The "turbulent flame speed" of wrinkled premixed flames

The determination of the turbulent flame speed is a central problem in combustion theory. Early studies by Damköhler and Shelkin resorted to geometrical and scaling arguments to deduce expressions for the turbulent flame speed and its dependence on turbulence intensity. A more rigorous approach was undertaken by Clavin and Williams who, based on a multi-scale asymptotic approach valid for weakly wrinkled flames, derived an expression that apart from a numerical factor recaptures the early result by Damköhler and Shelkin. The common denominator of the phenomenological and the more rigorous propositions is an increase in turbulent flame speed due solely to an increase in flame surface area. Various suggestions based on physical and/or experimental arguments have been also proposed, incorporating other functional parameters into the flame speed relation. The objective of this work is to extend the asymptotic results to a fully nonlinear regime that permits to systematically extract scaling laws for the turbulent flame speed that depend on turbulence intensity and scale, mixture composition and thermal expansion, flow conditions including effects of curvature and strain, and flame instabilities. To this end, we use a hybrid Navier-Stokes/front-capturing methodology, which consistently with the asymptotic model, treats the flame as a surface of density discontinuity separating burned and unburned gases. The present results are limited to positive Markstein length, corresponding to lean hydrocarbon-air or rich hydrogen-air mixtures, and to wrinkled flames of vanishingly small thickness, smaller that the smallest fluid scales. For simplicity we have considered here two-dimensional turbulence, which although lacks some features of real three-dimensional turbulence, is not detrimental when using the hydrodynamic model under consideration, because the turbulent flame retains its laminar structure and its interaction with turbulence is primarily advective/kinematic in nature.