Onset of centrifugal instability at a rotating cylinder in a stratified fluid

Abstract

In this experimental note, we consider the centrifugal instability of a laminar shear layer, generated by the impulsive start of the rotation of a circular solid cylinder about its vertical axis immersed in a linearly stratified fluid. The flow is determined by the Reynolds number, Re, based on the cylinder rotation rate and the cylinder radius, and the Froude number, F r , represented by the ratio of the rotation frequency Ω over the buoyancy frequency N. The onset of the instability starts when the boundary layer reaches a certain thickness. We show for this boundary layer that there is a transition from the centrifugally unstable regime to a wave-like regime at Fr ≈ 1 and a stable flow below a critical Reynolds number. We focus on the centrifugally unstable regime F r 1, for which the onset time and wavelength are predicted by scaling laws that depend on the Reynolds number. Agreement with the theoretical prediction of Kim and Choi ["The onset of instability in the flow induced by an impulsively started rotating cylinder," Chem. Eng. Sci. 60, 599-608 (2005)] in a homogeneous fluid confirms that the instability of this boundary layer is not modified by the presence of stratification. These results therefore show that the centrifugal instability of the spin-up boundary is dominated by inertial motions, suggesting that close lateral boundaries, as in thin-gap stratified Taylor-Couette flow, increase the effects of buoyancy on the instability and wavelength.

Impact of centrifugal buoyancy on strato-rotational instability

“…As such, the base flow is not unidirectional. This meridional flow will be shown here to be critical in determining the helical instability reported in the experiments of Flór et al (2018).…”

“…4). Flór et al (2018). The flow conditions correspond to salt-stratified Taylor-Couette flow in the very wide gap regime with radius ratio η = Ri/Ro = 1/15, Froude number Fr = 0.53 and inner cylinder Reynolds number Rei = 521.…”

“…where η = R i /R o is the radius ratio and γ = H/D is the aspect ratio. We shall fix the annular geometry to η = 0.07 and γ = 1, very close to those of the experimental apparatus used in Flór et al (2018).…”

“…The experiments of Flór et al (2018) used water with salt as the stratifying agent. Instead, we shall use temperature and fix the Prandtl number σ = 6, nominally corresponding to water at approximately 25 • C. Flór et al (2018) used Re i = Re η/(1 − η) and Fr to describe the parameter regimes of their experiments. With the non-dimensionalization we have used, Re and Gr are the non-dimensional groups that naturally appear.…”

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In a recent experiment on the flow between two concentric cylinders with the inner cylinder rotating and the fluid being stably stratified, Flór, Hirschberg, Oostenrijk & van Heijst (2018) found helical wave structures confined to the inner cylinder in an annulus with small inner-to-outer radius ratio (very large gap) in regimes where the Froude number (ratio of cylinder rotation frequency to buoyancy frequency) is less than one. These helical waves were reported to originate at the corners where the inner cylinder meets the top and bottom boundaries, and were found to be asymmetric with the lower helical wave being more intense.

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