Lattice Boltzmann Flow Simulation of Porous Medium-clear Fluid Interface Regions

The effects of a porous medium on the flow in the interface region between the porous wall and clear fluid are discussed. Laminar flows in the interface regions of foamed porous walls are microscopically simulated by the lattice Boltzmann method. The chosen porous structure is the body-centered-cubic or the unit cube structure whose porosity ranges . The velocity distribution in the interface regions show that the flow penetration is very little and it decays until one porediameter depth from the interface. The results also show that the coefficients of the stress jump condition across the interface are negative and the total friction of the porous interface is slightly lower than that of a solid smooth surface. 0.58 0.98 − INTRODUCTION Viscous fluid flows through porous media have been studied for many years after Darcy (1856) since porous materials are so common in engineering fields as well as in the earth science. Once such porous materials are applied to fluid flows for guiding, filtering or controlling the fluid, the characteristics of flows over/around porous surfaces are also very important for designing the flowdevices. Accordingly, boundary layer flows over permeable walls have been studied by many researchers as well. Beavers and Joseph (1967) studied on the effects of a porous medium on the flow in the interface region between the porous wall and clear fluid (named “interface region” hereafter). They measured mass flow rates over permeable beds in laminar flow conditions and found that the mass flow rates increased compared with those in impermeable cases. Many other following studies (e.g. Rudraiah, 1985; Larson and Higdon, 1986; Sahraoui and Kaviany, 1992; Ochoa-Tapia and Whitaker, 1995; Gupte and Advani, 1997; James and Davis, 2001; Breugem et al.,2005) also focused on the laminar flow regime experimentally and numerically since the flow physics is rather complicated and controversial even in laminar flows. In fact, the effects of flow penetration into porous media are still not fully understood. Gupte and Advani (1997) performed mean flow measurements in the interface region of fibrous mats using a laser Doppler anemometry. They reported that the flow penetration into the porous layer was stronger than that estimated semitheoretically in the laminar flow regime. On the contrary, James and Davis (2001) reported numerically that the external flow penetrated the fibrous porous medium very little. The slip velocity at the interface is also one of the issues. On the porous interface, the fluid motion is not totally damped having non-zero statistical tangential velocity component. Since this slip velocity is required to solve the momentum equations over the interface as a boundary condition, Beavers and Joseph (1967) proposed a condition that is a function of the permeability and the Darcy velocity of the porous media. This condition was supported by many studies such as Saffman (1971), Larson and Higdon (1986) and Sahraoui and Kaviany (1992). Ochoa-Tapia and Whitaker (1995) proposed another relation introducing a jump condition of the stress distribution across the interface. Both of the conditions require experimental measurements for determining their coefficients. Another important issue is the friction of the porous surfaces. In laminar flow regimes, Beavers and Joseph (1967) reported that the friction over the porous walls reduced due to the porosity. However, Zagni and Smith (1976) and Zippe and Graf (1983) experimentally found that the friction factors of turbulent flows over permeable beds became higher than those over impermeable walls with the same surface roughness. Kong and Schetz (1982) observed that the increase of the skin friction was due to the combined effects of roughness and porosity. In order to understand the flow physics causing the difference between the laminar and the turbulent regimes, it is necessary to discuss the detailed flow physics in the interface region. However, measuring flow profiles across the interface region is almost impossible even with the latest laser techniques. Thus, we expect that numerical analyses can provide useful information instead. Recently, the LBM (lattice Boltzmann method) (McNamara and Zanetti, 1988; Higuera and Jimenez, 1989) has become powerful tool for simulating flows inside porous media (e.g. Martys and Chen, 1996; Keehm, et al., 2004; Pan et al., 2006) due to its coding simplicity of the method for complicated flow geometries. Therefore, in order to understand the flow characteristics such as the flow penetration, the slip velocity and the friction of the porous interfaces in laminar flow regimes, the present study has performed numerical simulations using the LBM. LATTICE BOLTZMANN METHOD The presently used LBM employs the single-relaxationtime (SRT) and multiple-relaxation-time (MRT) Bhatnagar-Gross-Krook (BGK) models (Bhatnagar et al., 1954) which are briefly described below. Single relaxation time LBM The lattice Boltzmann method for flow simulation by the SRT BGK model may be written as ( , ) ( , ) ( , ) ( , )