Mechanics of Fluids

424 Flow with a free surface
Fig. 10.7
factor for turbulent flow in rough pipes.
C
(2g) 1/2 = f −1/2 = 4 log 10 (d/k) + 2.28
Substituting m 1/6 /n for C (as in eqn 10.9), and 4m for d, then gives (with
metre-second units)
whence
m1/6 (19.62) 1/2n = 4 log10 (4m/k) + 2.28
n = 0.0564m1/6
log 10 (14.86m/k)
Because an open channel lacks axial symmetry, if for no other reason, the
numerical factors in this expression are hardly trustworthy. Nevertheless the
expression does suggest that, with the logarithmic type of velocity profile
to be expected in turbulent flow, n is not very sensitive to changes in k,
and even less so to changes in m. Many attempts have been made to apply
theories developed for turbulent flow in pipes to that in open channels. The
difficulties, however, are great; in addition, the effects of the free surface and
of non-uniform shear stress round the wetted perimeter are uncertain; and
as yet no conclusive theory has emerged. Even so, the qualitative conclusions
drawn from Fig. 10.7 about the relation of C to the Reynolds number and
to the roughness size are valid – at least for rigid channel boundaries.
In alluvial channels, however, in which the surfaces are composed of movable
sand or gravel, the roughness elements are not permanent, but depend
on the flow. Ripples and dunes may form in the boundary material, and the
spacing of these humps may be much greater than the spacing of the irregularities
on the walls of pipes or on rigid boundaries of open channels. Under
such conditions as Nikuradse investigated – where the roughness projections
are close together – the wake behind one projection interferes with the flow
around those immediately downstream. The larger irregularities formed in
alluvial channels, however, have a different kind of effect, which, in turn,
results in a much larger friction loss than the size of the individual particles
alone would suggest.