ρV 2
τo =
(2.100)
2
y
12.27 o
5.75 log
ks
2.4.6 Energy and Momentum Coefficients for Rivers
In prismatic or constructed channels it is common to assume that the energy coefficient α and the
momentum coefficient β are unity. In river channels, this is usually not the case. From Equations
2.45 and 2.22:
1
v 3 dA
α=
(2.45)
A
V3A
and
1
v 2 dA
β=
(2.22)
A
V2A
The velocity distribution in wide channels for turbulent flow over a rough boundary is given by
Equation 2.78 with X = 1.0
v
= 2.5 In (30.2 y / k s )
(2.101)
V*
The average velocity in the vertical is
2.5V*
1 yo
0 v dy ~ y  y′
y
y′ In (y / y′) dy
V=

(2.102)
o
yo
o
Here, the upper limit of integration is yo, the depth of flow and the lower limit is
ks
y′ =
(2.103)
30.2
the value of y for which Equation 2.78 gives a zero velocity. The integration of Equation 2.102
yields
y o y o
V
In  1
= 2.5
(2.104)
y o  y′ y′
V*
For a vertical section of unit width, the momentum coefficient β′ is
2.28