Spiral vortices traveling between two rotating defects in the Taylor-Couette system

Abstract

Numerical calculations of vortex flows in Taylor-Couette systems with counter rotating cylinders are presented. The full, time-dependent Navier-Stokes equations are solved with a combination of a finite difference and a Galerkin method. Annular gaps of radius ratio eta=0.5 and of several heights are simulated. They are closed by nonrotating lids that produce localized Ekman vortices in their vicinity and that prevent axial phase propagation of spiral vortices. The existence and spatiotemporal properties of rotating defects, modulated Ekman vortices, and the spiral vortex structures in the bulk are elucidated in quantitative detail.

Spatio-temporal behavior of spiral vortex flow

“…However, slightly asymmetric SW can appear within the transition sequence from SW to spiral vortices [16,17,23]. This behavior is also in agreement with recent results of numerical simulation of the Navier-Stokes equation by Hoffmann et al [1]. Analogue to the analysis of SW, also the axial scan of a downward traveling spirals R-SPI(as already shown in figure 1(c)) is analyzed in the following.…”

“…But the amplitudes of both types of spiral modes are almost equal in the vicinity of the end plates and decay directly at the end plates. Especially at the bottom end plate, where the phase of R-SPI-mode is annihilated, the amplitudes of both modes are almost equal and form together with the Ekman vortex structure a 'wavy-like' flow in this regime as numerically observed by Hoffmann et al [1]. This structure can even be observed in (a) and a nodal point, as typically observed for SW, exist at z ≈ 3d.…”

“…As a further consequence non-rotating rigid end plates at top and bottom subcritically innervate axisymmetric Ekman vortices in the annulus. Recent numerical simulations have shown that 'pure' spiral vortices only exist in a bulk region which is bound by a 'wavy-like' rotating defects in the vicinity of the axisymmetric Ekman vortices near the end plates [1]. Spiral vortices in finite systems are therefore predicted to differ qualitatively in their spatio-temporal properties from 'pure' spirals in infinite systems.…”

Bifurcation behavior of standing waves

“…Spiral vortices have been first calculated theoretically by Krueger et al [12] and observed experimentally by Snyder [13]. Subsequent experimental and numerical studies on spiral vortices have been carried out by Andereck et al [14], Langford et al [11] as well as Sanchez et al [15] and Hoffmann et al [16], respectively, in a wide Reynolds number regime for different radius and aspect ratios.…”

“…Non-rotating rigid end plates at top and bottom as often used in experimental systems change the properties of spirals in counter-rotating Taylor-Couette flow due to the presence of rotating defects in the vicinity of the axisymmetric Ekman vortices near the end plates [16]. As a consequence of broken translational invariance in experimental systems two different types of standing waves (denoted SW 0 and SW π ) are found to replace spiral vortices as the primary pattern that occur from a supercritical Hopf bifurcation of the basic laminar flow [1,2].…”