  We investigate neutrino-driven convection in core collapse supernovae and its
ramifications for the explosion mechanism. We begin with an ``optimistic'' 15
solar mass precollapse model, which is representative of the class of stars
with compact iron cores. This model is evolved through core collapse and bounce
in one dimension using multigroup (neutrino-energy--dependent) flux-limited
diffusion (MGFLD) neutrino transport and Lagrangian hydrodynamics, providing
realistic initial conditions for the postbounce convection and evolution. Our
two-dimensional simulation begins at 106 ms after bounce at a time when there
is a well-developed gain region, and proceeds for 400 ms. We couple
two-dimensional (PPM) hydrodynamics to one-dimensional MGFLD neutrino
transport. At 225 ms after bounce we see large-scale convection behind the
shock, characterized by high-entropy, mushroom-like, expanding upflows and
dense, low-entropy, finger-like downflows. The upflows reach the shock and
distort it from sphericity. The radial convection velocities become supersonic
just below the shock, reaching magnitudes in excess of 10^9 cm/sec. Eventually,
however, the shock recedes to smaller radii, and at about 500 ms after bounce
there is no evidence in our simulation of an explosion or of a developing
explosion. Failure in our ``optimistic'' 15 solar mass Newtonian model leads us
to conclude that it is unlikely, at least in our approximation, that
neutrino-driven convection will lead to explosions for more massive stars with
fatter iron cores or in cases in which general relativity is included.
