Steady Quasi-One-Dimensional Detonations in Idealized Systems

Abstract

Steady, quasi-one-dimensional, shock-initiated reactive flow is investigated for an ideal-gas medium undergoing the reversible reaction A ⇄ B, having an Arrhenius rate constant. The lateral expansion is supposed to be given by A(x)/A(0) = [1 + βμx2/(β − 1)] [1 + μx2/(β − 1)]−1, where A denotes the cross-sectional area and x is the distance from the shock. Long-time solutions to the piston problem are constructed, where possible, from steady solutions having final mass velocity equal to the piston velocity. Solutions which can be plausibly amended by rarefaction waves so as to satisfy arbitrarily small piston velocities are regarded as quasi-one-dimensional Chapman-Jouguet detonations. Explicit numerical solutions are found for systems having fixed values of the heat of reaction, heat capacity ratio and activation energy for a wide range of values of the reaction rate k, relative to the expansion rate μ. Chapman-Jouguet waves are found to include equilibrium Chapman-Jouguet solutions, transonic solutions, and double-wave solutions involving an additional shock in the reaction zone.