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Planar flames are intrinsically unstable in open domains due to the thermalexpansion across the burning front--the Landau-Darrieus instability. Thisinstability leads to wrinkling and growth of the flame surface, andcorresponding acceleration of the flame, until it is stabilized by cuspformation. We look at the Landau-Darrieus instability for C/O thermonuclearflames at conditions relevant to the late stages of a Type Ia supernovaexplosion. Two-dimensional direct numerical simulations of both single-mode andmulti-mode perturbations using a low Mach number hydrodynamics code arepresented. We show the effect of the instability on the flame speed as afunction of both the density and domain size, demonstrate the existence of thesmall scale cutoff to the growth of the instability, and look for the proposedbreakdown of the non-linear stabilization at low densities. The effects ofcurvature on the flame as quantified through measurements of the growth rateand computation of the corresponding Markstein number. While accelerations of afew percent are observed, they are too small to have any direct outcome on thesupernova explosion.Comment: accepted to ApJ. Figure 1 degraded for size constraints. replaced  with accepted version (one new figure added

Direct Numerical Simulations of Type Ia Supernovae Flames. I. The Landau‐Darrieus Instability