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The cooling box problem: convection with a quadratic equation of state

Published online by Cambridge University Press:  06 May 2021

Jason Olsthoorn Open the ORCID record for Jason Olsthoorn [Opens in a new window] ,
Edmund W. Tedford  and
Gregory A. Lawrence
Jason Olsthoorn*
Affiliation:
Department of Civil Engineering, University of British Columbia, 2002-6250 Applied Science Lane, Vancouver, BC, CanadaV6T 1Z4
Edmund W. Tedford
Affiliation:
Department of Civil Engineering, University of British Columbia, 2002-6250 Applied Science Lane, Vancouver, BC, CanadaV6T 1Z4
Gregory A. Lawrence
Affiliation:
Department of Civil Engineering, University of British Columbia, 2002-6250 Applied Science Lane, Vancouver, BC, CanadaV6T 1Z4
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the convective cooling of a fluid with a quadratic equation of state (EOS) by performing three-dimensional direct numerical simulations of a flow with a fixed top-boundary temperature, which is lower than the initial fluid temperature. We consider fluid temperatures near the density maximum, where the nonlinearity is expected to be important. When the EOS is nonlinear, the resultant vertical transport of heat is fundamentally different and significantly lower than the predictions derived for a linear EOS. Further, three dimensionless groups parameterise the convective system: the Rayleigh number (${Ra}_0$), the Prandtl number (Pr) and the dimensionless bottom water temperature $(T_B)$. We further define an effective Rayleigh number (${Ra}_{eff} = {Ra}_0 \ T_B^2$), which is equivalent to the traditional Rayleigh number used with a linear EOS. We present a predictive model for the vertical heat flux, the top boundary-layer thickness, and the turbulent kinetic energy (TKE) of the system. We show that this model agrees well with the direct numerical simulations. This model could be used to understand how quickly freshwater lakes cool in high-latitude environments.

JFM classification

Convection: Buoyancy-driven instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 918 , 10 July 2021 , A6
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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