Coupling of acoustic and intrinsic modes in 1-D combustor models

Abstract

The work is concerned with the theoretical examination of a new type of thermo-acoustic instability in combustors not reported in the literature. The instability results from linear coupling between the conventional acoustic mode and the recently discovered “flame intrinsic modes.” Within the framework of a 1D model of a quarter wave resonator with the standard n-t model of flame heat release, intrinsic-acoustic mode coupling occurs when the real parts of the frequencies of neighboring acoustic and flame-intrinsic modes at small interaction index n are close. While at small n the Eigen-functions of close acoustic and flame intrinsic modes clearly exhibit their distinctive identities, with increase of n the mode identities become blurred and the Eigen-functions of acoustic modes resemble more and more those of flame intrinsic modes and at a certain n become indistinguishable. We refer them as coupled intrinsic-acoustic modes or coupled modes. When the “Rayleigh index” for a coupled mode behaving as an acoustic mode at small n is negative, at a larger n such a mode can nevertheless become unstable at one of the nearby intrinsic mode frequencies. We find analytically the instability domain due to coupling in the parameter space. Near the instability boundary, we reduce the transcendental dispersion relation to a quadratic or, if higher accuracy is desired, to a quartic equation. These models capture well all four possible coupling scenarios.