Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T18:29:30.236Z Has data issue: false hasContentIssue false
  • English
  • Français

Pressure-driven flow in a channel with porous walls*

Published online by Cambridge University Press:  04 May 2011

QIANLONG LIU  and
ANDREA PROSPERETTI
QIANLONG LIU
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore MD 21218, USA
ANDREA PROSPERETTI*
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore MD 21218, USA Faculty of Science & Technology and Burgerscentrum, University of Twente, AE 7500 Enschede, The Netherlands
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The finite-Reynolds-number three-dimensional flow in a channel bounded by one and two parallel porous walls is studied numerically. The porous medium is modelled by spheres in a simple cubic arrangement. Detailed results on the flow structure and the hydrodynamic forces and couple acting on the sphere layer bounding the porous medium are reported and their dependence on the Reynolds number illustrated. It is shown that, at finite Reynolds numbers, a lift force acts on the spheres, which may be expected to contribute to the mobilization of bottom sediments. The results for the slip velocity at the surface of the porous layers are compared with the phenomenological Beavers–Joseph model. It is found that the values of the slip coefficient for pressure-driven and shear-driven flow are somewhat different, and also depend on the Reynolds number. A modification of the relation is suggested to deal with these features. The Appendix provides an alternative derivation of this modified relation.

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow Geophysical and Geological Flows: Sediment transport
Type
Papers
Information
Journal of Fluid Mechanics , Volume 679 , 25 July 2011 , pp. 77 - 100
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

