The source of viscosity in astrophysical accretion flows is still a hotly debated issue. We investigate the contribution of convective turbulence to the total viscosity in a self-consistent approach, where the strength of convection is determined from the vertical disc structure itself. Additional sources of viscosity are parametrized by a beta-viscosity prescription, which also allows an investigation of self-gravitating effects. In the context of accretion discs around stellar mass and intermediate mass black holes, we conclude that convection alone cannot account for the total viscosity in the disc, but significantly adds to it. For accretion rates up to 10 per cent of the Eddington rate, we find that differential rotation provides a sufficiently large underlying viscosity. For higher accretion rates, further support is needed in the inner disc region, which can be provided by a magnetorotational instability (MRI)-induced viscosity. We briefly discuss the interplay of MRI, convection and differential rotation. We conduct a detailed parameter study of the effects of central masses and accretion rates on the disc models, and find that the threshold value of the supporting viscosity is determined mostly by the Eddington ratio with only little influence from the central black hole mass.