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

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-I𝑠) or through a longwave stationary bifurcation (also known as type-II𝑠). The latter type-II𝑠 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.