1932

Abstract

Turbulent flows in porous media occur in a wide variety of applications, from catalysis in packed beds to heat exchange in nuclear reactor vessels. In this review, we summarize the current state of the literature on methods to model such flows. We focus on a range of Reynolds numbers, covering the inertial regime through the asymptotic turbulent regime. The review emphasizes both numerical modeling and the development of averaged (spatially filtered) balances over representative volumes of media. For modeling the pore scale, we examine the recent literature on Reynolds-averaged Navier–Stokes (RANS) models, large-eddy simulation (LES) models, and direct numerical simulations (DNS). We focus on the role of DNS and discuss how spatially averaged models might be closed using data computed from DNS simulations. A Darcy–Forchheimer-type law is derived, and a prior computation of the permeability and Forchheimer coefficient is presented and compared with existing data.

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Erratum: Modeling Turbulent Flows in Porous Media
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2020-01-05
2024-10-06
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Literature Cited

  1. Antohe B, Lage J 1997. A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int. J. Heat Mass Transf. 13:3013–24
    [Google Scholar]
  2. Apte S, He X, Wood B 2018. Volume-averaged continuum approach for turbulent flows in porous media: an a-priori DNS analysis. Proceedings of the Summer Program 2018 P Moin, J Urzay75–84 Stanford, CA: Cent. Turbul. Res.
    [Google Scholar]
  3. Apte S, Martin M, Patankar N 2009. A numerical method for fully resolved simulation FRS of rigid particle-flow interactions in complex flows. J. Comput. Phys. 228:2712–38
    [Google Scholar]
  4. Atmakidis T, Kenig E 2009. CFD-based analysis of the wall effect on the pressure drop in packed beds with moderate tube/particle diameter ratios in the laminar flow regime. Chem. Eng. J. 155:404–10
    [Google Scholar]
  5. Battiato I, Ferrero PT, O'Malley D, Miller CT,, Takhar PS 2019. Theory and applications of macroscale models in porous media. Transp. Porous Media 1305–76
    [Google Scholar]
  6. Bear J 1972. Dynamics of Flow in Porous Media New York: Am. Elsevier
    [Google Scholar]
  7. Belcher SE 2005. Mixing and transport in urban areas. Philos. Trans. R. Soc. Lond. A 363:2947–68
    [Google Scholar]
  8. Belcher SE, Harman IN, Finnigan JJ 2012. The wind in the willows: flows in forest canopies in complex terrain. Annu. Rev. Fluid Mech. 44:479–504
    [Google Scholar]
  9. Bennethum LS, Giorgi T 1997. Generalized Forchheimer equation for two-phase flow based on hybrid mixture theory. Transp. Porous Media 26:261–75
    [Google Scholar]
  10. Berselli LC, Grisanti CR, John V 2007. Analysis of commutation errors for functions with low regularity. J. Comput. Appl. Math. 206:1027–45
    [Google Scholar]
  11. Blake F 1922. The resistance of packing to fluid flow. Trans. Am. Inst. Chem. Eng. 14:415–21
    [Google Scholar]
  12. Blois G, Best JL, Sambrook Smith GH, Hardy RJ 2014. Effect of bed permeability and hyporheic flow on turbulent flow over bed forms. Geophys. Res. Lett. 41:6435–42
    [Google Scholar]
  13. Boomsma K, Poulikakos D, Zwick F 2003. Metal foams as compact high performance heat exchangers. Mech. Mater. 35:1161–76
    [Google Scholar]
  14. Breugem W, Boersma B 2005. Direct numerical simulations of turbulent flow over a permeable wall using a direct and a continuum approach. Phys. Fluids 17:025103
    [Google Scholar]
  15. Burns J, Jamil J, Ramshaw C 2000. Process intensification: operating characteristics of rotating packed beds—determination of liquid hold-up for a high-voidage structured packing. Chem. Eng. Sci. 55:2401–15
    [Google Scholar]
  16. Calis H, Nijenhuis J, Paikert B, Dautzenberg F, Van Den Bleek C 2001. CFD modelling and experimental validation of pressure drop and flow profile in a novel structured catalytic reactor packing. Chem. Eng. Sci. 56:1713–20
    [Google Scholar]
  17. Chau K, Gaffney J, Baird C, Church G 1985. Resistance to air flow of oranges in bulk and in cartons. Trans. ASAE 28:2083–88
    [Google Scholar]
  18. Christakos G 2000. Modern Spatiotemporal Geostatistics Oxford: Oxford Univ. Press
    [Google Scholar]
  19. Chu X, Weigand B, Vaikuntanathan V 2018. Flow turbulence topology in regular porous media: from macroscopic to microscopic scale with direct numerical simulation. Phys. Fluids 30:065102
    [Google Scholar]
  20. Chukwudozie C, Tyagi M 2013. Pore scale inertial flow simulations in 3-D smooth and rough sphere packs using lattice Boltzmann method. AIChE J. 59:4858–70
    [Google Scholar]
  21. Craft T, Launder B 1996. A Reynolds stress closure designed for complex geometries. Int. J. Heat Fluid Flow 17:245–54
    [Google Scholar]
  22. Dave A, Sun K, Hu L 2018. Numerical simulations of molten salt pebble-bed lattices. Ann. Nucl. Energy 112:400–10
    [Google Scholar]
  23. de Carvalho TP, Morvan HP, Hargreaves D, Oun H, Kennedy A 2015. Experimental and tomography-based CFD investigations of the flow in open cell metal foams with application to aero engine separators. 2015 Proceedings of the ASME Turbo Expo 2015: Turbine Technical Conference and Exposition Pap. GT2015-43509. New York: Am. Soc. Mech. Eng.
