Page 257 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
The above simplified equation has been derived by substituting the equation of conti-
nuity. Thus, the conservation of mass is satisfied indirectly without explicitly solving for
the mass conservation Equation 8.23. 249
We have a total of three steps to obtain a solution for the momentum and continuity
equations. As discussed in Chapter 7, Equation 8.35 is solved at the first step, followed
by Equation 8.37 in the second step and Equation 8.36 in the third step. Additional steps,
such as temperature or concentration calculations, can be added as an addition to the above
three steps.
In problems in which non-isothermal and mass transfer effects are involved, additional
equations will be solved, after velocity correction. If no coupling exists between the veloc-
ities and the other variables, such as temperature and concentration and the steady state
solution is only of interest, the steady velocity and pressure fields can be established first,
and the rest of the variables can be calculated using the steady state velocity and pressure
values.

8.3.2 Spatial discretization

Once a temporal discretization of the equations has been achieved, then a spatial discretiza-
tion may be carried out. In this text, the finite element discretization will be carried out
using linear triangular elements. Assuming a Galerkin approximation, the variables can be
expressed as

u i = [N]{u i }; u i = [N]{ u i }; ˜u i = [N]{ ˜ u i }; p = [N]{p}; = [N]{ } (8.38)

where [N] are the shape functions. We assume that the equations are solved in the order
mentioned before, that is, first the intermediate velocity components, then the pressure field
and, finally, the velocity correction. On considering the intermediate velocity calculation,
we have the following weak form, in which porosity is assumed to be an averaged quantity
over an element and body forces are neglected for the sake of simplicity:

n
1 T t T ∂ u i
[N] ˜u i d
= − [N] u j d

∂x j
T n
1 1 ∂[N] ∂u i
− d
Re
∂x i ∂x i
n
t C |V|
− [N] T u i + √ 3/2 u i d
+ b.t (8.39)

ReDa Da
where b.t. represents the boundary integral resulting from an integration by parts of the
second-order terms (Green’s lemma, Appendix 1). The weak form of the Step 2 calculation
for the pressure field can be written (assuming θ = 1) as

T
1 ∂[N] ∂( p) n+1 1 T ∂ ˜u i
− d
= [N] d
(8.40)

∂x i ∂x i t
∂x i