Environmental Engineering Reference
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of magnitude as the shear velocity , u * (LT −1 ), which is
defined by
where d is the depth of flow in the channel (L). The
theoretical expression for the vertical turbulent diffu-
sion coefficient, ε v , given by Equation (4.6), has been
confirmed experimentally in a laboratory flume (Jobson
and Sayre, 1970). The coefficient in Equation (4.6)
(= 0.067) is sometimes given as 0.1 (Martin and
McCutcheon, 1998). vertical mixing can be enhanced
locally by secondary currents (notably at sharp bends)
and by obstacles in the flow such as bridge piers. Equa-
tion (4.6) has been shown to adequately describe verti-
cal mixing of an effluent downstream of a river outfall
(Zhang and Zhu, 2011a).
Experimental results in straight rectangular channels
show that the transverse turbulent diffusion coefficient,
ε t , can be estimated by the relation
τ
ρ
0
u *
=
(4.1)
where τ 0 is the mean shear stress on the (wetted) perim-
eter of the stream (FL −2 ), and ρ is the density of the fluid
(ML −3 ). The boundary shear stress, τ 0 , can be expressed
in terms of the Darcy-Weisbach friction factor, f (dimen-
sionless), and the mean (longitudinal) flow velocity, V
(LT −1 ), by Chin (2013).
= f
V
(4.2)
τ
ρ
2
0
8
ε t = 0.15
du
(
straight uniform channels (4.7)
)
Combining Equations (4.1) and (4.2) leads to the fol-
lowing expression for the shear velocity:
*
where the coefficient of 0.15 can be taken to have an
error bound of ±50%, since experimental results yielded
results in the range ε t / du * = 0.08-0.24 (Lau and Krish-
nappan, 1977). Sidewall irregularities and variations in
channel shape both serve to increase the transverse tur-
bulent diffusion coefficient over that given by Equation
(4.7). A more typical estimate of the transverse diffu-
sion coefficient in natural streams is
f V
(4.3)
u
=
*
8
The friction factor, f , can generally be estimated from
the channel roughness, hydraulic radius, and Reynolds
number using the Colebrook equation, which for open
channels can be approximated by
ε t
= 0.6
du
(
natural streams
)
(4.8)
*
1
k
2.5
s
= −
2
log
+
(4.4)
where the coefficient of 0.6 can be taken to have an
error bound of ±50% and is sometimes approximated
as 1 (Martin and McCutcheon, 1998). Experimental
studies continue to provide data on which more precise
methods of estimating transverse turbulent diffusion
coefficients are being developed.
Bends in natural streams cause secondary (trans-
verse) currents that can increase the transverse turbu-
lent diffusion coefficient considerably beyond that given
by Equation (4.8) (Albers and Steffler, 2007). In these
cases, the term “transverse dispersion” more accurately
describes the mixing process rather than “transverse
diffusion”; however, quantification of the process by a
(pseudo-)transverse diffusion coefficient, ε t , will be
retained for comparison with transverse diffusion in
straight channels. Yotsukura and Sayre (1976) estimated
ε t = 3.4 du * in a stretch of the Missouri River containing
90° and 180° bends, and Seo et al., (2008) estimated ε t
values as high as 2.6 du * in the bends of a meandering
(S-shaped) channel. Yotsukura and Sayre (1976) sug-
gested the following analytic relation for estimating ε t
in meandering channels
10
f
12
R
Re
f
where k s is the roughness height in the channel (L), R
is the hydraulic radius (L), defined as the flow area
divided by the wetted perimeter, and Re is the Reynolds
number (dimensionless), defined by
Re = 4 VR
ν
(4.5)
where ν is the kinematic viscosity of the fluid in the
channel (L2T−1), 2 T −1 ), usually water. Equation (4.4) should be
used with caution in estimating the friction factor, f , in
mountain streams, where the presence of transient
storage zones (pools), stagnant zones due to bed irregu-
larity, and the existence of hyporheic zones in the gravel
bed all require special consideration (Meier and
Reichert, 2005). The hyporheic zone is the area below
the streambed between the rocks and cobbles.
Based on a theoretical analysis of turbulent mixing,
Elder (1959) showed that the average value of the verti-
cal turbulent diffusion coefficient, ε v (L 2 T −1 ), in a wide
open channel can be estimated by the relation
2
2
K U
u
B
r
ε t
=
du
(4.9)
*
ε v = 0.067
du
(4.6)
*
*
c
 
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