Environmental Engineering Reference
In-Depth Information
the longitudinal dispersion coefficients in rivers, an
alternative measurement technique is to use combine
ield measurements of velocity and bathymetry profiles
at river cross sections with a theoretical relationship
between
K
L
and the velocity and bathymetry profile.
Field measurements of velocity and bathymetry profiles
across rivers can be done by a variety of methods, with
the most effective methods utilizing acoustic Doppler
current profilers (ADCPs) to measure the flow depths
and longitudinal velocity distribution at a dense array
of points within a river transect. A commonly used theo-
retical relationship between
K
L
and the velocity and
bathymetry profiles was proposed by Fischer (1967) as
Consequently, vertical variations in the mean velocity
are usually neglected in deriving expressions relating
the longitudinal dispersion coefficient to the velocity
distribution in natural streams. Several empirical and
semiempirical formulas have been developed to esti-
mate the longitudinal dispersion coefficient,
K
L
, in
open-channel flow, and se
ve
ral of these formulas are
listed in Table 4.1, where
d
is the mean depth of the
stream,
u
*
is the shear velocity given by Equation (4.3),
and
w
is the top width of the stream. In applying the
equations in Table 4.1, it is important to keep in mi
n
d
that these equations apply to wide streams (
w d
),
where longitudinal dispersion is dominated by trans-
verse variations in the mean velocity, and the dispersion
caused by vertical variations in mean velocity is rela-
tively small. If only vertical variations in the mean veloc-
ity are considered and the velocity distribution is
assumed to be logarithmic, the longitudinal dispersion
coefficient is given by (Elder, 1959)
1
W
y
1
(
∫
∫
K
= −
u y h y dy
′
( ) ( )
dy
′
L
A
ε
h y
′
)
0
0
t
(4.35)
y
′
∫
u y h y dy
′
(
′′
) (
′′
)
′′
0
where
y
(L) is the transverse coordinate that runs from
0 at one bank to the width
W
(L) at the other bank,
A
is the total cross-sectional area of the stream (L
2
),
h
is
the depth (L) at location
y
, ε
t
is
th
e transverse mixing
coefficient (LT
−2
),
K
du
L
=
5.93
(4.36)
*
Therefore, for the equations given in Ta
bl
e 4.1 to be
applicable, the calculated values of
K du
( ) ( )
is the longitudinal
velocity deviation (LT
−1
),
u
(
y
) i
s
the depth-averaged
longitudinal velocity (LT
−1
), and
U
is the average longi-
tudinal velocity over the cross section (LT
−1
). The deri-
vation of Equation (4.35) along with an example
application is given in Section 3.6. Assumptions associ-
ated with using Equation (4.35) to estimate
K
L
include:
(1) the flow is one-dimensional (no secondary currents);
and (2) the width of the river is much larger than the
depth, so that the transverse shear, and not the vertical
shear, controls the dispersion. Comparisons between
ADCP-based estimates of
K
L
using Equation (4.35)
with dye studies indicate that ADCP-based estimates
are in fair agreement with the results of dye studies, and
this agreement is at least as good as empirical estimates
of
K
L
(Carr and Rehmann, 2007). Some of this discrep-
ancy can be attributed to the inability of ADCPs to
accurately measure flows velocities close to the water
surface and channel boundaries.
u y
′
=
u y U
−
L
/
*
must be
greater than 5.93; otherwise, vertical variations in veloc-
ity dominate the dispersion process and Equation (4.36)
should be used to estimate the
K
L
. Equation (4.36)
was derived by assuming a vertical (logarithmic)
velocity profile generated by roughness elements on the
bottom of the channel; however, in cases where bottom
TABLE 4.1. Estimates of Longitudinal Dispersion
Coefficient in Rivers
Formula
Reference
2
2
K
du
w
d
V
u
Fischer et al.
(1979)
Liu (1977)
L
=
0.011
*
*
2
0.5
K
du
w
d
V
u
L
=
0.18
*
*
2
K
du
w
d
Koussis and
Rodríguez-
Mirasol
(1998)
Iwasa and Aya
(1991)
Seo and Cheong
(1998)
Deng et al.
(2001)
L
*
=
0.6
4.2.2.2 Empirical Estimates of K
L
.
It has been dem-
onstrated that the longitudinal dispersion coefficient is
proportional to the square of the distance over which a
velocity shear flow profile extends (Fischer et al., 1979).
Since natural streams typically have widths that are at
least 10 times the depth, the longitudinal dispersion
coefficient associated with transverse variations in the
mean velocity can be expected to be on the order of 100
times larger than the longitudinal dispersion coefficient
associated with vertical variations in the mean velocity.
1.5
K
du
w
d
L
*
=
2.0
1.428
0.620
K
du
w
d
V
u
L
=
5.915
*
*
5
3
2
K
du
0.01875
w
d
V
u
L
*
=
1.38
1
3520
V
u
w
d
*
0.145
+
*
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