Sediment Movement in Alluvial Rivers - River Dynamics and Integrated River Management - page 286

velocity.
Equation (5.46) is the formula Shields used for the drag force. In his original derivation he neglected
the effect of uplift. Actually, the basic structure of the formula is not changed whether uplift is included
or not. Hence, the formula shows that, when grains start to move, the ratio of the drag force acting on a
grain to its weight is a function of the grain Reynolds number, as shown in Fig. 5.25.
After the work of Shields, a number of other researchers studied the incipient motion of sediment,
these include White (1970), Mantz (1977), Tison (1948), and Li and Sun (1964). Their results are included
in Fig. 5.25. A belt for the incipient drag force can be drawn to represent the data. This belt differs from
the original Shields curve in two important ways.
(1) Shields concluded that if grain Reynolds number is less than 2, the curve would be a straight line
with a 45 e slope. However, he had no experimental data in that region. He may have relied on an
analogy with the relation between the drag coefficient and the Reynolds number, and simply decided that

W JJ D should be inversely proportional to the grain Reynolds number if the latter is sufficiently

small. The later experiments, shown in Fig. 5.25, indicate that this concept does not agree with reality. In
this range of Reynolds number,

( )

c s

D is proportional to Re * with an exponent of -0.3.
(2) If Re * is quite large, Shields proposed that

W JJ ( )

c s

D would have the value 0.06. The results
now available show that this value should be taken as an upper limit; and the lower limit should be about
0.04. In Fig. 5.25, most of the results fall within this range. From the experimental data of Paintal (1971),
Miller et al. (1977) recommended that the ratio approaches the value 0.045, not 0.06, if Re * is quite large.

W JJ ( )

c s

Fig. 5.25 Shields curve for incipient motion for non-cohesive sediment (Shields curve and its modification)

In the Shields diagram, the parameter U (W o = U U 2 ) occurs in both the abscissa and the ordinate. Hence,
in determining the incipient drag force one must proceed by trial and error. To simplify the process, a set
of lines for constant values of the parameter of

D

0.1 JJ gD , with a slope of 2, has been drawn in

s

Q J

Fig. 5.25. The intersections of these lines with the Shields curve are the corresponding drag forces for
incipient motion.

5.2.2.2 Bed Load Transportation
In the last part of the 19th century, the French scientist DuBoys advanced a theory for bed load motion
based on shear stress. Since then, many scientists have studied the phenomenon and they have proposed