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Librational forcing of a rapidly rotating fluid-filled cube

Published online by Cambridge University Press:  13 March 2018

Ke Wu ,
Bruno D. Welfert  and
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window]
Ke Wu
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ 85287, USA
Bruno D. Welfert
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ 85287, USA
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ 85287, USA
*
†Email address for correspondence: [email protected]
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Abstract

The flow response of a rapidly rotating fluid-filled cube to low-amplitude librational forcing is investigated numerically. Librational forcing is the harmonic modulation of the mean rotation rate. The rotating cube supports inertial waves which may be excited by libration frequencies less than twice the rotation frequency. The response is comprised of two main components: resonant excitation of the inviscid inertial eigenmodes of the cube, and internal shear layers whose orientation is governed by the inviscid dispersion relation. The internal shear layers are driven by the fluxes in the forced boundary layers on walls orthogonal to the rotation axis and originate at the edges where these walls meet the walls parallel to the rotation axis, and are hence called edge beams. The relative contributions to the response from these components is obscured if the mean rotation period is not small enough compared to the viscous dissipation time, i.e. if the Ekman number is too large. We conduct simulations of the Navier–Stokes equations with no-slip boundary conditions using parameter values corresponding to a recent set of laboratory experiments, and reproduce the experimental observations and measurements. Then, we reduce the Ekman number by one and a half orders of magnitude, allowing for a better identification and quantification of the contributions to the response from the eigenmodes and the edge beams.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Waves in rotating fluids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 842 , 10 May 2018 , pp. 469 - 494
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Copyright
© 2018 Cambridge University Press 

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Wu et al. supplementary movie 1

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=1/6 (right), for libration frequencies 0.6484 (top) and 0.6742 (bottom), over one libration period, both at Ekman number 1e-4.6 and libration amplitude 0.04.

Download Wu et al. supplementary movie 1(Video)
Video 7.9 MB

Wu et al. supplementary movie 2

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=0.45 (right), of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.677 (top), and of the eigenmode at eigenfrequency 0.6772 (bottom).

Download Wu et al. supplementary movie 2(Video)
Video 1.6 MB

Wu et al. supplementary movie 3

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=0.45 (right), of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.699 (top), and of the eigenmode at eigenfrequency 0.6962 (bottom).

Download Wu et al. supplementary movie 3(Video)
Video 1.5 MB

Wu et al. supplementary movie 4

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=0.45 (right), of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.705 (top), and of the eigenmode at eigenfrequency 0.7022 (bottom).

Download Wu et al. supplementary movie 4(Video)
Video 1.7 MB

Wu et al. supplementary movie 5

Contours of x-vorticity in the vertical plane x=0 (left) and of z-vorticity in the horizontal plane z=0.45 (right), of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.708 (top), and of the eigenmode at eigenfrequency 0.7045 (bottom).

Download Wu et al. supplementary movie 5(Video)
Video 1.8 MB

Wu et al. supplementary movie 6

Contours of z-vorticity in the horizontal planes z=0.45, 0.30, 0.15 and 0.0 of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.699.

Download Wu et al. supplementary movie 6(Video)
Video 1.8 MB

Wu et al. supplementary movie 7

Contours of x-vorticity in the vertical planes x=0.45, 0.30, 0.15 and 0.0 of the nonlinear simulation over one libration period at Ekmnan number 1e-6, libration amplitude 1e-6, and libration frequency 0.699.

Download Wu et al. supplementary movie 7(Video)
Video 2.1 MB

Wu et al. supplementary movie 8

Contours of x-vorticity in the vertical plane z=0 at the oscillation phase pi/2 of the nonlinear simulations at Ekman number 1e-6 and libration amplitude 1e-6, animating 999 snapshots at libration frequencies 0.001 to 0.999.

Download Wu et al. supplementary movie 8(Video)
Video 21.3 MB