Skip to main content
SpringerLink
Log in

A Numerical Study of the Flow Past an Impermeable Sphere Embedded in a Porous Medium

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

The flow past an impermeable sphere embedded in a fluid-saturated porous medium was investigated numerically. The flow is considered viscous, laminar, axisymmetric, steady and incompressible. The generalized model for fluid flow through an isotropic, rigid and homogeneous porous medium was used. The stream function–vorticity equations were solved numerically in spherical coordinates system by a defect correction multigrid method. The computations were focused on the influence of the Darcy number (Da) and Forchheimer constant on the flow field for two boundary conditions on the surface of the sphere: slip and no-slip. The conditions when the macroscopic inertial terms of the generalized model can be neglected are presented. For high-porosity porous media, the macroscopic inertial terms of the generalized model can be neglected if Da \(Re< 1\) (Re is the sphere Reynolds number).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Abbreviations

\(a\) :

Radius of the sphere

\(C_{D}\) :

Total drag coefficient

\(C_{D,F}\) :

Friction drag coefficient

\(C_{D,P}\) :

Pressure drag coefficient

\(C_{F}\) :

Forchheimer constant

\(d\) :

Diameter of the sphere, \(d = 2a\)

Da :

Darcy number, \(K / d^{2}\)

\(K\) :

Permeability of the porous medium

\(p\) :

Dimensionless local average pressure

\(r\) :

Dimensionless radial coordinate, \(r^{*} / a\), in spherical coordinate system

\(r^{*}\) :

Radial coordinate in spherical coordinate system

Re :

Reynolds number based on the diameter of the sphere, \(Re = U_{0}d\rho / \mu \)

\(U_{0}\) :

Darcy free-stream velocity

\(V_{R}\) :

Darcy dimensionless radial velocity component

\(V_{\theta }\) :

Darcy dimensionless tangential velocity component

\(x\) :

Transformed dimensionless radial coordinate

\(\beta \) :

Dimensionless slip coefficient

\(\varepsilon \) :

Porosity of the porous medium

\(\mu \) :

Dynamic viscosity of the fluid

\(\theta \) :

Polar angle in spherical coordinate system

\(\rho \) :

Density of the fluid

\(\psi \) :

Dimensionless stream function

\(\zeta \) :

Dimensionless vorticity

\(s\) :

Refers to the surface of the sphere

References

  • Bagchi, P., Ha, M.Y., Balachandar, S.: Direct numerical simulation of flow and heat transfer from a sphere in a uniform cross-flow. ASME J. Fluids Eng. 123, 347–358 (2001)

    Google Scholar 

  • Bear, J., Bachmat, Y.: Introduction to Modeling of Transport Phenomena in Porous Media. Kluiwer Academic, Dordrecht (1990)

    Book  Google Scholar 

  • Berman, B.: Flow of a Newtonian fluid past an impervious sphere embedded in a porous medium. Indian J. Pure Appl. Math 27, 1249–1256 (1996)

    Google Scholar 

  • Brabston, D.C.: Numerical solution of steady viscous flow past spheres and gas bubbles. Ph. D. Thesis, California Institute of Technology (1974)

  • Brabston, D.C., Keller, H.B.: Viscous flow past spherical gas bubbles. J. Fluid Mech. 69, 179–189 (1975)

    Google Scholar 

  • Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1, 27–34 (1947)

    Google Scholar 

  • Charya, D.S., Murthy, J.V.R.: Flow past an axisymmetric body embedded in a saturated porous medium. C. R. Mec. 330, 417–423 (2002)

    Google Scholar 

  • Cherepanov, G.P.: Two-dimensional convective heat/mass transfer for low Prandtl and any Peclet numbers. SIAM J. Appl. Math. 58, 942–960 (1998)

    Google Scholar 

  • Clift, R., Grace, J.R., Weber, M.E.: Bubbles, Drops and Particles. Academic Press, New York (1978)

    Google Scholar 

  • Coulaud, O., Morel, P., Caltagirone, J.P.: Numerical modeling of nonlinear effects in laminar flow through a porous medium. J. Fluid Mech. 190, 393–407 (1988)

    Google Scholar 

  • Cvetkovic, V.D.: A continuum approach to high velocity flow in a porous medium. Transp. Porous Media 1, 63–97 (1986)

    Google Scholar 

  • Darcy, H.P.G.: Les Fontaines Publiques de la Ville de Dijon. Victor Dalmont, Paris (1856)

    Google Scholar 

  • Ene, H.I., Polisevski, D.: Thermal Flow in Porous Media. Rudel, Dordrecht (1987)

    Book  Google Scholar 

  • Feng, Z.-G., Michaelides, E.E.: Unsteady mass transport from a sphere immersed in a porous medium at finite Peclet numbers. Int. J. Heat Mass Transf. 42, 535–546 (1999)

    Google Scholar 

  • Ganapathy, R.: Creeping flow past a solid sphere in a porous medium. ZAMM 77, 871–875 (1997)

    Google Scholar 

  • Hassanizadeh, S.M., Gray, W.C.: High velocity flow in porous media. Transp. Porous Media 2, 521–531 (1987)

