  We investigated the structure of the spherically symmetric accretion flows
through the standing shock wave onto the proto-neutron star in the post-bounce
phase of the collapse-driven supernova. We assume that the accretion flow is in
a steady state controlled by the neutrino luminosity and mass accretion rate
that are kept constant. We obtain solutions of the steady Euler equations for a
wide range of neutrino luminosity and mass accretion rate. We employ a
realistic EOS and neutrino-heating rates. More importantly, we take into
account the effect of convection phenomenologically. For each mass accretion
rate, we find the critical neutrino luminosity, above which there exists no
steady solution. These critical points are supposed to mark the onset of the
shock revival. As the neutrino luminosity increases for a given mass accretion
rate, there appears a convectively unstable region at some point before the
critical value is reached. We introduce a phenomenological energy flux by
convection so that the negative entropy gradient should be canceled out. We
find that the convection lowers the critical neutrino luminosity substantially.
We also consider the effect of the self-gravity. It is found that the
self-gravity is important only when the neutrino luminosity is high. The
critical luminosity, however, is little affected if the energy transport by
convection is taken into account.