    [Google Scholar]
  24. de Carvalho TP, Morvan H, Hargreaves D, Oun H, Kennedy A 2017. Pore-scale numerical investigation of pressure drop behaviour across open-cell metal foams. Transp. Porous Media 117:311–36
    [Google Scholar]
  25. de Lemos MJS 2006. Turbulence in Porous Media: Modeling and Applications New York: Elsevier
    [Google Scholar]
  26. de Lemos MJS 2009. Numerical simulation of turbulent combustion in porous materials. Int. Commun. Heat Mass Transf. 36:996–1001
    [Google Scholar]
  27. de Wasch A, Froment G 1972. Heat transfer in packed beds. Chem. Eng. Sci. 27:567–76
    [Google Scholar]
  28. Deen NG, Kriebitzsch SH, van der Hoef MA, Kuipers J 2012. Direct numerical simulation of flow and heat transfer in dense fluid–particle systems. Chem. Eng. Sci. 81:329–44
    [Google Scholar]
  29. Deen NG, Peters E, Padding JT, Kuipers J 2014. Review of direct numerical simulation of fluid–particle mass, momentum and heat transfer in dense gas–solid flows. Chem. Eng. Sci. 116:710–24
    [Google Scholar]
  30. Dixon AG, Nijemeisland M, Stitt EH 2006. Packed tubular reactor modeling and catalyst design using computational fluid dynamics. Adv. Chem. Eng. 31:307–89
    [Google Scholar]
  31. Dixon AG, Nijemeisland M, Stitt EH 2013. Systematic mesh development for 3D CFD simulation of fixed beds: contact points study. Comput. Chem. Eng. 48:135–53
    [Google Scholar]
  32. Dixon AG, Walls G, Stanness H, Nijemeisland M, Stitt EH 2012. Experimental validation of high Reynolds number CFD simulations of heat transfer in a pilot-scale fixed bed tube. Chem. Eng. J. 200:344–56
    [Google Scholar]
  33. Dolamore F, Fee C, Dimartino S 2018. Modelling ordered packed beds of spheres: the importance of bed orientation and the influence of tortuosity on dispersion. J. Chromatogr. A 1532:150–60
    [Google Scholar]
  34. Duraisamy K, Iaccarino G, Xiao H 2019. Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51:357–77
    [Google Scholar]
  35. Dybbs A, Edwards R 1984. A new look at porous media fluid mechanics—Darcy to turbulent. Fundamentals of Transport Phenomena in Porous Media J Bear, Y Corapcioglu199–254 Dordrecht, Neth.: Martinus Nijhof
    [Google Scholar]
  36. Elenbaas JR, Katz DL 1948. A radial turbulent flow formula. Trans. AIME 174:25–40
    [Google Scholar]
  37. Ergun S 1952. Fluid flow through packed columns. J. Chem. Eng. Prog. 48:89–94
    [Google Scholar]
  38. Eshghinejadfard A, Daróczy L, Janiga G, Thévenin D 2016. Calculation of the permeability in porous media using the lattice Boltzmann method. Int. J. Heat Fluid Flow 62:93–103
    [Google Scholar]
  39. Fancher GH, Lewis JA 1933. Flow of simple fluids through porous materials. Ind. Eng. Chem. 25:1139–47
    [Google Scholar]
  40. Ferdos F, Dargahi B 2016. A study of turbulent flow in large-scale porous media at high Reynolds numbers. Part I: numerical validation. J. Hydraul. Res. 54:663–77
    [Google Scholar]
  41. Finn J 2013. A numerical study of inertial flow features in moderate Reynolds number flow through packed beds of spheres Ph.D. Thesis, Oregon State Univ., Corvallis, OR
    [Google Scholar]
  42. Finn J, Apte SV 2012. Characteristics of vortical structures in random and arranged packed beds of spheres. ASME 2012 Fluids Engineering Division Summer Meeting, pp. 129–38 New York: Am. Soc. Mech. Eng.
