A Model for the Thermo-Acoustic Feedback of Transverse Acoustic Modes and Periodic Oscillations in Flame Position in Cylindrical Flame Tubes

In the past decades, several feedback mechanisms for longitudinal acoustic modes in gas turbine combustors have been investigated. These mechanisms are successfully used in predictive tools like acoustic network models to analyze low-frequency instabilities in combustion systems. In contrast, little is known about high-frequency oscillations — fluctuations at several kHz. Most theories are derived from experimental investigations of afterburners in the 1950s and 1960s, indicating an interaction of vortex shedding, fluctuating vorticity and heat release. In this work a different feedback mechanism for high-frequency oscillations in cylindrical flame tubes related to transverse acoustic modes is suggested and analysed: Transverse acoustic pressure fluctuations are linked to an oscillating velocity field. A time-dependent but periodic displacement field can be derived from these velocity fluctuations. The model assumes that the zone of heat release is displaced by the velocity fluctuations. Pressure oscillations and periodically deflected heat release lead to a contribution to the Rayleigh criterion without fluctuations in the global heat release. This effect is studied in a circular cross section presuming a circular zone of heat release. Expressions for the displacement of the flame front are derived from the analytical solution of the wave equation in cylindrical geometries assuming a quiescent medium, constant density and speed of sound. The Rayleigh criterion is integrated and growth rates are evaluated whereas damping effects are neglected as they are not subject to this study. Characteristics of the model are assessed and compared to experimental observations to check the validity and the applicability of the theory.