Geoscience Reference
In-Depth Information
Q
i
−
Q
i+
1
/
2
1
/
2
C
i
−
1
C
i
C
i+1
ν
i
−
1
/
2
ν
i
−
1
/
2
V
i
−
1
V
i
V
i+1
∆
x
i
∆
x
i+1
∆
x
i
−
1
A
i+
1
/
2
A
i
−
1
/
2
FIGURE 6.3
Generic control volume in a 1D discretization.
In these equations,
t
*
is a time interval between
t
and
t
+ ∆
t
, to be defined
according to criteria outlined in the next paragraph.
C
i
−
1
2
is the concentration on
the interface between elements
i
and
i
- 1 and will be specified later. Combining
the three equations, we obtain:
t
+
∆
t
t
(
)
(
VC
)
−
(
VC
)
*
tt
=
ii
ii
=
QC
−
QC
∆
t
i
−
1
2
i
−
1
2
i
+
1
2
i
+
1
2
(6.1)
*
*
tt
=
tt
=
( )
( )
CC
x
−
+
CC
x
−
−
ν
A
i
i
−
1
+
ν
A
i
+
1
i
i
−
1
2
i
−
1
2
(
∆∆
x
)
i
+
1
2
i
+
1
2
(
∆
+
∆
x
i
1
)
1
2
1
2
i
i
−
1
i
+
In order to introduce the Taylor series discretization methods and to analyze
stability and accuracy concepts, let us consider a simplified version of Equation (6.1).
Consider the particular case of a channel with uniform and permanent geometry and
regular discretization. The cross section (
), and discharge are constant.
Assume that diffusivity can be considered constant. Under these conditions,
Equation (6.1) becomes
A
), volume (
V
*
tt
=
*
tt
=
CC
x
−
t
+
∆
t
t
CC
t
−
CCC
x
−+
2
i
−
1
2
i
+
1
2
i
i
=
U
+
ν
i
−
1
i
i
+
1
(6.2)
2
∆
∆
∆
where
is the ratio between
the volume and the average cross section. This is the most popular form of the
transport equation but, as shown above, it is applicable only to particular conditions.
Additional approaches are required to calculate the advective flux, because the
concentration is defined at the center of the control volumes and not at the faces. These
approaches and their numerical consequences are described in the next sections.
U
is the constant cross-section average velocity and
∆
x
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