The laminar and turbulent natural-convection flow in a two-dimensional square cavity heated from the vertical side is numerically calculated up to a Rayleigh number of 1014 for air and up to 1015 for water. Three different turbulence models are compared: the standard k-ε model with logarithmic wall functions and the low-Reynolds-number models of Chien, and Jones and Launder. The position of the laminar-turbulent transition in the vertical boundary layer strongly depends on the turbulence model used. Moreover, multiple solutions for the transition position can occur for a fixed Rayleigh number at the same numerical grid. The thermal stratification in the core of the cavity breaks up when the flow becomes turbulent. Comparison of the averaged wall-heat transfer with experiments for the hot vertical plate and for tall vertical cavities shows that the standard k-ε model gives a too high prediction, whereas the low-Reynolds-number models are reasonably close to the experiment.