and from Equation 2.74
τo / ρ
V*
dv
=
=
(4.9)
κy
κy
dy
Thus
ε s = βεm = βκV* y(1 - y / y o )
(4.10)
where:
=
β
Coefficient relating εs to εm
=
Von Karman's velocity coefficient assumed equal to 0.4
κ
=
V*
Shear velocity equal to
gRS in steady uniform flow
Equation 4.10 indicates that εm and εs are zero at the bed and at the water surface, and have a
maximum value at mid-depth. The substitution of Equation 4.10 into Equation 4.4 gives
-ω
dc
dy
=
(4.11)
y
βκV*
c
y(1 -
)
yo
and after integration
Z
y - y a
c
= o
(4.12)
ca y yo - a
where:
c
=
Concentration at a distance y from the bed
ca
=
Concentration at a point a above the bed
z
=
ω/βκV*, the Rouse number, named after the engineer who developed the
equation in 1937
Figure 4.4 shows a family of curves obtained by plotting Equation 4.12 for different values of
the Rouse number z. An evaluation of the Rouse number, z, shows that for small values, the
sediment distribution is nearly uniform. For large z values, little sediment is found near the
water surface. The value of z is small for large shear velocities V* or small fall velocities ω.
Thus, for small particles or for extremely turbulent flows, the concentration profiles are uniform.
The values of β and κ have been investigated. For fine particles, β is approximately 1.0.
Also, it is well known that in clear water κ = 0.4, but apparently decreases with increasing
sediment concentration.
Using the logarithmic velocity distribution for steady uniform flow and Equation 4.12, the
equation for suspended sediment transport becomes
4.9
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