with increase in amplitude of the acoustic field. Maximum variation of 30% in phase averaged droplet diameter is observed in an acoustic cycle. The harmonics in the phase averaged droplet diameter are observed when there is reversal of droplet velocity. Smaller diameter droplets are entrained by the acoustic velocity oscillations.