Velocity measurements and free-shear-layer instabilities in a rotating liquid metal

Abstract

The dynamics of rotating flows in magnetohydrodynamics (MHD) are

expected to play an important role in astrophysical accretion disks,

specifically via the generation of the magnetorotational instability

(MRI). The Princeton MRI experiment, a Taylor-Couette device with

split axial endcap rings and with a gallium-indium-tin working fluid,

was constructed to study rotating MHD flows in the laboratory. This

work uses an ultrasound Doppler velocimetry (UDV) diagnostic to

measure internal fluid velocities of the mean background flow and also of fluctuations driven by instabilities in this experiment.

Mean azimuthal velocities have been measured to show the effect of an applied axial magnetic field. For moderate magnetic field strengths, with the Elsasser number less than 1,

the azimuthal velocity at a point, normalized to the inner cylinder velocity, is constant for a given Alfven speed normalized to the inner cylinder velocity. In the Elsasser much greater than 1 regime,

the shear at the split in the endcaps extends axially as a free

Shercliff layer through the full height of the experiment.

When the inner endcap ring rotates faster than the outer endcap ring,

the free shear layer produces a Kelvin-Helmholtz instability with an

eigenmode that fills the experimental volume when Elsasser is greater than 1, or when there is sufficient background rotation so that the Rossby number is greater than 2.35. When the inner endcap ring rotates slower than the outer endcap ring, the Kelvin-Helmholtz instability is present in the absence of an applied magnetic field or background rotation.

Using a linear eigenmode analysis of a free shear layer, the

threshold for the appearance of the Kelvin-Helmholtz instability is

presented as a competition between Rayleigh's centrifugal instability, which is driven by a negative radial gradient in the specific angular momentum, and the Kelvin-Helmholtz instability, which is driven by the velocity shear. Both instabilities act to smooth out the shear layer. While the eigenmodes of the centrifugal instability are

confined to the free shear layer, allowing quiet flow in the bulk of

the fluid when that instability dominates, the eigenmodes of the

Kelvin-Helmholtz instability fill the fluid volume, creating

large-amplitude coherent velocity fluctuations throughout.