Geoscience Reference
In-Depth Information
Similarly, integrating Eq. (2.67) over the flow depth leads to the depth-integrated
y -momentum equation:
hU y )
∂(
∂(
hU y )
+ ∂(
hU x U y )
[
(
T yx +
D yx ) ]
gh
z s
1
ρ
h
+
=−
+
t
x
y
y
x
1
ρ
[
h
(
T yy
+
D yy
) ]
1
ρ
+
+
τ by )
sy
y
(2.83)
where T yx and T yy are the depth-averaged shear and normal stresses; D yx and D yy
account for the dispersion momentum transports due to the vertical non-uniformity of
velocity, defined as D yx
=− h z s
2 dz ;
sy is the y -component
of wind shear force per unit horizontal area at the water surface; and
=
D xy and D yy
z b (
u y
U y
)
τ
τ by is the y -
component of bed shear force per unit horizontal area.
The depth-averaged stresses T ij (
=
)
can be related to the gradients of the
depth-averaged velocities by the Bossinesq assumption similar to Eq. (2.45) in a tur-
bulence model, such as the depth-averaged k -
i , j
x , y
turbulence model proposed by Rastogi
and Rodi (1978). However, there is not a general method to handle the dispersion
terms D ij . D ij are not related to turbulence, but both D ij and T ij represent momentum
transports as effective stresses. In nearly straight channels, the dispersion transports
are usually combined with the turbulent stresses. In curved channels, secondary flows,
especially the helical flow, play an important role in fluvial processes, and thus the
dispersion transports become important and should be taken into account through
additional model closures. This is discussed in Section 6.3.
ε
Depth-averaged sediment transport equations
Unlike the depth-averaged quantities defined by Eq. (2.76),
the depth-averaged
suspended-load concentration, C , is defined by
z s
1
C
=
u s cdz
(2.84)
(
h
δ)
U s
z b + δ
where U s is the streamwise depth-averaged velocity, and u s is the local flow velocity
projected to the streamwise direction. By definition, U s
= z s
z b + δ
u s dz
/(
h
δ)
, but U s
U x +
is approximately set as the resultant depth-averaged velocity U
=
U y at each
horizontal point.
Integrating the three-dimensional sediment transport equation (2.72) over the
suspended-load zone leads to
z s
z s
z s
z s
z s
c
∂(
u x c
)
∂(
u y c
)
∂(
u z c
)
∂(ω
s c
)
t dz
+
dz
+
dz
+
dz
dz
x
y
z
z
z b + δ
z b + δ
z b + δ
z b + δ
z b + δ
z s
dz
z s
dz
z s
dz 2.85
s
c
s
c
s
c
=
ε
+
ε
+
ε
x
x
y
y
z
z
z b + δ
z b + δ
z b + δ
 
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