Justin Tang successfully defends his Masters thesis on Fickett detonations..

and gets nominated for another one of those fancy awards...i guess it's a trend by now.

photo courtesy of A. Bellerive

Justin Tang thesis now shows that Fickett's model, i.e., a reactive form of the Burgers' equation, reproduces the same type of pulsating instabilities as real detonations! It also recovers the route to chaos via period doubling bifurcations.

Space-time diagram showing the modulated amplification of forward facing pressure waves by the location of the reaction zone (controlled by the lead shock strength).  In-phase and out-of-phase passage of pressure waves through the reaction zone modulate the leading shock strength. Figure adapted from his Masters thesis.

More reading available in our publications:

Radulescu M.I. and Tang, J., Non-linear dynamics of self-sustained supersonic reaction waves: Fickett’s detonation analogue. Physical Review Letters 107(16):164503 (2011) DOI:10.1103/PhysRevLett.107.164503

Tang, J. and Radulescu, M.I., Dynamics of shock induced ignition in Fickett's model: influence of x. Proceedings of the Combustion Institute 34(2):2035-2041 (2013) DOI:10.1016/j.proci.2012.05.079

and in his thesis:

Study of the Instability and Dynamics of Detonation Waves using Fickett's Analogue to the Reactive Euler Equations 

Thesis Abstract

The instability behaviour and dynamics that are seen in detonation waves are studied using Fickett's model with a 2-step reaction model with separately controlled induction and reaction zones. This model acts as a simplified toy-model to the reactive Euler equations allowing for more clarity of the detonation phenomenon. The key simplification in Fickett's model is that its characteristic description for wave-interaction is reduced and more transparent when compared with the reactive Euler equations. The dynamics of the system can be described by the coupling between only two families of characteristics: the forward traveling pressure wave characteristics and the particle path characteristics, along which the reaction takes place.

We numerically simulate a self-supported detonation propagating down a 1D channel and investigate the pulsating instability behaviour. We are able to clarify the governing mechanism behind the pulsations through a characteristic analysis describing the coupling that takes place between the amplification of the compressions waves and the alteration to the induction timing. The precompression of the induction zone controls the onset of the energy release, which becomes shifted in or out of phase with the compression waves causing a feedback response.

The acceleration phase of the pulsations and the amplification behaviour are further clarified through a parametric study of the reaction parameters $\epsilon$ (the inverse activation energy controlling the induction timing) and $K$ (the ratio of induction to reaction times controlling the rate of energy release in relation to the induction delay time). An analytical solution for the acceleration of the path of the onset of reactions, representing the strength of the amplification, is also found using high-activation energy asymptotics.

A sequence of instability modes is shown to develop with increasing sensitivity of the induction rate. Fickett's model is shown to reproduce the same period doubling bifurcation and route to chaos as seen in the full reactive Euler equations. Additionally, a galloping mode and quench behaviour were found to be reproducible in Fickett's model.