Triggering, bypass transition and the effect of noise on a linearly stable thermoacoustic system

Abstract

This paper explores the analogy between triggering in thermoacoustics and bypass transition to turbulence in hydrodynamics. These are both mechanisms through which a small perturbation causes a system to develop large self-sustained oscillations, despite the system being linearly stable. For example, it explains why round pipe flow (Hagen–Poiseuille flow) can become turbulent, even though all its eigenvalues are stable at all Reynolds numbers. In hydrodynamics, bypass transition involves transient growth of the initial perturbation, which arises due to linear non-normality of the stability operator, followed by attraction towards a series of unstable periodic solutions of the Navier–Stokes equations, followed by repulsion either to full turbulence or re-laminarization. This paper shows that the triggering process in thermoacoustics is directly analogous to this. In thermoacoustics, the linearized stability operator is also non-normal and also gives rise to transient growth. The system then evolves towards an unstable periodic solution of the governing equations, followed by repulsion either to a stable periodic solution or to the zero solution. This paper demonstrates that initial perturbations that have higher amplitudes at low frequencies are more effective at triggering self-sustained oscillations than perturbations that have similar amplitudes at all frequencies. This paper then explores the effect that different types of noise have on triggering. Three types of noise are considered: pink noise (higher amplitudes at low frequencies), white noise (similar amplitudes at all frequencies) and blue noise (higher amplitudes at high frequencies). Different amplitudes of noise are applied, both as short bursts and continuously. Pink noise is found to be more effective at causing triggering than white noise and blue noise, in line with the results found in the first part of the paper. In summary, this paper investigates the triggering mechanism in thermoacoustics and demonstrates that some types of noise cause triggering more effectively than others.