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An important question is what fraction of gas available at the outer boundary can finally fall on to the black hole. This determines the observational appearance of accretion flows, and is also related to the evolution of black hole mass and spin. Previous two‐dimensional hydrodynamical simulations of hot accretion flows have found 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 on to the black hole), while the rest circulates in the convective eddies or is 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 the convective instability to become weaker or completely disappear 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, and 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.

On the convective instability of hot radiative accretion flows