Planar flames are intrinsically unstable in open domains because of thermal expansion across the burning front, i.e., the Landau-Darrieus instability. This instability leads to wrinkling and growth of the flame surface and corresponding acceleration of the flame, until it is stabilized by cusp formation. We look at the Landau-Darrieus instability for C/O thermonuclear flames at conditions relevant to the late stages of a Type Ia supernova explosion. Two-dimensional direct numerical simulations of both single-mode and multimode perturbations using a low Mach number hydrodynamics code are presented. We show the effect of the instability on the flame speed as a function of both the density and domain size, demonstrate the existence of the small-scale cutoff to the growth of the instability, and look for the proposed breakdown of the nonlinear stabilization at low densities. The effects of curvature on the flame are quantified through measurements of the growth rate and computation of the corresponding Markstein number. While accelerations of a few percent are observed, they are too small to have any direct outcome on the supernova explosion.