The study is concerned with theoretical examination of thermo-acoustic instabilities in combustors and focuses on recently discovered 'flame intrinsic modes'. These modes differ qualitatively from the acoustic modes in a combustor. Although these flame intrinsic modes were intensely studied, primarily numerically and experimentally, the instability properties and dependence on the characteristics of the combustor remain poorly understood. Here we investigate analytically the properties of intrinsic modes within the framework of a linearized model of a quarter wave resonator with temperature and cross-section jump across the flame, and a linear n model of heat release. The analysis of dispersion relation for the eigen-modes of the resonator shows that there are always infinite numbers of intrinsic modes present. In the limit of small interaction index n the frequencies of these modes depend neither on the properties of the resonator, nor on the position of the flame. For small n these modes are strongly damped. The intrinsic modes can become unstable only if n exceeds a certain threshold. Remarkably, on the neutral curve the intrinsic modes become completely decoupled from the environment. Their exact dispersion relation links the intrinsic mode eigen-frequency i with the mode number i m and the time lag : 21 ii mm , where m =0, +/-1. The main results of the study follow from the mode decoupling on the neutral curve and include explicit analytic expressions for the exact neutral curve on the n plane, and the growth/decay rate dependence on the parameters of the combustor in the vicinity the neutral curve. The instability domain in the parameter space was found to have a very complicated shape, with many small islands of instability, which makes it difficult, if not impossible, to map it thoroughly numerically. The analytical results have been verified by numerical examination.
This paper is concerned with the theoretical study of thermo-acoustic instabilities in combustors and focuses upon recently discovered flame intrinsic modes. Here, a complete analytical description of the salient properties of intrinsic modes is provided for a linearized one-dimensional model of open-open combustors with temperature and crosssection jump across the flame taken into account. The standard n À model of heat release is adopted, where n is the interaction index and is the time lag. We build upon the recent key finding that for a closed-lopen combustor, on the neutral curve, the intrinsic mode frequencies become completely decoupled from the combustor parameters like crosssection jump, temperature jump and flame location. Here, we show that this remarkable decoupling phenomenon holds not only for closed-open combustors but also for all combustors with the ideal boundary conditions (i.e. closed-open, open-open and closed-closed). Making use of this decoupling phenomenon for the open-open combustors, we derive explicit analytic expressions for the neutral curve of intrinsic mode instability on the n À plane as well as for the linear growth/decay rate near the neutral curve taking into account temperature and cross-section jumps. The instability domain on the n À plane is shown to be qualitatively different from that of the closed-open combustor; in openopen combustors it is not confined for large. To find the instability domain and growth rate characteristics for non-ideal open-open boundaries the combustor end boundaries are perturbed and explicit analytical formulae derived and verified by numerics.
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