Premixed flame stability under shear-enhanced diffusion: Effect of the flow direction

In the presence of shear-enhanced diffusion (Taylor dispersion), flame propagation is effectively anisotropic. This study focuses on the influence of the direction of a shear flow relative to the direction of propagation on the diffusional-thermal instabilities of premixed flames. The problem is addressed analytically using large activation energy asymptotics, complemented by numerical simulations, in the framework of a constant density two-dimensional model. The model, obtained by depth averaging of the governing equations in a Hele-Shaw configuration, accounts for shear-enhanced diffusion. A linear stability analysis is carried out analytically, leading to a dispersion relation involving three parameters: the Lewis number Le; the Taylor-dispersion coefficient p, which is proportional to the Péclet number; and the angle φ between the direction of propagation of the unperturbed planar flame and the flow direction. Based on the dispersion relation, stability diagrams are determined in terms of the parameters, along with bifurcations curves identifying the nature of the instabilities observed. It is shown that cellular instabilities expected when Le < 1 can now occur as a result of Taylor dispersion in Le > 1 mixtures, provided the angle φ exceeds a critical value approximately equal to 75◦. In general, it is found that an increase in φ from 0◦ to 90◦ has a stabilizing effect in subunity Lewis number mixtures Le < 1 and a destabilizing effect when Le > 1. Particular attention is devoted to the cellular long-wave instability encountered, which is found to be described by a modified Kuramoto-Sivashinsky equation. The equation involves the three aforementioned parameters and includes a dispersion term (a third-order spatial derivative) as well a drift term (first-order derivative) whenever φ = 0◦ and φ = 90◦, which is whenever the direction of the shear flow is neither parallel nor perpendicular to the direction of flame propagation.