Linear Stability of a Detonation Wave with a Model Three-Step Chain-Branching Reaction

The linear stability of a planar detonation wave with a three-step chain-branching reaction is studied by a normal mode approsch. The reaction model consists of a chain-initiation step and a chain-branching step governed by Arrhenius kinetics, with a chain-termination step which is independent of temperature. It mimics the essential reaction dynamics of a real chain-branching chemical system. The linear stability of the steady detonation wave to two-dimensional disturbances is studied with the chain-branching crossover temperature, i.e., the temperature at which chain-branching and chain-termination rates are equal, ss a bifurcation parameter. This parameter determines the ratio of the length of the chain-branching induction zone to the chain-termination zone within the steady detonation wave. The effect of linear transverse disturbances is considered for two values of the chain-branching crossover temperature: in one the planar steady detonation wave is stable to one-dimensional disturbances, while in the other it is unstable to such disturbances. Keywords-Detonation, Stability, Chain-branching reactions. P. INTRODUCTION The problem of the one-dimensional linear instability of a detonation wave in a chemical mixture which was assumed to react via an idealised one-step Arrhenius reaction was first studied by Erpenbeck [1,2]. A Laplace transform technique was used to analyse the behaviour of small amplitude disturbances from the plane steady detonation wave. Later, Lee and Stewart [3] used a normal mode approach to address the linear instability problem. Their numerical shooting technique provided a straightforward way of calculating the stability spectra. Experimental studies on one-dimensional pulsating detonation instabilities such as [4] and on multidimensional transverse instabilities such as [5] demonstrate a distinct sensitivity to the type of the chemical mixture used. In particular, a majority of the mixtures used in these experiments involved the reaction of hydrogen and oxygen, which occurs through a chain-branching reaction mechanism. This involves a sequence of chain-initiation, chain-branching, and chain-termination stages. A small amount of reactant is converted into chain-carriers, which may be either free radicals or atoms, by means of the slow chain-initiation reactions. The chain-carriers are then rapidly multiplied through chain-branching reactions, while the rise in concentration of chainradicals is retarded by chain-termination steps which occur either through absorption at, the vessel walls or through three-body collisions in the interior. In the following, both the oneand twedimensional linear instabilities of a planar detonation wave with a model three-step chain-branching reaction are considered. The model consists of an Arrhenius type chain-initiation step with a large activation energy and an Arrhenius type M. Short was supported in this work by an E.P.S.R.C. research grant and J. W. Dold was supported by an E.P.S.R.C. Advanced Fellowship. t Present address: Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL 61801, U.S.A.