ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

2. METHOD AND NUMERICAL MODEL.
2.1 Method
Studying the flow of fluids through out porous media is a current problem in
engineering and industry as demonstrated by numerous investigations covering a wide
range applications in biomechanics, ceramic engineering, chemical engineering,
geophysics etc. 7 The flow of fluids are described by the Navier-Stoke's equations.
Unfortunately, there are no analytical general solutions of this system. So we chose a
numerical method to obtain a numerical solution. Through tools of computational fluid
dynamic (CFD), the fields of velocity and pressure for a considered bulk domain are
determined. However, it is nearly impossible for a flow of fluid in a porous medium
because of the complex internal and external geometry of such a macroporous structure.
Add to this the limits due to computers such as power, calculation time and cost. So we
propose to use a known solution: to solve the Navier-Stoke's equations in one small
volume then to generalize the results to a bulk domain. The concept “Representative
Elementary Volume”(REV) is often used but a problem still remains which consists in
the determination of this REV (Fig.1a). For example, BACHTMAT 8 used the variation
of porosity adjacent to a point as a function of the average volume (Fig.1b).
(a)
(b)
Fig.1. Example of REV (a) and REV as function of the packing spheres (b).
In several particular cases, porous media have an isotropic and periodical structure. So
we will use another small volume called “Unit Cell” (UC) instead of a REV (Fig.2). The
volume of a unit cell is generally smaller than a REV so the solving of Navier-Stoke's
equations is simplified.
(a) (b)
Fig.2. Example of UC (a: 2 dimensions, b: 3 dimensions) in a porous medium (cubical
system).
2.2 Numerical model to determine the fields of velocity and pressure of REV or UC.
To solve directly the Navier-Stoke's equations in one domain through numerical
methods and tools of computational fluid dynamic, a numerical model must be
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