On the sublimit solution branches of the stripe patterns formed in counterflow diffusion flames by diffusional-thermal instability

Abstract

The nonlinear dynamics of striped diffusion flames, formed in a two-dimensional counterflow by diffusional-thermal instability with Lewis numbers sufficiently less than unity, is investigated numerically by examining various two-dimensional flame-structure solutions bifurcating from the one-dimensional steady solution. The Lewis numbers for fuel and oxidizer are identically set to be 0.3, and an overall single-step Arrhenius-type chemical reaction with a Zel'dovich number of 7 is employed as the chemistry model. Particular attention is focused on the flame-stripe solution branches in the sub-extinction regime and on the hysteresis encountered during the transition between different solution branches. In the numerical simulations, a nonlinear solution with eight stripes is first realized from the one-dimensional solution at a Damkohler number slightly greater than the extinction Damkohler number. The eight-stripe solution survives Damkohler numbers much smaller than the extinction Damkohler number until successive bifurcations, leading to the doubling of the pattern wavelength, occur at the subsequent forward-transition conditions. At the first forward-transition Damkohler number occurs the transition to a four-stripe solution, which in turn transits to a two-stripe solution at the second forward-transition Damkohler number, a value somewhat smaller than the first. However, further transition from a two-stripe solution to a one-stripe solution is not always possible even if a one-stripe solution can be accessed independently for particular initial conditions. The Damkohler-number ranges and shapes for the two-stripe and one-stripe solutions are found to be virtually identical, implying that each stripe could be an independent structure if the distance between stripes is sufficiently large. By increasing the Damkohler number, backward transitions can be observed. In comparison with the forward-transition Damkohler numbers, the corresponding backward-transition Damkohler numbers are always much greater, thereby indicating significant hysteresis between the stripe patterns of strained diffusion flames.