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Transport in Porous Media

, Volume 92, Issue 1, pp 101–118 | Cite as

Analysis of Momentum Transfer in a Lid-Driven Cavity Containing a Brinkman–Forchheimer Medium

  • Duoxing YangEmail author
  • Ziqiu Xue
  • Simon A. Mathias
Article

Abstract

The CE/SE (the space-time conservation element and solution element method) scheme with the second-order accuracy has been proposed. And the pretreatment method has been introduced to convert the parabolic equations to the hyperbolic equations, which are accurately solved by the CE/SE method. The lid-driven rectangular cavity containing a porous Brinkman–Forchheimer medium is studied in this numerical investigation. The Brinkman–Forchheimer equation is used such that both the inertial and viscous effects are incorporated. The governing equations are solved by the improved CE/SE approach. The characteristics of the flow are analyzed with emphasis on the influence of the Darcy number and the cavity depth. It is found that the porous medium effect decreases both the strength and the number of eddies, especially for deep cavities.

Keywords

Lid-driven cavity flow Porous medium CE/SE method CO2 storage Forchheimer 

List of symbols

u

Dimensionless volume-averaged velocity vector [–]

P

Dimensionless pressure [–]

α = μe/μ

Viscosity ratio [–]

μ

Shear viscosity of the fluid [ML−1T−1]

μe

Effective viscosity [ML−1T−1]

Re = LU/μ

Reynolds number [–]

\({\varphi}\)

Porosity [–]

Da

Darcy number [–]

t

Dimensionless time [–]

L

Characteristic length [L]

U

Characteristic velocity [LT−1]

b

Depth ratio [–]

Fb

Forchheimer parameter (Geometric function) [L−1]

K

Permeability [L2]

u

Dimensionless horizontal velocity [–]

v

Dimensionless vertical velocity [–]

τ

Dimensionless virtual time [–]

C2

Artificial compressibility coefficient [–]

ρ

Dimensionless density [–]

(j, k, n)

Space-time mesh points

Q

Vectors of primary variable

E

Flux in x-direction

F

Flux in y-direction

S

Source term vector

Sm

Components of the source term vector

S (V)

Boundary of an arbitrary space-time region

V

Space-time region

Area of a surface element

n

Outward unit normal of a surface element

P′(j, k, n)

