Interaction of Weakly Unstable Auto-Oscillatory Modes in a Solid Propellant Combustion Model

Abstract

In the presence of instability a planar flame wave steadily propagating through a solid reagent may become unstable to oscillatory modes. This scenario has previously been examined in models of strictly condensed combustion where the laminar wave is shown to lose stability via a Hopf bifurcation. In this study we examine the dynamics of an asymptotic model of homogeneous solid propellant combustion where a planar wave loses stability to a pair of oscillatory modes. After re-introducing a free-interface solid propellant combustion model, the form and linear stability of small perturbations to a laminar flame sheet are described. It is shown that for certain geometries two or more modes lose stability simultaneously as a physical parameter related to the solid state activation energy is varied. Multiple time scales are then introduced in order to explore the interaction of this pair of weakly unstable modes and to describe the evolution of their amplitude and phase for large times. The dependence of the dynamics on physical parameters is also investigated and the analytical predictions are found to closely match numerical predictions over much of the parameter space.