An asymptotic derivation of the linear stability of the square-wave detonation using the Newtonian limit

Abstract

The two-dimensional linear stability of a detonation wave characterized by a one-step irreversible Arrhenius reaction is examined by a two-parameter asymptotic approach. The first is the limit of high activation energy in which the underlying steady detona­tion structure tends to the classical square-wave profile. The second is due to Blythe & Crighton ( Proc. R. Soc. Lond . A 426, 189-209 (1989)) and assumes the Newtonian limit in which the ratio of the isotropic sound speed to the isothermal sound speed is close to unity. It is found that under two possible choices of distinguished limit between the two parameters, analytical forms for the perturbation variables in the induction zone and equilibrium zones of the perturbed detonation can be found in normal mode form. A dispersion relation describing the growth rate of the perturba­tions is then obtained in one case through a compatibility condition on the structure of the perturbation eigenfunctions at the flame front, and in the other through a matching of the perturbation variables across the flame front into the equilibrium zone, where an acoustic radiation condition is imposed.