In a gas turbine, ingestion of hot gas into the high-pressure turbine disc cavities could cause metal overheat. To prevent this, cool air is taken from the compressor and ejected through the cavities. However, this sealing flow also reduces the overall efficiency, and a compromise has to be found between the level of ingestion tolerated and the losses. Recent advances made in applying Computational Fluid Dynamics to such configurations are presented, with the aim of better understanding the physical phenomena and providing reliable design tools. First, results showing the pumping effect of the rotating disc are presented, including the influence of flow instabilities observed in both computational and experimental results. Second, the influence of the main annulus pressure asymmetries are analysed on a simplified representation of an available experiment, showing the combined influence of asymmetries generated by vanes and struts. Finally, a rim seal geometry representative of aero-engine design is studied in comparison to experiment, exhibiting the coupled influence of the cavity instabilities and annulus asymmetries.