  How many fraction of gas available at the outer boundary can finally fall
onto the black hole is an important question. It determines the observational
appearance of accretion flows, and is also related with the evolution of black
hole mass and spin. Previous two-dimensional hydrodynamical simulations of hot
accretion flows find that the flow is convectively unstable because of its
inward increase of entropy. As a result, the mass accretion rate decreases
inward, i.e., only a small fraction of accretion gas can fall onto the black
hole, while the rest circulates in the convective eddies or lost in convective
outflows. Radiation is usually neglected in these simulations. In many cases,
however, radiative cooling is important. In the regime of the luminous hot
accretion flow (LHAF), radiative cooling is even stronger than the viscous
dissipation. In the one dimensional case, this implies that the inward increase
of entropy will become slower or the entropy even decreases inward in the case
of an LHAF. We therefore expect that convective instability becomes weaker or
completely disappears when radiative cooling is important. To examine the
validity of this expectation, in this paper we perform two-dimensional
hydrodynamical simulations of hot accretion flows with strong radiative
cooling. We find that compared to the case of negligible radiation, convection
only becomes slightly weaker. Even an LHAF is still strongly convectively
unstable, its radial profile of accretion rate correspondingly changes little.
We find the reason is that the entropy still increases inward in the
two-dimensional case.
