Geoscience Reference
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fluid property but strongly depends on the state of turbulence, perhaps varying largely
in time and space. In most flow regions, the eddy viscosity is much larger than the
molecular viscosity; thus, the latter usually is negligible.
In direct analogy to the turbulent momentum transport, the turbulent sediment flux
is assumed to be proportional to the gradient of sediment concentration:
s ¯
c
u i c = ε
(2.46)
x i
where
ε
s is the turbulent diffusivity of sediment.
ε
s is often related to the eddy viscosity
by
c being the Schmidt number (between 0.5 and 1.0), which is
discussed in detail in Section 3.5.1.
Note that the last term of Eq. (2.45) can be combined with the pressure-gradient
term when Eq. (2.45) is inserted into the momentum equation (2.43). Therefore, the
appearance of k in Eq. (2.45) does not necessitate the determination of it, and the set of
equations (2.42)-(2.44) is closed using relations (2.45) and (2.46). Then the problem
becomes how to determine the eddy viscosity. The eddy viscosity is usually assumed to
be proportional to the velocity scale
ε
= ν
c , with
σ
s
t
u and the length scale L m of (large-scale) turbulent
ˆ
motions:
ν
∝ˆ
uL m
(2.47)
t
One of the zero-equation turbulence models widely used for the eddy viscosity is
the Prandtl mixing length model, which postulates that the velocity scale
u for two-
dimensional shear flows is equal to the mean-velocity gradient times the mixing length,
thus yielding
ˆ
l m
¯
u
ν
=
(2.48)
t
z
where l m is the mixing leng th. Comm only used relations are: l m = κ
z for boundary
z (
layer flows, and l m
is the von
Karman constant, z is the distance to the wall boundary or the bed, and h is the
flow depth.
The mixing length model is suitable for flows where turbulence is in local equi-
librium, rather than where the convective and/or diffusive transport of turbulence is
important. Generally, the mixing length model is often used for simple shear-layer
flows where l m can be specified empirically. It is rarely used for rapidly varied flows,
such as recirculating flows, in channels with complex geometry, due to difficulties in
specifying l m .
Another often used zero-equation turbulence model is the parabolic eddy visco-
sity model:
= κ
1
z
/
h
)
for open-channel flows, in which
κ
U z 1
h
z
ν t = κ
(2.49)
where U
is the bed shear velocity.
 
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