WATER FLOW 1287
force, A e t s , in the direction of the flow and the gravity force
component, W s sin w which attempts to cause the particle to
roll down the side slope, where t s unit tractive force on the
side of the channel, W s submerged weight of the particle,
w angle of the channel side. The resultant of these two
forces, F, is:FW(sinscAs).(^22) wt 2212 / (86)
The motion of the article is resisted by its frictional force ( R ):
RW scos tan ,wu (87)
where tan u is the coefficient of friction and u is the angle of
repose of the material.
Equating Eq. (86) and (87) for the condition of impend-
ing motion and solving for ts:
twu
w
s u
r
e
W
A
cos tan
tan
tan
.1
2
2
⎛
⎝⎜
⎞
⎠⎟
(88)
A similar equation can be written for the case of a particle on
a level bed when motion is impending, thus:
tl r
e
W
A
tan ,q (89)
where t (^) l denotes the unit tractive force on the level bed.
The tractive force ratio, K, is defined as, ts/tl is obtained
by dividing Eq. (87) to Eq. (88) and simplifying:
K(sin sin).1
(^2212) wu/ / (90)
From Eq. (89) it can be seen that the tractive force ratio,
K is a function of the side slope and angle of repose of the
material only.
Critical Tractive Force The permissible tractive force is
the maximum unit tractive force that will not cause signifi-
cant scour of the material lining the channel bed on a level
surface. It is often found from laboratory observations and
is known as the critical tractive force. It is influenced by the
amount of organic matter and fine suspended sediment in
the water. The effect of the fine sediment is to increase the
allowable critical tractive force. Figure 13 shows curves of
permissible tractive forces as recommended by the United
States Bureau of Reclamation.
River Engineering
In river flows, a greater number and range of factors have
to be considered in addition to those parameters used in the
analysis of canals. These variables include bigger size bed
materials, large suspended and bed sediment loads, unsteady
and a wide variation of flood flows, meandering and braid-
ing, large changes in stream channel cross-sections, obstruc-
tions to flow, and other factors involved. An analysis of river
engineering is, therefore, beyond the scope of this chapter.
Readers are recommended to consult the works of Blench
(1966), Shen (1971, 1972), Inglis (1949) and Leopold,
Wolman and Miller (1964). More specialized treatment of
sediment transport, bedforms and stream geometry can be
found in the publications of Einstein (1972), Leopold and
Maddock (1953), Richardson and Simons (1967), Yalin
(1971), Kennedy (1963), Christensen (1972), and Ackers
(1964). Standard texts which cover the subject more formally
include those of Graf (1971), Henderson (1966), Raudkivi
(1967) and Leliavsky (1955).
FLOW WITH AN ICE COVER
A river flowing with an ice cover has, in addition to the bed
and side frictional forces, the shear resistance imposed by
a buoyant boundary represented by the floating ice cover.
Chee and Haggag (1984) have developed equations con-
cerning floating boundary stream flow which are repro-
duced here. The essential concepts and assumptions are first
discussed.
A channel with a buoyant cover can be divided into two
subsections as shown in Figure 14. The flow in subsection
(1) is influenced by the bed and sides while subsection (2) is
controlled by the cover. The two subsections are divided by
a separation surface which represents the locus of no shear
and maximum velocity. The equations of energy, continuity,
Coarse non-
cohesive
material
25% larger
Large
amount
of fine
sediment
Small amount
of fine
sediment
Clear
water
Mean diameter, mm
Critical tractive force, 1bs./ft
2
1.0
1.0
0.1
0.1
0.01
10 100
FIGURE 13 Critical tractive force for canals.
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