Page 260 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
252
Note that ij is assumed to be the boundary edge of an element. The forcing vector of
the x 2 component of the momentum equation is
 n

1 b i u 2i + b j u 2j + b k u 2k
{f 2 }=  b i u 2i + b j u 2j + b k u 2k  n 1
4A Re
0
 n

1 c i u 2i + c j u 2j + c k u 2k
+  c i u 2i + c j u 2j + c k u 2k  n 2 (8.55)
4A Re
0
The forcing vector, arising from the discretization of the second-order pressure terms
in Step 2, is
 n

b i p i + b j p j + b k p k
{f 3 }=  b i p i + b j p j + b k p k  n 1
4A
0
 n

c i p i + c j p j + c k p k
+  c i p i + c j p j + c k p k  n 2 (8.56)
4A
0
The implementation of the flux and other boundary conditions is very similar to the
method discussed in the previous chapter.
8.3.3 Semi- and quasi-implicit forms
Single-phase incompressible fluid flow problems can be solved in a fully explicit form,
which is quite popular in fluid dynamics calculations (Malan et al. 2002; Nithiarasu 2003).
However, a solution for the generalized porous medium equations using a fully explicit
form has been less successful. This is mainly due to the large values of the solid matrix
drag terms, especially at smaller Darcy numbers. In order to eliminate some of the time-
step restrictions imposed by these terms, schemes other than the fully explicit forms are
discussed below.
In the semi-implicit (SI) form (Nithiarasu and Ravindran 1998), the porous medium source
terms and pressure equation are treated implicitly. In other words, θ = θ 3 = 1and θ 1 =
θ 2 = 0. Although this scheme has good convergence characteristics, further complications
are introduced by the scheme. The split in the momentum equation (Equation 8.35) will be
different, that is,

2
˜ u i − u n 1 |V| ˜ u i u j ∂ u i n 1 ∂ u i n

i
+ ˜ u i + C √ =− + (8.57)
t ReDa Da 3/2 ∂x j Re ∂x i 2
or

n n 2 n
1 1 |V| 1 u i u j ∂ u i 1 ∂ u i
˜ u i + + C √ = − + (8.58)
t ReDa Da 3/2 t ∂x j Re ∂x i 2