Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Improved Eddy-Viscosity Modelling of Turbulent Flow around Porous–Fluid Interface Regions


The RANS modelling of turbulence across fluid-porous interface regions within ribbed channels has been investigated by applying double (both volume and Reynolds) averaging to the Navier–Stokes equations. In this study, turbulence is represented by using the Launder and Sharma (Lett Heat Mass Transf 1:131–137, 1974) low-Reynolds-number \(k-\varepsilon \) turbulence model, modified via proposals by either Nakayama and Kuwahara (J Fluids Eng 130:101205, 2008) or Pedras and de Lemos (Int Commun Heat Mass Transf 27:211–220, 2000), for extra source terms in turbulent transport equations to account for the porous structure. One important region of the flow, for modelling purposes, is the interface region between the porous medium and clear fluid regions. Here, corrections have been proposed to the above porous drag/source terms in the k and \(\varepsilon \) transport equations that are designed to account for the effective increase in porosity across a thin near-interface region of the porous medium, and which bring about significant improvements in predictive accuracy. These terms are based on proposals put forward by Kuwata and Suga (Int J Heat Fluid Flow 43:35–51, 2013), for second-moment closures. Two types of porous channel flows have been considered. The first case is a fully developed turbulent porous channel flow, where the results are compared with DNS predictions obtained by Breugem et al. (J Fluid Mech 562:35–72, 2006) and experimental data produced by Suga et al. (Int J Heat Fluid Flow 31:974–984, 2010). The second case is a turbulent solid/porous rib channel flow to examine the behaviour of flow through and around the solid/porous rib, which is validated against experimental work carried out by Suga et al. (Flow Turbul Combust 91:19–40, 2013). Cases are simulated covering a range of porous properties, such as permeability and porosity. Through the comparisons with the available data, it is demonstrated that the extended model proposed here shows generally satisfactory accuracy, except for some predictive weaknesses in regions of either impingement or adverse pressure gradients, associated with the underlying eddy-viscosity turbulence model formulation.

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

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


\(c_\mathrm{F} \) :

Forchheimer coefficient

\( c_{\varepsilon 1}, c_{\varepsilon 2}, c_k\) :

Non-dimensional turbulence model constants

\( c_{\mu }\) :

Coefficient in the eddy-viscosity


Darcy number, \( \hbox {Da}=K/H^2 \)

\( d_\mathrm{p} \) :

Pore diameter

\( f^{\phi }_{U}, f^{\phi }_{k}, f^{\phi }_{\varepsilon } \) :

Damping functions for source terms of porous media

\( G_{\varepsilon } \) :

Generation rate of \( \varepsilon \) due to porous media

\( G_k\) :

Generation rate of k due to porous media

H :

Channel height

h :

Rib height

K :

Permeability of porous media

k :

Turbulent kinetic energy

P :


\( P_k \) :

Production term

\( \hbox {Re}_\mathrm{b} \) :

Bulk Reynolds number, \(\hbox {Re}_\mathrm{b}=U_\mathrm{b} H/(2\nu ) \)

\( R_\mathrm{t} \) :

Turbulent Reynolds number

\( U_\mathrm{D} \) :

Darcy or superficial velocity

\( \Delta V \) :

Total volume

\( \Delta V_\mathrm{f} \) :

Fluid volume

\( y^{\prime } \) :

Normal distance from the nearest porous surface

\(\delta _{ij}\) :

Kronecker delta unit symbol

\(\tilde{\varepsilon }\) :

‘Quasi-homogeneous’ dissipation rate of the turbulent kinetic energy, k

\(\nu \) :

Kinematic viscosity

\(\nu _\mathrm{t}\) :

Kinematic turbulent viscosity

\(\phi \) :

Porosity of inhomogeneous porous media \( (=\Delta V_\mathrm{f}/\Delta V) \)

\(\phi _{\infty }\) :

Porosity of homogeneous porous media

\(\rho \) :

