Numerical and experimental study of the time-dependent states and the slow dynamics in a von Ka´rma´n swirling flow

Abstract

The characteristics of time-dependent swirling flows in a von Ka´rma´n configuration are investigated numerically up to Reynolds number Re¼3000 (based on the angular velocity and the radius of the cylinder), and experimentally within turbulent regimes. Experimental results are analyzed together with the periodic and aperiodic flow obtained numerically. In the present configuration, the fluid is contained in a cylindrical cavity with aspect ratio (height to radius) !¼2 and the motion is driven by the exact counter rotation of the end walls while the sidewall is at rest. Spectral direct numerical simulations show that for this geometry the axisymmetric base flow becomes unstable to non-axisymmetric perturbations with azimuthal wavenumber m¼1 through a subcritical bifurcation, and the corresponding flow exhibits a pattern with one cat’s eye in the axial-azimuthal planes. Increasing the Reynolds number the flow becomes unstable to non-axisymmetric steady perturbations with even azimuthal wavenumbers, and the corresponding flows exhibits a two cat’s eyes pattern. The occurrence of cat’s eye pattern in radial-azimuthal surfaces was observed in this and other aspect ratio cavities and is associated with vortices in 3D steady flows with characteristic azimuthal modes [Nore, C., Tuckerman, L.S., Daube, O. and Xin, S., The 1 : 2 mode interaction in exactly counter-rotating von Ka´rma´n swirling flow, J. Fluid Mech., 2003, 477, 51–88; Lackey, T.C. and Sotiropoulos, F., Relationship between stirring rate and Reynolds number in the chaotically advected steady flow in a container with exactly counter-rotating lids, Phys. Fluids, 2006, 18, 1–14]. Timedependent regimes are obtained numerically when the value of the Reynolds number is Re"1500. The time dependency is associated with a pulsation of the two vortices found in the steady regime. Experimental visualizations and measurements show that in turbulent regimes the flow also exhibits two vortices, but in this case they travel in the azimuthal direction with a frequency compatible with the frequency obtained in the numerical simulations at much lower Reynolds number. The azimuthal drift of these vortices is associated with the asymmetry of the mean azimuthal flow with respect to the equatorial plane.