Minimal energy thresholds for triggering in the Rijke tube: the effect of the time delay

Abstract

Triggering is the process by which a linearly stable thermoacoustic system can reach self-sustained oscillations. This nonlinear phenomenon is activated only for sufficiently large amplitudes of perturbations to the equilibrium state. In this work, using a nonlinear variational optimisation method coupled with energy bisection, we compute the minimal thresholds for triggering in the Rijke tube. In particular, extending previous works, we take into account the effect of the time delay by optimising not only the perturbations at initial time, but also the velocity at the hot-wire position in the time-delay interval. We found that, for sufficiently large time delays, the nonlinearity linked to the delayed flow velocity bears a strong potential for energy growth, leading to transient amplifications of the energy reaching ${O}(10^{2})$, two orders of magnitude larger than those reported in previous studies. Notably, the gain increases with the time delay, but decreases with the initial energy of the perturbation, thus reaching very high values close to the triggering threshold of the system. The minimal energy for triggering self-sustained oscillations achieves energy values as low as ${O}(10^{-4})$, two orders of magnitude smaller than previous estimates. This indicates that, for thermoacoustic systems characterised by a non-negligible time delay, taking into account the effect of the time-delayed variables, as well as the system nonlinearity, is crucial for correctly evaluating the triggering energy thresholds.