Effect of a shear flow on the Darrieus–Landau instability in a Hele-Shaw channel

The Darrieus–Landau instability of premixed flames propagating in a narrow Hele-Shaw channel in the presence of a strong shear flow is investigated, incorporating also the Rayleigh–Taylor and diffusive-thermal instabilities. The flow induces shear-enhanced diffusion (Taylor dispersion) in the two-dimensional depth averaged equations. Since the diffusion enhancement is in the streamwise direction, but not in the spanwise direction, this leads to anisotropic diffusion and flame propagation. To understand how such anisotropies affect flame stability, two important cases are considered. These correspond to initial unperturbed conditions pertaining to a planar flame propagating in the streamwise or spanwise directions. The analysis is based on a two-dimensional model derived by asymptotic methods and solved numerically. Its numerical solutions comprise the computation of eigenvalues of a linear stability problem as well as time-dependent simulations. These address the influence of the shear-flow strength (or Peclet number Pe), preferential diffusion (or Lewis number Le) and gravity (or Rayleigh number Ra). Dispersion curves characterising the perturbation growth rate are computed for selected values of Pe, Le and Ra. Taylor dispersion induced by strong shear flows is found to suppress the Darrieus–Landau instability and to weaken the flame wrinkling when the flame propagates in the streamwise direction. In contrast, when the flame propagates in the spanwise direction, the flame is stabilised in Le1 mixtures. In the latter case, Taylor dispersion coupled with gas expansion facilitates flame wrinkling in an unusual manner. Specifically, stagnation points and counter-rotating vortices are encountered in the flame close to the unburnt gas side. More generally, an original finding is the demonstration that vorticity can be produced by a curved flame in a Hele-Shaw channel even in the absence of gravity, whenever Pe≠0, and that the vorticity remains confined to the flame preheat and reaction zones.