Steady axisymmetric flow in an open cylindrical container with a partially rotating bottom wall

The steady motion of a viscous fluid in a cylindrical container with a partially rotating bottom wall and a free surface is investigated by means of axisymmetric Navier–Stokes simulations. The flow above the spinning disk at the center of the bottom wall is dominated by an Ekman boundary layer that drives the fluid radially outward. In contrast, an inward flow ensues along the outer, stationary part of the bottom wall, where the radially increasing pressure distribution set up by the rotating fluid motion near the free surface is not balanced by a corresponding centrifugal force. As a result, flow separation occurs at an intermediate radial location close to the outer edge of the rotating disk. Thus a flow configuration results that is dominated by a meridional vortex above the spinning disk, and a counterrotating vortex above the stationary part of the bottom wall. Simulations are conducted for various aspect ratios and Reynolds numbers, in order to evaluate the resulting changes in the vortex breakdown configurations. As the ratio of container radius to disk radius increases above a value of about 2.3, the influence of the lateral container wall on the features of the central flow in the neighborhood of the spinning disk becomes insignificant. By means of a simplified model problem, it is demonstrated that this rapid loss of influence is due to the exponential decay of the azimuthal surface velocity beyond the edge of the disk. This exponential decay is confirmed by the numerical data, and it reflects the fact that as the lateral wall moves outward, the stationary part of the end wall becomes the main sink for the azimuthal momentum of the fluid.