Geoscience Reference
In-Depth Information
i
; and
r
is the repose angle. The positive sign on the right-hand side of Eq. (6.99)
is used in the case of downslope from point
i
to
P
, i.e.,
z
bi
>
φ
r
,
while the negative sign is used in the case of upslope from point
i
to
P
, i.e.,
z
bi
<
z
bP
−
z
bP
+
l
Pi
tan
φ
r
.
Suppose that
I
points out of the eight adjacent points exceed the sliding criterion.
Eq. (6.99) should be applied to each of these
I
points. These
I
points may be divided
into two groups according to whether they form downslopes or upslopes to point
P
. Denote
I
up
as the number of points that form upslopes to point
P
. Mass balance
between cell
P
and these
I
up
cells reads
l
Pi
tan
φ
I
up
z
bP
A
P
+
1
z
bi
A
i
=
0
(6.100)
i
=
where
A
i
are the areas of the cells centered at points
P
and
i
, respectively.
The equation set consisting of Eq. (6.99) for all
I
up
points and the mass balance
equation (6.100) provides a unique solution for
A
P
and
z
bi
(
i
=
1, 2,
...
,
I
up
)
and
z
bP
,
which can be derived as follows.
Eq. (6.99) is rearranged as
z
bi
=
z
bP
+
z
bP
−
z
bi
−
φ
r
l
Pi
tan
(6.101)
Inserting Eq. (6.101) into Eq. (6.100) yields
⎡
⎤
⎛
⎝
⎞
I
up
I
up
⎣
⎦
⎠
(6.102)
z
bP
=
1
A
i
(
z
bi
−
z
bP
+
l
Pi
tan
φ
)
A
P
+
1
A
i
r
i
=
i
=
Once
z
bP
has been determined using Eq. (6.102),
z
bi
(
i
=
1, 2,
...
,
I
up
)
can then
be obtained from Eq. (6.101).
In the same way, an equation similar to (6.100) can be obtained for mass balance
among the group of points that form downslopes to point
P
, and the solution for
z
bi
and
z
bP
for this group can then be found.
The above algorithm should be performed by sweeping over the entire domain or
region of interest. Because one sweep may not get the solution for the entire domain,
it must be repeated until all slopes are gentler than the repose angle. Moreover, the
sweeping sequence should be alternated between the positive and negative
x
and
y
directions.
Because of bank erosion, the wetted area varies with time during the channel mean-
dering process. The methods of handling drying and wetting processes described in
Section 6.1.4 can be used here. Either a fixed grid covering the entire area in which
the channel may migrate, or a moving grid that conforms to the migrating channel can
be used. Nagata
et al
. (2000) and Duan
et al
. (2001) adopted moving grids. In their
approaches, flow, sediment transport, bed change, and bank erosion are computed
on the old mesh at each time step. After the bank lines have been moved by erosion
and deposition, a new mesh conforming to the new bank lines is created, and the
