Page:EB1911 - Volume 14.djvu/84

Values of Coefficient c/(c + 25.4) in the Formula v_m = cv₀/(c + 25.4).

Hydraulic  Mean Depth  = m.	Very Smooth  Channels.  Cement.	Smooth Channels. Ashlar or  Brickwork.	Rough  Channels.  Rubble Masonry.	Very Rough  Channels. Canals in Earth.	Channels  encumbered  with 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	..	..	..	..	..
∞	..	..	..	..	.79

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.

 

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.

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 Ω = (b + nd) d = (b₀ − nd) d, and the wetted perimeter χ = b + 2d √ (n² + 1).

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 = κr, the hydraulic mean radius is μr and the area of section of the waterway νr ², where κ, μ, and ν have the following values:—

Depth of water in terms of radius	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$κ =	.01	.05	.10	.15	.20	.25	.30	.35	.40	.45	.50	.55	.60	.65	.70	.75	.80	.85	.90	.95	1.0
Hydraulic mean depth in terms of radius	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$μ =	.00668	.0321	.0523	.0963	.1278	.1574	.1852	.2142	.242	.269	.293	.320	.343	.365	.387	.408	.429	.449	.466	.484	.500
Waterway in terms of square of radius	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ν =	.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