    [Google Scholar]
  43. Finn J, Apte SV 2013a. Integrated computation of finite-time Lyapunov exponent fields during direct numerical simulation of unsteady flows. Chaos: Interdiscip. J. Nonlinear Sci. 23:013145
    [Google Scholar]
  44. Finn J, Apte SV 2013b. Relative performance of body fitted and fictitious domain simulations of flow through fixed packed beds of spheres. Int. J. Multiphase Flow 56:54–71
    [Google Scholar]
  45. Finnigan J 2000. Turbulence in plant canopies. Annu. Rev. Fluid Mech. 32:519–71
    [Google Scholar]
  46. Forchheimer P 1901. Wasserbewegung durch Boden. Z. Ver. dtsch. Ing. 45:1736–41
    [Google Scholar]
  47. Forward EA 1945. Communications in response to “On the resistance coefficient-Reynolds number relationship for fluid flow through a bed of granular material” by H.E. Rose. Proc. Inst. Mech. Eng. 153:163
    [Google Scholar]
  48. Fourar M, Radilla G, Lenormand R, Moyne C 2004. On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Adv. Water Resour. 27:669–77
    [Google Scholar]
  49. Germano M 1992. Turbulence: the filtering approach. J. Fluid Mech. 238:325–36
    [Google Scholar]
  50. Getachew D, Minkowycz W, Lage J 2000. A modified form of the κ-model for turbulent flows of an incompressible fluid in porous media. Int. J. Heat Mass Transf. 43:2909–15
    [Google Scholar]
  51. Glowinski R, Pan T, Hesla T, Joseph D, Periaux J 2001. A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169:363–426
    [Google Scholar]
  52. Gray WG, Miller CT 2014. Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems Cham, Switz.: Springer Int.
    [Google Scholar]
  53. Guardo A, Coussirat M, Recasens F, Larrayoz M, Escaler X 2006. CFD study on particle-to-fluid heat transfer in fixed bed reactors: convective heat transfer at low and high pressure. Chem. Eng. Sci. 61:4341–53
    [Google Scholar]
  54. Gunjal P, Ranade V, Chaudhari R 2005. Computational study of a single-phase flow in packed beds of spheres. AIChE J. 51:365–78
    [Google Scholar]
  55. Haeri S, Shrimpton J 2012. On the application of immersed boundary, fictitious domain and body-conformal mesh methods to many particle multiphase flows. Int. J. Multiphase Flow 40:38–55
    [Google Scholar]
  56. Hanjalić K, Launder B 1972. A Reynolds stress model of turbulence and its application to thin shear flows. J. Fluid Mech. 52:609–38
    [Google Scholar]
  57. He X, Apte S, Finn J, Wood B 2019. Characteristics of unsteady inertial to turbulent flows in a periodic porous unit cell. J. Fluid Mech. 873608–45
    [Google Scholar]
  58. He X, Apte S, Schneider K, Kadoch B 2018. Angular multiscale statistics of turbulence in a porous bed. Phys. Rev. Fluids 3:084501
    [Google Scholar]
  59. Hester ET, Cardenas MB, Haggerty R, Apte SV 2017. The importance and challenge of hyporheic mixing. Water Resour. Res. 53:3565–75
    [Google Scholar]
  60. Hill RJ, Koch DL 2002a. Moderate-Reynolds-number flow in a wall-bounded porous medium. J. Fluid Mech. 453:315–44
    [Google Scholar]
  61. Hill RJ, Koch DL 2002b. The transition from steady to weakly turbulent flow in a close-packed ordered array of spheres. J. Fluid Mech. 465:59–97
    [Google Scholar]
  62. Hill RJ, Koch DL, Ladd AJC 2001a. Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448:243–78
    [Google Scholar]
  63. Hill RJ, Koch DL, Ladd AJC 2001b. The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448:213–42
    [Google Scholar]
  64. Howell J, Hall M, Ellzey J 1996. Combustion of hydrocarbon fuels within porous inert media. Prog. Energy Combust. Sci. 22:121–45
    [Google Scholar]
  65. Hutter C, Zenklusen A, Kuhn S, von Rohr PR 2011. Large eddy simulation of flow through a streamwise-periodic structure. Chem. Eng. Sci. 66:519–29
