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Numerical simulation of convective motion in an anisotropic porous medium and cosymmetry conservation

Abstract

The onset of convection in a porous anisotropic rectangle occupied by a heat-conducting fluid heated from below is analyzed on the basis of the Darcy–Boussinesq model. It is shown that there are combinations of control parameters for which the system has a nontrivial cosymmetry and a one-parameter family of stationary convective regimes branches off from the mechanical equilibrium. For the two-dimensional convection equations in a porous medium, finite-difference approximations preserving the cosymmetry of the original system are developed. Numerical results are presented that demonstrate the formation of a family of convective regimes and its disappearance when the approximations do not inherit the cosymmetry property.

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References

  1. D. A. Nield and A. Bejan, Convection in Porous Media, 4th ed. (Springer, New York, 2013).

    Book  MATH  Google Scholar 

  2. L. Storesletten, “Effects of anisotropy on convection in horizontal and inclined porous layers,” in Emerging Technologies and Techniques in Porous Media, Ed. by D. B. Ingham (Kluwer, Dordrecht, 2004), pp. 285–306.

    Google Scholar 

  3. A. P. Tyvand and L. Storesletten, “Onset of convection in an anisotropic porous layer with vertical principal axes,” Trans. Porous Med. 108, 581–593 (2015).

    MathSciNet  Article  Google Scholar 

  4. D. V. Lyubimov, “Convective motions in a porous medium heated from below,” J. Appl. Mech. Tech. Phys. 16 (2), 257–262 (1975).

    Article  Google Scholar 

  5. V. I. Yudovich, “Cosymmetry, degeneration of solutions of operator equations, and onset of filtration convection,” Math. Notes 49 (5), 540–545 (1991).

    MathSciNet  Article  MATH  Google Scholar 

  6. V. I. Yudovich, “Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it,” Chaos 5 (2), 402–411 (1995).

    MathSciNet  Article  MATH  Google Scholar 

  7. V. N. Govorukhin, “Numerical simulation of the loss of stability for secondary steady regimes in the Darcy plane-convection problem,” Dokl. Phys. 43 (12), 806–808 (1998).

    Google Scholar 

  8. B. Karasözen and V. G. Tsybulin, “Finite-difference approximation and cosymmetry conservation in filtration convection problem,” Phys. Lett. A 262, 321–329 (1999).

    MathSciNet  Article  MATH  Google Scholar 

  9. B. Karasözen and V. G. Tsybulin, “Mimetic discretization of two-dimensional Darcy convection,” Comput. Phys. Commun. 167, 203–213 (2005).

    Article  MATH  Google Scholar 

  10. O. Yu. Kantur and V. G. Tsibulin, “A spectral–difference method for computing convective fluid motions in a porous medium and cosymmetry preservation,” Comput. Math. Math. Phys. 42 (6), 878–888 (2002).

    MathSciNet  MATH  Google Scholar 

  11. B. D. Moiseenko and I. V. Fryazinov, “Fully neutral scheme for the Navier–Stokes equations,” in Numerical Study of Hydrodynamic Instability (Inst. Prikl. Mat. Akad. Nauk SSSR, Moscow, 1980), pp. 186–209 [in Russian].

    Google Scholar 

  12. A. Arakawa, “Computational design for long-term numerical integration of the equations of fluid motion: Twodimensional incompressible flow, Part 1,” J. Comput. Phys. 1, 119–143 (1966).

    Article  MATH  Google Scholar 

  13. Y. Morinishi, T. S. Lund, O. V. Vasilyev, and P. Moin, “Fully conservative higher order finite difference schemes for incompressible flow,” J. Comput. Phys. 143, 90–124 (1998).

    MathSciNet  Article  MATH  Google Scholar 

  14. F. H. Harlow and J. E. Welch, “Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface,” Phys. Fluids 8, 2182–2189 (1965).

    MathSciNet  Article  MATH  Google Scholar 

  15. A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).

    MATH  Google Scholar 

  16. V. I. Yudovich, “On bifurcations under cosymmetry-breaking perturbations,” Dokl. Phys. 49 (9), 522–526 (2004).

    MathSciNet  Article  Google Scholar 

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Correspondence to M. A. Abdelhafez.

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Original Russian Text © M.A. Abdelhafez, V.G. Tsybulin, 2017, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 10, pp. 1734–1748.

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Abdelhafez, M.A., Tsybulin, V.G. Numerical simulation of convective motion in an anisotropic porous medium and cosymmetry conservation. Comput. Math. and Math. Phys. 57, 1706–1719 (2017). https://doi.org/10.1134/S0965542517100025

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  • DOI: https://doi.org/10.1134/S0965542517100025

Keywords

  • convection
  • porous medium
  • anisotropy
  • cosymmetry
  • finite-difference method
  • staggered grids