Unity maximum transient energy growth of heat-driven acoustic oscillations

Abstract

Transient energy growth of acoustic disturbances may trigger thermoacoustic instability in a non-normal thermoacoustic system. In this work, minimizing transient energy growth of heat-driven acoustic oscillations in an open-ended thermoacoustic system is considered. For this, a state-space thermoacoustic model with an acoustically compact heat source and distributed monopole-like actuators is developed. The heat source gives rise to the mean temperature jump, as experimentally measured. It is modeled with a modified King’s Law. Coupling the unsteady heat release model with a Galerkin series expansion of the acoustic waves present enables the time evolution of flow disturbances and acoustical energy to be calculated, thus providing a platform on which to gain insight on the system’s transient stability behaviors and the non-normal response of the system to the dynamic actuators. It is first shown that implementing a linear-quadratic regulator (LQR) leads to the system being asymptotically stabilized. However, the LQR optimization strategy fails in eliminating the transient growth. This finding is consistent with Pseudospectra analysis of the present system. In order to achieve unity maximum transient growth, a Lyapunov-based optimization strategy is systematically designed. It is found that this optimization strategy achieves both exponential decay of the acoustical energy and unity maximum transient growth. Furthermore, the sound pressure level is reduced by approximately 25dB. In addition, the number of the actuators K is shown to be related to the mode number N as K=N.