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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