Environmental Engineering Reference
In-Depth Information
αα
=
()
Pe
[13.51]
For the one-dimensional problem presented above:
Pe
2
⎛⎞
α
=
coth
−
⎜
⎝⎠
[13.52]
opt
2
Pe
13.6.4.
Transient problems
Above we considered a steady state problem. In transient conditions, the space
decentering correction is no longer satisfactory to ensure a correct solution. An
additional correction can solve this problem. For example, [YU 86] proposes:
()
∂
w
τ
( )
() ()
() ()
()
W
ξτ
,
=
N
ξ τ α ξ τ β ξ
.
w
+
.
W
.
w
+
.
W
.
[13.53]
L
L
L
L
∂
τ
() (
τ
where
=
w
is the time weighting function, depending on a dimensionless
time
τ
. The dimensionless decentering parameter has the optimal value:
τ
τ
−
α
β =−
Cr
opt
2
[13.54]
opt
3
Pe Cr
.
The Courant's number:
.
ut
Δ
Cr
=
[13.55]
h
measures the relative displacement during a time step of a particle in advection
compared to the size of an element. It is equal to one if the element is totally
crossed.
This method provides a spatial and temporal coupled decentering. It is called the
full upwind Petrov-Galerkin method
. It provides an excellent resolution for many
practical problems, provided that the Péclet number remains moderate (
Pe
<2) and
the Courant's number is significantly lower than unity (
Cr
≅
0.25 gives a good
result). Otherwise oscillations and an excessive numerical diffusion can always
appear.
Thus, in practical applications related to aquifers in which water velocities
u
and
mechanical dispersion
a
L
are known, limiting the Péclet number to a maximum
value around two automatically sets the maximum size of the elements. Once this