On Diffusion Flames in Turbulent Shear Flows: Modeling Reactant Consumption in a Mixing Layer

A model for the time-averaged burning of two parallel-flowing gaseous streams forming an isobaric, subsonic, fully turbulent, two-dimensional mixing layer downstream of a flat, perfectly slippery splitter plate is formulated and solved. One stream contains fuel, the other contains oxidant, with which the fuel is highly reactive on a molecular scale. Hence, the rate-controlling phenomenon for the formation of product gas is the inviscidly controlled macroscale mixing. In view of the extremely limited experimental data for this aerothermochemical flow, the mean rate of reactant consumption is taken proportional to the product of the mean mass fraction of fuel, the mean mass fraction of oxidant, and the mean principal element of the rate-of-strain tensor. The model is along the lines of the eddy break-up model proposed by Spalding (for the mean description of the burning of premixed reactants in a turbulent shear flow). The model is motivated here by the authors' previous studies, in which a strain rate tangential to a diffusion flame is shown to increase the rate of product generation by increasing the area of exposure of fuel to oxidant and by inducing more reactant to the vicinity of the flame. For a consistent level of closure, Görtler's solution, based on Prandtl's (exchange coefficient) eddy viscosity, is adopted for the flowfield, and the Reynolds analogy (with constant but nonunity turbulent Schmidt and Prandtl numbers) is used to characterize the eddy diffusivity for heat and species. By asymptotic analysis, numerical integration, and approximate solution by generalized Oseen linearization, the rate of evolution to, and the quantitative character of, the downstream asymptotic (self-similar) state is studied as a function of the problem parameters. The net amount of product gas generated, as indicated by integration across the layer of the mean rate of reactant consumption, increases with larger difference in speed between the two streams, and also with smaller values of the turbulent Schmidt number. Increasing the richness of the slower stream above stoichiometric proportion is predicted to give a slightly greater increase in net product gas generation than comparably increasing the richness of the faster stream above stoichiometric proportion.