Exploring how a stable stratification can destabilize Taylor-Couette flow

Significance 

The flow between rotating concentric cylinders is known as Taylor-Couette flow. As the rotation rate of the inner cylinder is increased, the circular Couette flow becomes unstable and eventually becomes turbulent. The fluid between the cylinders can be made vertically stably stratified by adding a salt in such a way that the density of the fluid at the bottom is greater than that at the top. Surprisingly, this stable stratification can destabilize the flow, so that instabilities onset at rotation rates where the unstratified Couette flow would be stable. This non-axisymmetric strato-rotational instability was discovered early in the twenty-first century, and has since been seen in experiments. When the stratification is strong, that is the buoyancy frequency is much larger than the rotation rate of the inner cylinder, researchers used Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) methods to show the inviscid problem is always unstable, even when the unstratified Rayleigh stability criterion is satisfied. The stable stratification allows types of wave motion not seen in the unstratified problem, and these waves can release the available kinetic energy to drive new instabilities.

Building on this previous work, the University of Leeds researchers Dr. Evy Kersalé and Professor Chris Jones, together with their postgraduate student Luke Robins, investigated the problem in which the inner cylinder rotation rate and the buoyancy frequency are comparable, so their ratio, the Froude number, is order unity. Their research is currently published in the Journal of Fluid Mechanics. They extended the inviscid WKBJ analysis to finite Froude number, and also investigated the onset of instability when viscosity is included. They examined numerically the range of cylinder radius ratios 0.05 < η < 0.95, and the range of cylinder rotation ratios 0.05 < µ < 0.95, where µ is the rotation rate of the outer cylinder divided by that of the inner cylinder, so both cylinders rotate in the same direction with the inner rotating faster than the outer. A remarkable variation in the structure of the onset modes and critical Reynolds numbers was found. In some parts of the η-µ plane, the modes previously found by inviscid analysis are the first to become unstable, but in others the critical modes can only be found using viscous theory. The recently discovered radiative instability, a mode by which vortices can lose energy by radiating it away, is shown to be the first viscous mode to onset for certain values of η and µ. This opens up the possibility of exploring the radiative instability experimentally in the highly controlled environment of Taylor-Couette flow. At a Froude number of unity, there is even a particular point in the η-µ plane, called the triple point, where three different modes of instability onset at the same critical Reynolds number. One of these modes is the classical mode previously found by inviscid studies, another is the vortex radiative instability mode, and the third is a new mode of instability, not previous seen, which only occurs for wide gaps (low η).

In summary, this investigation has brought to light new instabilities and found a connection between radiative instability and the classical problem of the onset of instability in stratified Taylor-Couette flow. Professor Chris Jones, in a statement to Advances in Engineering, said that their study would hopefully encourage experimenters to look at stratified Taylor-Couette flow in wide gaps, with a view to exploring the radiative instability and to verify the existence of the new instabilities. The study will also benefit future numerical investigations of the nonlinear development of these newly discovered instabilities.

stable stratification can destabilize Taylor-Couette flow-Advances in Engineering
Figure legend: At radius ratio η=0.1 and cylinder rotation ratio µ=0.15 radiative instability is the dominant form of instability at onset. Strong flow occurs near the inner cylinder, which triggers waves radiating towards the outer cylinder. Colors and contours denote flow in the azimuthal direction, arrows flow in a vertical cross-section.

Reference

Robins, L., Kersalé, E., & Jones, C. (2020). Viscous and inviscid strato-rotational instability. Journal of Fluid Mechanics, 894 (A13), 1-35.

Go To Journal of Fluid Mechanics

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