v_m = c [root](mi),
where c is a coefficient, the values of which have been already given
in the table in § 98. Hence
v_m = cv0/(c + 25.4).
_Values of Coefficient c/(c + 25.4) in the Formula v_m = cv0/(c +
25.4)._
+----------+---------+----------+---------+----------+----------+
|Hydraulic | Very | Smooth | Rough |Very Rough| Channels |
|Mean Depth| Smooth |Channels. |Channels.| Channels.|encumbered|
| = m. |Channels.|Ashlar or | Rubble | Canals in| with |
| | Cement. |Brickwork.| Masonry.| Earth. | Detritus.|
+----------+---------+----------+---------+----------+----------+
| | | | | | |
| 0.25 | .83 | .79 | .69 | .51 | .42 |
| 0.5 | .84 | .81 | .74 | .58 | .50 |
| 0.75 | .84 | .82 | .76 | .63 | .55 |
| 1.0 | .85 | .. | .77 | .65 | .58 |
| 2.0 | .. | .83 | .79 | .71 | .64 |
| 3.0 | .. | .. | .80 | .73 | .67 |
| 4.0 | .. | .. | .81 | .75 | .70 |
| 5.0 | .. | .. | .. | .76 | .71 |
| 6.0 | .. | .84 | .. | .77 | .72 |
| 7.0 | .. | .. | .. | .78 | .73 |
| 8.0 | .. | .. | .. | .. | .. |
| 9.0 | .. | .. | .82 | .. | .74 |
| 10.0 | .. | .. | .. | .. | .. |
| 15.0 | .. | .. | .. | .79 | .75 |
| 20.0 | .. | .. | .. | .80 | .76 |
| 30.0 | .. | .. | .82 | .. | .77 |
| 40.0 | .. | .. | .. | .. | .. |
| 50.0 | .. | .. | .. | .. | .. |
| [oo] | .. | .. | .. | .. | .79 |
+----------+---------+----------+---------+----------+----------+
[Illustration: FIG. 107.]
§ 107. _River Bends._--In rivers flowing in alluvial plains, the
windings which already exist tend to increase in curvature by the
scouring away of material from the outer bank and the deposition of
detritus along the inner bank. The sinuosities sometimes increase till
a loop is formed with only a narrow strip of land between the two
encroaching branches of the river. Finally a "cut off" may occur, a
waterway being opened through the strip of land and the loop left
separated from the stream, forming a horseshoe shaped lagoon or marsh.
Professor James Thomson pointed out (_Proc. Roy. Soc._, 1877, p. 356;
_Proc. Inst. of Mech. Eng._, 1879, p. 456) that the usual supposition
is that the water tending to go forwards in a straight line rushes
against the outer bank and scours it, at the same time creating
deposits at the inner bank. That view is very far from a complete
account of the matter, and Professor Thomson gave a much more
ingenious account of the action at the bend, which he completely
confirmed by experiment.
[Illustration: FIG. 108.]
When water moves round a circular curve under the action of gravity
only, it takes a motion like that in a free vortex. Its velocity is
greater parallel to the axis of the stream at the inner than at the
outer side of the bend. Hence the scouring at the outer side and the
deposit at the inner side of the bend are not due to mere difference
of velocity of flow in the general direction of the stream; but, in
virtue of the centrifugal force, the water passing round the bend
presses outwards, and the free surface in a radial cross section has a
slope from the inner side upwards to the outer side (fig. 108). For
the greater part of the water flowing in curved paths, this difference
of pressure produces no tendency to transverse motion. But the water
immediately in contact with the rough bottom and sides of the channel
is retarded, and its centrifugal force is insufficient to balance the
pressure due to the greater depth at the outside of the bend. It
therefore flows inwards towards the inner side of the bend, carrying
with it detritus which is deposited at the inner bank. Conjointly with
this flow inwards along the bottom and sides, the general mass of
water must flow outwards to take its place. Fig. 107 shows the
directions of flow as observed in a small artificial stream, by means
of light seeds and specks of aniline dye. The lines CC show the
directions of flow immediately in contact with the sides and bottom.
The dotted line AB shows the direction of motion of floating particles
on the surface of the stream.
§ 108. _Discharge of a River when flowing at different Depths._--When
frequent observations must be made on the flow of a river or canal,
the depth of which varies at different times, it is very convenient to
have to observe the depth only. A formula can be established giving
the flow in terms of the depth. Let Q be the discharge in cubic feet
per second; H the depth of the river in some straight and uniform
part. Then Q = aH + bH², where the constants a and b must be found by
preliminary gaugings in different conditions of the river. M. C.