Fluid density

\(\left\langle \,\,\right\rangle ^f\) :

Intrinsic average

\(\langle \,\,\rangle \) :

Volume average


Direct numerical simulation


Large eddy simulation


Launder Sharma modified by Nakayama and Kuwahara (2008)


Launder Sharma modified by Pedras and de Lemos (2001b)


Pore per inch


Reynolds-averaged Navier–Stokes


Representative elementary volume


Semi-implicit method for pressure-linkage equations




Upstream monotonic interpolation for scalar


Volume averaging theory


  1. Antohe, B., Lage, J.: A general two-equation macroscopic turbulence model for incompressible flow in porous media. Int. J. Heat Mass Transf. 40(13), 3013–3024 (1997)

  2. Beavers, G., Joseph, D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(01), 197–207 (1967)

  3. Breugem, W., Boersma, B., Uittenbogaard, R.: The influence of wall permeability on turbulent channel flow. J. Fluid Mech. 562, 35–72 (2006)

  4. Chan, H., Zhang, Y., Leu, J., Chen, Y.: Numerical calculation of turbulent channel flow with porous ribs. J. Mech. 26(1), 15–28 (2010)

  5. Chan, H.-C., Huang, W., Leu, J.-M., Lai, C.-J.: Macroscopic modeling of turbulent flow over a porous medium. Int. J. Heat Fluid Flow 28(5), 1157–1166 (2007)

  6. Chandesris, M., Jamet, D.: Boundary conditions at a planar fluid-porous interface for a poiseuille flow. Int. J. Heat Mass Transf. 49(13–14), 2137–2150 (2006)

  7. Chandesris, M., Jamet, D.: Derivation of jump conditions for the turbulence \( k-\varepsilon \) model at a fluid/porous interface. Int. J. Heat Fluid Flow 30(2), 306–318 (2009)

  8. De Lemos, M.J.: Turbulent kinetic energy distribution across the interface between a porous medium and a clear region. Int. Commun. Heat Mass Transf. 32(1–2), 107–115 (2005)

  9. De Lemos, M.J.: Turbulent flow around fluid-porous interfaces computed with a diffusion-jump model for k and \(\varepsilon \) transport equations. Transp. Porous Media 78(3), 331–346 (2009)

  10. Dunn, C., Lopez, F., Garcia, M.H.: Mean flow and turbulence in a laboratory channel with simulated vegatation (hes 51). Technical report (1996)

  11. Dybbs, A., Edwards, R.V.: A new look at porous media fluid mechanics–Darcy to turbulent. In: Bear, J., Corapcioglu, M.Y. (eds.) Fundamentals of Transport Phenomena in Porous Media, pp. 199–256. Springer, Dordrecht (1984)

  12. Finnigan, J.: Turbulence in plant canopies. Annu. Rev. Fluid Mech. 32(1), 519–571 (2000)

  13. Getachew, D., Minkowycz, W., Lage, J.: A modified form of the \( k-\varepsilon \) model for turbulent flows of an incompressible fluid in porous media. Int. J. Heat Mass Transf. 43(16), 2909–2915 (2000)

  14. Hwang, J.: Heat transfer-friction characteristic comparison in rectangular ducts with slit and solid ribs mounted on one wall. Trans. Am. Soc. Mech. Eng. J. Heat Transf. 120, 709–716 (1998)

  15. Iacovides, H., Launder, B.: Internal blade cooling: the cinderella of computational and experimental fluid dynamics research in gas turbines. Proc. Inst. Mech. Eng. Part A J. Pow. Energy 221(3), 265–290 (2007)

  16. Iacovides, H., Raisee, M.: Recent progress in the computation of flow and heat transfer in internal cooling passages of turbine blades. Int. J. Heat Fluid Flow 20(3), 320–328 (1999)

