Geoscience Reference
In-Depth Information
Substituting Eq. (2.116) into Eqs. (2.113) and (2.114) and assuming a constant
p
a
yields
g
z
s
u
x
)
∂(ρ
u
x
)
+
∂(ρ
+
∂(ρ
u
y
u
x
)
+
∂(ρ
u
z
u
x
)
0
g
∂
z
s
z
∂ρ
=−
ρ
x
−
x
dz
∂
t
∂
x
∂
y
∂
z
∂
∂
+
∂τ
+
∂τ
+
∂τ
xy
xx
xz
(2.117)
∂
x
∂
y
∂
z
g
z
s
u
y
)
∂(ρ
∂(ρ
u
y
)
+
∂(ρ
u
x
u
y
)
∂
+
∂(ρ
u
z
u
y
)
∂
0
g
∂
z
s
z
∂ρ
+
=−
ρ
−
y
dz
∂
t
x
∂
y
z
∂
y
∂
+
∂τ
+
∂τ
+
∂τ
yx
yy
yz
(2.118)
∂
x
∂
y
∂
z
ρ
0
is the flow density at the water surface.
Integrating Eqs. (2.112), (2.117), and (2.118) over the flow depth leads to the
depth-integrated 2-D flow equations:
∂
ρ
where
h
+
∂(ρ
hU
x
)
∂
+
∂(ρ
hU
y
)
+
ρ
b
∂
z
b
∂
=
0
(2.119)
∂
t
x
∂
y
t
hU
x
)
∂
∂(ρ
hU
x
)
∂
+
∂(ρ
+
∂(ρ
hU
y
U
x
)
gh
∂
2
gh
2
∂ρ
x
+
∂
[
(
T
xx
+
D
xx
)
]
z
s
1
h
=−
ρ
x
−
t
x
∂
y
∂
∂
∂
x
+
∂
[
h
(
T
xy
+
D
xy
)
]
+
τ
−
τ
bx
(2.120)
sx
∂
y
hU
y
)
∂
∂(ρ
∂(ρ
hU
y
)
+
∂(ρ
hU
x
U
y
)
gh
∂
z
s
∂
1
2
gh
2
∂ρ
y
+
∂
[
h
(
T
yx
+
D
yx
)
]
+
=−
ρ
−
∂
t
∂
x
y
y
∂
∂
x
+
∂
[
h
(
T
yy
+
D
yy
)
]
+
τ
sy
−
τ
by
(2.121)
∂
y
where
b
is the density of the water-sediment mixture in the bed surface layer, deter-
mined by
ρ
, with
p
m
being the porosity of the surface-layer bed
material. Note that in the derivation of Eqs. (2.119)-(2.121), it is assumed that the
flow density is constant along the flow depth but varies horizontally.
Integrating Eqs. (2.112), (2.113), and (2.115) over the flow width yields the width-
integrated 2-D equations of flow with a density varying in the longitudinal section:
ρ
b
=
ρ
f
p
m
+
ρ
p
m
)
(
1
−
s
bU
x
)
∂
bU
z
)
∂
∂(ρ
b
)
+
∂(ρ
+
∂(ρ
=
0
(2.122)
∂
t
x
z
bU
x
bU
x
)
∂
bU
z
U
x
(
T
xx
+
D
xx
b
∂
p
∂
∂(ρ
)
+
∂(ρ
+
∂(ρ
)
x
+
∂
[
b
)
]
=−
∂
t
x
∂
z
∂
x
(
T
xz
+
D
xz
+
∂
[
b
)
]
−
(
m
1
τ
+
m
2
τ
)
x
1
x
2
∂
z
(2.123)
