V 2 2W
1
∆Z =
(2.163)
2g rc W 2
1 
12rc2
By assuming that the maximum velocities are close to the centerline of the channel in the bend
and that the flow pattern inward and outward from the centerline can be represented as forced
and free vortices, respectively, then:
1 rc C 2 r 2
2
1 ro C1
∆Z = ri
dr + rc 3 dr
2
(2.164)
g
g
r
r
and when r = rc, V = Vmax
Vmax
Therefore, C2 =
and C1 = Vmax rc
rc
and Equation 2.164 becomes:
rc
2
2
2
ri
Vmax
2 

∆Z =
(2.165)
r
r
2g
c
o
The differences in superelevation that are obtained by using the different equations are small,
and in alluvial channels the resulting erosion of the outside bank and deposition on the inside
bank leads to further error in computing superelevation. Therefore, it is recommended that
Equation 2.158 be used to compute superelevation in alluvial channels. For lined canals with
strong curvature, superelevation should be computed using Equations 2.162 or 2.165.
An example showing how to calculate superelevation in bends from velocity measurements is
presented in Section 2.14 (SI) and 2.15 (English) at the end of this chapter. The example also
compares the various approximate equations included in this section.
2.7.4 Supercritical Flow in Bends
Rapid flow or supercritical flow in a curved prismatic channel produces cross wave disturbance
patterns which persist for long distances in a downstream direction. These disturbance patterns
are the result of nonequilibrium conditions which persist because the disturbances cannot
propagate upstream or even propagate directly across the stream. Therefore, the turning effect
of the walls is not felt on all filaments of the flow at the same time and the equilibrium of the flow
is destroyed. The waves produced form a series of troughs and crests in the water surface along
the channel walls.
Two methods have been used in the design of curves for rapid flow in channels. One method is
to bank the floor of the channel and the other is to provide curved vanes in the flow. Banking on
the floor produces lateral forces which act simultaneously on all filaments and causes the flow to
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