Spatial point

Hm

Space-time flux vector

Qm

Components of vector Q

Em

Components of vector E

Fm

Components of vector F

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References

  1. Al-amiri A.M.: Analysis of momentum and energy transfer in a lid-driven cavity filled with a porous medium. Int. J. Heat Mass Transf. 43, 3513–3527 (2000)CrossRefGoogle Scholar
  2. Chang S.C.: The method of space-time conservation element and solution element—a new approach for solving the Navier–Stokes and Euler equations. J. Comput. Phys. 119, 295–324 (1995)CrossRefGoogle Scholar
  3. Chorin A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)CrossRefGoogle Scholar
  4. Costa V.A.F., Oliveira M.S.A., Sousa A.C.M.: Numerical simulation of non-Darcian flows through spaces partially filled with a porous medium. Comput. Struct. 82, 1535–1541 (2004)CrossRefGoogle Scholar
  5. Funaro D., Giangi M., Mansutti D.: A splitting method for unsteady incompressible viscous fluids imposing no boundary conditions on pressure. J. Sci. Comput. 13(1), 95–104 (1998)CrossRefGoogle Scholar
  6. Ghia U., Ghia K.N., Shin C.T.: High Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J. Comput. Phys. 48, 387–411 (1982)CrossRefGoogle Scholar
  7. Guo Z., Zhao T.S.: Lattice Boltzmann model for incompressible flows through porous media. Phys. Rev. E 66, 036304-1–036304-9 (2002)Google Scholar
  8. Guo Y., Hsu A.T., Wu J. et al.: Extension of CE/SE method to 2D viscous flows. Comput. Fluids 33, 1349–1361 (2004)CrossRefGoogle Scholar
  9. Howells I.D.: Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J. Fluid Mech. 64, 449–475 (1974)CrossRefGoogle Scholar
  10. Jue T.C.: Analysis of oscillatory flow with thermal convection in a rectangle cavity filled with porous medium. Int. Commun. Heat Mass Transf. 27(7), 985–994 (2000)CrossRefGoogle Scholar
  11. Kenta S., Takashi F., Yuichi N. et al.: Numerical simulation of supercritical CO2 injection into subsurface rock masses. Energy Convers. Manag. 49, 54–61 (2008)CrossRefGoogle Scholar
  12. Khanafer K.M., Chamkha A.J.: Mixed convection flow in a lid-driven enclosure filled with a fluid-saturated porous medium. Int. J. Heat Mass Transf. 42, 2465–2481 (1999)CrossRefGoogle Scholar
  13. Khanafer K., Vafai K.: Double-diffusive mixed convection in a lid-driven enclosure filled with a fluid-saturated porous medium. Numer. Heat Trans. A 42, 465–486 (2002)CrossRefGoogle Scholar
  14. Oztop H.F., Dagtekin I.: Mixed convection in a two-sided lid-driven differentially heated square cavity. Int. J. Heat Mass Transf. 47, 1761–1769 (2004)CrossRefGoogle Scholar
  15. Shankar P.N., Deshpande M.D.: Fluid mechanics in the driven cavity. Ann. Rev. Fluid Mech. 32, 93–136 (2000)CrossRefGoogle Scholar
  16. Shi J.Q., Xue Z.Q., Durucan S.: Supercritical CO2 core flooding and imbibition in Tako sandstone-influence of sub-core scale heterogeneity. Int. J. Greenh. Gas Control 5, 75–87 (2011)CrossRefGoogle Scholar
  17. Teh C.I., Nie X.: Coupled consolidation theory with non-Darcian flow. Comput. Geotech. 29, 169–209 (2002)CrossRefGoogle Scholar
  18. Wang C.Y.: The recirculating flow due to a moving lid on a cavity containing a Darcy–Brinkman medium. Appl. Math. Model. 33, 2054–2361 (2009)CrossRefGoogle Scholar
  19. Wang J.T., Liu K.X., Zhang D.L.: An improved CE/SE scheme for multi-material elastic-plastic flows and its applications. Comput. Fluid 38(3), 544–551 (2008)CrossRefGoogle Scholar
  20. Whitaker S.: The Forchheimer equation: A theoretical development. Transp. Porous Media 25, 27–61 (1996)CrossRefGoogle Scholar
  21. Xu X.F., Chen S.Y., Zhang D.X.: Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers. Adv. Water Resour. 29, 397–407 (2006)CrossRefGoogle Scholar
  22. Yamada H., Nakamura F., Watanabe Y. et al.: Measuring hydraulic permeability in a streambed using the packer test. Hydrol. Process. 19, 2507–2524 (2005)CrossRefGoogle Scholar
  23. Yang D.X., Yang Y.P., Costa V.A.F.: Numerical simulation of non-Darcian flow through a porous medium. Particuology 7, 193–198 (2009)CrossRefGoogle Scholar
  24. Yang D.X., Li G.M., Zhang D.L.: A CE/SE scheme for flows in porous media and its application. Aerosol Air Qual. Res. 9(2), 266–276 (2009)Google Scholar
  25. Zhang, Z., Yu, S.T., Chang, S.C. et al.: A Modified space-time CE/SE method for Euler and Navier–Stokes equations. AIAA, 99–3277 (1999)Google Scholar
  26. Zhang W., Huang G., Zhan H. et al.: Two-region non-Darcian flow toward a well in a confined aquifer. Adv. Water Resour. 31, 818–827 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institute of Crustal Dynamics, CEABeijingPeople’s Republic of China
  2. 2.Research Institute of Innovative Technology for the Earth (RITE)KyotoJapan
  3. 3.Department of Earth SciencesDurham UniversityDurhamUK

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