    [Google Scholar]
  66. Irvine D, Jayas D, Mazza G 1993. Resistance to airflow through clean and soiled potatoes. Trans. ASAE 36:1405–10
    [Google Scholar]
  67. Jafari A, Zamankhan P, Mousavi S, Pietarinen K 2008. Modeling and CFD simulation of flow behavior and dispersivity through randomly packed bed reactors. Chem. Eng. J. 144:476–82
    [Google Scholar]
  68. Jayaraju S, Roelofs F, Komen E, Dehbi A 2016. RANS modeling of fluid flow and dust deposition in nuclear pebble-beds. Nucl. Eng. Des. 308:222–37
    [Google Scholar]
  69. Jeong J, Hussain F 1995. On the identification of a vortex. J. Fluid Mech. 285:69–94
    [Google Scholar]
  70. Jin Y, Kuznetsov A 2017. Turbulence modeling for flows in wall bounded porous media: an analysis based on direct numerical simulations. Phys. Fluids 29:045102
    [Google Scholar]
  71. Jin Y, Uth MF, Kuznetsov A, Herwig H 2015. Numerical investigation of the possibility of macroscopic turbulence in porous media: a direct numerical simulation study. J. Fluid Mech. 766:76–103
    [Google Scholar]
  72. Jouybari N, Maerefat M, Nimvari M 2016. A pore scale study on turbulent combustion in porous media. Heat Mass Transf. 52:269–80
    [Google Scholar]
  73. Kaviany M 2012. Principles of Heat Transfer in Porous Media New York: Springer
    [Google Scholar]
  74. Kazerooni RB, Hannani SK 2007. Simulation of turbulent flow through porous media employing a v2f model. AIP Conf. Proc. 963:1257–60
    [Google Scholar]
  75. Khayamyan S, Lundström TS, Gren P, Lycksam H, Hellström JGI 2017. Transitional and turbulent flow in a bed of spheres as measured with stereoscopic particle image velocimetry. Transp. Porous Media 117:45–67
    [Google Scholar]
  76. Koch DL, Hill RJ 2001. Inertial effects in suspension and porous-media flows. Annu. Rev. Fluid Mech. 33:619–47
    [Google Scholar]
  77. Kołodziej A, Krajewski W, Dubis A 2001. Alternative solution for strongly exothermal catalytic reactions: a new metal-structured catalyst carrier. Catal. Today 69:115–20
    [Google Scholar]
  78. Kopanidis A, Theodorakakos A, Gavaises E, Bouris D 2010. 3D numerical simulation of flow and conjugate heat transfer through a pore scale model of high porosity open cell metal foam. Int. J. Heat Mass Transf. 53:2539–50
    [Google Scholar]
  79. Kundu P, Kumar V, Mishra I 2014. Numerical modeling of turbulent flow through isotropic porous media. Int. J. Heat Mass Transf. 75:40–57
    [Google Scholar]
  80. Kuroki M, Ookawara S, Ogawa K 2009. A high-fidelity CFD model of methane steam reforming in a packed bed reactor. J. Chem. Eng. Jpn. 42:73–78
    [Google Scholar]
  81. Kuwahara F, Kameyama Y, Yamashita S, Nakayama A 1998. Numerical modeling of turbulent flow in porous media using a spatially periodic array. J. Porous Media 1:47–55
    [Google Scholar]
  82. Kuznetsov AV 2017. What we can learn from DNS of turbulence in porous media: modeling turbulent flow in composite porous/fluid domains. Proceedings of the 13th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics HEFAT2017 JP Meyer932–39 Leonia, NJ: EDAS
    [Google Scholar]
  83. Ladd AJ 1994a. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech. 271:285–309
    [Google Scholar]
  84. Ladd AJ 1994b. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271:311–39
    [Google Scholar]
  85. Lage JL 1998. The fundamental theory of flow through permeable media from Darcy to turbulence. Transport Phenomena in Porous Media I 1 DB Ingham, I Pop1–30 Oxford: Elsevier Sci.
    [Google Scholar]
  86. Lage J, De Lemos M, Nield D 2002. Modeling turbulence in porous media. Transport Phenomena in Porous Media II DB Ingham, I Pop198–230 Oxford: Elsevier Sci.
