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Convectively driven shear and decreased heat flux

Published online by Cambridge University Press:  31 October 2014

David Goluskin*
Affiliation:
Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA
Hans Johnston
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA
Glenn R. Flierl
Affiliation:
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Edward A. Spiegel
Affiliation:
Department of Astronomy, Columbia University, New York, NY 10027, USA New York University, New York, NY, USA
*
Present address: Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA. Email address for correspondence: goluskin@umich.edu

Abstract

We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh–Bénard convection between free-slip boundaries. We focus on the ability of the convection to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers ($\mathit{Pr}$) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number ($\mathit{Ra}$) sufficiently, and we explore the resulting convection for $\mathit{Ra}$ up to $10^{10}$. When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as $\mathit{Ra}\rightarrow \infty$. The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with $\mathit{Ra}$. When the large-scale shear is present with $\mathit{Pr}\lesssim 2$, the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with $\mathit{Ra}$ for $\mathit{Pr}=1$. When the shear is present with $\mathit{Pr}\gtrsim 3$, the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of $\mathit{Ra}$, but the growth rates are slower than any previously reported for Rayleigh–Bénard convection without large-scale shear. We find that the Nusselt numbers grow proportionally to $\mathit{Ra}^{0.077}$ when $\mathit{Pr}=3$ and to $\mathit{Ra}^{0.19}$ when $\mathit{Pr}=10$. Analogies with tokamak plasmas are described.

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Papers
Copyright
© 2014 Cambridge University Press 

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Goluskin et al. supplementary movie

Supplement to figure 2(a): Temperature in non-shearing convection with Ra=2•105 and Pr=10. The hottest fluid (red) is one dimensionless degree warmer than the coldest fluid (blue). The dimensionless time span is 0.04.

Download Goluskin et al. supplementary movie(Video)
Video 2.1 MB

Goluskin et al. supplementary movie

Supplement to figure 2(b): Temperature in sustained shearing convection with Ra=2•106 and Pr=10. The hottest fluid (red) is one dimensionless degree warmer than the coldest fluid (blue). The dimensionless time span is 6•10-3.

Download Goluskin et al. supplementary movie(Video)
Video 3.1 MB

Goluskin et al. supplementary movie

Supplement to figure 2(c): Temperature in sustained shearing convection with Ra=2•107 and Pr=10. The hottest fluid (red) is one dimensionless degree warmer than the coldest fluid (blue). The dimensionless time span is 1.2•10-3.

Download Goluskin et al. supplementary movie(Video)
Video 3.1 MB

Goluskin et al. supplementary movie

Supplement to figure 2(d): Temperature in sustained shearing convection with Ra=2•108 and Pr=10. The hottest fluid (red) is one dimensionless degree warmer than the coldest fluid (blue). The dimensionless time span is 2.4•10-4.

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

Goluskin et al. supplementary movie

Supplement to figure 8: Temperature in bursting shearing convection with Ra=2•108 and Pr=1. The hottest fluid (red) is one dimensionless degree warmer than the coldest fluid (blue). The time span matches that of the time series shown in the figure.

Download Goluskin et al. supplementary movie(Video)
Video 10 MB