A Green’s function approach to model the pseudo-active control of thermoacoustic instabilities

In this paper, we take advantage of using a Green's function approach as an analytical tool to study linear and nonlinear aspects of thermoacoustic systems.Green's function is defined as the impulse response of a system; it has a clear physical meaning in combustion systems, and it provides a fast and flexible tool to predict thermoacoustic instabilities in both time and frequency domains.In this study, we consider a tube, which houses two sources: a heat source and an external noise source.The heat source is modelled by an amplitude-dependent n-law, the noise source is assumed to emit two types of noise (pink noise or white noise).We use the Green's function approach to derive an integral equation for the acoustic field in the Rijke tube, and we also derive an algebraic equation for the thermoacoustic eigenfrequency.Both equations are validated.The results that we found, show that the presence of noise (whether pink or white noise) results in "triggering" an instability and in accelerating the growth of the amplitude.These effects become more pronounced as the level of noise increases.The influence of pink noise is stronger than that of white noise.We also studied how the hysteresis behavior (a nonlinear effect in dynamical systems) is affected by the noise, using the heater power as bifurcation parameter.Our study reveals that the width of the bistable region decreases as the strength of noise increases.

Large eddy simulations (LES) of premixed hydrogen-enriched swirling flames were performed to investigate the flame topology and combustion instabilities with different hydrogen concentrations. A compressible LES approach is utilised to account for the self-excited combustion dynamics. A transported probability density function (pd f) approach is adopted to account for sub-grid scale (sgs) turbulence-chemistry interaction, and the solution to the joint sgs – pd f evolution equation of the scalars is obtained by the stochastic field method. The chemistry is represented using a reduced chemical reaction mechanism containing 15 reaction steps and 19 species. The results revealed that as the concentration of hydrogen increases, the flame is shortened in the injecting direction and more confined in the cross-sectional direction, which is consistent with experimental observations. The self-excited limit-cycle oscillations for all considered cases were successfully reproduced, with the predicted peak frequencies of the chamber pressure spectra in excellent agreement with the measured values. The feedback loop of the oscillations is successfully captured and analysed with the temporal evolution of axial velocity and heat release presented.

The prediction of thermoacoustic instabilities is fundamental for combustion systems such as domestic burners and industrial gas turbine engines. High-amplitude pressure oscillations cause thermal and mechanical stress to the equipment, leading to premature wear or even critical damage. In this paper we present a new approach to produce nonlinear (i.e. amplitude-dependent) stability maps of a combustion system as a function of various parameters. Our approach is based on the tailored Green’s function of the combustion system, which we calculate analytically. To this end, we assume that the combustor is one-dimensional, and we describe its boundary conditions through reflection coefficients. The heat release is modelled by a generalised $n{\it\tau}$ law. This includes a direct-feedback term in addition to the usual time-lag term; moreover, its parameters (time lag, coupling coefficients) depend on the oscillation amplitude. The model provides new insight into the physical mechanism of the feedback between heat release rate and acoustic perturbations. It predicts the key nonlinear features of the thermoacoustic feedback, such as limit cycles, bistability and hysteresis. It also explains the frequency shift in the acoustic modes.