The thermo-acoustic Fant equation

Abstract

A theoretical analysis is made of combustion instabilities of three combustor configurations. The equations governing aeroacoustics and combustion are derived, arriving at an acoustic analogy in terms of the pressure and total enthalpy. A solution for the acoustic analogy is determined in terms of a Green's function and initial instability results are presented for the pressure Green's function. These predictions are limited by assumptions made about the combustion zone. Finally a `reduced complexity' equation is derived accounting for a generalised combustion zone. The equation is nonlinear and furnishes limit cycle solutions for finite amplitude burner modes. It is a generalisation to combustion flows of the Fant equation used to investigate the production of voiced speech (G Fant. Acoustic Theory of Speech Production. Mouton, The Hague, 1960). The Fant equation governs the unsteady volume ow past the flame holder which, in turn, determines the acoustics of the entire system. The equation includes a fully determinate part that depends on the geometry of the flame-holder and the thermo-acoustic system, and terms defined by integrals involving thermo-aerodynamic sources, such as the flame and vortex sound sources. Illustrative numerical results are presented for both the linearised equation and the full nonlinear equation. The linearised equation governs the growth rate of the natural acoustic modes, which are excited into instability by unsteady heat release from the flame and damped by large scale vorticity production and radiation losses. The full nonlinear equation, however, governs the `limit cycle' formation when absorption of sound by vortex shedding at trailing edges equally opposes sound generation by the flame. Limit cycle modes are of particular interest because they cannot be captured in linear predictions and are the primary source of combustor instabilities.