Basic flow regimes in the differentially-heated rotating annulus

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Sloping convection exhibits a wide range of different types of flow regime, depending upon the strength of heating and background rotation rate Ω. We can map the behaviour of the flow on a regime diagram as illustrated below. In order to be able to compare experiments carried using different sized apparatus and different working fluids, and to generalize results to other systems, we represent the strength of heating and background rotation rate Ω in terms of dimensionless parameters. The main parameters usually used are:

1. Taylor number - representing the (squared) ratio of Coriolis to viscous forces (ν is the kinematic viscosity)
2. Thermal Rossby or Hide number - representing the ratio of buoyancy to Coriolis forces
3. Prandtl number - characterising the working fluid (κ is the thermal diffusivity)

Regime Diagram

The regime diagram maps the regions of parameter space over which different types of flow are found.

The hotspots marked on the diagram above link to the following clips:

• clip z02b
• clip z02c
• clip z02d
• clip z02e
• clip z02f
• clip z02g
• clip z02h

• Axisymmetric flow occurs at low values of Ω for a given temperature contrast
• Regular, coherent waves occur at moderate values of Ω, with a tendency for higher azimuthal wavenumbers at larger Ω and Taylor number.
• Within each regular wavenumber regime there may be transitions/bifurcations from steady wave states to time-dependent states (periodic or chaotic) known as
• Irregular, more chaotic flows occur at high Ω and large Taylor number/small Hide number.

Some other, very impressive, flow visualisations of some of the main regimes in the thermally-driven rotating annulus at the top surface (using Kalliroscope rheoscopic fluid) were obtained by Richard Pfeffer, George Buzyna and Robin Kung of Florida State University in Tallahassee (USA) in the early 1980s. Links are provided here to the clips:

• Non-rotating flow: clip z41a
• Axisymmetric flow: clip z41b
• m = 3AV: clip z41c
• m = 4MAV: clip z41d
• m = 5SV: clip z41e
• m = 6SV: clip z41f
• m = 7/irr: clip z41g
• m = irr: clip z41h

Note that these experiments were carried out with a free upper surface, using silicone oil as working fluid. The temperature was kept fixed and rotation rate gradually increased. As the rotation rate increases, the free upper surface becomes increasingly inclined because of the centrifugal distortion. This introduces a beta-effect, much as found when the horizontal boundaries are deliberately sloped.