Geoscience Reference
In-Depth Information
Figure 6.12
(a) Coordinate systems and (b) secondary flows in a curved channel.
compound cross-sections; however, they are usually much smaller in size and less
important than the helical flow in a channel bend. Therefore, only the helical flow is
considered here.
Though the main flow is affected by the aforementioned secondary flows, the vertical
distribution of the streamwise flow velocity in curved channels can still be assumed
to follow the logarithmic law (Rozovskii, 1957) or the power law (Zimmermann and
Kennedy, 1978). A function for the vertical distribution of the helical flow velocity
was derived by Rozovskii (1957), but its original formulation is complex and incon-
venient to use. If the Chezy coefficient is larger than 50, the Rozovskii distribution
can be simplified to a linear distribution, which is used here to evaluate the transverse
velocity (Odgaard, 1986):
b
s
I
2
z
1
u
n
=
U
n
+
h
−
(6.78)
where
U
n
is the depth-averaged cross-stream velocity,
u
n
is the local cross-stream
velocity at height
z
,
b
s
is a coefficient with a value of about 6.0, and
I
is the intensity
of helical flow. Note that for the sake of simplicity,
z
is defined here as the vertical
coordinate above the channel bed rather than an arbitrary datum.
Theoretically
I
r
at the channel centerline (Rozovskii, 1957; de Vriend,
1977), in which
r
is the local radius of curvature. De Vriend (1981a) proposed an
equation for approximately determining the helical flow intensity
I
in the entire bend:
=
U
s
h
/
D
I
h
∂
D
I
h
∂
∂(
hI
)
+
∂(
hU
x
I
)
+
∂(
hU
y
I
)
=
∂
∂
I
+
∂
∂
I
∂
t
∂
x
∂
y
x
∂
x
y
∂
y
I
h
T
a
−
β
I
hU
s
r
−
(6.79)
where
D
I
is a coefficient representing the diffusion and dispersion of
I
,
T
a
is the
adaptation time of
I
, and
I
is a coefficient that is 1.0 in de Vriend's original equation
but usually ranges from 1.0 to 2.0.
Eq. (6.79) needs to be solved numerically. This means that one partial differen-
tial equation is added to the set of shallow water equations, and the computational
effort is increased. To avoid this, by neglecting the time-derivative term, convection
terms, longitudinal diffusion term, and other high-order terms, Wu and Wang (2004a)
β
