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An FFT method for the computation of thermal diffusivity of porous periodic media

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Abstract

In this paper, we provide fast Fourier transform (FFT) iterative schemes to compute the thermal diffusivity of a periodic porous medium. We consider the fluid flow through a porous rigid solid due to a prescribed macroscopic gradient of pressure and a macroscopic gradient of temperature. As already proved in the literature, the asymptotic homogenization procedure is reduced to the resolution of two separated problems for the unit cell: (i) the fluid flow governed by the Stokes equations with an applied gradient of pressure, and (ii) the heat transfer by both convection and conduction due to an applied macroscopic gradient of temperature. We develop new numerical approaches based on FFT for the implementation of the cell problems. In a first approach, a simple iterative method based on the primal variable (gradient of temperature) is provided to solve the heat transfer problem. In order to improve the convergence in the range of high values of the prescribed gradient of pressure, we propose a more sophisticated iterative scheme based on the polarization. In order to evaluate their capacities, these FFT algorithms are applied to some specific microstructures of interest including flows past parallel pores (Poiseuille flows) and periodically or randomly distributed cylinders.

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References

  1. Auriault, J.-L., Geindreau, C., Boutin, C.: Filtration law in porous media with poor separation of scales. Transp. Porous Med. 60, 89–108 (2005)

    Article  MathSciNet  Google Scholar 

  2. Alcocer, F.J., Kumar, V., Singh, P.: Permeability of periodic porous media. Phys. Rev. E. 59, 711–714 (1999)

    Article  Google Scholar 

  3. Alcocer, F.J., Singh, P.: Permeability of periodic arrays of cylinders for viscoelastic flows. Phys. Fluids 14, 2578–2581 (2002)

    Article  MATH  Google Scholar 

  4. Allaire, G., Raphael, A.L.: Homogenization of a convection–diffusion model with reaction in a porous medium. C.R. Acad. Sci. Paris Ser. I 344, 523–528 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alshare, A.A., Strykowski, P.J., Simon, T.W.: Modeling of unsteady and steady fluid flow, heat transfer and dispersion in porous media using unit cell scale. Int J Heat Mass Transfer 53, 2294–2310 (2010)

    Article  MATH  Google Scholar 

  6. Arbogast, T., Brunson, D.S.: A computatonal method for approximating a Darcy-Stokes system governing a vuggy porous, ICES Report 03–47. University of Texas, Austin (2003)

    Google Scholar 

  7. Arnold, D.N., Brezzi, F., Fortin, M.: A stable finite element for the Stokes equations. Calcolo. 21, 337–344 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Auriault, J.L.: Upscaling heterogeneous media by asymptotic expansions. J. Eng. Mech. 128, 817–822 (2002)

    Article  Google Scholar 

  9. Auriault, J.L., Boutin, C., Geindreau, C.: Homogenization of Coupled Phenomena in Heterogenous Media. Wiley, London (2010)

    Google Scholar 

  10. Bahloff, M., Mikelic, A., Wheeler, M.F.: Polynomial filtration law for low Reynolds number flows through porous media. Transp. Porous Media 81, 36–60 (2010)

    MathSciNet  Google Scholar 

  11. Barrere, J ., Caltagirone, J.P., Gipouloux, O.: Détermination numérique de la perméabilité en milieu poreux périodique tridimensionnel. C. R. Acad. Sci. Paris Sér. II 310, 347–352 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Berdichevsky, A.L., Cai, Z.: Perform permeability predictions by self consistent method and finite element simulation. Polym. Compos. 14, 132–143 (1993)

    Article  Google Scholar 

  13. Bloch, J.F., Auriault, J.L.: Heat transfer in nonsaturated porous media. Modelling by homogenisation. Transp. Porous Media 30, 301–321 (1998)

    Article  MathSciNet  Google Scholar 

  14. Bonnet, G.: Effective properties of elastic periodic composite media with fibers. J. Mech. Phys. Solids 55, 881–899 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Carbonell, R.G., Whitaker, S.: Heat and mass transfer in porous media. In: Fundamentals of Transport Phenomena in Porous Media, pp. 121–198. Springer, Berlin (1984)

  16. Auriault, J.L., Skjetne, E.: New insights on steady, non-linear flow in porous media. Eur. J. Mech. B Fluids 18, 131–145 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Edwards, D.A., Shapiro, M., Bar Yoseph, P., Shapira, M.: The influence of Reynolds number upon the apparent permeability of spatially periodic arrays of cylinders. Phys. Fluids A 2, 45–55 (1990)

    Article  MATH  Google Scholar 

  18. Forchheimer, P.: Wasserbewegung durch Boden. Z. Ver. Deutsch. Ing 45, 1782–1788 (1901)

    Google Scholar 

  19. Fortin, M.: Old and new finite element for incompressible flows. Int. J. Numer. Methods Fluids 1, 347–364 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Giorgi, T.: Derivation of the Forchheimer law via matched asymptotic expansions. Transp. Porous Media 29, 191–206 (1997)

    Article  Google Scholar 

  21. 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 (1949)

    Article  MATH  Google Scholar 

  22. Brinkman, H.C.: On the permeability of media consisting of closely packed porous particles. Appl. Sci. Res. 1, 81–86 (1949)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  25. Idris, Z., Orgeas, L., Geindreau, C., Bloch, J.F., Auriault, J.L.: Microstructural effects on the flow law of power law fluids through fibrous media. Model. Simul. Mater. Sci. Eng. 12, 317–330 (2004)

