We consider the case of a tall differentially heated cavity (containing air) for which the buoyancy flow is in the so-called boundary-layer regime (displaying thin regions featured by strong temperature gradients adjacent to the vertical walls and thermally stratified fluid in the centre). By using a numerical method able to account for compressibility and variable viscosity effects, we investigate the response of such a system to the application of temperature differences in a range of characteristic numbers for which the resulting flow is expected to be time-periodic or highly time-dependent. More precisely, the problem is examined by allowing both the Rayleigh number and the inclination (0) of this configuration with respect to the vertical direction to span relatively wide intervals (1.8 x 10(5) <= Ra <= 3 x 10(6), -90 degrees <= theta <= 90 degrees). Insights into the influence of variable thermodynamic properties are obtained by comparing such numerical results with equivalent simulations based on the classical Boussinesq approximation. The evolution of the considered system towards a fully turbulent state is also discussed to a certain extent. (C) 2018 Elsevier Ltd. All rights reserved.