Stratified circular Couette flow: instability and flow regimes

Abstract

The stability conditions of the flow between two concentric cylinders with the inner one rotating (circular Couette flow) have been investigated experimentally and theoretically for a fluid with axial, stable linear density stratification. The behaviour of the flow, therefore, depends on the Froude number Fr = Ω/N (where Ω is the angular velocity of the inner cylinder and N is the buoyancy frequency of the fluid) in addition to the Reynolds number and the non-dimensional gap width ε, here equal to 0.275.Experiments show that stratification has a stabilizing effect on the flow with the critical Reynolds number depending on N, in agreement with linear stability theory. The selected, most amplified, vertical wavelength at onset of instability is reduced by the stratification effect and is for the geometry considered only about half the gap width. Furthermore, the observed instability is non-axisymmetric. The resulting vortex motion causes some mixing and this leads to layer formation, clearly visible on shadowgraph images, with the height of the layer being determined by the vertical vortex size. This regime of vertically reduced vortex size is referred to as the S-regime.For larger Reynolds and Froude numbers the role of stratification decreases and the most amplified vertical wavelength is determined by the gap width, giving rise to the usual Taylor vortices (we call this the T-regime). The layers which emerge are determined by these vortices. For relatively small Reynolds number when Fr ≈ 1 the Taylor vortices are stable and the layers have a height h equal to the gap width; for larger Reynolds number or Fr ≈ 2 the Taylor vortices interact in pairs (compacted Taylor vortices, regime CT) and layers of twice the gap width are predominant. Stratification inhibits the azimuthal wavy vortex flow observed in homogeneous fluid. By further increasing the Reynolds number, turbulent motions appear with Taylor vortices superimposed like in non-stratified fluid.The theoretical analysis is based on a linear stability consideration of the axisymmetric problem. This gives bounds of instability in the parameter space (Ω, N) for given vertical and radial wavenumbers. These bounds are in qualitative agreement with experiments. The possibility of oscillatory-type instability (overstability) observed experimentally is also discussed.

Stability of density-stratified viscous Taylor-Couette flows

“…2 we compare the calculated marginal stability line (i.e. (ω) = 0) for axisymmetric disturbances with experimental values by Boubnov et al (1995). The agreement is rather good except for small values of the Froude number…”

“…Further experimental and theoretical studies by Boubnov et al (1995) and Caton et al (2000) of flows between concentric rotating cylinders with a stable axial density stratification confirmed the stabilizing role of the density stratification and showed that i) the critical Reynolds number depends on the buoyancy frequency (or Brunt-Väisälä frequency) of the fluid; and ii) the stratification reduces the vertical extension of the Taylor vortices. The numerical results of Hua et al (1997) have reproduced the experimental results.…”

Experimental Analysis of the Stratorotational Instability in a Cylindrical Couette Flow

“…The first experiment on stratified Taylor-Couette flow was performed by Thorpe [3] in 1966. His results were further extended by the experiments of Withjack and Chen [4], Boubnov et al [5], Caton et al [6] and by the numerical simulations of Hua et al [7]. Most of these studies were performed with a stationary outer cylinder and all concluded that stable vertical stratification stabilizes the flow and reduces the axial wavelength of the unstable modes which bifurcate through Hopf bifurcations.…”

“…their pulsating axisymmetric modes) within the explored range, presumably because its unstable region is too narrow and our experimental device is not precise enough to excite this very marginal state (see the comparison between the results of [5] and [6]). Nevertheless, in experiments at µ = 0 and large Reynolds number, we observed the reappearance of stationary axisymmetric Taylor vortices in agreement with the regime diagram of Boubnov et al [5]. This stationary axisymmetric mode was never excited when decreasing µ for µ > 0, but once excited at µ = 0, they could then be maintained up to µ = 0.2.…”

Primary and Secondary Hopf Bifurcations in Stratified Taylor-Couette Flow

“…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.…”