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The relaxation time for viscous and porous gravity currents following a change in flux

Published online by Cambridge University Press:  24 May 2017

Thomasina V. Ball Open the ORCID record for Thomasina V. Ball [Opens in a new window] ,
Herbert E. Huppert ,
John R. Lister  and
Jerome A. Neufeld
Thomasina V. Ball*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CMS Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Cambridge CB3 0EZ, UK Department of Earth Sciences, Bullard Laboratories, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
Herbert E. Huppert
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CMS Wilberforce Road, Cambridge CB3 0WA, UK Faculty of Science, University of Bristol, Bristol BS8 1UH, UK School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
John R. Lister
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CMS Wilberforce Road, Cambridge CB3 0WA, UK
Jerome A. Neufeld
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CMS Wilberforce Road, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Cambridge CB3 0EZ, UK Department of Earth Sciences, Bullard Laboratories, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: tvb21@cam.ac.uk
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Abstract

The equilibration time $\unicode[STIX]{x1D70F}$ in response to a change in flux from $Q$ to $\unicode[STIX]{x1D6EC}Q$ after an injection period $T$ applied to either a low-Reynolds-number gravity current or one propagating through a porous medium, in both axisymmetric and one-dimensional geometries, is shown to be of the form $\unicode[STIX]{x1D70F}=Tf(\unicode[STIX]{x1D6EC})$ , independent of all the remaining physical parameters. Numerical solutions are used to investigate $f(\unicode[STIX]{x1D6EC})$ for each of these situations and compare very well with experimental results in the case of an axisymmetric current propagating over a rigid horizontal boundary. Analysis of the relaxation towards self-similarity provides an illuminating connection between the excess (deficit) volume from early times and an asymptotically equivalent shift in time origin, and hence a good quantitative estimate of $\unicode[STIX]{x1D70F}$ . The case $\unicode[STIX]{x1D6EC}=0$ of equilibration after ceasing injection at time $T$ is a singular limit. Extensions to high-Reynolds-number currents and to the case of a constant-volume release followed by constant-flux injection are discussed briefly.

JFM classification

Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Lubrication theory Low-Reynolds-number flows: Porous media

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Papers
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Journal of Fluid Mechanics , Volume 821 , 25 June 2017 , pp. 330 - 342
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© 2017 Cambridge University Press 

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