This paper discusses the application of RANS models to the computation of two-dimensional natural convection flows in different types of differentially heated cavities. They include a square cavity with differentially heated vertical walls and tall rectangular cavities, with differentially heated long walls, which are either vertical, or inclined. For the modelling of the turbulent stresses, eddy-viscosity-based two-equation models, as well as second-moment closures, have been employed. The eddy-diffusivity approach has been used for modelling the turbulent heat fluxes in the former, while second-moment-closure computations have employed both the generalised gradient diffusion hypothesis and a more elaborate algebraic model, which involves solving additional transport equations for the scalar variance and its dissipation rate. For the modelling of the near-wall turbulence, the eddy-viscosity computations involved the use of either a low-Reynolds-number model with fine near-wall grids, or a high-Reynolds-number version with wall functions. The standard log-law-based wall function (SWF) and also the more advanced analytical wall function (AWF), which does not rely on a prescribed near-wall velocity profile, have been tested. Only high-Reynolds number versions of the second-moment closures have been employed, with the two types of wall function mentioned above. For the square cavity, where the flow is turbulent in only the boundary layer regions, use of a low-Reynolds-number k-epsilon model resulted in the prediction of entirely laminar flow, while of the two high-Re versions, the one with the analytical wall function produced the more satisfactory predictions. Introduction of second-moment closures improved the predictions of the turbulence field. For the tall cavities, the vertical and the inclined cavity at 60 degrees to the horizontal, with the upper side heated, produce similar mean fields, and most of the models tested perform reasonably well over much of the domain. The k-epsilon model with all the near-wall treatments tested produces reasonable predictions of the local Nusselt number variation with levels which are slightly lower than the experimental data. The stress transport schemes tend to over-predict the mixing near the end walls, although this can be improved by adopting a more complex model for the turbulent heat fluxes. This more elaborate thermal model has little effect on the predicted local Nusselt number, which is already in close agreement with the measurements. There are subtle differences in the turbulence fields between the 60 degrees and 90 degrees cases, some of which are captured by the stress transport models. When the cavity is further inclined to an angle of 5 degrees, still under stable heating arrangements, the turbulence levels are quite low and, as a result, most of the models predict fairly similar mean flow profiles, in reasonable agreement with available LES data. (C) 2014 Elsevier Ltd. All rights reserved.