condition of validity of equation (8.19) is that Re(wd/v) 103. As, from
equation (8.1), 0 / U^2 *, equation (8.19) can also be written as
Fr^2 d=U^2 */gd∆c. (8.20)
For a sediment particle on a slope (e.g. the side slope of a canal)
inclined at an angle to the horizontal, the critical shear stress is reduced
by a factor of {1 (sin^2 /sin^2 !)}1/2, where !is the natural angle of stability
of the non-cohesive material. (For stability, naturally !.) The average
value of !is about 35°.
On the other hand, the maximum shear stress induced by the flow on
a side slope of the canal is usually only about 0.75 gyS(instead of gRS
applicable for the bed of wide canals – equation (8.1) and Worked
example 8.2). Thus, in designing a stable canal in alluvium it is necessary
to ascertain whether the bed or side slope stability is the critical one for
channel stability. In a channel which is not straight the critical shear
stresses are further reduced by a factor between 0.6 and 0.9 (0.6 applies to
very sinuous channels).
Investigations into bedload transport have been going on for decades
without a really satisfactory all-embracing equation being available to
connect the fluid and sediment properties. This is due mainly to the complex-
ity of the problem including the effect of different bed forms on the mode
and magnitude of bedload transport, the stochastic nature of the problem
and the difficulty of verifying laboratory investigation in prototype. Never-
theless, substantial advances have been made. Most of the approaches used
can be reduced to a correlation between the sediment transport parameter,
qs/d3/2(g∆)1/2, where qs is the sediment transport (in m^3 s^1 m^1 ) and
Fr^2 d1/U^2 */∆gd, where is called the flow parameter (can also contain
an additional parameter – the ripple factor – to account for the effect of bed
form (Graf, 1984)). The power of Fr^2 din many correlations varies between 2
and 3, i.e. qsvaries as Vnwith 4n6, demonstrating the importance of a
good knowledge of the velocity field in the modelling and computation of
bedload transport, particularly when using two- or three-dimensional models.
Examples of simplified andcorrelations are the Meyer–Peter
and Muller equation
(4/ 0.188)3/2, (8.21)
the Einstein–Brown equation
40 ^3 (8.22)
and the Engelund–Hansen equation
0.4^ 5/2/. (8.23)
326 RIVER ENGINEERING
