Linear analysis of thermo-diffusive instability from edge flames to fully-premixed laminar flames with a wide range of Damköhler number and Lewis number greater than unity

A model problem for thermo-diffusive instabilities on planar laminar flames is considered, which studies flames stabilized in the proximity of a cold burner in a co-flow mixing layer between fuel and oxidizer streams. It accounts for the driving mechanisms for the onset of thermo-diffusive instability, but assumes constant density and neglects flow or pressure gradients. A premixedness parameter characterizes the transverse gradient of the mixture fraction at the upstream boundary. By varying the premixedness parameter from zero to unity, different flame structures are recovered covering the complete spectrum from non-premixed edge flames to fully-premixed planar flames, through partially-premixed triple flames. A modal linear stability analysis is presented. The equations governing small-amplitude flame fluctuations are recast as an eigenvalue problem, in which the eigenfunctions describe two-dimensional flow field variables with arbitrary spatial dependence, and the eigenvalues describe the oscillation frequency and temporal growth rate. For all partially-premixed flames, a pair of complex eigenvalues are found, corresponding to upstream–downstream pulsations of the flame leading edge. These eigenmodes are unstable for a bounded region in the Premixedness-Damköhler space. Fully-premixed flames present multiple pairs of unstable eigenmodes; the most unstable pair corresponds to an upstream–downstream oscillation of the flame without distortion along the transverse direction, while the subsequent ones describe wavy deformations of the flame structure consistent with the formation of cellular patterns. The predictions of the linear stability analysis compare well with results from nonlinear simulations. Novelty and significance statement A novel methodology is presented for the instability analysis of laminar flames that considers linear eigenmodes with arbitrary dependence on two spatial directions. The computationally-inexpensive approach allows to perform vast parametric studies of the influence of the physical parameters, thus providing new physical insights. Here, it is applied to a model problem for thermo-diffusive instability, that allows studying the complete range of flames possible in a co-flow configuration in a unified set up. A wide range of Damköhler numbers a premixedness parameters are analyzed and the instability maps are reported, which is a novelty in the literature. The methodology proposed can be directly applied to other 2D configurations and can incorporate more complex phenomena like the coupling between the thermodiffusive instability and flow and pressure gradients, and differential diffusivity.