Chain-branching explosions in mixing layers

Abstract

The chain-branching process leading to ignition in the high-temperature hydrogen-oxygen mixing layer is studied by application of a novel WKB-like method when, as is typically the case, two branching radicals cannot be assumed to maintain steady state. It is shown that the initiation reactions, responsible for the early radical buildup, cease being important when the radical mass fractions reach values of the order of the ratio of the characteristic branching time to the characteristic initiation time, a very small quantity at temperatures of practical interest. The autocatalytic character of the chain-branching reactions causes the radical concentrations to grow exponentially with downstream distance in the process that follows. It is shown that the transverse radical profiles that emerge can be described by exponential series of the WKB type in inverse powers of the streamwise coordinate. The analysis reveals that, because of the effect of radical diffusion, the rate of radical growth is uniform across the mixing layer in the first approximation, with the exponential growth in distance having the same nondimensional streamwise variation as that of a premixed branching explosion evaluated at the transverse location where the effective Damkoher number based on the flow velocity and branching rate is maximum. This functional streamwise variation, as well as the leading-order representation of the radical profiles, is obtained by imposing a condition of bounded, nonoscillatory behavior on the solution. The resulting radical profiles peak at the location of maximum local Damkohler number and decay exponentially to the sides. Analysis of the solution in the vicinity of the maximum, which is a turning point of second order in the WKB expansion, yields the second-order correction to the growth rate as an eigenvalue in a linear eigenvalue problem. The method developed can be extended to the analysis of chain-branching explosions in laminar, self-similar mixing layers with an arbitrary number of branching steps adopted for describing the chemistry.