  The large range of time and length scales involved in type Ia supernovae (SN
Ia) requires the use of flame models. As a prelude to exploring various options
for flame models, we consider, in this paper, high-resolution three-dimensional
simulations of the small-scale dynamics of nuclear flames in the supernova
environment in which the details of the flame structure are fully resolved. The
range of densities examined, 1 to $8 \times 10^7$ g cm$^{-3}$, spans the
transition from the laminar flamelet regime to the distributed burning regime
where small scale turbulence disrupts the flame. The use of a low Mach number
algorithm facilitates the accurate resolution of the thermal structure of the
flame and the inviscid turbulent kinetic energy cascade, while implicitly
incorporating kinetic energy dissipation at the grid-scale cutoff. For an
assumed background of isotropic Kolmogorov turbulence with an energy
characteristic of SN Ia, we find a transition density between 1 and $3 \times
10^7$ g cm$^{-3}$ where the nature of the burning changes qualitatively. By $1
\times 10^7$ g cm$^{-3}$, energy diffusion by conduction and radiation is
exceeded, on the flame scale, by turbulent advection. As a result, the
effective Lewis Number approaches unity. That is, the flame resembles a laminar
flame, but is turbulently broadened with an effective diffusion coefficient,
$D_T \sim u' l$, where $u'$ is the turbulent intensity and $l$ is the integral
scale. For the larger integral scales characteristic of a real supernova, the
flame structure is predicted to become complex and unsteady. Implications for a
possible transition to detonation are discussed.
