The Darrieus–Landau instability is an intrinsicflame instability that occurs in premixed flames due to the thermal expansion of the gas produced by the combustion process. It was predicted independently by Georges Jean Marie Darrieus and Lev Landau. The instability analysis behind the Darrieus–Landau instability considers a planar, premixed flame front subjected to very small perturbations. It is useful to think of this arrangement as one in which the unperturbed flame is stationary, with the reactants directed towards the flame and perpendicular to it with a velocity u1, and the burnt gases leaving the flame also in a perpendicular way but with velocity u2. The analysis assumes that the flow is an incompressible flow, and that the perturbations are governed by the linearized Euler equations and, thus, are inviscid. With these considerations, the main result of this analysis is that, if the density of the burnt gases is less than that of the reactants, which is the case in practice due to the thermal expansion of the gas produced by the combustion process, the flame front is unstable to perturbations of any wavelength. Another result is that the rate of growth of the perturbations is inversely proportional to their wavelength; thus small flame wrinkles grow faster than larger ones. In practice, however, diffusive and buoyancy effects that are not taken into account by the analysis of Darrieus and Landau may have a stabilizing effect. Amable Liñán and Forman A. Williams quote in their book that it was indeed courageous of Darrius and Landau to publish - in the face of laboratory evidence of the existence of stable planar laminar flames - analyses showing the instability of a plane flame sheet to disturbances of all wavenumbers.
Dispersion relation
If the disturbances to the steady planar flame sheet are of the form, where is the transverse coordinate system perpendicular to the steady flame sheet, is the time, is the wavevector of the disturbance and is the temporal growth rate of the disturbance. Then the dispersion relation is given by where is the laminar burning velocity, and is the ratio of unburnt to burnt gas density. Since always in combustion due to the thermal expansion by heat release, the growth rate is also always positive for all wavenumbers. The whole analysis becomes invalid when the wavelength becomes comparable to the flame thickness, i.e.,, where is the thermal diffusivity. If the buoyancy forces are taken into account for vertically propagating planar flames so that the gravity vector with magnitude points from the unburnt side to burnt gas side, the dispersion relation becomes This implies that gravity introduces stability when. The range of instability region due to the density jump across the flame becomes. If, then there is no hydrodynamic instability.
