Geoscience Reference
In-Depth Information
bU
z
)
∂
bU
z
bU
x
U
z
(
T
zx
+
D
zx
b
∂
˜
∂(ρ
)
+
∂(ρ
)
+
∂(ρ
p
z
+
∂
[
b
)
]
=−
ρ
bg
−
∂
t
∂
x
z
∂
∂
x
(
T
zz
+
D
zz
)
]
∂
+
∂
[
b
−
(
m
1
τ
+
m
2
τ
)
z
1
z
2
z
(2.124)
By applying the hydrostatic pressure assumption, the vertical momentum equation
(2.124) can be simplified to Eq. (2.116), and then the streamwise momentum
equation (2.123) is turned to
x
dz
gb
z
s
bU
x
bU
x
)
∂
bU
z
U
x
∂(ρ
)
+
∂(ρ
+
∂(ρ
)
0
bg
∂
˜
z
s
z
∂ρ
=−
ρ
x
+
∂
t
x
∂
z
∂
∂
(
T
xx
+
D
xx
(
T
xz
+
D
xz
+
∂
[
b
)
]
+
∂
[
b
)
]
∂
x
∂
z
−
(
m
1
τ
+
m
2
τ
)
(2.125)
x
1
x
2
Integrating Eqs. (2.112) and (2.117) over the cross-section yields the general 1-D
equations of flow with a longitudinally variable density:
AU
∂(ρ
A
)
+
∂(ρ
)
+
ρ
b
∂
A
b
∂
=
0
(2.126)
∂
t
∂
x
t
x
AU
2
∂
∂
)
+
∂
∂
gA
∂
˜
z
s
1
2
gAh
p
∂ρ
AU
t
(ρ
ρβ
=−
ρ
x
−
x
−
χ
ˆ
τ
bx
(2.127)
∂
∂
=
0
h
2
d
dy
where
h
p
A
, with
h
2
d
being the local flow depth.
Note that the effect of sediment concentration on the flow field is taken into account
in Eqs. (2.112)-(2.127) through the density of the water-sediment mixture defined in
Eq. (2.23). The effect of bed change is considered in the 1-D and depth-averaged 2-D
models by including the bed change terms in Eqs. (2.119) and (2.126), whereas this
is done in the width-averaged 2-D and 3-D models by specifying the near-bed fluxes
and changing the computational domains at the bed boundary.
/
2.5 NET EXCHANGE FLUX OF SUSPENDED LOAD NEAR BED
2.5.1 Exchange model using near-bed capacity
formula
In the depth-averaged 2-D (or 1-D) model, the near-bed sediment exchange flux
D
b
−
E
b
in the suspended-load transport equation must be modeled, because the near-
bed concentration
c
b
is not a dependent variable to be solved. The deposition flux
