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

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 used to describe such systems, a high dimensional model which takes into account internal degrees of freedom of the flame is employed. A Linear Quadratic (LQ) Regulator is formulated for this model. The LQ Regulator is able to control the energy fluctuations in the model, including the transient energy growth. Favourable actuator and sensor locations for the LQ Regulator are sought using Linear Matrix Inequalities (LMI) optimisation techniques. According to the LMI analysis, locations closer to the flame are found to result in lower bounds on the transient energy of the model. Sensor location closer to the flame also result in better estimation. When the plant noise is concentrated in the acoustic variables, the half duct length is an unfavourable location for measurement. Measures of controllability/ observability based on Hankel singular values (HSVs) are also used to determine suitable actuator-sensor location pairs. The Linear Quadratic Gaussian (LQG) framework is then formulated for this model using the favoured locations. Classical balanced truncation is used to obtain reduced order models of this full order model to achieve faster control. The controller based on the reduced order model is able to control instabilities in the nonlinear model of the system.

Department Of Chemical Engineering

Department of Aerospace Engineering

18th International Congress on Sound and Vibration 2011, ICSV 2011

Control of thermoacoustic instabilties in a ducted premixed flame: Actuator-sensor placement and model reduction

Combustors with fuel-spray atomizers are particularly susceptible to pressure or velocity oscillations resulting from the occurrence of thermoacoustic oscillations. Experimental investigations of distilled water spray - swirl - acoustic interactions are presented. The presence of a swirl flow field alters the behavior of water spray drastically. Investigations showed complex behavior of water sprays in an acoustic cycle. In the presence of acoustic oscillations the phase averaged droplet diameter variation with in an acoustic cycle is more at the axial locations compared with off axis locations. The phase averaged axial velocity variation of the droplets is high compared to that of phase averaged radial and tangential velocities with in an acoustic cycle. Periodic clustering of droplets in the spray field is observed in the presence of acoustic field. A maximum particle count of 14 times the minimum count in a phase angle bin is observed within an acoustic cycle for the acoustic pressure amplitude of 3000 Pa. The droplet diameter distribution in the spray field is affected in the presence of acoustic oscillations.

Fulltext

International Journal of Spray and Combustion Dynamics

Dynamics of spray - Swirl - acoustics interactions

In this paper, the effect of the non-normal nature of premixed flame-acoustic interaction and the effect of internal flame dynamics on thermoacoustic instability are investigated. The thermoacoustic system is modeled by considering the acoustic momentum and energy equations together with the equation for the evolution of the flame front obtained from the kinematic G-equation. The linearised operator for this thermoacoustic system is non-normal leading to non-orthogonality of its eigenvectors. Non-orthogonal eigenvectors can cause transient growth in evolutions even when all the eigenvectors are decaying for a linearly stable case. Therefore, classical linear stability theory cannot predict the finite time transient growth observed in non-normal systems. A parametric study of the variation in transient growth due to change in parameters such as flame location and flame angle is performed. In addition to projections along the acoustic variables of velocity and pressure, the optimal initial condition for the self evolving system has significant projections along the variables for heat release rate. Nonlinear simulations show subcritical transition to instability and dominant mode shift for the present model. It is shown that low order models such as a time lag model or a lumped model for the heat release rate dynamics as a function of acoustic velocity cannot adequately capture the non-normal effects. The notion of phase between acoustic pressure and heat release rate as an indicator of stability is examined. Copyright © 2010 by Subramanian P. and Sujith R. I.

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

Non-normality and internal flame dynamics in premixed flame - Acoustic interaction

Analysis of thermoacoustic instability is performed in a horizontal Rijke tube with an electrical resistance heater as the heat source. The analysis starts with the governing equations for the fluid flow. The governing equations become stiff and are difficult to solve by CFD, as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore asymptotic analysis is performed in the limit of small Mach number and compact heat source to eliminate the above stiffness problem. The unknown variables are expanded in powers of Mach number. Two systems of governing equations are obtained: one for the acoustic field and the other for the unsteady flow field in the hydrodynamic zone around the heater. The coupling between the acoustic field and the unsteady heat release rate from the heater appears naturally from asymptotic analysis. Further, a non-trivial additional term appears in the momentum equation of the hydrodynamic zone, which has serious consequences on the stability of the system. This additional term is the pseudo-acceleration term, which appears due to the change in the frame of reference from an inertial frame to an accelerating frame of reference, performed in the hydrodynamic zone during the asymptotic analysis. The presence of pseudo-acceleration term is generic, when there are two length scales involved in the problem. Subcritical transition to instability is simulated in the present paper and is related to the hysteresis in the asymptotic state of the system with the heater power, observed in Rijke tube experiments. The global asymptotic stability of the system with the system parameters is investigated using bifurcation diagram. Present numerical simulations confirm the importance of the pseudo-acceleration term. Bifurcation diagrams obtained from the simulations with and without pseudo-acceleration term are completely different. The transition to instability occurs via different types of bifurcations in the above two cases. Simulations performed with the response function for the unsteady heat release rate are compared with the simulation with pseudo-acceleration term. Numerical simulations are performed using the Galerkin technique for the acoustic zone and CFD for the hydrodynamic zone. Acoustic streaming is shown to happen during limit cycle and its effect on the unsteady heat release rate is discussed. Copyright © 2010 by Mariappan S. and Sujith R. I.

