Geoscience Reference
In-Depth Information
bU
z
bU
x
U
z
bU
z
)
∂
(
T
zx
+
D
zx
b
∂
p
∂
∂(
)
+
∂(
)
+
∂(
1
ρ
1
ρ
∂
[
b
)
]
=−
bg
−
z
+
∂
t
∂
x
z
∂
x
(
T
zz
+
D
zz
1
ρ
∂
[
b
)
]
1
ρ
(
+
−
m
1
τ
+
m
2
τ
)
z
1
z
2
∂
z
(2.97)
p
and
T
ij
where
are the width-averaged pressure and stresses, respec-
tively;
D
ij
are the dispersion momentum transports due to the lateral non-uniformity
of flow velocity, defined as
D
xx
(
i
,
j
=
x
,
z
)
=−
b
b
2
=−
b
b
2
−
U
x
2
dy
,
D
xz
=
D
zx
b
1
(
u
x
)
b
1
=−
b
b
2
u
x
−
U
x
)(
u
z
−
U
z
)
dy
, and
D
zz
u
z
−
U
z
)
2
dy
;
(
b
1
(
τ
xl
and
τ
zl
(
=
)
are the
shear stresses in the
x
- and
z
-directions on the two bank surfaces; and
m
l
are the bank
slope coefficients, defined as
m
l
=[
l
1, 2
2
.
For gradually varied flows, the effects of inertia, diffusion, and dispersion in the
vertical momentum equation (2.97) can be neglected, yielding the hydrostatic pressure
equation (2.65). The
x
-momentum equation (2.96) is then turned to
2
2
1
/
1
+
(∂
b
l
/∂
x
)
+
(∂
b
l
/∂
z
)
]
bU
x
)
∂
bU
x
)
∂
bU
z
U
x
)
∂
(
T
xx
+
D
xx
)
]
∂
∂(
+
∂(
+
∂(
gb
∂
˜
z
s
1
ρ
∂
[
b
=−
x
+
t
x
z
∂
x
(
T
xz
+
D
xz
1
ρ
∂
[
b
)
]
1
ρ
(
+
−
m
1
τ
+
m
2
τ
)
x
1
x
2
∂
z
(2.98)
where
z
s
is the laterally-averaged water surface elevation.
Integrating Eq. (2.72) over the flow width leads to the width-integrated suspended-
load transport equation:
˜
bC
bU
x
C
bU
z
C
s
C
∂(
)
+
∂(
)
+
∂(
)
−
∂(
b
ω
)
∂
t
∂
x
∂
z
∂
z
b
x
+
D
sx
b
+
D
sz
s
∂
C
∂
s
∂
C
∂
=
∂
∂
+
∂
∂
ε
ε
+
S
c
(2.99)
x
z
z
where
C
is the width-averaged concentration of suspended load;
D
sx
and
D
sz
are
the dispersion fluxes, defined as
D
sx
b
b
2
−
U
x
−
C
dy
and
D
sz
1
1
b
=−
b
1
(
u
x
)(
c
)
=−
b
2
u
z
−
U
z
)(
−
C
dy
; and
S
c
includes the sediment exchange at banks and the side
discharge from tributaries.
The bed-load zone is so thin that it is not necessary to consider the vertical variation
of sediment concentration in this zone. The width-integrated bed-load transport is
determined using the 1-D transport equation introduced in the next subsection.
b
1
(
c
)