    [Google Scholar]
  87. Langford JA, Moser RD 1999. Optimal LES formulations for isotropic turbulence. J. Fluid Mech. 398:321–46
    [Google Scholar]
  88. Larsson H, Schjøtt Andersen PA, Byström E, Gernaey KV, Krühne U 2017. CFD modeling of flow and ion exchange kinetics in a rotating bed reactor system. Ind. Eng. Chem. Res. 56:3853–65
    [Google Scholar]
  89. Lasseux D, Arani A, Ahmadi A 2011. On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media. Phys. Fluids 23:7
    [Google Scholar]
  90. Lasseux D, Valdés-Parada FJ, Bellet F 2019. Macroscopic model for unsteady flow in porous media. J. Fluid Mech. 862:283–311
    [Google Scholar]
  91. Lee KB, Howell JR 1991. Theoretical and experimental heat and mass transfer in highly porous media. Int. J. Heat Mass Transf. 34:2123–32
    [Google Scholar]
  92. Leonhard A 1974. Energy cascade in large eddy simulation of turbulent fluid flow. Adv. Geophys. A 18:237–48
    [Google Scholar]
  93. Lian Y, Dallmann J, Sonin B, Roche K, Liu W et al. 2018. Large eddy simulation of turbulent flow over and through a rough permeable bed. Comput. Fluids 180:128–38
    [Google Scholar]
  94. Ling J, Kurzawski A, Templeton J 2016. Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807:155–66
    [Google Scholar]
  95. Link J, Cuypers L, Deen N, Kuipers J 2005. Flow regimes in a spout–fluid bed: a combined experimental and simulation study. Chem. Eng. Sci. 60:3425–42
    [Google Scholar]
  96. Logtenberg S, Nijemeisland M, Dixon AG 1999. Computational fluid dynamics simulations of fluid flow and heat transfer at the wall–particle contact points in a fixed-bed reactor. Chem. Eng. Sci. 54:2433–39
    [Google Scholar]
  97. Lu J, Das S, Peters EAJF, Kuipers JAM 2018. Direct numerical simulation of fluid flow and mass transfer in dense fluid-particle systems with surface reactions. Chem. Eng. Sci. 176:1–18
    [Google Scholar]
  98. Lucci F, Della Torre A, Montenegro G, Kaufmann R, Eggenschwiler PD 2017. Comparison of geometrical, momentum and mass transfer characteristics of real foams to Kelvin cell lattices for catalyst applications. Int. J. Heat Mass Transf. 108:341–50
    [Google Scholar]
  99. Lumley JL 1970. Toward a turbulent constitutive relation. J. Fluid Mech. 41:413–34
    [Google Scholar]
  100. Lumley JL 1979. Computational modeling of turbulent flows. Advances in Applied Mechanics 18 C-S Yih123–76 New York: Academic
    [Google Scholar]
  101. Magnico P 2003. Hydrodynamic and transport properties of packed beds in small tube-to-sphere diameter ratio: pore scale simulation using an Eulerian and a Lagrangian approach. Chem. Eng. Sci. 58:5005–24
    [Google Scholar]
  102. Magnico P 2009. Pore-scale simulations of unsteady flow and heat transfer in tubular fixed beds. AIChE J. 55:849–67
    [Google Scholar]
  103. Mahesh K, Constantinescu G, Apte S, Iaccarino G, Ham F, Moin P 2006. Large-eddy simulation of reacting turbulent flows in complex geometries. J. Appl. Mech. 73:374–81
    [Google Scholar]
  104. Maier RS, Kroll D, Kutsovsky Y, Davis H, Bernard RS 1998. Simulation of flow through bead packs using the lattice Boltzmann method. Phys. Fluids 10:60–74
    [Google Scholar]
  105. Malico I, Ferreira de Sousa P 2012. Modeling the pore level fluid flow in porous media using the immersed boundary method. Numerical Analysis of Heat and Mass Transfer in Porous Media JMPQ Delgado, AG Barbosa de Lima, MV da Silva229–51 Berlin: Springer
    [Google Scholar]
  106. Mansour N, Wray A 1994. Decay of isotropic turbulence at low Reynolds number. Phys. Fluids 6:808–14
    [Google Scholar]
  107. Masuoka T, Takatsu Y 1996. Turbulence model for flow through porous media. Int. J. Heat Mass Transf. 39:2803–9
    [Google Scholar]
  108. Mei C, Auriault JL 1991. The effect of weak inertia on flow through a porous medium. J. Fluid Mech. 222:647–63
    [Google Scholar]
  109. Mittal R, Iaccarino G 2005. Immersed boundary methods. Annu. Rev. Fluid Mech. 37:239–61
    [Google Scholar]
  110. Moin P, Mahesh K 1998. Direct numerical simulation: a tool in turbulence research. Annu. Rev. Fluid Mech. 30:539–78
    [Google Scholar]
  111. Nakayama A, Kuwahara F 1999. A macroscopic turbulence model for flow in a porous medium. J. Fluids Eng. 121:427–33
    [Google Scholar]
  112. Nguyen T, Muyshondt R, Hassan Y, Anand N 2019. Experimental investigation of cross flow mixing in a randomly packed bed and streamwise vortex characteristics using particle image velocimetry and proper orthogonal decomposition analysis. Phys. Fluids 31:025101
    [Google Scholar]
  113. Nield DA 1991. The limitations of the Brinkman-Forchheimer equation in modeling flow in a saturated porous medium and at an interface. Int. J. Heat Fluid Flow 12:269–72
    [Google Scholar]
  114. Nield DA 2001. Alternative models of turbulence in a porous medium, and related matters. J. Fluids Eng. 123:928–31
    [Google Scholar]
