Intrinsic flame instabilities in combustors: Analytic description of a 1-D resonator model

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−τ 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 ωi with the mode number mi and the time lag τ: ωi=(2mi+1)(π/τ)+mπ/τ, 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−τ 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.