Nonlinear dynamics of cellular flames

The problem of a flame stabilized by a line source of fuel in the cellular regime, i.e., L < 1, where L is the Lewis number, is solved numerically. It is found that as L is decreased, transitions from stationary axisymmetric to stationary cellular to nonstationary cellular flames occur. The nonstationary cellular flames can exhibit both periodic and quasi-periodic dynamics. In particular, as L is decreased successive transitions from stationary axisymmetric solutions, to stationary four-cell solutions, to spinning four-cell solutions are computed. The spinning four-cell solutions are very slowly traveling waves that arise due to an infinite period, symmetry breaking bifurcation, in which the reflection symmetry of the stationary four-cell solution is broken. Near the transition point, the traveling wave solution branch is unstable and perturbations evolve to either a stationary five-cell or a nonstationary mixed-mode solution exhibiting apparently quasi-periodic dynamics. If L is further decreased beyond a critical value L_x, the traveling wave solution branch becomes stable. Beyond another critical value L_H, the traveling wave branch loses stability to a branch of mixed mode, apparently quasi-periodic, solutions that appear to arise due to the interaction of unstable three- and four-cell traveling wave solutions.