Geoscience Reference
In-Depth Information
The average suspended-load concentration,
C
, at the cross-section is defined as
1
A
s
U
C
=
ucdA
(2.105)
A
s
where
c
is the local suspended-load concentration, and
A
s
is the flow area in the
suspended-load zone, as shown in Fig. 2.7. It is often assumed that
A
s
A
. Integrating
Eq. (2.72) over the suspended-load zone leads to the 1-D suspended-load transport
equation:
≈
AC
β
A
s
∂
C
∂
∂
∂
+
∂
∂
)
=
∂
∂
AUC
(
E
b
−
D
b
)
x
(
ε
+
B
(2.106)
t
x
x
s
where
E
b
and
D
b
are the width-averaged sediment entrainment and deposition fluxes at
the interface between the suspended-load and bed-load zones, and
β
s
is the correction
factor for suspended load:
ucdA
cdA
U
A
s
β
=
(2.107)
s
A
s
Note that no dispersion term appears in Eq. (2.106), due to the definition of
C
in
Eq. (2.105). However, if
C
is defined by Eq. (2.100), a dispersion term should appear
in Eq. (2.106). Normally, the diffusion term in Eq. (2.106) is ignored, yielding
AC
β
AUC
∂
∂
+
∂(
)
(
E
b
−
D
b
)
=
B
(2.108)
t
∂
x
s
Integrating Eq. (2.72) over the bed-load zone yields the 1-D bed-load mass balance
equation:
p
m
)
∂
A
b
∂
+
∂(
A
δ
C
δ
)
∂
+
∂
Q
b
∂
(
D
b
−
E
b
(
1
−
=
B
)
(2.109)
t
t
x
where
t
is the rate of change in bed area;
A
b
is the cross-sectional area of the
bed above a reference datum, as shown in Fig. 2.7;
A
δ
is the cross-sectional area of
the bed-load zone;
Q
b
is the bed-load transport rate at the cross-section; and
C
δ
is the
laterally-averaged bed-load concentration.
In analogy to Eq. (2.90), by using
C
δ
=
∂
A
b
/∂
Q
b
/(
A
δ
U
b
)
, Eq. (2.109) can be rewritten as
Q
b
U
b
p
m
)
∂
+
∂
∂
+
∂
A
b
∂
Q
b
∂
(
D
b
−
E
b
)
(
1
−
=
B
(2.110)
t
t
x
where
U
b
is the laterally-averaged velocity of bed load.
