Sathesh Mariappan

Bifurcation analysis of thermoacoustic instability in a horizontal Rijke tube

A bifurcation analysis of the dynamical behavior of a horizontal Rijke tube model is performed in this paper. The method of numerical continuation is used to obtain the bifurcation plots, including the amplitude of the unstable limit cycles. Bifurcation plots for the variation of nondimensional heater power, damping coefficient and the heater location are obtained for different values of time lag in the system. Subcritical bifurcation was observed for variation of parameters and regions of global stability, global instability and bistability are characterized. Linear and nonlinear stability boundaries are obtained for the simultaneous variation of two parameters of the system. The validity of the small time lag assumption in the calculation of linear stability boundary has been shown to fail at typical values of time lag of system. Accurate calculation of the linear stability boundary in systems with explicit time delay models, must therefore, not assume a small time lag assumption. Interesting dynamical behavior such as co-existing multiple attractors, quasiperiodic behavior and period doubling route to chaos have been observed in the analysis of the model. Comparison of the linear stability boundaries and bifurcation behavior from this reduced order model are shown to display trends similar to experimental data.

Analysis of dynamic stall using dynamic mode decomposition technique

Dynamic mode decomposition is applied to investigate the unsteady flowfield around a pitching airfoil. The extracted flow structures are termed as dynamic mode decomposition modes. Analyses are performed for both attached flow and dynamic stall cases. Initially, flowfield data generated from numerical simulations are investigated. The effect of exclusion of the flowfield near the surface of the airfoil on the structure of the dynamic mode decomposition modes is examined. This is of vital importance for the experimental measurements, as the flowfield near the airfoil’s surface is difficult to measure using particle image velocimetry. In the latter part of the paper, dynamic mode decomposition modes extracted from the experimental data are analyzed. The structure of these modes are compared with the modes obtained from the proper orthogonal decomposition technique. Finally, effects of phase averaging on the modes are discussed.

Thermoacoustic instability in a 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.

Non-normality of thermoacoustic interactions: an experimental investigation

An experimental investigation of the non-normal nature of thermoacoustic interactions in an electrically heated horizontal Rijke tube is performed. Since non-normality and the associated transient growth are linear phenomena, the experiments have to be confined to the linear regime. The bifurcation diagram for the subcritical Hopf bifurcation into a limit cycle behavior has been determined, after which the amplitude levels, for which the system acts linearly, have been identified for dierent power inputs to the heater. There are two main objectives for this experimental investigation. The first one deals with the extraction of the linear eigenmodes associated with the acoustic pressure from experimental data. This is accomplished by the Dynamic Mode Decomposition (DMD) technique applied in the linear regime. The non-orthogonality between the eigenmodes is determined for various values of heater power. The second objective is to identify evidence of transient perturbation growth in the system. The total acoustic energy in the duct has been monitored as the thermoacoustic system has been initialized by linear combinations of the two dominant eigenmodes. Transient growth, on the order of previous theoretical studies, has been found, and its parameter dependence on amplitude ratio and phase angle of the initial eigenmode components has been determined. This study represents the first experimental confirmation of non-normality in thermoacoustic systems.

Thermoacoustic instability in solid rocket motor: n on- normality and nonlinear instabilities

An analytical frame work is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into acc ount 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, thermoa coustic 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 a symptotic (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 sy stem is stable according to classical linear stability theory. The terms contributing to the no n-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 a nalysis. 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 respons e 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 in cluding second order acoustics and a physics based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to ‘pulsed instability’. P ulsed instability is shown to occur with pressure coupling for burn rate response. Transien t growth is shown to play an important role in pulsed instability.

