Geoscience Reference
In-Depth Information
which applies in the range of 2
<
B
/
h
<
20 for laboratory channels and 10
<
B
/
h
<
100 in waterways.
Fischer
et al
. (1979) proposed the following relation for natural waterways:
0.011
B
2
U
2
hU
∗
E
=
(12.24)
Eqs. (12.22)-(12.24) give the magnitude of the longitudinal effective diffusivity,
which may be used in the 1-D model. However, the effective diffusivity usually is
anisotropic. In particular, the dispersion effect is strongly related to the flow direc-
tion. Sladkevich
et al
. (2000) proposed a method to convert the longitudinal effective
diffusivity to the depth-averaged 2-D components in the Cartesian coordinate axes.
The effective diffusivity is treated as a tensor,
E
ij
, which is determined by
q
i
q
j
q
2
E
ij
=
ε
δ
+
ε
d
(12.25)
t
ij
where
q
is the flow discharge per unit width,
q
i
is the component of
q
in the
i
-direction,
and
δ
ij
is the Kronecker delta.
An alternative is the relation suggested by Holly and Usseglio-Polatera (1984):
E
s
cos
2
E
n
sin
2
E
xx
=
θ
+
θ
E
xy
=
(
E
s
−
E
n
)
sin
cos
(12.26)
θ
θ
E
s
sin
2
E
n
cos
2
E
yy
=
θ
+
θ
where
E
s
and
E
n
are the longitudinal and transverse effective diffusivities, respectively;
and
is the angle (positive counter-clockwise) of the longitudinal direction from the
x
-axis. Note that
E
xy
is applied in the cross-derivative diffusion terms, which need to
be added in Eqs. (12.8) and (12.9) for more general applications.
As described in Section 6.3, the helical flowmotion in curved channels induces a kind
of dispersion and significantly affects the main flow, sediment transport, and channel
morphological change. It might also affect heat and salinity transport. Therefore, the
application of formulas (12.22)-(12.24) in strongly curved channels should be done
with caution. Estimation using on-site measurement data is recommended.
θ
12.1.4 Heat transfer across water and bed surfaces
Surface heat exchange plays an important role in the thermodynamic processes in
aquatic systems. As shown in Fig. 12.1, heat generally transfers across the water surface
by short-wave radiation, long-wave radiation, evaporation, condensation, convection,
and conduction. The short-wave radiation is a penetrative effect that distributes its heat
through a significant range of the water column, while the others occur only at the
water surface.
