Environmental Engineering Reference
In-Depth Information
uh
Pe
=
is Péclet's dimensionless number compared to the size of the mesh. It
2
D
represents the ratio of the effects of advection and diffusion. For a purely diffusive
problem,
Pe
= 0. For a purely advective problem
Pe
=∞.
According to the amplitude of the advection term with respect to the diffusion
term, the solution changes. With no advection, the concentration varies linearly and
the finite element model gives the exact solution. If we impose a flow velocity from
the zero concentration end towards the unit concentration end, we reduce the
importance of the polluted area. Unfortunately, space oscillations of the response
appear with a higher amplitude when the advection is important.
Let us put the one-dimensional problem in equations. If
h
is the length of a finite
element, then:
x
N
=− =−
1
ξ
1
[13.39]
i
h
x
N
==
j
[13.40]
h
with the isoparametric coordinate ξ∈[0,1]. The gradient is:
1
∇= ∇=
N
N
j
i
[13.41]
h
CC
−
∂
C
x
j
i
=
[13.42]
∂
h
The stiffness matrix related to an element is thus written as:
+−
11
++
11
D
⎡
⎤ ⎡
u
⎤
K
=
+
[13.43]
⎢
⎥ ⎢
⎥
ij
h
−+
11
2
−−
11
⎣
⎦ ⎣
⎦
The assembly of a one-dimensional problem discretized by a set of uniform
length finite elements for each degree of freedom gives the following equation:
D
u
(
) (
)
−+− +−+ =
CCC
2
CCQ
[13.44]
i
−
1
i
i
+
1
i
−
1
i
+
1
p i
,
h
2
where
Q
p,i
is an imposed nodal flow. This equation can be put into dimensionless
form:
