Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T11:52:20.575Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous lock-exchange in rectangular channels

Published online by Cambridge University Press:  14 February 2011

J. MARTIN,
N. RAKOTOMALALA,
L. TALON and
D. SALIN
J. MARTIN
Affiliation:
Université Pierre et Marie Curie, Université Paris-Sud, CNRS, Laboratoire FAST, Bâtiment 502, Rue du Belvedere, Campus Universitaire, F-91405 Orsay CEDEX, France
N. RAKOTOMALALA
Affiliation:
Université Pierre et Marie Curie, Université Paris-Sud, CNRS, Laboratoire FAST, Bâtiment 502, Rue du Belvedere, Campus Universitaire, F-91405 Orsay CEDEX, France
L. TALON
Affiliation:
Université Pierre et Marie Curie, Université Paris-Sud, CNRS, Laboratoire FAST, Bâtiment 502, Rue du Belvedere, Campus Universitaire, F-91405 Orsay CEDEX, France
D. SALIN*
Affiliation:
Université Pierre et Marie Curie, Université Paris-Sud, CNRS, Laboratoire FAST, Bâtiment 502, Rue du Belvedere, Campus Universitaire, F-91405 Orsay CEDEX, France
*
†Email address for correspondence: salin@fast.u-psud.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy-driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a two-dimensional (2D) Stokes flow between two parallel horizontal boundaries separated by a vertical height H and, recently, for a cylindrical tube. In the present paper, we calculate it, analytically, for a rectangular channel (horizontal thickness b and vertical height H) of any aspect ratio (H/b) and compare our results with experiments in horizontal rectangular channels for a wide range of aspect ratios (1/10 to 10). We also discuss the 2D Stokes–Darcy model for flows in Hele-Shaw cells and show that it leads to a rather good approximation, when an appropriate Brinkman correction is used.

JFM classification

Geophysical and Geological Flows: Gravity currents

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 673 , 25 April 2011, pp. 132 - 146
Copyright
Copyright © Cambridge University Press 2011

References

REFERENCES

Bear, J. 1988 Dynamics of Fluids in Porous Media. Elsevier.Google Scholar
Benjamin, T. J. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
Birman, V. K., Martin, J. E. & Meiburg, E. 2005 The non-Boussinesq lock-exchange problem. Part 2. High-resolution simulations. J. Fluid Mech. 537, 125–144.CrossRefGoogle Scholar
Bizon, C., Werne, J., Predtechensky, A., Julien, K., McCormick, W., Swift, J. & Swinney, H. 1997 Plume dynamics in quasi-2D turbulent convection. Chaos 7, 107–124.Google Scholar
Bonometti, T., Balachandar, S. & Magnaudet, J. 2008 Wall effects in non-Boussinesq density currents. J. Fluid Mech. 616, 445–475.CrossRefGoogle Scholar
Brinkman, H. 1947 A calculation of the viscous forces exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–39.Google Scholar
Cantero, M., Lee, J., Balachandar, S. & Garcia, M. 2007 On the front velocity of gravity currents. J. Fluid Mech. 586, 1–39.CrossRefGoogle Scholar
Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 27–42.CrossRefGoogle Scholar
Fernandez, J., Kurowski, P., Petitjeans, P. & Meiburg, E. 2002 Density-driven unstable flows of miscible fluids in a Hele-Shaw cell. J. Fluid Mech. 451, 239–260.CrossRefGoogle Scholar
Gondret, P., Rakotomalala, N., Rabaud, M., Salin, D. & Watzky, P. 1997 Viscous parallel flows in finite aspect ratio Hele-Shaw cell: analytical and numerical results. Phys. Fluids 9, 1841–1843.CrossRefGoogle Scholar
Graf, F., Meiburg, E. & Härtel, C. 2002 Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations. J. Fluid Mech. 451, 261–282.CrossRefGoogle Scholar
Gratton, J. & Minotti, F. 1990 Self-similar viscous gravity currents: phase-plane formalism. J. Fluid Mech. 210, 155–182.CrossRefGoogle Scholar
Hallez, Y. & Magnaudet, J. 2009 A numerical investigation of horizontal viscous gravity currents. J. Fluid Mech. 630, 71–91.Google Scholar
Huppert, H. H. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
Huppert, H. H. & Woods, A. W. 1995 Gravity-driven flows in porous layers. J. Fluid Mech. 292, 55–69.CrossRefGoogle Scholar
Lowe, R. J., Rottman, J. W. & Linden, P. F. 2005 The non-Boussinesq lock-exchange problem. Part 1. Theory and experiments. J. Fluid Mech. 537, 101–124.CrossRefGoogle Scholar
Martin, J., Rakotomalala, N. & Salin, D. 2002 a Gravitational instability of miscible fluids in a Hele-Shaw cell. Phys. Fluids 14, 902–905.CrossRefGoogle Scholar
Martin, J., Rakotomalala, N., Salin, D. & Böckmann, M. 2002 b Buoyancy-driven instability of an autocatalytic reaction front in a Hele-Shaw cell. Phys. Rev. E 65, 051605.CrossRefGoogle Scholar
Neufeld, J. A. & Huppert, H. E. 2009 Modelling carbon dioxide sequestration in layered strata. J. Fluid Mech. 625, 353–370.CrossRefGoogle Scholar
Ruyer-Quil, C. 2001 Inertial corrections to the Darcy law in a Hele-Shaw cell. C. R. Acad. Sci. Paris 329, Serie IIb 337–342.Google Scholar
Séon, T., Znaien, J., Salin, D., Hulin, J. P., Hinch, E. J. & Perrin, B. 2007 Transient buoyancy-driven front dynamics in nearly horizontal tubes. Phys. Fluids 19, 123603.Google Scholar
Shin, J. O., Dalziel, S. B. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1–34.Google Scholar
Simpson, J. E. 1972 Effects of the lower boundary on the head of a gravity current. J. Fluid Mech. 53, 759–768.CrossRefGoogle Scholar
Taghavi, S. M., Seon, T., Martinez, D. M. & Frigaard, I. A. 2009 Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 1–35.CrossRefGoogle Scholar
Takagi, D. & Huppert, H. 2007 The effect of confining boundaries on viscous gravity currents. J. Fluid Mech. 577, 495–505.CrossRefGoogle Scholar
Talon, L., Martin, J., Rakotomalala, N., Salin, D. & Yortsos, Y. 2003 Lattice BGK simulations of macrodispersion in heterogeneous porous media. Water Resour. Res. 39, 1135–1142.CrossRefGoogle Scholar
Zeng, J., Yortsos, Y. C. & Salin, D. 2003 On the Brinkman correction in unidirectional Hele-Shaw flows. Phys. Fluids 15, 3829–3836.CrossRefGoogle Scholar