Geoscience Reference
In-Depth Information
By applying the Leibniz integral rule and boundary conditions (2.68), (2.71), (2.73),
and (2.75) to Eq. (2.85) and assuming that the bed-load zone is very thin, i.e.,
δ
h
,
the depth-integrated suspended-load transport equation is obtained:
hC
β
∂
∂
+
∂(
)
+
∂(
hU
y
C
)
hU
x
C
t
∂
x
∂
y
s
h
D
sx
h
D
sy
=
∂
∂
ε
s
∂
C
+
∂
∂
ε
s
∂
C
x
+
y
+
+
E
b
−
D
b
(2.86)
∂
∂
x
y
where
β
s
is a correction factor for suspended load:
z
s
u
s
cdz
U
s
z
s
z
b
+
δ
cdz
β
=
(2.87)
s
z
b
+
δ
Note that the coefficient
β
s
in Eq. (2.86) should not be zero; otherwise, no suspended
load moves.
s
should also appear in the diffusion terms in Eq. (2.86), but it is lumped
into the diffusivity
β
ε
s
for simplicity. However, if the depth-averaged sediment con-
centration
C
is defined using Eq. (2.76) rather than (2.84),
β
s
should appear in the
convection terms rather than the storage term, i.e.,
∂(
hC
)
+
∂(β
s
hU
x
C
)
+
∂(β
s
hU
y
C
)
∂
t
∂
x
∂
y
h
D
sx
h
D
sy
=
∂
∂
s
∂
C
+
∂
∂
s
∂
C
ε
x
+
ε
y
+
+
E
b
−
D
b
(2.88)
x
∂
y
∂
It should be clarified that defining the depth-averaged suspended-load concentration
C
by Eq. (2.76) results in the unit suspended-load discharge
q
s
=
β
s
UhC
, while the
definition (2.84) yields
q
s
UhC
. If mass balance is respected, either definition can
be used. The definition (2.84) is adopted in this topic, except where stated otherwise.
The coefficient
=
s
is actually the ratio of the depth-averaged sediment and flow
velocities and accounts for the temporal lag between flow and suspended-load trans-
port in the depth-averaged 2-D model. As demonstrated later,
β
s
also appears in the
1-D model. However, this lag due to difference between the depth-averaged flow and
sediment velocities can be automatically taken into account in the 3-D (or vertical
2-D) model, which directly uses the local flow velocity and sediment concentration as
dependent variables. The evaluation of
β
s
is discussed in Section 3.8.
D
sx
and
D
sy
in Eq. (2.86) are called dispersion sediment fluxes, which account for
the dispersion effect due to the non-uniformdistributions of flow velocity and sediment
concentration over the flow depth, defined as
D
sx
β
h
z
s
1
=−
z
b
(
u
x
−
U
x
)(
c
−
C
)
dz
and
h
z
s
1
D
sy
dz
. In nearly straight channels, the dispersion fluxes
may be combined with the (turbulent) diffusion fluxes, with
=−
z
b
(
u
y
−
U
y
)(
c
−
C
)
ε
s
replaced by a mixing
