k–e Macro-scale modeling of turbulence based on a two scale analysis in porous media Franc ¸ois Pinson a , Olivier Gre ´goire a, * , Olivier Simonin b a CEA Saclay, DEN/DM2S/SFME/LETR, 91191 Gif sur Yvette Cedex, France b IMFT, UMR 5502, CNRS/INPT/UPS, Alle ´e du Pr. Camille Soula, 31400 Toulouse, France Received 27 September 2005; received in revised form 11 January 2006; accepted 4 March 2006 Available online 19 June 2006 Abstract In this paper, turbulent flows in media laden with solid structures are considered. Following previous studies, we apply both statistical and spatial averages. The solid matrix action on turbulence is then put forward as a sub-filter production. To model this term, we per- form a two-scale analysis that highlights energy transfers between the mean motion, the macroscopic and the sub-filter turbulent kinetic energies. Within this framework, we show that the sub-filter production is an energy transfer between the mean motion kinetic energy and the turbulent kinetic energy. We propose to model this sub-filter production using the wake dissipation and the work performed by the mean macroscopic flow against the mean specific drag. From this analysis, a macroscopic k–e model is then derived for a stratified porous media that includes a friction factor model accounting for turbulence non-equilibrium. Comparisons between this model and a former model [Nakayama, A., Kuwahara, F., 1996. A macroscopic turbulence model for flow in a porous medium. J. Fluid Eng.-T ASME 121, 427–433] are carried out. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Porous media; k–e Model; Wake dissipation; Friction factor; Channel flow; Volume average 1. Introduction The macroscopic modeling of turbulent flows passing through porous media or media laden with solid structures concerns many practical applications such as nuclear reac- tors, heat exchangers or canopy flows. In such flows, vari- ous study scales coexist. The challenge of the macroscopic modeling is not to reproduce the fine structure dynamics of the flow but to take into account information embedded in smaller scale for large scale modelization. With this aim, we choose to use two average operators: the statistical average that is practical for turbulence study and the spatial aver- age, well adapted for the porous media approach. It has been shown that constrains apply when using spa- tial average (Quintard and Whitaker, 1994a,b). If macro- scopic quantities length scales are large with respect to the filter size then the spatial average is assumed idempo- tent. In what follows, hÆi denotes the volume average, while hÆi f is the fluid volume average (hÆi /hÆi f ), where / is the porosity of the medium. The symbol d denotes the devia- tion of a quantity from its fluid averaged value. The statis- tical average and the fluctuation of some quantity f are respectively denoted  f and f 0 . In a strict mathematical way, both averages commute (Pedras and de Lemos, 2001). However, each modeling step, related to an average application, involves simplifications. Hence, the macro- scopic turbulence modelization necessarily depends on the order of application of these two averages (Nield, 2001; Travkin, 2001). Following Pedras and de Lemos (2001), de Lemos and Braga (2003) and also Nakayama and Kuwahara (1996), we choose first to apply the statistical 0142-727X/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.ijheatfluidflow.2006.03.018 * Corresponding author. Tel.: +33 1 69 08 22 85; fax: +33 1 69 08 85 68. E-mail addresses: francois.pinson@cea.fr (F. Pinson), olivier.gregoire @cea.fr (O. Gre ´goire), olivier.simonin@imft.fr (O. Simonin). www.elsevier.com/locate/ijhff International Journal of Heat and Fluid Flow 27 (2006) 955–966