Detonation waves exhibit a very complex dynamics, with both longitudinal and transversal instabilities. Because of the complexity of the equations, much of the theoretical understanding relies on extensive (and costly) numerical simulations of the reactive compressible Navier-Stokes (or Euler) equations. Simplified theories of combustion are desirable to help better understand the various physical processes involved.
The first attempt at a reduced qualitative description is due to Fickett (1979), who introduced a toy model as a vehicle to understand the intricacies of detonation theory. Several others followed on his steps [Clavin, He, Ludford, Majda, Rosales, Williams, etc]. However, these earlier models lacked the correct physics behind the observed complex dynamics — their traveling waves are stable. Recently new and improved models (yet still simple) were introduced, and these have been shown to incorporate the effects needed to reproduce the observed dynamics. First in 1-D [Radulescu-Tang, 2011] and [Kasimov-Faria-Rosales, 2013], and then in 2-D [Faria 2014] and [Faria-Kasimov-Rosales, 2015].
The models can be asymptotically justified, under the assumptions of: small heat release, large activation energy, weak nonlinearity, and the Newtonian approximation. In this talk I will introduce some of these models, sketch its derivations and the stability analysis associated with them, and illustrate the complex dynamics they produce with numerical simulations.