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

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.

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Abbreviations

\(c_\mathrm{F} \) :

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

\(\delta _{ij}\) :

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

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

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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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Correspondence to Qahtan Al-Aabidy.

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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). https://doi.org/10.1007/s11242-019-01357-0

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Keywords

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