Diffusive-thermal instabilities of a planar premixed flame aligned with a shear flow

Abstract

The stability of a thick planar premixed flame, propagating steadily in a direction transverse to that of unidirectional shear flow, is studied. A linear stability analysis is carried out in the asymptotic limit of infinitely large activation energy, yielding a dispersion relation. The relation characterises the coupling between Taylor dispersion (or shear-enhanced diffusion) and the flame thermo-diffusive instabilities, in terms of two main parameters, namely, the reactant Lewis number L e and the flow Peclet number P e . The implications of the dispersion relation are discussed and various flame instabilities are identified and classified in the L e - P e plane. An important original finding is the demonstration that for values of the Peclet number exceeding a critical value, the classical cellular instability, commonly found for L e < 1 , exists now for L e > 1 but is absent when L e < 1 . In fact, the cellular instability identified for L e > 1 is shown to occur either through a finite-wavelength stationary bifurcation (also known as type-I s ) or through a longwave stationary bifurcation (also known as type-II s ). The latter type-II s bifurcation leads in the weakly nonlinear regime to a Kuramoto-Sivashinsky equation, which is determined. As for the oscillatory instability, usually encountered in the absence of Taylor dispersion in L e > 1 mixtures, it is found to be absent if the Peclet number is large enough. The stability findings, which follow from the dispersion relation derived analytically, are complemented and examined numerically for a finite value of the Zeldovich number. The numerical study involves both computations of the eigenvalues of a linear stability boundary-value problem and numerical simulations of the time-dependent governing partial differential equations. The computations are found to be in good qualitative agreement with the analytical predictions.