Asymptotic flame shapes and speeds of hydrodynamically unstable laminar flames

The self-induced baroclinic instability of flames, the Landau-Darrieus instability, is studied numerically in the nonlinear range. A level set (G equation) based approach accounting for heat-release effects is used to follow the flame response to initial perturbation and shape evolution. It is shown that the instability leads to the development of product bubbles moving into the unburned mixture and cold mixture spikes penetrating into the burned gases similar to the bubble-spike configuration produced by the nonlinear Rayleigh-Taylor instability of interfaces in a gravitational field. Independently of the initial perturbation, the flame bubble approaches asymptotically a shape close to a paraboloid. The substantial growth of the flame-surface area due to the instability increases the flame propagation speed to the asymptotic value;$+0.29\sqrt{\alpha -1)}{S}_l$where α is the density ratio and S_l the laminar burning velocity. The asymptotic amplitude of the flame is approximately$A=0.37d\sqrt{\alpha -1}$, where d is the flame width. The burning velocity has a minor effect on the asymptotic shape of the flames. When the turbulence scale is much smaller than the size of the combustion apparatus, the results can be directly applied to turbulent flames by replacing S₁ by the turbulent burning velocity.