Geoscience Reference
In-Depth Information
simplified Eq. (6.79) to an ordinary differential equation for the helical flow intensity
in the developed regions:
I
d
2
I
d
D
I
B
2
1
T
a
−
β
I
hU
s
r
=
(6.80)
η
2
is the dimensionless transverse coordinate (
y
/
η
η
=
where
0 at the inner bank
and 1 at the outer bank; and
B
is the channel width at the water surface.
By assuming constant
D
I
,
T
a
, and source term in Eq. (6.80) and applying boundary
conditions
I
B
), with
=
η
=
η
=
0at
0 and
1, a solution for Eq. (6.80) was then derived as
e
−
B
/
√
T
a
D
I
e
B
/
√
T
a
D
I
e
B
/
√
T
a
D
I
rI
1
−
e
−
B
/
√
T
a
D
I
e
B
η/
√
T
a
D
I
e
−
B
/
√
T
a
D
I
e
−
B
η/
√
T
a
D
I
(6.81)
−
1
I
hU
s
=
1
−
−
e
B
/
√
T
a
D
I
β
−
−
Eq. (6.81) shows that the cross-stream profile of
I
is determined by parameters
B
,
T
a
,
D
I
, and
I
. Among these parameters, the channel width
B
is predefined.
D
I
can
be determined using
D
I
β
α
d
is an empirical coefficient. If the radius
of curvature at the channel centerline
r
c
and the average flow velocity
U
are used to
represent the adaptation length and velocity scales of
I
, respectively, it can be assumed
that the adaptation time scale
T
a
=
α
d
U
∗
h
, in which
a
is a dimensionless coefficient.
Therefore, the product of
T
a
and
D
I
can be determined by
=
α
a
r
c
/
U
. Here,
α
t
r
c
n
√
gh
5
/
6
T
a
D
I
=
λ
(6.82)
where
d
.
Therefore, the helical flow intensity profile along the cross-section is determined
by two parameters:
λ
t
is the product of
α
a
and
α
β
I
and
λ
t
. Usually,
β
I
determines the magnitude of
I
, while
λ
t
determines its lateral distribution. According to calibrations using many laboratory
and field measurements,
t
has a value of about 3.0.
Fig. 6.13 compares the secondary flow intensity calculated using Eq. (6.81) and
that measured by de Vriend (1981b) in an 180
◦
bend with a rectangular cross-section
β
I
is in the range of 1.0-2.0, and
λ
Figure 6.13
Transverse distribution of secondary flow intensity in de Vriend's bend
(Wu and Wang, 2004a).
