A parabolic linear evolution equation for cellular detonation instability

Using the combined limits of a large activation energy and a ratio of specific heats close to unity, a dispersion relation has recently been derived which governs the stability of a steady Chapman - Jouguet detonation wave to two-dimensional linear disturbances. The analysis considers instability evolution time scales that are long on the time scale of fluid particle passage through the main reaction layer. In the following, a simplified polynomial form of the dispersion relation is derived under an appropriate choice of a distinguished limit between an instability evolution time scale that is long on the time scale of particle passage through the induction zone and a transverse disturbance wavelength that is long compared to the hydrodynamic thickness of the induction zone. A third order in time, sixth order in space, parabolic linear evolution equation is derived which governs the initial dynamics of cellular detonation formation. The linear dispersion relation is shown to have the properties of a most unstable wavenumber, leading to a theoretical prediction of the initial detonation cell spacing and a wavenumber above which all disturbances decay, eliminating the growth of small-wavelength perturbations. The role played by the curvature of the detonation front in the dynamics of the cellular instability is also highlighted.