Numerical study of axisymmetric vortex breakdown in an annulus

Summary

The steady axisymmetric Navier-Stokes equations describing the flow in a cylindrical container and in an annulus with one or two rotating endwalls were solved by the continuation method. The unsteady flow regime was obtained from steady solution by following the temporal development of a small axisymmetric disturbance introduced into the initial conditions for either the full or the linearized system of equations. By variation of the boundary condition for vorticity at the axis the transition of the flow in a conventional container to that in an annulus with infinitesimal inner radius was continuously followed; the flow in the latter case has been found to differ only insignificantly from that in the container. By further continution in the radii ratio (the ratio between the inner and the outer radii), flow patterns in an annulus were obtained. An annulus with the stress-free (slip) boundary condition at the inner wall was also considered in order to examine the influence of viscous effects on the formation of separation bubble. While for the solid inner cylinder the recirculation region turned out to persist even at very large values of the radii ratio, the stress-free condition imposed at this boundary leads to vanishing of the separation bubble already at relatively small values of this parameter. The transition to unsteady motion (which, in application to a cylindrical container, was shown by Tsitverblit [14] to be due to the supercritical Hopf bifurcation) was studied in comparison with the results for a cylindrical container.