Environmental Engineering Reference
In-Depth Information
13.7.2.
Two-dimensional reference problem
The second example, proposed by Hughes [HUG 82], has become a reference
criterion for the various decentering methods. It concerns the modeling of a square
aquifer (Figure 13.7) where the concentration is imposed along the contour (
C
= 1
along S1, and
C
= 0 along S2). The convective velocity is uniform (1 m/s) and is
oriented with θ angle with respect to the mesh.
c =1
c =0
θ
= 30°
S
1
S
2
c =0
Figure 13.7.
Description of Hughes' problem
Figure 13.8 shows the equal concentration curves obtained under steady state
conditions for several Courant and Péclet numbers. Figures 13.8a and 13.8b present
a dispersion case (
a
L
=
a
T
= 0.6 m) for a Courant number smaller and larger than
unity respectively.
Figure 13.8c is related to a purely convective case (
Pe
= ∞) and Figure 13.8d
concerns the same problem as that presented in Figure 13.8a but with an irregular
finite element mesh (the Courant and Péclet numbers of the mesh are slightly
different).
We note once again that if the Courant number
Cr
is greater than unity, space
oscillations appear and the solution is highly degraded. This is a characteristic of the
Petrov-Galerkin decentering methods. The fully convective solution is not perfect
and presents small oscillations. The irregular mesh gives similar results to those
obtained with a regular mesh.
