Asymptotic Analysis of Plane Steady Detonations

Abstract

Steady-state solutions for a first order, one-step, irreversible unimolecular reaction of the Arrhenius kind are analyzed on the basis of activation-energy asymptotics. In this limit, it is found that steady solutions exist only when the flame temperature $T_ * $ lies in an interval $[ T_{ * \min } ,T_{ * \max } )$. Here $T_{ * \min } $ corresponds to the weak detonation and $T_{ * \max } $ corresponds to the ZND limit of strong detonation. In particular, when $T_ * $ is sufficiently large (and hence for the ZND limit), not all the reactant is consumed at the flame, a small remainder being burnt over large distances behind the flame. The analysis covers both the weak detonation and all strong detonations for general and Chapman–Jouguet conditions. Typical velocity and temperature profiles are computed and plotted for $T_ * $ ranging from the ZND detonation limit to the shock-after-weak detonation limit.