Experimental investigation of non-normality of thermoacoustic interaction in an electrically heated Rijke tube

An experimental investigation of the non-normal nature of thermoacoustic interactions in an electrically heated horizontal Rijke tube is performed. Since non-normality and the associated transient growth are linear phenomena, the experiments have to be confined to the linear regime. The bifurcation diagram for the subcritical Hopf bifurcation into a limit cycle behavior has been determined, after which the amplitude levels, for which the system acts linearly, have been identified for different power inputs to the heater. There are two main objectives for this experimental investigation. The first one deals with the extraction of the linear eigenmodes associated with the acoustic pressure from experimental data. This is accomplished by the Dynamic Mode Decomposition (DMD) technique applied in the linear regime. The non-orthogonality between the eigenmodes is determined for various values of heater power. The second objective is to identify evidence of transient perturbation growth in the system. The total acoustic energy in the duct has been monitored as the thermoacoustic system evolves from its initial condition. Transient growth, on the order of previous theoretical studies, has been found, and its parameteric dependence on amplitude ratio and phase angle of the initial eigenmode components has been determined. This study represents the first experimental confirmation of non-normality in thermoacoustic systems.

Role of transient growth in subcritical transition to thermoacoustic instability in a horizontal 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 thermoacoustic 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 » 10 4 ) 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.

Onset of flame-intrinsic thermoacoustic instabilities in partially premixed turbulent combustors:

We investigate the onset of thermoacoustic instabilities in a turbulent combustor terminated with an area contraction. Flow speed is varied in a swirl-stabilized, partially premixed combustor and the system is observed to undergo a dynamical transition from combustion noise to instability via intermittency. We find that the frequency of thermoacoustic oscillations does not lock-on to any of the acoustic modes. Instead, we observe that the dominant mode in the dynamics of combustion noise, intermittency and thermoacoustic instability is a function of the flow speed. We also find that the observed mode is insensitive to the changes in acoustic field of the combustor, but it varies as a function of upstream flow time scale. This new kind of thermoacoustic instability was independently discovered in the recent theoretical analysis of premixed flames. They are known as intrinsic thermoacoustic modes. In this paper, we report the experimental observation and the route to flame intrinsic thermoacoustic instabilities in partially premixed flame combustors. A simplified low-order network model analysis is performed to examine the driving mechanism. Frequencies predicted by the network model analysis match well with the experimentally observed dominant frequencies. Intrinsic flame-acoustic coupling between the unsteady heat release rate and equivalence ratio fluctuations occurring at the location of fuel injection is found to play a key role. Further, we observe intrinsic thermoacoustic modes to occur only when the acoustic reflection co-efficients at the exit are low. This result indicates that thermoacoustic systems with increased acoustic losses at the boundaries have to consider the possibility of flame intrinsic thermoacoustic oscillations.

Theoretical Formulation for the Investigation of Acoustic and Entropy-Driven Combustion Instabilities in Gas Turbine Engines

In gas turbine combustors, the unsteady flame is a source of acoustic and entropy waves, leading to combustion noise. When these fluctuations couple with the flame and establish a positive feedback mechanism, they grow in amplitude resulting in combustion instability. Combustion instability can be driven either by the acoustic waves or acceleration of entropy waves. Entropy-driven instability is one of the dominant cause of low frequency combustion instability in industrial gas turbines, where the flow exiting the combustor is accelerated by the turbine nozzle guide vanes. This chapter presents the theoretical framework to model the generation, convection, acceleration and reflection of acoustic and entropy waves in gas turbine combustors with variable area geometry. We also discuss in detail the procedure to solve the equations and present a comparison between acoustic-driven instability and entropy-driven instability.

Influence of non perfect impedance boundary on the bistable region in thermoacoustic interactions

We investigate the influence of non perfect impedance boundary on the bistable zone in thermoacoustic interactions of a horizontal Rijke tube. A wave based approach is used to obtain the nonlinear dispersion relation with frequency dependent impedance boundary condition. The location and the time delay in the response of the heater are considered as bifurcation parameters to obtain the stability boundaries. In the presence of non perfect impedance boundary condition, we find that the extent of globally unstable regime reduces and the bistable zone significantly increases. The quantitative changes in the stability boundaries and the bistable zone are investigated for different time lags. However, the nature of bifurcation remains sub critical and unaltered for the range of time delays considered in the present study.