Nonlinear development of hydrodynamically-unstable flames in three-dimensional laminar flows

The hydrodynamic instability, which results from the large density variations between the fresh mixture and the hot combustion products, was discovered by Darrieus and Landau over seventy years ago, and has been named after its inventors. The instability, which prevents flames from being too flat, was thought to lead immediately to turbulent flames. Recent studies, initiated by weakly nonlinear analyses and extended by two-dimensional simulations suggest that this is not the case. It was established that the flame beyond the onset of instability, develops into a cusp-like structure pointing towards the burned gas region that propagates at a speed substantially larger than the laminar flame speed. In this work, we present for the first time a systematic study of the bifurcation phenomena in the more realistic three-dimensional flow. The computations are carried out within the context of the hydrodynamic theory where the flame is treated as a surface of density discontinuity separating burned gas from the fresh mixture, and propagates at a speed that depends on the local curvature and hydrodynamic strain rate. A low Mach-number Navier-Stokes solver modified by an appropriate source term is used to determine the flow field that results from the gas expansion and the flame is tracked using a level-set methodology with a surface parameterization method employed to accurately capture the local velocity and stretch rate. The numerical scheme is shown to recover the known exact solutions predicted in the weak gas expansion limit and corroborates the bifurcation results from linear stability analysis. The new conformations that evolve beyond the instability threshold have sharp crest pointing towards the burned gas with ridges along the troughs, and propagate nearly 40% faster than planar flames. Indeed, the appearance of sharp folds and creases, which are some manifestations of the Darrieus-Landau instability, have been observed on the surface of premixed flames in various laminar and turbulent settings.