Page 266 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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1
2
u 1 = u = 0 CONVECTION IN POROUS MEDIA

u 1 = u 2 = 0 u = u = 0
2
1
T = 1 T = 0 10

u = u = 0
2
1
Figure 8.7 Natural convection in a fluid-saturated variable porosity medium. Problem
boundary conditions

8.6.1 Constant porosity medium

Problems in which the variation in porosity is less significant normally occur in porous
media, which have small, solid particle sizes. For instance, thermal insulation is one such
example in which the variation in porosity near the solid walls is not important but
the uniform free stream porosity value can be very high. In order to investigate such
media, a benchmark problem involving buoyancy-driven convection in a square cavity has
been solved.
The problem definition is similar to the one shown in Figure 8.7, the difference being
that the aspect ratio is unity. The square enclosure is filled with a fluid-saturated porous
medium, with constant and uniform properties except for the fluid density, which is again
incorporated via the Boussinesq approximation. A 51 × 51 non-uniform mesh (Figure 8.8),
is employed for this problem.
The Darcy and non-Darcy flow regime classifications and the Darcy number limits have
been discussed by many researchers. One important suggestion was given in the paper by
Tong and Subramanian (Tong and Subramanian 1985). In Figure 8.9, we show the velocity
and temperature distribution at different Darcy and Rayleigh numbers. In this case, the
product of the Darcy and Rayleigh numbers is kept at a constant value in order to amplify