The 1[ratio]2 mode interaction in exactly counter-rotating von Kármán swirling flow

Abstract

The flow produced in an enclosed cylinder of height-to-radius ratio of two by the counter-rotation of the top and bottom disks is numerically investigated. When the Reynolds number based on cylinder radius and disk rotation is increased, the axisymmetric basic state loses stability and different complex flows appear successively: steady states with an azimuthal wavenumber of 1; travelling waves; near-heteroclinic cycles; and steady states with an azimuthal wavenumber of 2. This scenario is understood in a dynamical system context as being due to a nearly codimension-two bifurcation in the presence of $O(2)$ symmetry. A bifurcation diagram is determined, together with the most dangerous eigenvalues as functions of the Reynolds number. Two distinct types of near-heteroclinic cycles are observed, with either two or four bursts per cycle. The physical mechanism for the primary instability could be the Kelvin–Helmholtz instability of the equatorial azimuthal shear layer of the basic state.

Slow Dynamics in a Turbulent von Kármán Swirling Flow

“…These vortices have a characteristic size D vortex = 5 cm and they appear simultaneously to the disymmetry of the mean flow. Previous works in similar configurations [7] showed the formation of a static vortex in the equatorial plane (z = 0) for much lower Re, inaccessible with the present configuration.…”

“…These flows have been studied analytically [3,4], numerically [5,6,7] and experimentally [8,9,10]. Recently it has been shown that they can present multistability and memory effects [11].…”

Numerical and experimental study of the time-dependent states and the slow dynamics in a von Kármán swirling flow

“…In the range 412 Re 452 unsteady states are predicted theoretically and observed experimentally, consisting in a travelling wave and a modulated wave of the previous one. Increasing again the rotation rate the stable solution is found to be the steady ''Pure mode'', for Re up to 500 by Nore et al (2003Nore et al ( , 2004.…”

“…At higher Reynolds number and s ¼ 0 the flow is unstable to azimuthal travelling waves (Gelfgat et al 1996(Gelfgat et al , 2001). The case of cavities with À $ O(1), with the end walls in exact counter rotation, s ¼ À1, has been studied both experimentally and theoretically by Gelfgat et al (1996), Nore et al (2003Nore et al ( , 2004Nore et al ( , 2005. The base flow is axisymmetric and the instability of the steady flow sets up due to free shear layer instabilities through a Kelvin Helmholtz instability.…”

“…The results of the stability analysis in this configuration, and the investigation of states at low Reynolds numbers, were provided by Nore et al (2003). At critical Reynolds number Re c,1 ¼ 349 the axisymmetric flow becomes unstable to non-axisymmetric perturbations leading to a steady mode called ''Mixed mode'' (Nore et al 2003(Nore et al , 2004(Nore et al , 2005.…”