This is a sequel to the previous paper, in which we investigated the structure and stability of the spherically symmetric accretion flows through the standing shock wave onto the proto-neutron star in the postbounce phase of the collapse-driven supernova. Following the prescription in the previous paper, 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 steady solutions for a wide range of neutrino luminosities and mass accretion rates. In so doing, as an extension to the previous models, we employ a realistic EOS and neutrino-heating rate. 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, which is in accord with the results of multidimensional numerical simulations done over the years. We also consider the effect of the self-gravity, which was neglected in the previous paper. 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.
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