Environmental Engineering Reference
In-Depth Information
If we could build a finite element mesh on the characteristic lines and could
move with them at water velocity, the problem to be solved would be purely
diffusive. Its resolution by Galerkin-type elements would be easy and precise. This
is inapplicable, however, because the mesh would become mis-shapen and boundary
conditions would be accounted for with significant difficulty. To solve this, we can
work in a fixed mesh of Galerkin's type, but should consider at each time step and
integration point that the concentration to consider is that located at an upstream
point on the same flow line at a distance given by characteristics equation [13.27].
Formally, the problem seems very simple. Nevertheless, a major difficulty remains
because it is necessary within a finite element code to write relations between points
that are not bound by the finite element mesh organization but by the field of flow
velocities. This removes the structure of the finite element code, adding a layer of
relationships that makes it considerably more complex. This is neither elegant nor
effective in terms of computing time. Explicit algorithms [LI 97] showed the
effectiveness on this type of problem, despite the limitations on time step size.
13.7. Examples and applications
Two- and three-dimensional finite elements, based on the principles presented in
the preceding sections, and particularly on the Petrov-Galerkin decentering method,
were developed and introduced into the LAGAMINE computer code, which was
developed at the University of Liege. We hereafter propose a few examples of
applications that, although relatively simple and academic, display some of the
numerical limitations of developed elements and therefore define their application
field. The model is then applied to two more concrete examples to study the
behavior of solutes (mainly nitrates) in a cretaceous chalk aquifer.
13.7.1. Pollutant pulse one-dimensional propagation
As a first example, we consider a very simple reference case: namely the spread
of a bell distribution of the pollutant concentration in a one-dimensional velocity
field. Figure 13.6 gives the evolution of concentration for three calculation times
and for five different cases. Figure 13.6a shows the perfectly convective case
( Pe = ∞), without any degradation or immobile water effect, with a Courant's
number Cr equal to 0.5, where the pollutant pulse moves almost perfectly parallel to
itself. We note, however, a small spreading due to numerical diffusion. The
parameters of the problem shown in Figure 13.6b are identical to those of the
previous problem, except that the dispersion is no longer null ( Pe = 2). The
pollutant pulse therefore moves with an increasingly marked flattening over time.
The only difference between the problem analyzed in Figure 13.6c and the previous
one is that the Courant's number Cr is equal to 2.5. This value exceeds unity and
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