In this research a spectral element method is used to perform direct numerical simulation (DNS) and implicit large-eddy simulation (LES) of flows induced by centrifugal buoyancy in rotating cavities. These flows occur, for instance, in the compressor cavities of gas turbines internal air systems, in which cooling air is used to extract heat from compressor disks. Buoyancy-induced flows are inherently challenging to study using computational fluid dynamics (CFD), since turbulence models based on the Reynolds-averaged Navier-Stokes (RANS) equations are not able to provide an accurate description of the phenomena induced by the interplay between buoyancy and rotation. For this reason, model-free approaches are desirable, since they can provide an accurate description of the flow physics. First, the method is applied to a rotor/stator configuration, in which regions of laminar, transitional and fully turbulent flow coexist, and the results are compared with experimental data from the literature. Subsequently, flow induced by centrifugal buoyancy in a sealed rotating annulus is investigated using linear stability analysis, DNS and LES. It is shown that the onset of convection for a rotating cavity is similar to that for the problem of Rayleigh-Bénard convection. Analysing flow statistics for different values of the Rayleigh number, it is shown that the disk boundary layer behaves as a laminar Ekman layer, both in terms of its thickness and of its velocity profiles. This is observed even when instantaneous profiles are considered, despite the unsteadiness of the solution. The results also show that the shroud thermal boundary layer scaling is consistent with that of natural convection under gravity. Introducing an axial throughflow of cooling air, some features observed in the sealed cavity are maintained, however a strong reduction in the core temperature and a corresponding increase in the shroud heat transfer occur. The axial throughflow also promotes a significant increase in the range of frequencies observed inside the cavity.