    Google Scholar 

  • Hsu, C.T., Cheng, P.: Thermal dispersion in a porous medium. Int. J. Heat Mass Transf. 33, 1587–1597 (1990)

    Google Scholar 

  • Johnson, T.A., Patel, V.C.: Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70 (1999)

    Google Scholar 

  • Juncu, G.: Brinkman–Forchheimer–Darcy flow past an impermeable sphere embedded in a porous medium. Presented at Xth Workshop on Mathematical Modelling of Environmental and Life Sciences Problems, to appear in Anal. Univ. Ovidius Constanta, Seria Matematica (2015)

  • Juncu, G., Mihail, R.: Numerical solution of the steady incompressible Navier–Stokes equations for the flow past a sphere by a multigrid defect correction technique. Int. J. Numer. Meth. Fluids 11, 379–395 (1990)

    Google Scholar 

  • Kaviany, M.: Principles of Heat Transfer in Porous Media, 2nd edn. Springer, Berlin (1995)

    Book  Google Scholar 

  • Lage, J.L.: The fundamental theory of flow through permeable media: from Darcy to turbulence. In: Ingham, D.B., Pop, I. (eds.) Transport Phenomena in Porous Media, pp. 1–30. Elsevier, Oxford (1998)

    Chapter  Google Scholar 

  • Le Bars, M., Worster, M.G.: Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173 (2006)

    Google Scholar 

  • Leont’ev, N.E.: Flow past a cylinder and a sphere in a porous medium within the framework of the Brinkman equation with the Navier boundary condition. Fluid Dyn. 49, 232–237 (2014)

    Google Scholar 

  • Levy, A., Levi-Hevroni, D., Sorek, S., Ben-Dor, G.: Derivation of Forchheimer terms and their verification by application to waves propagation in porous media. Int. J. Multiph. Flow 25, 683–704 (1999)

    Google Scholar 

  • Nield, D.A.: Closure statements on the Brinkman—Forchheimer—extended Darcy model. Int. J. Heat Fluid Flow 17, 34–35 (1996)

    Google Scholar 

  • Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)

    Google Scholar 

  • Nithiarasu, P., Seetharamu, K.N., Sundararajan, T.: Finite element modeling of flow, heat and mass transfer in fluid saturated porous media. Arch. Comput. Methods Eng. 9, 3–42 (2002)

    Google Scholar 

  • Oliver, D.L.R., Chung, J.N.: Steady flows inside and around a fluid sphere at low Reynolds numbers. J. Fluid Mech. 154, 215–230 (1985)

    Google Scholar 

  • Pop, I., Ingham, D.B.: Flow past a sphere embedded in a porous medium based on the Brinkman model. Int. Commun. Heat Mass Transf. 23, 865–874 (1996)

    Google Scholar 

  • Romero, L.A.: Low or high Peclet number flow past a sphere in a saturated porous medium. SIAM J. Appl. Math. 54, 42–71 (1994)

    Google Scholar 

  • Romero, L.A.: Low or high Peclet number flow past a prolate spheroid in a saturated porous medium. SIAM J. Appl. Math. 55, 952–974 (1995a)

    Google Scholar 

  • Romero, L.A.: Forced convection past a slender body in a saturated porous medium. SIAM J. Appl. Math. 55, 975–985 (1995b)

    Google Scholar 

  • Rudraiah, N., Shivakumaran, I.S., Palaniappan, D., Chandrashekar, D.V.: Flow past an impervious sphere embedded in a porous medium based on non-Darcy model. Adv. Fluid Mech. 2, 253–256 (2004)

    Google Scholar 

  • Sorek, S., Levi-Hevroni, D., Levy, A., Ben-Dor, G.: Extensions to the macroscopic Navier–Stokes equation. Transp. Porous Media 61, 215–233 (2005)

    Google Scholar 

  • Sorek, S., Ronen, D., Gitis, V.: Scale-dependent macroscopic balance equations governing transport through porous media. Transp. Porous Media 81, 61–72 (2010)

    Google Scholar 

  • Straughan, B.: Structure of the dependence of Darcy and Forchheimer coefficients on porosity. Int. J. Eng. Sci. 48, 1610–1621 (2010)

    Google Scholar 

  • Vafai, K., Tien, C.L.: Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transf. 24, 195–203 (1981)

    Google Scholar 

  • Wang, C.Y.: Darcy–Brinkman flow with solid inclusions. Chem. Eng. Commun. 197, 261–274 (2009)

    Google Scholar 

  • Yu, P., Zeng, Y., Lee, T.S., Chen, X.B., Low, H.T.: Numerical simulation on steady flow around and through a porous sphere. Int. J. Heat Fluid Flow 36, 142–152 (2012)

    Google Scholar 

Download references

Acknowledgments

The author would like to express his sincere thanks to the two anonymous reviewers for helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Chemical and Biochemical Engineering, Politehnica University Bucharest, Polizu 1, 011061, Bucharest, Romania

    Gheorghe Juncu

Authors
  1. Gheorghe Juncu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gheorghe Juncu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Juncu, G. A Numerical Study of the Flow Past an Impermeable Sphere Embedded in a Porous Medium. Transp Porous Med 108, 555–579 (2015). https://doi.org/10.1007/s11242-015-0488-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-015-0488-7

Keywords

Search

Navigation