  17. Kaviany, M.: Principles of Heat Transfer in Porous Media. Springer, Berlin (1991)

  18. Kuwata, Y., Suga, K.: Modelling turbulence around and inside porous media based on the second moment closure. Int. J. Heat Fluid Flow 43, 35–51 (2013)

  19. Kuwata, Y., Suga, K., Sakurai, Y.: Development and application of a multi-scale \( k-\varepsilon \) model for turbulent porous medium flows. Int. J. Heat Fluid Flow 49, 135–150 (2014)

  20. Kuznetsov, A.: Numerical modeling of turbulent flow in a composite porous/fluid duct utilizing a two-layer k-\(\varepsilon \) model to account for interface roughness. Int. J. Therm. Sci. 43(11), 1047–1056 (2004)

  21. Kuznetsov, A., Xiong, M.: Development of an engineering approach to computations of turbulent flows in composite porous/fluid domains. Int. J. Therm. Sci. 42(10), 913–919 (2003)

  22. Launder, B., Sharma, B.: Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transf. 1(2), 131–137 (1974)

  23. Lee, K., Howell, J.: Forced convective and radiative transfer within a highly porous layer exposed to a turbulent external flow field. In: Proceedings of the 1987 ASME-JSME Thermal Engineering Joint Conference, vol. 2, pp. 377–386 (1987)

  24. Leu, J., Chan, H., Chu, M.: Comparison of turbulent flow over solid and porous structures mounted on the bottom of a rectangular channel. Flow Meas. Instrum. 19(6), 331–337 (2008)

  25. Lien, F., Leschziner, M.: Multigrid acceleration for recirculating laminar and turbulent flows computed with a non-orthogonal, collocated finite-volume scheme. Comput. Methods Appl. Mech. Eng. 118(3–4), 351–371 (1994a)

  26. Lien, F., Leschziner, M.: Upstream monotonic interpolation for scalar transport with application to complex turbulent flows. Int. J. Numer. Methods Fluids 19(6), 527–548 (1994b)

  27. Liou, T., Chen, S., Shih, K.: Numerical simulation of turbulent flow field and heat transfer in a two-dimensional channel with periodic slit ribs. Int. J. Heat Mass Transf. 45(22), 4493–4505 (2002)

  28. Masuoka, T., Takatsu, Y.: Turbulence model for flow through porous media. Int. J. Heat Mass Transf. 39(13), 2803–2809 (1996)

  29. Mößner, M., Radespiel, R.: Modelling of turbulent flow over porous media using a volume averaging approach and a reynolds stress model. Comput. Fluids 108, 25–42 (2015)

  30. Naaktgeboren, C., Krueger, P., Lage, J.: Inlet and outlet pressure-drop effects on the determination of permeability and form coefficient of a porous medium. J. Fluids Eng. 134(5), 051209 (2012)

  31. Nakayama, A., Kuwahara, F.: A macroscopic turbulence model for flow in a porous medium. Trans. Am. Soc. Mech. Eng. J. Fluids Eng. 121, 427–433 (1999)

  32. Nakayama, A., Kuwahara, F.: A general macroscopic turbulence model for flows in packed beds, channels, pipes, and rod bundles. J. Fluids Eng. 130(10), 101205 (2008)

  33. Nezu, I., Sanjou, M.: Turburence structure and coherent motion in vegetated canopy open-channel flows. J. Hydro Env. Res. 2(2), 62–90 (2008)

  34. Nikora, V., McEwan, I., McLean, S., Coleman, S., Pokrajac, D., Walters, R.: Double-averaging concept for rough-bed open-channel and overland flows: theoretical background. J. Hydraul. Eng. 133(8), 873–883 (2007)

  35. Nuntadusit, C., Wae-Hayee, M., Bunyajitradulya, A., Eiamsa-ard, S.: Thermal visualization on surface with transverse perforated ribs. Int. Commun. Heat Mass Transf. 39(5), 634–639 (2012)

