Annular and spiral patterns in flows between rotating and stationary discs

Different instabilities of the boundary layer flows that appear in the cavi ty between stationary and rotating discs are investigated using three-dimen sional direct numerical simulations. The influence of curvature and confine ment is studied using two geometrical configurations: (i) a cylindrical cav ity including the rotation axis and (ii) an annular cavity radially confine d by a shaft and a shroud. The numerical computations are based on a pseudo -spectral Chebyshev-Fourier method for solving the incompressible Navier-St okes equations written in primitive variables. The high level accuracy of t he spectral methods is imperative for the investigation of such instability structures, The basic how is steady and of the Batchelor type. At a critic al rotation rate, stationary axisymmetric and/or three-dimensional structur es appear in the Bodewadt and Ekman layers while at higher rotation rates a second transition to unsteady flow is observed. All features of the transi tions are documented. A comparison of the wavenumbers, frequencies, and pha se velocities of the instabilities with available theoretical and experimen tal results shows that both type II (or A) and type I (or B instabilities a ppear, depending on flow and geometric control parameters. Interesting patt erns exhibiting the coexistence of circular and spiral waves are found unde r certain conditions.