Analytic description of flame intrinsic instability in one-dimensional model of open–open combustors with ideal and non-ideal end boundaries

Abstract

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 cross-section jump across the flame taken into account. The standard [Formula: see text] model of heat release is adopted, where n is the interaction index and τ 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 cross-section 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 [Formula: see text] 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 [Formula: see text] plane is shown to be qualitatively different from that of the closed–open combustor; in open–open combustors it is not confined for large τ. 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.