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Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents

Published online by Cambridge University Press:  16 April 2014

Zhong Zheng ,
Ivan C. Christov Open the ORCID record for Ivan C. Christov [Opens in a new window]  and
Howard A. Stone
Zhong Zheng
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Ivan C. Christov
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, 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 report experimental, theoretical and numerical results on the effects of horizontal heterogeneities on the propagation of viscous gravity currents. We use two geometries to highlight these effects: (a) a horizontal channel (or crack) whose gap thickness varies as a power-law function of the streamwise coordinate; (b) a heterogeneous porous medium whose permeability and porosity have power-law variations. We demonstrate that two types of self-similar behaviours emerge as a result of horizontal heterogeneity: (a) a first-kind self-similar solution is found using dimensional analysis (scaling) for viscous gravity currents that propagate away from the origin (a point of zero permeability); (b) a second-kind self-similar solution is found using a phase-plane analysis for viscous gravity currents that propagate toward the origin. These theoretical predictions, obtained using the ideas of self-similar intermediate asymptotics, are compared with experimental results and numerical solutions of the governing partial differential equation developed under the lubrication approximation. All three results are found to be in good agreement.

JFM classification

Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Hele-Shaw flows Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 747 , 25 May 2014 , pp. 218 - 246
Copyright
© 2014 Cambridge University Press 

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