Geoscience Reference
In-Depth Information
Width-integrating the steady depth-averaged 2-D suspended-load transport equ-
ation leads to
B
B
0
αω
s
(
Uh
∂
C
x
dy
=−
C
−
C
∗
)
dy
(2.138)
∂
0
and the 1-D formulation of Eq. (2.138) is
UH
∂
C
∂
(
C
−
C
x
=−
α
1
d
ω
∗
)
(2.139)
s
where
H
,
U
,
C
, and
C
are the flow depth, velocity, actual and equilibrium suspended-
load concentrations averaged over the cross-section, respectively; and
∗
α
1
d
is the
adaptation coefficient in the 1-D model.
The equilibrium depth-averaged suspended-load concentration at each vertical line
may be determined using the Zhang (1961) formula introduced in Section 3.5.3:
U
3
gh
m
C
∗
=
K
∗
(2.140)
ω
s
where
K
∗
is a coefficient, and
m
is an exponent.
In analogy to Eq. (2.140), the actual depth-averaged suspended-load concentration
at each vertical line is assumed to have the relation:
K
U
3
gh
m
C
=
(2.141)
ω
s
where
K
is a coefficient similar to
K
∗
.
The depth-averaged flow velocity at each vertical line is assumed to be proportional
to the local flow depth:
h
r
U
∝
(2.142)
where
r
is an exponent and has a value of 2/3 if the Manning equation is used.
Substituting relations (2.140)-(2.142) into Eq. (2.138) and comparing the resulting
equation with Eq. (2.139) leads to (Zhou and Lin, 1998)
B
0
h
r
+
1
dy
B
h
(
3
r
−
1
)
m
dy
B
0
h
(
3
r
−
1
)
m
+
r
+
1
dy
α
0
α
1
d
=
(2.143)
Eq. (2.143) shows that
α
1
d
is related to the cross-sectional shape and varies with
exponents
m
and
r
. After
α
has been determined using Eq. (2.136),
α
1
d
can be calcu-
lated using Eq. (2.143). As an approximation,
α
may be assumed to be constant along
