Bifurcation phenomena in Taylor-Couette flow in a very short annulus with radial through-flow

Abstract

In this study, the non-linear dynamics of Taylor-Couette flow in a very small-aspect-ratio wide-gap annulus in a counter-rotating regime under the influence of radial through-flow are investigated by solving its full three-dimensional Navier-Stokes equations. Depending on the intensity of the radial flow, either an axisymmetric (pure m = 0 mode) pulsating flow structure or an axisymmetric axially propagating vortex will appear below the centrifugal instability threshold of the circular Couette flow. In contrast to earlier detected propagating vortices, these break Z2 symmetry and exist as a pair of degenerated solutions. Either pulsating or propagating solutions originate owing to the here considered small aspect ratio of the system, resulting in a constant competition between two-and one-cell states. With the variation of radial flow, a pulsating flow appears as a supercritical Hopf bifurcation from a stationary twin-pair state, persisting as the main feature. Thereafter, it undergoes period-doubling bifurcation and another Hopf bifurcation, leading to a quasi-periodic solution on a 2-torus. The propagating vortices are found to exist stably in two separate parameter regions, which feature different underlying dynamics. Although in one regime, the flow appears only as a limit cycle solution upon which saddle-node-invariant-circle bifurcation occurs, the other regime features more complex dynamics with richer Hopf bifurcation bifurcation sequences. The periods double to eventually introduce second and third incommensurate frequencies that change the flow to appear as 1-, 2-and 3-torus solutions.