The purpose of this article is the direct numerical simulation (DNS) of the complex phenomena that precede the transition to turbulence inside a cavity subjected to rotation. The configurations of cylindrical cavities subjected to a radial throughflow or to a differential rotation of the walls are relevant to rotating machinery devices. At a high rotation rate, the DNS exhibits instability patterns arising inside the thin layers close to the disks. The efficient spectral solver is based on a Chebyshev-Fourier approximation. For large aspect ratio and at high Reynolds number, an instability occurs inside the Ekman and Bödewadt layers in the form of annular and spiral vortices that are characteristic of type I and type II instabilities.