Recent Results on Stability of Planar Detonations

Abstract

Abstract. We describe recent analytical and numerical results on stability and behavior of viscous and inviscid detonation waves obtained by dynamical systems/Evans function techniques like those used to study shock and reaction diffusion waves. In the first part, we give a broad description of viscous and inviscid results for 1D perturbations; in the second, we focus on inviscid high-frequency stability in multi-D and associated questions in turning point theory/WKB expansion.

A minimal hyperbolic system for unstable shock waves

“…Analysis of detonations by means of the reactive Euler equations has received much attention and continues actively at present. See, for example, recent reviews [1,2,42,7,49,5]. One of the key properties of gaseous detonations is their manifestly time-dependent and spatially complex dynamics.…”

“…A similar model which also included diffusive effects was proposed independently by Majda [32]. The Majda model has received much attention in the mathematics literature [49,31,21,22,30] as a prototype to study existence and stability of traveling waves. It must be pointed out that in the analyses of the Majda model only stable traveling waves have been found so far, to the best of our knowledge.…”

Linear stability analysis of detonations via numerical computation and dynamic mode decomposition

“…For example, in a detonation problem considered in the present work, the challenge stems from the complexity of reaction rate equations that combined with the fluid equations, produce a system of ODEs that can be stiff and/or singular. As a result, performing accurate calculations of spectral properties becomes quite difficult [6,27,31,42,55,58,65,71,72].…”