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Interacting oscillatory boundary layers and wall modes in modulated rotating convection

Published online by Cambridge University Press:  14 April 2009

A. RUBIO ,
J. M. LOPEZ  and
F. MARQUES
A. RUBIO
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
J. M. LOPEZ*
Affiliation:
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
F. MARQUES
Affiliation:
Departament de Física Aplicada, Universitat Politècnica de Catalunya, Barcelona 08034, Spain
*
†Email address for correspondence: [email protected]
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Abstract

Thermal convection in a rotating cylinder near onset is investigated using direct numerical simulations of the Navier–Stokes equations with the Boussinesq approximation in a regime dominated by the Coriolis force. For thermal driving too small to support convection throughout the entire cell, convection sets in as alternating pairs of hot and cold plumes in the sidewall boundary layer, the so-called wall modes of rotating convection. We subject the wall modes to small amplitude harmonic modulations of the rotation rate over a wide range of frequencies. The modulations produce harmonic Ekman boundary layers at the top and bottom lids as well as a Stokes boundary layer at the sidewall. These boundary layers drive a time-periodic large-scale circulation that interacts with the wall-localized thermal plumes in a non-trivial manner. The resultant phenomena include a substantial shift in the onset of wall-mode convection to higher temperature differences for a broad band of frequencies, as well as a significant alteration of the precession rate of the wall mode at very high modulation frequencies due to the mean azimuthal streaming flow resulting from the modulations.

JFM classification

Convection: Buoyant boundary layers Geophysical and Geological Flows: Rotating flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 625 , 25 April 2009 , pp. 75 - 96
Copyright
Copyright © Cambridge University Press 2009

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References

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Rubio et al. supplementary movie

Movie 1. Wall-localized convection in Rayleigh-Bénard convection with constant rotation. In this Coriolis force dominated regime the onset of convection is to rotating waves of thermal plumes near the sidewall, the so-called wall modes of rotating convection. This solution has wave number m = 17 and becomes linearly stable at Ra = 4.23x104. The m = 17 wall mode loses stability to a mixed Küppers-Lortz/wall mode solution at about Ra=9.5x104, the convection threshold predicted by Chandrasekhar. This movie corresponds to figure 2 in the paper. Shown are isosurfaces of the temperature perturbation at Θ= +/- 0.05 for a solution with Ra=5x104, Ω0=625 & γ=4.

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Video 1.5 MB

Rubio et al. supplementary movie

Movie 2. The m = 17 wall mode from movie 1 is subjected to a modulated rotation rate given by Ω(t) = Ω0(1+A sin(Ωmt)) (in the laboratory frame). Shown are isosurfaces of the temperature perturbation at Θ = +/-0.05 for Ra=5x104, Ω0=625, A=0.0075, ΩM=101.75 & γ=4.

Download Rubio et al. supplementary movie(Video)
Video 5.7 MB

Rubio et al. supplementary movie

Movie 3. Quenching of the m = 17 wall mode from movie 1 using a modulation amplitude slightly larger than that used in movie 2. Shown are isosurfaces of the temperature perturbation at Θ = +/- 0.05 for Ra=5x104, Ω0=625, A=0.01, ΩM=101.75 & γ=4. The three movies are shown in the rotating reference frame.

Download Rubio et al. supplementary movie(Video)
Video 3.5 MB

Rubio et al. supplementary movie

Movie 4. The axisymmetric synchronous state of modulated Rayleigh-Bénard convection at Ra=4x104, Ω0=625, A=0.05, Ωm = 101.75 & γ=4. Here Ra < Rac and thermal convection is absent. However, the modulated rotation drives an alternating secondary flow by vortex line bending as the cylinder accelerates and decelerates. The top panel shows ten positive (red) and ten negative (blue) contours of the stream function Ψ ∈ [-1.5,1.5], in a meridional plane r ∈ [0,γ], z ∈ [-0.5,0.5]. The middle frame shows five positive (red) and five negative (blue) quadratically spaced color levels of the azimuthal vorticity η ∈ [-1000,1000] and ten quadratically spaced contours of the vortex lines rv+r2Ω0 ∈[0,104]. The bottom frame shows ten positive (red) and ten

Download Rubio et al. supplementary movie(Video)
Video 15.2 MB