Asymptotic Analysis of an Expanding Detonation

摘要

An expanding cylindrically or spherically symmetric detonation is analyzed in a regime in which the radius of the detonation is much greater than the width of the reaction zone. Under this assumption the fundamental equations may be approximated by a system of autonomous ordinary differential equations for the flow velocity and a reaction progress variable. The independent variable of this system is a radial variable in the rest frame of the detonation front. The radius of curvature and the detonation speed enter the system as parameters. At zero curvature this system reduces to the plane wave equations of Zeldovich, von Neumann and Doering. The plane wave equations possess a degenerate bifurcation point with a nilpotent linear part, which bifurcates into a saddle node when the radius is finite. Any smooth transonic solution must pass through the saddle node. This fact determines the wave speed implicitly as a function of radius. To leading order, the correction to the detonation speed as a function of curvature is proportional to the curvature, on the basis of formal and numerical considerations. 16 refs., 4 figs.