Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

the ¯uid and of the solid are coupled. For a survey of this problem the reader is
referred to the recent book by Zienkiewicz et al. 44
Here we use the averaged governing equations derived by many investigators to
solve buoyancy driven convection in a porous medium. 45;46 These equations can be
summarized for a variable porosity medium as 46
continuity
momentum
f
"
energy
@ui @
‡
@t @xj ˆÿ 1
"
@
@x i
u ju i
"
…p"†‡ 1
"
@
@x i
R h
@u i
@x i
@T
@t ‡ u @T
i
@ui ˆ 0 …5:17†
@xi ÿ u i ÿ C f
p
@x i
ˆ @
@x i
p
ukuk " 3=2 ui ‡ gi …T1 ÿ T† …5:18†
k @T
@x i
…5:19†
where u i are the averaged velocity components, " is the porosity of the medium, is
the medium permeability, C is a constant derived from experimental correlations and
here we use Ergun's relations 47 in our calculations (some investigators vary the nonlinear
term using a non-dimensional parameter called the Forchheimer number;
interested readers can consult reference 48), k is the thermal conductivity of the
porous medium and R h is the averaged heat capacity given as
R h ˆ "… c p† f ‡…1 ÿ "†… c p† s
…5:20†
In the above equations, subscripts f and s correspond to ¯uid and solid respectively.
The following relation for permeability can be used if the porosity and average
particle size are known
ˆ
" 3 d 2 p
150…1 ÿ " 2 †
Buoyancy driven ¯ows 159
…5:21†
where d p is the particle size. Some researchers use a value for di€erent from the ¯uid
viscosity. However, here we generally use the ¯uid viscosity. More details on the
derivation of the above equations can be found in the cited articles.
As the structure of the above governing equations is similar to that of the singlephase
¯ow equations, the application of the CBS algorithm is obvious. 49ÿ55 However
the fully explicit or semi-implicit forms cannot be used e ciently due to strong porous
medium terms. Here, to overcome the time step limitations imposed by these terms
(last two terms before the body force in the momentum equation) we need to solve
them implicitly, though quasi-implicit schemes 47;49 can be used. Although the CBS
algorithm is an obvious choice here, use of convection stabilizing terms can be
neglected in low Rayleigh number (Reynolds number) porous media ¯ows.