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Converging gravity currents over a permeable substrate

Published online by Cambridge University Press:  07 August 2015

Zhong Zheng ,
Sangwoo Shin  and
Howard A. Stone
Zhong Zheng
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Sangwoo Shin
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: hastone@princeton.edu
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Abstract

We study the propagation of viscous gravity currents along a thin permeable substrate where slow vertical drainage is allowed from the boundary. In particular, we report the effect of this vertical fluid drainage on the second-kind self-similar solutions for the shape of the fluid–fluid interface in three contexts: (i) viscous axisymmetric gravity currents converging towards the centre of a cylindrical container; (ii) viscous gravity currents moving towards the origin in a horizontal Hele-Shaw channel with a power-law varying gap thickness in the horizontal direction; and (iii) viscous gravity currents propagating towards the origin of a porous medium with horizontal permeability and porosity gradients in power-law forms. For each of these cases with vertical leakage, we identify a regime diagram that characterizes whether the front reaches the origin or not; in particular, when the front does not reach the origin, we calculate the final location of the front. We have also conducted laboratory experiments with a cylindrical lock gate to generate a converging viscous gravity current where vertical fluid drainage is allowed from various perforated horizontal substrates. The time-dependent position of the propagating front is captured from the experiments, and the front position is found to agree well with the theoretical and numerical predictions when surface tension effects can be neglected.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 669 - 690
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Copyright
© 2015 Cambridge University Press 

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