  115. Nield DA, Bejan A 2017. Convection in Porous Media Cham, Switz.: Springer Int. 5th ed.
    [Google Scholar]
  116. Nijemeisland M, Dixon A 2004. CFD study of fluid flow and wall heat transfer in a fixed bed of spheres. AIChE J. 50:906–21
    [Google Scholar]
  117. Nimvari ME, Maerefat M, El-Hossaini M, Jouybari NF 2014. Numerical study on turbulence effects in porous burners. J. Porous Media 17:129–42
    [Google Scholar]
  118. Nouri N, Martin A 2015. Three dimensional radiative heat transfer model for the evaluation of the anisotropic effective conductivity of fibrous materials. Int. J. Heat Mass Transf. 83:629–35
    [Google Scholar]
  119. Oberlack M 1997. Invariant modeling in large-eddy simulation of turbulence. Annual Research Briefs 1997, pp. 3–22 Stanford, CA: Cent. Turbul. Res.
    [Google Scholar]
  120. Ovaysi S, Piri M 2010. Direct pore-level modeling of incompressible fluid flow in porous media. J. Comput. Phys. 229:7456–76
    [Google Scholar]
  121. Packman AI, Salehin M, Zaramella M 2004. Hyporheic exchange with gravel beds: basic hydrodynamic interactions and bedform-induced advective flows. J. Hydraul. Eng. 130:647–56
    [Google Scholar]
  122. Panfilov M, Fourar M 2006. Physical splitting of nonlinear effects in high-velocity stable flow through porous media. Adv. Water Resour. 29:30–41
    [Google Scholar]
  123. Patil VA, Liburdy JA 2013a. Flow structures and their contribution to turbulent dispersion in a randomly packed porous bed based on particle image velocimetry measurements. Phys. Fluids 25:113303
    [Google Scholar]
  124. Patil VA, Liburdy JA 2013b. Turbulent flow characteristics in a randomly packed porous bed based on particle image velocimetry measurements. Phys. Fluids 25:043304
    [Google Scholar]
  125. Patil VA, Liburdy JA 2015. Scale estimation for turbulent flows in porous media. Chem. Eng. Sci. 123:231–35
    [Google Scholar]
  126. Pedras MH, de Lemos MJ 2001. Macroscopic turbulence modeling for incompressible flow through undeformable porous media. Int. J. Heat Mass Transf. 44:1081–93
    [Google Scholar]
  127. Pedras MH, de Lemos MJ 2003. Computation of turbulent flow in porous media using a low-Reynolds model and an infinite array of transversally displaced elliptic rods. Numer. Heat Transf. A 43:585–602
    [Google Scholar]
  128. Perot B, Moin P 1995. Shear-free turbulent boundary layers. Part 1. Physical insights into near-wall turbulence. J. Fluid Mech. 295:199–227
    [Google Scholar]
  129. Pokrajac D, Manes C 2009. Velocity measurements of a free-surface turbulent flow penetrating a porous medium composed of uniform-size spheres. Transp. Porous Media 78:367
    [Google Scholar]
  130. Pope S 1975. A more general effective-viscosity hypothesis. J. Fluid Mech. 72:331–40
    [Google Scholar]
  131. Pope S 2000. Turbulent Flows Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  132. Quintard M, Whitaker S 1994. Transport in ordered and disordered porous media I: the cellular average and the use of weighting functions. Transp. Porous Media 14:163–77
    [Google Scholar]
  133. Rose H 1945. On the resistance coefficient—Reynolds number relationship for fluid flow through a bed of granular material. Proc. Inst. Mech. Eng. 153:154–68
    [Google Scholar]
  134. Rosti ME, Brandt L, Pinelli A 2018. Turbulent channel flow over an anisotropic porous wall—drag increase and reduction. J. Fluid Mech. 842:381–94
    [Google Scholar]
  135. Rúa D, Hernández L 2016. Phenomenological evaluation of industrial reformers for glycerol steam reforming. Int. J. Hydrog. Energy 41:13811–19
    [Google Scholar]
