Koushik Balasubramanian

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

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.

Non-normality and nonlinearity in combustion–acoustic interaction in diffusion flames

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.

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

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.

Nonlinear Response of Diffusion Flames to Uniform Velocity Disturbances

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.

Revisiting a Model for Combustion Instability involving Vortex Shedding

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.

Relativistic shock formation in the presence of radial entropy gradients

The nonlinear steepening of relativistic acoustic waves is investigated. The nonlinear evolution of a planar wave in a homentropic flow field is understood well through relativistic simple waves. However, in situations where the wave is nonplanar and the flow field is nonhomentropic, the concept of simple waves cannot be used. In the present paper, effect of entropy gradients on the nonlinear distortion of a spherical wave is analyzed using the wave front expansion technique. It is shown that the behavior of a relativistic wave in nonhomentropic environment is slightly different from the nonrelativistic wave. A closed form solution is obtained for the slope at the wave front. A general criterion for a compression wave to steepen into a shock is obtained. The distortion of compression and rarefaction waves is examined for some well known equations of state. However, the method used in this paper is general and can be easily extended to analyze shock formation in any fluid. Also, expressions for time and lo...