Spectral solution of the Navier-Stokes equations for rotating flows

Direct numerical simulation by spectral methods are developed and used to study the instabilities in the Ekman and Bodewadt layers at the transition to the time-dependent regimes. The physical phenomena are characteristic of rotating flows with walls. The geometrical cavities are elementary geometries that are relevant of turbine applications and that also refer to typical configurations studied in fundamental investigations and in experiments. The three - dimensional Chebyshev - Fourier collocation method is based on a projection scheme to solve the coupling between the velocity and the pressure. The method devoted to annular domain is extended to fully cylindrical domain involving the axis of rotation. A special development has been required to deal with the singular behaviour of the coefficients when the radius tends to zero. The investigation was carried out in two types of generic configurations that are the forced throughflow in a rotating cavity and the confined flout driven by the differential rotation inside a rotor-stator cavity. Depending on the aspect ratio and on the Reynolds number, counter-rotating rolls can superimpose to the boundary layer flow near the disks under the form of annular and spiral structures. The characteristic parameters of the perturbations (wavelength, frequency, phase velocity, inclination of the spiral) are shown to be relevant of the types I and II instabilties in rotating flows.