Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-19T14:54:05.974Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical modelling of nonlinear effects in laminar flow through a porous medium

Published online by Cambridge University Press:  21 April 2006

O. Coulaud ,
P. Morel  and
J. P. Caltagirone
O. Coulaud
Affiliation:
UER de Mathématiques, Mathématiques Appliquées, LA au CNRS 226, 351, Cours de la Libération, 33405 Talence Cedex, France
P. Morel
Affiliation:
UER de Mathématiques, Mathématiques Appliquées, LA au CNRS 226, 351, Cours de la Libération, 33405 Talence Cedex, France
J. P. Caltagirone
Affiliation:
Laboratoire d'Energétique et Phénomènes de Transfert, UA CNRS 873, Esplanade des Arts et Métiers, 33405 Talence Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium. The method chosen is the numerical resolution of flow equations at a pore scale. The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base. For a given flow through this medium the pressure drop is evaluated numerically.

The Navier-Stokes equations are discretized by the mixed finite-element method. The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation. The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & Pironneau.

For Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law. For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's law.

On the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media. In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 190 , May 1988 , pp. 393 - 407
Copyright
© 1988 Cambridge University Press

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.)

References

Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.
Beavers, G. S. & Sparrow, E. M. 1969 Non Darcy flow through fibrous porous media. Trans. ASME E: J. Appl. Mech. 36, 711–714.Google Scholar
Beavers, G. S., Sparrow, E. W. & Rodenz, D. E. 1973 Influence of bed size on the flow characteristics and porosity of randomly packed beds of spheres. Trans. ASME E: J. Appl. Mech. 40, 655–660.Google Scholar
Beck, J. L. 1972 Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 1377–1383.Google Scholar
Bristeau, J., Glowinski, R., Mantel, B., Periaux, J., Perrier, P. & Pironneau, O. 1979 A Finite Element Approximation of Navier–Stokes Equations for Incompressible Viscous Fluids. Rautmann.
Catton, I. 1984 Natural convection heat transfer in porous media. NATO Advanced study Institute on Natural Convection: Fundamentals and Application. Hemisphere.
Chauveteau, G. 1965 Essai sur la loi de darcy et les écoulements laminaires. Thèse, Toulouse.
Conca, C. & Steer, D. 1983 Résolution des équations bidimensionelles de Navier–Stokes pour un fluide incompressible et visqueux en régime stationnaire, modules N.S.K.I.N.C. et COTABM; Brochure MODULEF 107.
Coulaud, O., Morel, P. & Caltagirone, J. P. 1986 Effets non linéaires dans les écoulements en milieu poreux. C.R. Acad. Sci. Paris 302, 263–266.Google Scholar
Darcy, H. P. G. 1856 Les Fontaines Publiques de la Ville de Dijon. Paris: Victor Dalmont.
de Vries, J. 1979 Prediction of non-Darcy flow in porous media; J. Irrigation Drainage Engng. 105, 147–162.Google Scholar
Dupuit, J. 1863 Etudes Théoriques et Pratiques sur le Mouvement des Eaux. Paris: Dunod.
Forchheimer, P. 1901 Wasserbegung dusch Baden. VDIZ. 45, 1782–1788.Google Scholar
Glowinski, R. 1984 Numerical Methods for Nonlinear Variational Problems. Springer.
Glowinski, R. & Pironneau, O. 1978 Approximation par éléments finis mixtes du problème de Stokes en formulation vitesse-pression. C.R. Acad. Sci. Paris 286, 181–183; 225–228.Google Scholar
Glowinski, R. & Pironneau, O. 1979 On a mixed finite element approximation of the Stokes problem (I): convergence of the approximate solutions. Numer. Maths 33, 397–494.CrossRefGoogle Scholar
Irmay, S. 1958 On the theoretical derivation of Darcy and Forchheimer formulas. Trans. Am. Geophys. Union 39, 702–707.Google Scholar
Joseph, D. D., Nield, D. A. & Papanicolaou, G. 1982 Non linear equation governing flow in a saturated porous medium. Wat. Resour. Res. 18, 1049–1052.Google Scholar
Modulef 1984 Un Code Modulaire d'Eléments Finis “Cours et Séminaires”. INRIA Rep.
Nield, D. A. & Joseph, D. D. 1985 Effects of quadratic drag on convection in a saturated porous medium. Phys. Fluids 28, 995–997.Google Scholar
Polak, E. 1971 Computational Methods in Optimization. Academic.
Thomasset, F. 1980 Implementation of Finite Element Methods for Navier–Stokes equation. Springer.
Whitaker, S. 1969 Advances in the theory of fluid motion in porous media. Indust. Engng Chem. 61, 14–28.Google Scholar