With an appendix by Daniel Lhuillier

References

REFERENCES

Alazmi, B. & Vafai, K. 2001 Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Intl J. Heat Mass Transfer 44, 1735–1749.CrossRefGoogle Scholar
Auriault, J.-L. 1991 Heterogeneous medium. Is an equivalent description possible? Intl J. Engng Sci. 29, 785–795.CrossRefGoogle Scholar
Auriault, J.-L. 2010 About the Beavers and Joseph boundary condition. Transp. Porous Med. 83, 257–266.CrossRefGoogle Scholar
Bagchi, P. & Balachandar, S. 2002 Effect of free rotation on the motion of a solid sphere in linear shear flow at moderate Re. Phys. Fluids 14, 2719–2737.CrossRefGoogle Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207.CrossRefGoogle Scholar
Breugem, W., Boersma, B. & Uittenbogaard, R. 2005 The laminar boundary layer over a permeable wall. Transp. Porous Med. 59, 267–300.CrossRefGoogle Scholar
Chandesris, M. & Jamet, D. 2009 Jump conditions and surface-excess quantities at a fluid/porous interface: A multi-scale approach. Transp. Porous Med. 78, 419–438.CrossRefGoogle Scholar
Cieszko, M. & Kubik, J. 1999 Derivation of matching conditions at the contact surface between fluid-saturated porous solid and bulk fluid. Transp. Porous Med. 34, 319–336.CrossRefGoogle Scholar
Furman, A. 2008 Modeling coupled surface-subsurface flow processes: A review. Vadose Zone J. 7, 741–756.CrossRefGoogle Scholar
Ghisalberti, M. 2009 Obstructed shear flows: similarities across systems and scales. J. Fluid Mech. 641, 51–61.CrossRefGoogle Scholar
Goharzadeh, A., Khalili, A. & Jørgensen, B. B. 2005 Transition layer thickness at a fluid-porous interface. Phys. Fluids 17, 057102.CrossRefGoogle Scholar
Goyeau, B., Lhuillier, D., Gobin, D. & Velarde, M. G. 2003 Momentum transport at a fluid-porous interface. Intl J. Heat Mass Transfer 46, 4071–4081.CrossRefGoogle Scholar
Hanspal, N. S., Waghode, A. N., Nassehi, V. & Wakeman, R. J. 2006 Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transp. Porous Med. 64, 73–101.CrossRefGoogle Scholar
Jäger, W. & Mikelic, A. 2009 Modelling effective interface laws for transport phenomena between an unconfined fluid and a porous medium using homogenization. Transp. Porous Med. 78, 489–508.CrossRefGoogle Scholar
James, D. & Davis, A. 2001 Flow at the interface of a fibrous porous medium. J. Fluid Mech. 426, 47–72.Google Scholar
Khakpour, M. & Vafai, K. 2008 Critical assessment of arterial transport models. Intl J. Heat Mass Transfer 51, 807–822.Google Scholar
Koplik, J., Levine, H. & Zee, A. 1983 Viscosity renormalization in the Brinkman equation. Phys. Fluids 26, 2864–2870.CrossRefGoogle Scholar
Larson, R. E. & Higdon, J. J. L. 1987 Microscopic flow near the surface of two-dimensional porous media. Part 2. Transverse flow. J. Fluid Mech. 178, 119–136.CrossRefGoogle Scholar
Le Bars, M. & Worster, M. G. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173.CrossRefGoogle Scholar
Lin, C. J., Peery, J. H. & Schowalter, W. R. 1970 Simple shear flow round a rigid sphere: inertial effects and suspension rheology. J. Fluid Mech. 44, 1–17.CrossRefGoogle Scholar
Liu, Q. & Prosperetti, A. 2010 Wall effects on a rotating sphere. J. Fluid Mech. 657, 1–21.CrossRefGoogle Scholar
Lobkowsy, A. E., Orpe, A. V., Molloy, R., Kudroll, A. & Rothman, D. H. 2008 Erosion of a granular bed driven by laminar fluid flow. J. Fluid Mech. 605, 47–58.Google Scholar
Martys, N., Bentz, D. & Garboczi, E. 1994 Computer simulation study of the effective viscosity in Brinkman's equation. Phys. Fluids 6, 1434–1439.CrossRefGoogle Scholar
Morad, M. & Khalili, A. 2009 Transition layer thickness in a fluid-porous medium of multi-sized spherical beads. Exp. Fluids 46, 323–330.CrossRefGoogle Scholar
Neale, G. & Nader, W. 1974 Practical significance of Brinkman's extension of Darcy's law: coupled parallel flows within a channel and a bounding porous medium. Can. J. Chem. Engng 52, 475–478.CrossRefGoogle Scholar
Nield, D. A. 2009 The Beavers-Joseph boundary condition and related matters: A historical and critical note. Transp. Porous Med. 78, 537–540.Google Scholar
Nield, D. A. & Kuznetsov, A. V. 2009 The effect of a transition layer between a fluid and a porous medium: shear flow in a channel. Transp. Porous Med. 78, 477–487.CrossRefGoogle Scholar
Ochoa-Tapia, J. & Whitaker, S. 1995 Momentum transfer at the boundary between a porous medium and a homogeneous fluid i: theoretical development. Intl J. Heat Mass Transfer 38, 2635–2646.CrossRefGoogle Scholar
Ouriemi, M., Aussillous, P. & Guazzelli, E. 2009 Sediment dynamics. Part 1. Bed-load transport by laminar shearing flows. J. Fluid Mech. 636, 295–319.CrossRefGoogle Scholar
Pokrajac, D. & Manes, C. 2009 Velocity measurements of a free-surface turbulent flow penetrating a porous medium composed of uniform-size spheres. Transp. Porous Med. 78, 367–383.CrossRefGoogle Scholar
Popov, P., Efendiev, Y. & Qin, G. 2009 Multiscale modeling and simulations of flows in naturally fractured karst reservoirs. Commun. Comput. Phys. 6, 162–184.CrossRefGoogle Scholar
Richardson, S. 1971 A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49, 327–336.CrossRefGoogle Scholar
Saffman, P. G. 1971 On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50, 959–964.CrossRefGoogle Scholar
Sahraoui, M. & Kaviany, M. 1992 Slip and no-slip velocity boundary conditions at interface of porous, plain media. Intl J. Heat Mass Transfer 35, 927–943.CrossRefGoogle Scholar
Sangani, A. & Acrivos, A. 1982 Slow flow through a periodic array of spheres. Intl J. Multiphase Flow 8, 343–360.CrossRefGoogle Scholar
Seminara, G. 2010 Fluvial sedimentary patterns. Annu. Rev. Fluid Mech. 42, 43–66.Google Scholar
Taylor, G. I. 1971 A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 49, 319–326.Google Scholar
Tilton, N. & Cortelezzi, L. 2008 Linear stability analysis of pressure-driven flows in channels with porous walls. J. Fluid Mech. 604, 411–445.Google Scholar
Vafai, K. & Kim, S. 1990 Fluid mechanics of the interface region between a porous medium and a fluid layer - an exact solution. Intl J. Heat Fluid Flow 11, 254–256.CrossRefGoogle Scholar
Valdés-Parada, F. J., Alvarez-Ramirez, J., Goyeau, B. & Ochoa-Tapia, J. A. 2009 Computation of jump coefficients for momentum transfer between a porous medium and a fluid using a closed generalized transfer equation. Transp. Porous Med. 78, 439–457.Google Scholar
Zhang, Q. & Prosperetti, A. 2009 Pressure-driven flow in a two-dimensional channel with porous walls. J. Fluid Mech. 631, 1–21.CrossRefGoogle Scholar
Zhang, Z. & Prosperetti, A. 2005 A method for three-dimensional particle flow simulations. J. Comput. Phys. 210, 292–324.CrossRefGoogle Scholar
Zick, A. A. & Homsy, G. M. 1982 Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115, 13–26.CrossRefGoogle Scholar