  Flame instabilities play a dominant role in accelerating the burning front to
a large fraction of the speed of sound in a Type Ia supernova. We present a
three-dimensional numerical simulation of a Rayleigh-Taylor unstable carbon
flame, following its evolution through the transition to turbulence. A low Mach
number hydrodynamics method is used, freeing us from the harsh timestep
restrictions imposed by sound waves. We fully resolve the thermal structure of
the flame and its reaction zone, eliminating the need for a flame model. A
single density is considered, $1.5\times 10^7 \gcc$, and half carbon\slash half
oxygen fuel--conditions under which the flame propagated in the flamelet regime
in our related two-dimensional study. We compare to a corresponding
two-dimensional simulation, and show that while fire-polishing keeps the small
features suppressed in two dimensions, turbulence wrinkles the flame on far
smaller scales in the three-dimensional case, suggesting that the transition to
the distributed burning regime occurs at higher densities in three dimensions.
Detailed turbulence diagnostics are provided. We show that the turbulence
follows a Kolmogorov spectrum and is highly anisotropic on the large scales,
with a much larger integral scale in the direction of gravity. Furthermore, we
demonstrate that it becomes more isotropic as it cascades down to small scales.
Based on the turbulent statistics and the flame properties of our simulation,
we compute the Gibson scale. We show the progress of the turbulent flame
through a classic combustion regime diagram, indicating that the flame just
enters the distributed burning regime near the end of our simulation.
