Primary and Secondary Hopf Bifurcations in Stratified Taylor-Couette Flow

Abstract

Experimental and theoretical results for the axially, stably stratified Taylor-Couette flow are presented. The primary instability is a direct Hopf bifurcation. It leads the system into an oscillatory state of confined internal waves, in good agreement with linear stability analysis. The secondary bifurcation, which leads the system to a pattern of drifting nonaxisymmetric vortices, is a subcritical Hopf bifurcation. This first experimental evidence of a global bifurcation is thought to be generic to dynamical systems with one destabilizing and one stabilizing control parameter.[S0031-9007(99)09248-0] PACS numbers: 47.20.Ky, 47.54. + r, 47.55.Hd The transition to chaos in extended nonlinear systems is still an open question [1][2][3]. The oscillatory nature of the primary bifurcation changes the scenario [4,5] and has not yet been fully explored. Such a primary Hopf bifurcation can be obtained by adding a stabilizing control parameter, allowing the existence of waves, to a dynamical system. Simply adding a stable, axial density stratification to the well-known Taylor-Couette flow [6], with only the inner cylinder rotating, allowed us to get such a rich dynamical system.In this Letter we present new experimental and theoretical results of the stratified Taylor-Couette flow (STC) that describe its first bifurcations and flow regimes. A novel bifurcation diagram is found, explaining the discrepancies between previous experimental studies [7,8] and numerical simulations [9]. The diagram enlightens the interaction between two branches and gives evidence to an unusual saddle-node transition. Such a diagram has never been observed experimentally. Our results apply to other systems such as rotating Rayleigh-Bénard convection [5], binary fluid convection [10], and convection subjected to a magnetic field, where it has been suggested theoretically [11].The apparatus, which is the same as in [7], consists of two coaxial cylinders of length L 573 mm. Only the inner cylinder of radius a 40 mm rotates, with the outer one b 52 mm, being at rest. The radius ratio of the cylinders is h a͞b 0.769 indicating a rather wide gap. The linear stratification is achieved with a salt solution. The control parameters are the rotation rate of the inner cylinder V (destabilizing parameter), and the Brunt-Vaïsälä frequency N 2 2͑g͞r 0 ͒≠r͞≠z . 0 (stabilizing parameter) which represents the stratification; z is the vertical direction. The Schmidt number (ratio of kinematic viscosity over molecular diffusivity) is 730. In all experiments, the observations and measurements were performed for times much smaller than the typical mixing time in the regimes of interest ͑ϳ5 h͒. In order to exhibit the structures in the different flow regimes, fluorescein dye, illuminated with a laser sheet cutting through the axis of the cylinders, is used. The temporal behavior of the flow was obtained by point measurements of the density fluctuations, using conductivity probes mounted flush with the outer cylinder. Two probes were fixed at the same height, at an ...

Non-axisymmetric instability of centrifugally stable stratified Taylor–Couette flow

“…They also show that the initial instability is axisymmetric, while non-axisymmetric motions appear only at a slightly higher Reynolds number than the critical one. Further studies of this problem are reported in Boubnov et al (1996), Boubnov & Hopfinger (1997), and Caton, Janiaud & Hopfinger (1999).…”

Flow instability and transitions in Taylor–Couette flow of a semidilute non-colloidal suspension

“…Taylor–Couette flow (TCF) between two coaxial cylinders has been one of the paradigmatic flows for a long time, since it allows the study of various physical phenomena in fluid dynamics, such as flow instability and transition (Coles 1965; DiPrima, Eagles & Ng 1984; Marques & Lopez 1997; Caton, Janiaud & Hopfinger 1999; Hristova et al. 2002), nonlinear dynamics (Pfister & Rehberg 1981; Mullin, Cliffe & Pfister 1987), pattern formation (Andereck, Liu & Swinney 1986; Boubnov, Gledzer & Hopfinger 1995; Lim, Chew & Xiao 1998; Guillerm et al.…”

“…Taylor-Couette flow (TCF) between two coaxial cylinders has been one of the paradigmatic flows for a long time, since it allows the study of various physical phenomena in fluid dynamics, such as flow instability and transition (Coles 1965;DiPrima, Eagles & Ng 1984;Marques & Lopez 1997;Caton, Janiaud & Hopfinger 1999;Hristova et al 2002), nonlinear dynamics (Pfister & Rehberg 1981;Mullin, Cliffe & Pfister 1987), pattern formation (Andereck, Liu & Swinney 1986;Boubnov, Gledzer & Hopfinger 1995;Lim, Chew & Xiao 1998;Guillerm et al 2015) and turbulence (Lathrop, Fineberg & Swinney 1992a,b;Huisman et al 2012;Grossmann, Lohse & Sun 2016). Moreover, flows of a dispersed phase (bubbles, drops or particles) and the migration behaviour of dispersions have been widely examined in the Taylor-Couette system due to numerous practical applications in chemical engineering, such as the oil industry, filtration and transformation of thermal energy (Chan & Leal 1981;Wereley & Lueptow 1999;Resende et al 2001;Rudman 2004;Climent, Simonnet & Magnaudet 2007).…”

Connections between centrifugal, stratorotational, and radiative instabilities in viscous Taylor-Couette flow

“…With the exception of [8], pre-2001 laboratory experiments on stratified Taylor-Couette flow were always carried out with a fixed outer cylinder [8][9][10][11][12] (so µ = 0 < η 2 ) and the relevance of the Rayleigh line was not questioned. The first experimental evidence of the SRI came in 2007 [13] where non-axisymmetric instability was clearly observed in the centrifugally-stable regime.…”