  136. Sagaut P 2006. Large Eddy Simulation for Incompressible Flows: An Introduction Berlin: Springer. 3rd ed.
    [Google Scholar]
  137. Schouten E, Borman P, Westerterp K 1994. Oxidation of ethene in a wall-cooled packed-bed reactor. Chem. Eng. Sci. 49:4725–47
    [Google Scholar]
  138. Shams A, Roelofs F, Komen E, Baglietto E 2013. Quasi-direct numerical simulation of a pebble bed configuration. Part I: flow (velocity) field analysis. Nucl. Eng. Des. 263:473–89
    [Google Scholar]
  139. Shams A, Roelofs F, Komen E, Baglietto E 2014. Large eddy simulation of a randomly stacked nuclear pebble bed. Comput. Fluids 96:302–21
    [Google Scholar]
  140. Sharma RK, Cresswell DL, Newson EJ 1991. Kinetics and fixed-bed reactor modeling of butane oxidation to maleic anhydride. AIChE J. 37:39–47
    [Google Scholar]
  141. Shaw RH, Schumann U 1992. Large-eddy simulation of turbulent flow above and within a forest. Bound. Layer Meteorol. 61:47–64
    [Google Scholar]
  142. Shayegan J, Hashemi M, Vakhshouri K 2008. Operation of an industrial steam reformer under severe condition: a simulation study. Can. J. Chem. Eng. 86:747–55
    [Google Scholar]
  143. Shimizu Y, Tsujimoto T, Nakagawa H 1990. Experiment and macroscopic modelling of flow in highly permeable porous medium under free-surface flow. J. Hydrosci. Hydraul. Eng. 8:69–78
    [Google Scholar]
  144. Smale N, Moureh J, Cortella G 2006. A review of numerical models of airflow in refrigerated food applications. Int. J. Refrig 29:911–30
    [Google Scholar]
  145. Smolarkiewicz P, Winter CL 2010. Pores resolving simulation of Darcy flows. J. Comput. Phys. 229:3121–33
    [Google Scholar]
  146. Soulaine C, Quintard M, Baudouy B, Van Weelderen R 2017. Numerical investigation of thermal counterflow of HE II past cylinders. Phys. Rev. Lett. 118:074506
    [Google Scholar]
  147. Spalart PR 2015. Philosophies and fallacies in turbulence modeling. Prog. Aerosp. Sci. 74:1–15
    [Google Scholar]
  148. Srikanth V, Huang C, Su T, Kuznetsov A 2018. Symmetry breaking in porous media as a consequence of the von Kármán instability. arXiv:1810.10141 [physics.flu-dyn]
  149. Suga K, Chikasue R, Kuwata Y 2017. Modelling turbulent and dispersion heat fluxes in turbulent porous medium flows using the resolved LES data. Int. J. Heat Fluid Flow 68:225–36
    [Google Scholar]
  150. Swift G, Kiel O 1962. The prediction of gas-well performance including the effect of non-Darcy flow. J. Pet. Technol. 14:791–98
    [Google Scholar]
  151. Taskin ME, Dixon AG, Nijemeisland M, Stitt EH 2008. CFD study of the influence of catalyst particle design on steam reforming reaction heat effects in narrow packed tubes. Ind. Eng. Chem. Res. 47:5966–75
    [Google Scholar]
  152. Tian FY, Huang LF, Fan LW, Qian HL, Gu JX et al. 2016. Pressure drop in a packed bed with sintered ore particles as applied to sinter coolers with a novel vertically arranged design for waste heat recovery. J. Zhejiang Univ. Sci. A 17:89–100
    [Google Scholar]
  153. Torabi M, Torabi M, Peterson G 2017. Heat transfer and entropy generation analyses of forced convection through porous media using pore scale modeling. J. Heat Transf. 139:012601
    [Google Scholar]
  154. Travkin V 2001. Discussion: alternative models of turbulence in a porous medium, and related matters. J. Fluids Eng. 123:931–34
    [Google Scholar]
  155. Uth MF, Jin Y, Kuznetsov A, Herwig H 2016. A direct numerical simulation study on the possibility of macroscopic turbulence in porous media: effects of different solid matrix geometries, solid boundaries, and two porosity scales. Phys. Fluids 28:065101
    [Google Scholar]
  156. Vafai K, Tien C 1981. Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transf. 24:195–203
    [Google Scholar]
  157. Vigneault C, Markarian NR, Da Silva A, Goyette B 2004. Pressure drop during forced-air ventilation of various horticultural produce in containers with different opening configurations. Trans. ASAE 47:807–14
    [Google Scholar]
  158. von Neumann J 1932. Physical applications of the ergodic hypothesis. PNAS 18:263–66
    [Google Scholar]
  159. Vortmeyer D, Winter R 1982. Impact of porosity and velocity distribution on the theoretical prediction of fixed-bed chemical reactor performance. Chemical Reaction Engineering—Boston J Wei, C Georgakis49–61 Washington, DC: Am. Chem. Soc.
