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 thoroughly studied acoustic modes in a combustor. Despite being intensely studied, primarily numerically and experimentally, their properties remain poorly understood. Here an analytical investigation produced a comprehensive picture of the properties of linear intrinsic modes within the framework of a one-dimensional model of closed-open and open-open combustors with temperature and cross-section jump across the flame, and a linear n - t law of heat release, where n is interaction index and t is the time lag. It has been shown there is always an infinite number of intrinsic modes present. In the limit of small n the frequencies of these modes depend neither on the properties of the combustor nor on the position of the flame. For small n these modes are strongly damped and they become unstable only if n exceeds a certain threshold. Remarkably, on the neutral curve the intrinsic modes become completely decoupled from the environment.
The main results of the study follow from the discovered decoupling on the neutral curve and include explicit analytic expressions for the exact neutral curve on the n - t plane, and for the growth/decay rate dependence on the parameters of the combustor. A new type of thermo-acoustic instability has been discovered. The instability occurs out due to coupling between intrinsic and conventional acoustic modes. Unstable or weakly decaying coupled acoustic modes behave exactly like an intrinsic mode, which increases the possible number of unstable intrinsic modes. The overall picture of intrinsic mode instabilities found for ideal boundary conditions has been shown to be robust with respect to modifications of the end boundary conditions. The explicit corrections to the position of the neutral curve and growth/decay rate have been found. The analytical results have been verified by numerics.