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

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## Abstract

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.

## Keywords

Turbulence in porous media Interface porous–fluid region Turbulent flow around a porous rib## List of Symbols

## Roman Symbols

- \(c_\mathrm{F} \)
Forchheimer coefficient

- \( c_{\varepsilon 1}, c_{\varepsilon 2}, c_k\)
Non-dimensional turbulence model constants

- \( c_{\mu }\)
Coefficient in the eddy-viscosity

- Da
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*Pressure

- \( P_k \)
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

## Greek Symbol

- \(\delta _{ij}\)
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

## Special Characters

- \(\left\langle \,\,\right\rangle ^f\)
Intrinsic average

- \(\langle \,\,\rangle \)
Volume average

## Acronyms / Abbreviations

- DNS
Direct numerical simulation

- LES
Large eddy simulation

- LSMNK
Launder Sharma modified by Nakayama and Kuwahara (2008)

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

- PPI
Pore per inch

- RANS
Reynolds-averaged Navier–Stokes

- REV
Representative elementary volume

- SIMPLE
Semi-implicit method for pressure-linkage equations

- TCL
Two-component-limit

- UMIST
Upstream monotonic interpolation for scalar

- VAT
Volume averaging theory

## Notes

### Acknowledgements

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.

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