The effect of physical boundaries on oscillatory bifurcation in counterrotating Taylor–Couette flow

Abstract

The results of an experimental study on the bifurcation structure of oscillatory modes in counterrotating Taylor–Couette flow with stationary end plates are presented. It is shown that the cylinder length L acts as an important geometric control parameter of the system. As a result of a supercritical Hopf bifurcation it is found that for an aspect ratio Γ=L/d>16 (d gap width) only spiral vortices appear in basic laminar flow. For Γ<10.5 spiral vortices are almost entirely replaced by two types of standing waves called SW0 and SWπ as supercritical oscillatory flow. Experimental evidence is presented that the mode exchange between standing waves SW0 and SWπ is governed by underlying Ekman induced vortices which appear as a result of stationary end plates in the flow. In this regime spiral vortices appear only from a sub- or supercritical symmetry breaking bifurcation of the standing waves. Within an “intermediate regime” between 10.5⩽Γ⩽16 spiral vortices are found to be the predominant primary oscillatory flow but small stability intervals of standing waves are also observed. Surprisingly, the experimentally determined critical Reynolds number is found to deviate not more than 2% from the numerical values for all values of aspect ratio even though they are calculated under the assumption of infinite axial length. Moreover, the critical oscillation frequency is also in agreement with the numerical values and is independent from Γ.

Bifurcation behavior of standing waves

“…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]. This has been predicted theoretically by Dangelmayr and Knobloch [17] and Landsberg and Knobloch [18] in a theory of Hopf bifurcation with broken translational invariance.…”

“…In previous studies [1,2] the appearance and the primary bifurcation of SW π and of another type of standing wave, i.e., SW 0 , have been investigated. Both types of standing waves appear in principle instead of spiral vortices as a consequence of broken translational invariance of the experimental system [17,18].…”

“…A Taylor-Couette apparatus with counter-rotating cylinders and non-rotating end plates is used for this study, as described in [1][2][3]. The inner cylinder of the apparatus is machined from stainless steel having a radius of r i = (12.50 ± 0.01) mm, while the outer cylinder is made from optically polished glass with a radius of r o = (25.00 ± 0.01) mm.…”

“…While the former part has been observed in experiments for both types of standing waves, i.e., SW 0 and SW π [1,2], the latter point has only been investigated for SW 0 [3]. In this work we focus on higher bifurcations from the other type of standing wave in counter-rotating Taylor-Couette flow, i.e., SW π .…”

“…Two different types of standing waves (SW0 and SWπ) can appear instead of spiral vortices from a supercritical Hopf bifurcation in counter-rotating Taylor-Couette flow for sufficiently small aspect ratios [1,2]. The bifurcation sequence from basic flow to spiral vortices via SW0 can include modulated waves, homoclinic bifurcations, and hysteresis as a consequence of broken translational invariance [3].…”