Modeling nonlinear thermoacoustic instability in an electrically heated Rijke tube

A theoretical framework has been developed to understand the non-normal nature of the thermoacoustic interaction in an electrically heated horizontal Rijke tube. The ther-moacoustic system considered here includes the dynamics of the chamber acoustic field and the heat source. The eigenmodes of the system are non-orthogonal due to the nonnormal nature of the linearised evolution operator, which leads to the transient growth in the perturbations even for a linearly stable system. The transient growth is measured in a norm, which in general physically represents the energy in the disturbance. The present framework allows one to obtain a norm systematically from the definition of disturbance energy. In the present paper, adjoint optimisation technique is used to obtain the optimum initial condition (suitable for system with large degrees of freedom ∼ 104) for maximum transient growth of the norm (disturbance energy). The optimum initial condition thus obtained has significant contributions from the variables governing the dynamics of the heat source. Some interesting flow structures are observed in the optimum initial condition near the heat source. The non-normal nature of the system is shown to be reflected in the reduction in the range of linearisation (threshold energy for the system to be nonlinearly unstable) of the system. © 2010 by S. Mariappan, P. J. Schmid &amp; R. I. Sujith.

16th AIAA/CEAS Aeroacoustics Conference (31st AIAA Aeroacoustics Conference)

Role of transient growth in subcritical transition to thermoacoustic instability in a horizontal Rijke tube

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.

World University root node

Thermoacoustic Instability in Solid Rocket Motor: Non-Normality and Nonlinear Instabilities

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

Journal of Sound and Vibration

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

Modeling thermoacoustic instabilities using inputs from experimental data usually employs a two-part approach: the response of the flame to perturbations in the flow is first obtained in terms of a flame transfer function, which then used as a source term in the acoustic equation. Within the framework of the two-part approach, an approximate modal analysis technique, known as describing function technique is used to model the nonlinearities in the flame dynamics. In this paper, we examine the consequences of employing a two-part approach in predicting the nonlinear behavior of a thermoacoustic system. Further, we examine the applicability of describing function technique for predicting the bifurcations of thermoacoustic systems. A toy model of thermoacoustic system was designed in order to analyze the effects of the two-part approach. It is found that the two-part approach modifies the dynamics of the system and cannot capture the intricate coupling between the combustion processes and acoustic field. A ducted diffusion flame model and a ducted premixed flame model are then studied as representative problems of thermoacoustic instability in combustors. The bifurcation plots obtained from the predictions using describing function technique were compared with the bifurcation plots obtained from the simulations in time domain for these models. The results show that the system behavior predicted by the describing function technique is quantitatively as well as qualitatively different from those obtained from time evolutions. © 2010 by Priya Subramanian, Vikrant Gupta, Bharat Tulsyan and R. I. Sujith.

Can describing function technique predict bifurcations in thermoacoustic systems?

Analyzing and quantifying parametric uncertainties numerically in thermoacoustic systems is a tedious task, even more so when the system exhibits triggering (pulsed instabilities). Here a novel interpolation based approach is presented and applied to a simplified model of a Rijke tube. It is seen that this integrated interpolation scheme is significantly faster than the traditional Monte Carlo simulations. The advantages of using this scheme and the reason for its success compared to other uncertainty quantification schemes are highlighted. The paper also discusses advantages of using an equi-probable node distribution which is seen to improve the accuracy of the scheme. The stochastic bifurcation diagrams and the probabilities of failure (POF) are obtained for various operating conditions and stability regions are plotted. © 2010 by Vineeth Nair, Sunetra Sarkar and R. I. Sujith.

Uncertainty quantification of subcritical bifurcation in a Rijke Tube

Thermoacoustic instability in solid rocket motor: Nonnormality and nonlinear instabilities

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?

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

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.

GENERAL

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

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.

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

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

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 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

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.