Environmental Engineering Reference
In-Depth Information
uh
h
(
)
(
)
−+− + −+ =
CCC
2
CCQ
[13.45]
i
−
1
i
i
+
1
i
−
1
i
+
1
p i
,
2
D
D
Qh
(
)
(
)
pi
,
−−
1
Pe
C
+ +−+
2
C
1
Pe
C
=
[13.46]
i
−
1
i
i
+
1
D
A similar equation to [13.42] is obtained if the problem is analyzed using the
finite differences method.
For a purely advection problem, in the absence of any diffusion and source term,
equation [13.42] is reduced to:
CC
=
[13.47]
i
+
1
i
−
1
If the number of nodes is odd, a solution is possible; it alternates the respective
concentrations of the two ends from node to node (Figure 13.4e). If the number of
nodes is even, the solution is impossible.
However, for a purely diffusive phenomenon, the equations are reduced to:
D
(
)
−+− =
CCC
2
0
[13.48]
i
−
1
i
i
+
1
h
the solution of which is a linear variation of the concentration between the values
imposed at the ends (Figure 13.4a). This latter problem is easily solved by the finite
element method.
So advection degrades the solution. Why is this so? The advective problem is, in
this form, totally different from the diffusive problem. In purely laminar advection,
the movement of pollutant can be described by the integration of velocities. A
particle follows its current line without ever spreading and its displacement is
achieved by temporal velocities integrating on this path. This type of solution can be
obtained by the
characteristics
method, as seen in section 13.5. On the other hand,
Galerkin's method − identical shape and weighting functions − is not adapted to this
problem, where diffusion is absent. While Péclet's number remains low (
Pe
< 1),
Galerkin-type finite elements give a correct answer. For larger values, an artificial
diffusion tends to appear and the results lose their accuracy.
13.6.3.
Petrov-Galerkin method
Historically, pollutant transport modeling was done by using the finite
differences method, which will not be explained here. Equation [13.42] should be
written in central differences. It is easy to transpose it to upstream or downstream
