On the two separate decay time scales of a detonation wave modelled by the Burgers equation and their relation to its chaotic dynamics

TL;DR

This paper models galloping-like pulsations in detonations using a reactive Burgers equation, revealing two distinct decay time scales linked to chaotic dynamics and characteristic wave interactions.

Contribution

It introduces a simplified detonation model that captures two separate decay time scales and their relation to chaotic behavior using wave characteristic analysis.

Findings

Two decay time scales are identified in the detonation velocity.

Characteristics from rarefaction waves influence the decay dynamics.

Wave amplification explains the distinct time scales.

Abstract

This study uses a simplified detonation model to investigate the behaviour of detonations with galloping-like pulsations. The reactive Burgers equation is used for the hydrodynamic equation, coupled to a pulsed source whereby all the shocked reactants are simultaneously consumed at fixed time intervals. The model mimics the short periodic amplifications of the shock front followed by relatively lengthy decays seen in galloping detonations. Numerical simulations reveal a saw tooth evolution of the front velocity with a period-averaged detonation speed equal to the Chapman-Jouguet velocity. The detonation velocity exhibits two distinct groups of decay time scales, punctuated by reaction pulses. At each pulse, a rarefaction wave is created at the reaction front's last position. A characteristic investigation reveals that characteristics originating from the head of this rarefaction take 1.57 periods to reach and attenuate the detonation front, while characteristics at the tail take an additional period. The leading characteristics are amplified twice, by passing through the reaction fronts of subsequent pulses, before arriving at the shock front, whilst the trailing characteristics are amplified three times. This leads to the two distinct groups of time scales seen in the detonation front speed.