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)
 
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