Page 256 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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Galerkin procedure, as discussed in the previous chapter, namely,
u n+1 − u n 1 ∂(p ) n+θ u j ∂ CONVECTION IN POROUS MEDIA

n+θ 1
u i
i
i
=− −
t ∂x i ∂x j

2 n+θ 2 n+θ 3
1 ∂ u i u i |V| u i
+ 2 − + C √ 3/2 + CG terms (8.33)
Re ∂x ReDa Da
i
The body force terms are neglected in the above equation in order to simplify the pre-
sentation. Additional dissipation, due to the characteristic Galerkin terms, may be neglected
here as we are dealing with very slow speed flow problems, especially at lower Rayleigh
or Reynolds numbers.
√
In Equation 8.33, the parameter ‘C’ is a constant equal to 1.75/ 150 (see
Equation 8.21). The parameters θ, θ 1 , θ 2 and θ 3 all vary between zero and unity and
with appropriate values, different schemes of interest can be established. The superscript θ
should be interpreted as
f n+θ = θf n+1 + (1 − θ)f n (8.34)
where the superscript n indicates the nth time iteration.
In the CBS scheme, the velocities are calculated by splitting Equation 8.33 into two
parts as below. In order to simplify the presentation, θ 1 , θ 2 and θ 3 areassumed tobeequal
to zero. It is important to note, however, that such an assumption severely restricts the time
step, which can be employed in the calculations. The semi- and quasi- implicit schemes,
as discussed in Section 8.3.3, are the schemes widely employed for porous media flow
calculations.
In Step 1, the pressure term is completely removed from Equation 8.33 and the interme-
diate velocity components ˜u i are calculated (similar to Step 1 of the CBS scheme discussed
in Chapter 7) as
2

˜u i ˜ u i − u n i u j ∂ u i n 1 ∂ u i n
= =− +
t t ∂x j Re ∂x 2
i
1 |V| u i
n
− u i + C √ 3/2 (8.35)
ReDa Da
The velocities can be corrected using the following equation, which has been derived by
subtracting Equation 8.35 from Equation 8.33, that is,

u i u n+1 − u n i ˜u i 1 ∂(p ) n+θ
i
= = − (8.36)
t t t ∂x i
However, the value of the pressure in the above equation is not known. In order to
establish the pressure field, a pressure Poisson equation can be derived from the above
equation and may be written as (see Section 7.6)

1 ∂ 2 1 ∂u ∗ i
(p ) n+θ = (8.37)
∂x 2 t ∂x i
i