On the dynamics and linear stability of one-dimensional steady detonation waves

Abstract

A detailed analysis of the dynamics and linear stability of a steady one-dimensional detonation wave propagating in a binary reactive system with an Arrhenius chemical kinetics of type is carried out. Starting from the frame of the kinetic theory, the binary reactive mixture is modelled at the mesoscopic scale by the reactive Boltzmann equation (BE), assuming hard sphere cross sections for elastic collisions and step cross sections with activation energy for reactive interactions. The corresponding hydrodynamic limit is based on a second-order non-equilibrium solution of the BE obtained in a previous paper, using the Chapman–Enskog method in a chemical regime for which the reactive interactions are less frequent than the elastic collisions. The resulting hydrodynamic governing equations are the reactive Euler equations, including a rate law which exhibits an explicit dependence on the reaction heat and forward activation energy of the chemical reaction. These equations are used to describe the spatial structure of the steady detonation wave solution and investigate how this structure varies with the reaction heat. The response of the steady solution to one-dimensional disturbances is studied using a normal-mode linear approach which leads to an initial-value problem for the state variable disturbances in the reaction zone. The stability problem is treated numerically, using an iterative shooting technique to determine the unstable modes. The analysis developed here emphasizes the influence of the chemical reaction heat and activation energy on the linear stability spectra.