Geoscience Reference
In-Depth Information
Figure 2.6
Configuration of flow and sediment transport.
p
=
p
a
+
ρ
g
(
z
s
−
z
)
(2.65)
where
z
s
is the water surface elevation, and
p
a
is the atmospheric pressure at the water
surface. A constant
p
a
is assumed here for a short river reach.
Substituting Eq. (2.65) into Eqs. (2.61) and (2.62) yields the
x
- and
y
-momentum
equations for gradually varied flows:
u
x
)
∂
∂
u
x
∂
+
∂(
+
∂(
u
y
u
x
)
+
∂(
u
z
u
x
)
g
∂
z
s
1
ρ
∂τ
1
ρ
∂τ
1
ρ
∂τ
xx
xy
xz
=−
x
+
+
+
t
x
∂
y
∂
z
∂
∂
x
∂
y
∂
z
(2.66)
u
y
)
∂
+
∂(
∂
u
y
∂
t
+
∂(
u
x
u
y
)
+
∂(
u
z
u
y
)
∂τ
∂τ
∂τ
g
∂
z
s
1
ρ
1
ρ
1
ρ
yx
yy
yz
=−
y
+
x
+
y
+
∂
x
y
∂
z
∂
∂
∂
∂
z
(2.67)
Because the channel bed and banks generally vary in much lower speed than the
flow, the following non-slip condition is applied at these solid boundaries:
u
bx
=
0,
u
by
=
0,
u
bz
=
0
(2.68)
The water surface is a free moving boundary, the location of which is part of the
solution. For a particle on the free surface, its location (
x
,
y
,
z
) can be described by
z
=
z
s
(
x
,
y
,
t
)
(2.69)