Chain-branching explosions in mixing layers

The chain-branching process leading to ignition in the high-temperature hyd rogen-oxygen mixing layer is studied by application of a novel WKB-like met hod when, as is typically the case, two branching radicals cannot be assume d to maintain steady state. It is shown that the initiation reactions, resp onsible for the early radical buildup, cease being important when the radic al mass fractions reach values of the order of the ratio of the characteris tic branching time to the characteristic initiation time, a very small quan tity at temperatures of practical interest. The autocatalytic character of the chain-branching reactions causes the radical concentrations to grow exp onentially with downstream distance in the process that follows. It is show n that the transverse radical profiles that emerge can be described by expo nential series of the WKB type in inverse powers of the streamwise coordina te. 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 nond imensional streamwise variation as that of a premixed branching explosion e valuated at the transverse location where the effective Damkoher number bas ed on the flow velocity and branching rate is maximum. This functional stre amwise variation, as well as the leading-order representation of the radica l profiles, is obtained by imposing a condition of bounded, nonoscillatory behavior on the solution. The resulting radical profiles peak at the locati on 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 correc tion to the growth rate as an eigenvalue in a linear eigenvalue problem. Th e method developed can be extended to the analysis of chain-branching explo sions in laminar, self-similar mixing layers with an arbitrary number of br anching steps adopted for describing the chemistry.