    Article  Google Scholar 

  26. Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40, 3647–3679 (2003)

    Article  MATH  Google Scholar 

  27. Koutsourelakis, P.S.: Stochastic upscaling in solid mechanics: an exercise in machine learning. J. Comput. Phys. 226, 301–325 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kuwahara, F., Nakayama, A., Koyama, H.: A numerical study of thermal dispersion in porous media. J. Heat. Transf. 118, 756–761 (2007)

    Article  Google Scholar 

  29. Ly, H.-B., Monchiet, V., Grande, D.: Computation of permeability with fast Fourier transform from 3d digital images of microstructures. Int. J. Numer. Methods Heat Fluid Flow 26, 1328–1345 (2016)

    Article  MATH  Google Scholar 

  30. Malinouskaya, I., Mourzenko, V.V., Thovert, J.-F., Adler, P.M.: Wave propagation through saturated porous media. Phys. Rev. E 7, 066302 (2008)

    Article  MathSciNet  Google Scholar 

  31. Mei, C.C., Auriault, J.-L.: The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222, 647–663 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Michel, J.-C., Moulinec, H., Suquet, P.: A computational scheme for linear and non-linear composites with arbitrary phase contrast. Int. J. Numer. Methods Eng. 52, 139–160 (2001)

    Article  Google Scholar 

  33. Michel, J.C., Moulinec, H., Suquet, P.: Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Eng. 172, 109–143 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  35. Moghari, M.: A numerical study of non-equilibrium convective heat transfer in porous media. J. Enhanced Heat Transfer. 18, 81–99 (2008)

    Article  Google Scholar 

  36. Monchiet, V.: Combining FFT methods and variational principles to compute bounds and estimates for the properties of elastic composites. Comput. Methods Appl. Mech. Eng. 283, 454–473 (2015)

    Article  MathSciNet  Google Scholar 

  37. Monchiet, V., Bonnet, G.: A polarization-based fft iterative scheme for computing the effective properties of elastic composites with arbitrary contrast. Int. J. Numer. Methods Eng. 89, 1419–1436 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Monchiet, V., Bonnet, G.: A polarization-based fast numerical method for computing the effective conductivity of composites. Int. J. Numer. Methods Heat Fluid Flow 23, 1256–1271 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Monchiet, V., Bonnet, G., Lauriat, G.: A fft-based method to compute the permeability induced by a Stokes slip flow through a porous medium. CR Mecanique 337, 192–197 (2009)

  40. Moulinec, H., Suquet, P.: A fast numerical method for computing the linear and nonlinear mechanical properties of composites. C. R. Acad. Sci. 318, 1417–1423 (1994)

    MATH  Google Scholar 

  41. Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput. Methods Appl. Mech. Eng. 157, 69–94 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  42. Moyne, C.: Two-equation model for a diffusive process in porous media using the volume averaging method with an unsteady-state closure. Adv. Water Resour. 20, 63–76 (1997)

    Article  Google Scholar 

  43. Moyne, C., Didierjean, S., Souto, H.P.A., Da Silveira, O.T.: Thermal dispersion in porous media: one-equation model. Int. J. Heat Mass Transf. 43, 3853–3867 (2000)

    Article  MATH  Google Scholar 

  44. Nguyen, T.-K., Monchiet, V., Bonnet, G.: A Fourier based numerical method for computing the dynamic permeability of porous media. Eur. J. Mech. B/Fluids. 37, 90–98 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. Pedras, M.H.J., De Lemos, M.J.S.: Thermal dispersion in porous media as a function of the solid-fluid conductivity ratio. Int. J. Heat Mass Transf. 51, 5359–5367 (2008)

    Article  MATH  Google Scholar 

  46. Sanchez-Palencia, E., Zaoui, A.: Homogenization Techniques for Composite Media. Springer, London (1987)

    Book  MATH  Google Scholar 

  47. Vu, T.L., Lauriat, G., Manca, O.: Forced convection of air through networks of Square rods or cylinders embedded in microchannels. Microfluid. Nanofluid. 16, 287–304 (2014)

    Article  Google Scholar 

  48. Whitaker, S.: Diffusion and dispersion in porous media. AIChE J. 13, 420–427 (1967)

    Article  Google Scholar 

  49. Wodie, J.-C., Levy, T.: Correction non lineaire de la loi de darcy. C. R. Acad. Sci. Paris Serie II 312, 157–161 (1991)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Modélisation et Simulation Multi Echelle, LMSME UMR8208 CNRS, Université Paris-Est, 5 Boulevard Descartes, 77454, Marne la Vallée Cedex, France

    Viet-Thanh To, Vincent Monchiet & Quy Dong To

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  1. Viet-Thanh To
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  2. Vincent Monchiet
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  3. Quy Dong To
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Correspondence to Vincent Monchiet.

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To, VT., Monchiet, V. & To, Q.D. An FFT method for the computation of thermal diffusivity of porous periodic media. Acta Mech 228, 3019–3037 (2017). https://doi.org/10.1007/s00707-017-1885-5

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  • DOI: https://doi.org/10.1007/s00707-017-1885-5

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