Weakly nonlinear reactive shocks with lateral divergence

Abstract

Asymptotic equations are obtained in a weakly nonlinear limit of the Wood-Kirkwood equations for slightly divergent reactive flow. Specifically, in cylindrical symmetry the equations of reactive flow are restricted to the central stream tube, and the radial component of the velocity is assumed to be a function of wavespeed of a signal propagating through the medium. Asymptotic equations are obtained in the limit where the Mach number is close to unity and the components of the velocity are small in comparison; the heat release is assumed to be small, and two cases with different chemical kinetics are examined. In one case an Arrhenius-type reaction rate with large activation energy is considered, and in another case a depletion-type rate with no Arrhenius factor is considered. The asymptotic equations are compared to the Fickett (1985) analogue of detonation. Finally, the existence of ZND type waves within the context of the model asymptotic equations is examined in special cases.