Moquerey found for part of the upper Saône, Q = 64.7H + 8.2H² in
metric measures, or Q = 696H + 26.8H² in English measures.
§ 109. _Forms of Section of Channels._--The simplest form of section
for channels is the semicircular or nearly semicircular channel (fig.
109), a form now often adopted from the facility with which it can be
executed in concrete. It has the advantage that the rubbing surface is
less in proportion to the area than in any other form.
[Illustration: FIG. 109.]
Wooden channels or flumes, of which there are examples on a large
scale in America, are rectangular in section, and the same form is
adopted for wrought and cast-iron aqueducts. Channels built with
brickwork or masonry may be also rectangular, but they are often
trapezoidal, and are always so if the sides are pitched with masonry
laid dry. In a trapezoidal channel, let b (fig. 110) be the bottom
breadth, b0 the top breadth, d the depth, and let the slope of the
sides be n horizontal to 1 vertical. Then the area of section is
[Omega] = (b + nd)d = (b0 - nd)d, and the wetted perimeter [chi] = b +
2d[root](n² + 1).
[Illustration: FIG. 110.]
When a channel is simply excavated in earth it is always originally
trapezoidal, though it becomes more or less rounded in course of time.
The slope of the sides then depends on the stability of the earth, a
slope of 2 to 1 being the one most commonly adopted.
Figs. 111, 112 show the form of canals excavated in earth, the former
being the section of a navigation canal and the latter the section of
an irrigation canal.
§ 110. _Channels of Circular Section._--The following short table
facilitates calculations of the discharge with different depths of
water in the channel. Let r be the radius of the channel section; then
for a depth of water = [kappa]r, the hydraulic mean radius is [mu]r
and the area of section of the waterway [nu]r², where [kappa], [mu],
and [nu] have the following values:--
+---------------------------------+------+-----+-----+-----+-----+-----+-----+-----+----+----+----+----+----+----+----+-----+-----+-----+-----+-----+-----+
| Depth of water in \ [kappa] = |.01 |.05 |.10 |.15 |.20 |.25 |.30 |.35 |.40 |.45 |.50 |.55 |.60 |.65 |.70 |.75 |.80 |.85 |.90 |.95 |1.0 |
| terms of radius / | | | | | | | | | | | | | | | | | | | | | |
| Hydraulic mean depth\ [mu] = |.00668|.0321|.0523|.0963|.1278|.1574|.1852|.2142|.242|.269|.293|.320|.343|.365|.387|.408 |.429 |.449 |.466 |.484 |.500 |
| in terms of radius/ | | | | | | | | | | | | | | | | | | | | | |
| Waterway in terms of\ [nu] = |.00189|.0211|.0598|.1067|.1651|.228 |.294 |.370 |.450|.532|.614|.709|.795|.885|.979|1.075|1.175|1.276|1.371|1.470|1.571|
| square of radius / | | | | | | | | | | | | | | | | | | | | | |
+---------------------------------+------+-----+-----+-----+-----+-----+-----+-----+----+----+----+----+----+----+----+-----+-----+-----+-----+-----+-----+
[Illustration: FIG. 111.--Scale 20 ft. = 1 in.]
[Illustration: FIG. 112.--Scale 80 ft. = 1 in.]
§ 111. _Egg-Shaped Channels or Sewers._--In sewers for discharging
storm water and house drainage the volume of flow is extremely
variable; and there is a great liability for deposits to be left when
the flow is small, which are not removed during the short periods when
the flow is large. The sewer in consequence becomes choked. To obtain
uniform scouring action, the velocity of flow should be constant or
nearly so; a complete uniformity of velocity cannot be obtained with
any form of section suitable for sewers, but an approximation to
uniform velocity is obtained by making the sewers of oval section.
Various forms of oval have been suggested, the simplest being one in
which the radius of the crown is double the radius of the invert, and
the greatest width is two-thirds the height. The section of such a
sewer is shown in fig. 113, the numbers marked on the figure being
proportional numbers.
[Illustration: FIG. 113.]
§ 112. _Problems on Channels in which the Flow is Steady and at
Uniform Velocity._--The general equations given in §§ 96, 98 are
[zeta] = [alpha](1 + [beta]/m); (1)
[zeta]v²/2g = mi; (2)