    [Google Scholar]
  160. Wallin S, Johansson AV 2000. An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows. J. Fluid Mech. 403:89–132
    [Google Scholar]
  161. Wang-Kee, Lee WJ, Hassan YA 2008. CFD simulation of a coolant flow and a heat transfer in a pebble bed reactor. Proceedings of the 4th International Topical Meeting on High Temperature Reactor Technology (HTR2008)171–75 New York: Am. Soc. Mech. Eng.
    [Google Scholar]
  162. Warhaft Z 1980. An experimental study of the effect of uniform strain on thermal fluctuations in grid-generated turbulence. J. Fluid Mech. 99:545–73
    [Google Scholar]
  163. Weatheritt J, Sandberg R 2017. The development of algebraic stress models using a novel evolutionary algorithm. Int. J. Heat Fluid Flow 68:298–318
    [Google Scholar]
  164. Wehinger GD, Eppinger T, Kraume M 2015. Detailed numerical simulations of catalytic fixed-bed reactors: heterogeneous dry reforming of methane. Chem. Eng. Sci. 122:197–209
    [Google Scholar]
  165. Whitaker S 1996. The Forchheimer equation: a theoretical development. Transp. Porous Media 25:27–61
    [Google Scholar]
  166. Whitaker S 1999. The Method of Volume Averaging Dordrecht, Neth. Springer Sci. Bus. Media
    [Google Scholar]
  167. Wood BD 2007. Inertial effects in dispersion in porous media. Water Resour. Res. 43:W12S16
    [Google Scholar]
  168. Wood BD 2008. The role of scaling laws in upscaling. Adv. Water Resour. 32:723–36
    [Google Scholar]
  169. Wood BD 2013. Revisiting the geometric theorems for volume averaging. Adv. Water Resour. 62:340–52
    [Google Scholar]
  170. Wood BD, Apte SV, Liburdy JA, Ziazi RM, He X et al. 2015. A comparison of measured and modeled velocity fields for a laminar flow in a porous medium. Adv. Water Resour. 85:45–63
    [Google Scholar]
  171. Wood BD, Valdés-Parada FJ 2013. Volume averaging: local and nonlocal closures using a Green's function approach. Adv. Water Resour. 51:139–67
    [Google Scholar]
  172. Xiang P, Kuznetsov AV, Seyam A 2008. A porous medium model of the hydro entanglement process. J. Porous Media 11:35–49
    [Google Scholar]
  173. Xu J, Froment GF 1989. Methane steam reforming: II. Diffusional limitations and reactor simulation. AIChE J. 35:97–103
    [Google Scholar]
  174. Yaglom AM 2012. Correlation Theory of Stationary and Related Random Functions: Supplementary Notes and References New York: Springer-Verlag
    [Google Scholar]
  175. Yang X, Scheibe TD, Richmond MC, Perkins WA, Vogt SJ et al. 2013. Direct numerical simulation of pore-scale flow in a bead pack: comparison with magnetic resonance imaging observations. Adv. Water Resour. 54:228–241
    [Google Scholar]
  176. Yu Z, Cao E, Wang Y, Zhou Z, Dai Z 2006. Simulation of natural gas steam reforming furnace. Fuel Process. Technol. 87:695–704
    [Google Scholar]
  177. Zeschky J, Goetz-Neunhoeffer F, Neubauer J, Lo SJ, Kummer B et al. 2003. Preceramic polymer derived cellular ceramics. Compos. Sci. Technol. 63:2361–70
    [Google Scholar]
  178. Zhang Z, Jia P, Feng S, Liang J, Long Y, Li G 2018. Numerical simulation of exhaust reforming characteristics in catalytic fixed-bed reactors for a natural gas engine. Chem. Eng. Sci. 191:200–7
    [Google Scholar]
  179. Zhou J, Adrian RJ, Balachandar S, Kendall TM 1999. Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387:353–96
    [Google Scholar]
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