  36. Ochoa-Tapia, J., Whitaker, S.: Momentum transfer at the boundary between a porous medium and a homogeneous fluid—i. theoretical development. Int. J. Heat Mass Transf. 38(14), 2635–2646 (1995)

  37. Panigrahi, P., Schröder, A., Kompenhans, J.: PIV investigation of flow behind surface mounted permeable ribs. Exp. Fluids 40(2), 277–300 (2006)

  38. Panigrahi, P., Schröder, A., Kompenhans, J.: Turbulent structures and budgets behind permeable ribs. Exp. Therm. Fluid Sci. 32(4), 1011–1033 (2008)

  39. Patankar, S., Spalding, D.: A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transf. 15(10), 1787–1806 (1972)

  40. Pedras, M., de Lemos, M.: On the definition of turbulent kinetic energy for flow in porous media. Int. Commun. Heat Mass Transf. 27(2), 211–220 (2000)

  41. Pedras, M., de Lemos, M.: Macroscopic turbulence modeling for incompressible flow through undeformable porous media. Int. J. Heat Mass Transf. 44(6), 1081–1093 (2001a)

  42. Pedras, M., de Lemos, M.: Simulation of turbulent flow in porous media using a spatially periodic array and a low-Re two-equation closure. Num. Heat Trans. Part A Appl. 39(1), 35–59 (2001b)

  43. Pokrajac, D., Manes, C.: Interface between turbulent flows above and within rough porous walls. Acta Geophys. 56(3), 824 (2008)

  44. Prinos, P., Sofialidis, D., Keramaris, E.: Turbulent flow over and within a porous bed. J. Hydraul. Eng. 129(9), 720–733 (2003)

  45. Rhie, C., Chow, W.: Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21(11), 1525–1532 (1983)

  46. Silva, R., de Lemos, M.: Numerical analysis of the stress jump interface condition for laminar flow over a porous layer. Numer. Heat Transf. Part A Appl. 43(6), 603–617 (2003a)

  47. Silva, R., de Lemos, M.: Turbulent flow in a channel occupied by a porous layer considering the stress jump at the interface. Int. J. Heat Mass Transf. 46(26), 5113–5121 (2003b)

  48. Straatman, A., Gallego, N., Yu, Q., Thompson, B.: Characterization of porous carbon foam as a material for compact recuperators. J. Eng. Gas Turbines Pow. 129(2), 326–330 (2007)

  49. Suga, K., Matsumura, Y., Ashitaka, Y., Tominaga, S., Kaneda, M.: Effects of wall permeability on turbulence. Int. J. Heat Fluid Flow 31(6), 974–984 (2010)

  50. Suga, K., Tominaga, S., Mori, M., Kaneda, M.: Turbulence characteristics in flows over solid and porous square ribs mounted on porous walls. Flow Turbul. Combust. 91(1), 19–40 (2013)

  51. Vafai, K., Tien, C.: Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transf. 24(2), 195–203 (1981)

  52. Wang, L., Salewski, M., Sundén, B.: Piv measurements of turbulent flow in a channel with solid or perforated ribs. In: ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels, pp. 1995–2003. American Society of Mechanical Engineers (2010)

  53. Whitaker, S.: The Method of Volume Averaging, vol. 13. Springer, Berlin (1999)

Download references


The financial support from the Ministry of Higher Education and Scientific Research of Iraq and the University of Kufa (Grant No. 754) is gratefully acknowledged. The authors would like to acknowledge the assistance given by IT Services and the use of the Computational Shared Facility at The University of Manchester.

Author information

Correspondence to Qahtan Al-Aabidy.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Al-Aabidy, Q., Craft, T.J. & Iacovides, H. Improved Eddy-Viscosity Modelling of Turbulent Flow around Porous–Fluid Interface Regions. Transp Porous Med 131, 569–594 (2020).

Download citation


  • Turbulence in porous media
  • Interface porous–fluid region
  • Turbulent flow around a porous rib