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Energetics and mixing of thermally driven flows in Hele-Shaw cells

Published online by Cambridge University Press:  17 November 2021

Hugo N. Ulloa Open the ORCID record for Hugo N. Ulloa [Opens in a new window]  and
Juvenal A. Letelier Open the ORCID record for Juvenal A. Letelier [Opens in a new window]
Hugo N. Ulloa
Affiliation:
Department of Earth and Environmental Science, University of Pennsylvania, Philadelphia, PA 19104-6316, USA Physics of Aquatic Systems Laboratory (APHYS)-Margaretha Kamprad Chair, École Polytechnique Fédérale de Laussane, CH-1015 Lausanne, Switzerland
Juvenal A. Letelier*
Affiliation:
Departamento de Ingeniería Civil, Universidad de Chile, Avenida Blanco Encalada 2002, Santiago 8370449, Chile Centro de Excelencia en Geotermia de los Andes (ANID-FONDAP), Plaza Ercilla 803, Santiago 8370450, Chile
*
†Email address for correspondence: juvenal.letelier@uchile.cl
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Abstract

Thermally driven flows in fractures play a key role in enhancing the heat transfer and fluid mixing across the Earth's lithosphere. Yet the energy pathways in such confined environments have not been characterised. Building on Letelier et al. (J. Fluid Mech., vol. 864, 2019, pp. 746–767), we introduce novel expressions for energy transfer rates – energetics – of geometrically constrained Rayleigh–Bénard convection in Hele-Shaw cells (HS-RBC) based on two different conceptual frameworks. First, we derived the energetics following the well-established framework introduced by Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128), in which the gravitational potential energy, $E_{\textit {p}}$, is decomposed into its available, $E_{\textit {ap}}$, and background, $E_{\textit {bp}}$, components. Secondly, we derived the energetics considering a new decomposition for $E_{\textit {p}}$, named dynamic, $E_{\textit {dp}}$, and reference, $E_{\textit {rp}}$, potential energies; $E_{\textit {dp}}$ is defined as the departure of the system's potential energy from the reference state $E_{\textit {rp}}$, determined by the ‘energy’ of the scalar fluctuations. For HS-RBC, both frameworks lead to the same energy transfer rates at a steady state, satisfying the relationship $\langle E_{\textit {ap}} \rangle _{\tau } = \langle E_{\textit {dp}} \rangle _{\tau } + 1/6$. Consistent with the work by Hughes et al. (J. Fluid Mech., vol. 729, 2013) on three-dimensional Rayleigh–Bénard convection, we report analytical expressions for the energetics and efficiencies of HS-RBC in terms of the Rayleigh number and the global Nusselt number. Additionally, we performed numerical experiments to illustrate the application of the energetics for the analysis of HS-RBC. Finally, we discuss the impact of the thermal forcing and the geometrical control exerted by Hele-Shaw cells on the development of boundary layers, protoplumes and the self-organisation of large-scale flows.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Flow Control: Mixing enhancement
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 930 , 10 January 2022 , A16
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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