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First results of three-dimensional simulations of a thermonuclear flame in Type~Ia supernovae are obtained using a new flame-capturing algorithm, and a PPM hydrodynamical code. In the absence of gravity, the flame is stabilized with respect to the Landau (1944) instability due to the difference in the behaviour of convex and concave portions of the perturbed flame front. The transition to turbulence in supernovae occurs on scales $\simeq 0.1 - 10$ km in agreement with the non-linear estimate $\lambda \simeq 2\pi D^2_l/g_{eff}$ based on the Zeldovich (1966) model for a perturbed flame when the gravity acceleration increases; $D_l$ is the normal speed of the laminar flame, and $g_{eff}$ is the effective acceleration.
The turbulent flame is mainly spread by large scale motions driven by the Rayleigh-Taylor instability. Small scale turbulence facilitates rapid incineration of the fuel left behind the front. The turbulent flame speed $D_t$ approaches $D_t \simeq U'$, where $U'$ is the root mean square velocity of turbulent motions, when the turbulent flame forgets initial conditions and reaches a steady state. The results indicate that in a steady state the turbulent flame speed should be independent of the normal laminar flame speed $D_l$. The three-dimensional results are in sharp contrast with the results of previous two-dimensional simulations which underestimate flame speed due to the lack of turbulent cascade directed in three dimensions from big to small spatial scales. The work was supported by the NSF grants AST 92-18035 and AST 93-005P.
