Volume 7 Issue 9 - March 6, 2009
Macroscopic modeling of turbulent flow over a porous medium
Jan-Mou Leu*, H. C. Chan, W. C. Huang

Department of Hydraulic and Ocean Engineering, College of Engineering, National Cheng Kung University
jmleu@mail.ncku.edu.tw

International Journal of Heat and Fluid Flow, 2007, 28(5), 1157-1166.

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Transfer phenomena at the fluid/porous interface have served an important topic of research. Concerning the practical applications, researchers are interested in estimating the momentum and mass fluxes between porous surface and the overlying fluid. Flows over a porous medium can be characterized by a hybrid clear fluid-porous medium domain that flows both over and through the porous medium. The internal flow field of the porous medium remains coupled with the overlying fluid. The flow interaction above and inside the porous medium, and the transfer mechanisms across the interface, deserve in-depth investigation.

Depending on the length scale used, analysis of porous medium flow can be divided into microscopic approach and macroscopic approach (Fig. 1). The microscopic approach uses the Reynolds-Averaged Navier-Stokes (RANS) equations to compute the detailed flow structures inside the porous medium. This requires knowing the geometry of each porous particle a priori. The tremendous computational requirements have deterred many practical applications because of the geometry of porous particles is usually unknown in real–life porous problems. The macroscopic approach utilizes macroscopic variables and transport equations to study the flow in a porous medium and obtained the averaged fluid properties. The geometry of porous particles is represented by porous properties (i.e. porosity and permeability), which simplifies the modeling procedure. On the basis of the pore Reynolds number, Rep, the literature recognizes distinct flow regimes in macroscopic modeling porous medium spanning from creeping flow (Rep<1) to fully turbulent regime (Rep>300). Turbulence in a porous medium is still a controversial issue. Nevertheless, recent engineering applications have come out with the interaction between turbulent flow and the porous medium. High-speed flow over or through such a highly permeable medium can lead to turbulent flow therein. Some investigations start to elucidate the effect of turbulent flow above and through the porous medium. Questionable aspects can be clarified via growing computational and numerical data.
Fig. 1. Definition sketch for the macroscopic approach and microscopic approach.

Therefore, this study presents a numerical implementation for examining such a hybrid domain. The turbulent flow in the clear fluid region describes by the RANS equations and standard k-ε model. The macroscopic turbulent quantities within the porous region are calculated using the well-known Brinkman–Forchheimer–extended Darcy model with the Double-decomposition turbulent model proposed by Pedras and de Lemos (2001). The transport equations for Double-decomposition turbulent model are as following:

where and are the extra terms due to the presence of the porous medium, compared to that of standard k-ε model. These extra terms (GkandGε) vanish in the limiting case in which no porous medium is present, or the porosity and the permeability are extremely high (Φ→1andK→∞), meaning that the transport equations of k and ε are recovered to standard k-ε model. The suitable interface conditions for turbulent flow over a porous medium domain are, however, still not well established. The present study proposes continuity of velocity, pressure, shear stress, turbulent kinetic energy, dissipation rate of turbulent kinetic energy and their fluxes across the interface. The present model treated the hybrid domain problem with a single domain approach by adopting these continuity interface conditions.

Our numerical results were used to compare with the experimental data available in the literature for two cases. Fig. 2 plots the turbulent velocities above the porous medium. It can be seen the present model indicates superior agreement with the measured data over the previous model. Fig. 3 compares the computed velocity distributions within the porous medium with the experimental data. Notably, the present numerical model accurately represents the general trend of the velocity distributions. Importantly, the computed velocity distributions agree with the experimental data at the fluid/porous interface and Darcy velocity region, but a slight difference is evident in the interface boundary layer region. This fact can be explained by both the vertical mean velocities and the vertical momentum fluxes demonstrate strong inhomogeneity of the turbulence pattern in the interface area. For a 2-D fully developed flow, the effects of vertical fluid motion between the gaps among the glass breads are not taken into account.
Fig. 2. Velocity distribution above porous medium. Symbols: Measured data (Prinos et al. 2003), dotted lines: computed data (Prinos et al. 2003); lines: present model.

Fig. 3. Velocity distribution within porous medium. Symbols: Measured data (Shimizu et al. 1990); lines: present model.

In order to further contribute to understand the fundamental characteristics of turbulence over and within the porous medium, the effects of the Reynolds number, the Darcy number (dimensionless permeability), and the porosity are investigated and discussed. Fig. 4 presents the variation of the computed velocity and turbulent kinetic energy profiles for the different values of Reynolds number. Within the range of analyzed Reynolds number, the flow characteristics remain independent of the Reynolds number. Figs. 5 and 6 show the effects of varying Darcy number and porosity on the flow characteristics. It demonstrates that greater Darcy number or porosity values correspond to more flow through the porous medium. The penetration of the turbulent kinetic energy into the porous medium is proportional to those two parameters. This is caused by the reduction of the damping effect of the porous material. These findings on the behavior of turbulent flow over a porous medium clearly indicate that the level of turbulent penetration depends strongly on the damping effect of the porous medium itself. The distributions of turbulent kinetic energy show that, in the simulated case of high permeability and porosity, the existence of extra terms in the Double-decomposition turbulent model is important in the porous region.
Fig. 4. Effect of Reynolds number on: (a) Velocity distribution and (b) Turbulent kinetic energy distribution.

Fig. 5. Effect of Darcy number on: (a) Velocity distribution and (b) Turbulent kinetic energy distribution.

Fig. 6. Effect of porosity on: (a) Velocity distribution and (b) Turbulent kinetic energy distribution.
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