Numerical, two-dimensional, time-dependent hydrodynamical models of geometrically thick accretion discs around black holes are presented. Accretion flows with non-effective radiation cooling (ADAFs) can be both convectively stable or unstable depending on the value of the viscosity parameter α. The high-viscosity flows (α ≃ 1) are stable and have a strong equatorial inflow and bipolar outflows. The low-viscosity flows (α ≤ 0.1) are convectively unstable and this induces quasi-periodic variability, with typical time-scales in the observed range of quasi-periodic oscillations. [ABSTRACT FROM AUTHOR]