FTC. 102. practical errors. Formulae have
been obtained based on less restricted hypotheses, but at present they are not practically so reliable, and are more complicated than the formulae obtained in the manner described above.
§ 96. Steady Flow of Water with Uniform Velocity in Channels of Constant Section.Let aa', bb' (fig. 103) be two cross sections normal to the direction of motion at a distance dl. Since the mass aa'bb' moves uniformly, the external forces acting on it are in equilibrium. Let St be the area of the cross sections, x the wetted perimeter,
In using this value of when v is not known, it is best to proceed by approximation.
§ 98. Darcy and Bazin's Expression for the Coefficient of Friction.Darcy and Bazin's researches have shown that varies very greatly for different degrees of roughness of the channel bed, and that it also varies with the dimensions of the channel. They give for i an empirical expression (silnilar to that for pipes) of the form
3 = a (I +R/m) ; (6) where m is the hydraulic mean depth. For different kinds of channels they give the following values of the coefficient of friction:
Kind of Channel. I a
I. Very smooth channels, sides of smooth
cement or planed timber . . . 0.00294 0.10
II. Smooth channels, sides of ashlar, brick 0.00373 0.23
work, planks
pitched with stone
IV. Very rough canals in earth . 0.00549 4.10
V. Torrential streams encumbered with detritus 0.00785 5.74
The last values (Class V.) are not Darcy and Bazin's, but are taken from experiments by Ganguillet. and Kutter on Swiss streams.
The following table very much facilitates the calculation of the mean velocity and discharge of channels, when Darcy and Bazin's value of the coefficient of friction is used. Taking the general formula for the mean velocity already given in equation (2a) above,
v=c. (mi),
where c=1 (2g/1'), the following table gives values of c for channels of different degrees of roughness, and for such values of the hydraulic mean depths as are likely to occur in practical calculations:
Values of c in v =c'( (mi), deduced from Darcy and Basin's Values.
o II U .t C .mac U W O Ga . II 1 J 5 e A~r y a A
a Vl 'j, U'~ UZ FIG 7 o'.0 3~,, g V~ ()A O. U `A
'D a ,g ° ~ 7A 4 C m.b tea .a °O 7•n 4 G 6 N.b
A E. >U WU kt'; U E 44 ~C~ 't e,
c.) a`i .o C 9 a
•25 125 95 57 26 18.5 8.5 147 130 112 89
•5 135 110 72 36 25.6 9.0 147 130 112 90 71
.75 139 116 81 42 30.8 9.5 147 130 112 90 ..
PO 141 119 87 48 34.9 10.0 147 130 112 91 72
1.5 143 122 94 56 41.2 11 147 130 113 92 ..
2.0 144 124 98 62 46.0 12 147 130 113 93 74
2.5 145 126 Tot 67 .. 13 147 130 113 94 ••
3.0 145 126 104 70 53 14 147 130 113 95 ..
3.5 146 127 105 73 15 147 130 114 96 77
4.0 146 128 ,o6 76 58 16 147 130 114 97 •.
4.5 146 128 107 78 17 147 130 114 97 •
5.0 146 128 108 8o 62 18 147 130 114 98
5.5 146 129 109 82 20 147 131 114 98 8o
6•o 147 129 110 84 65 25 148 131 115 100
6.5 147 129 110 85 30 148 131 115 102 83
7.0 147 129 110 86 67 40 148 131 116 103 85
7'5 147 129 111 87 .50 148 131 116 104 86
8.0 147 130 111 88 69 co 148 131 117 Io8 91
w
pq+gr+rs, of a section. Then the quantity m=S2/x is termed the hydraulic mean depth of the section. Let v be the mean velocity of the stream, which is taken as the common velocity of all the particles, i, the slope or fall of the stream in feet, per foot, being the ratio bc/ab.
The external forces acting on aa'bb' parallel to the direction of motion are three:(a) The pressures on aa' and bb', which are equal and opposite since the sections are equal and similar, and the mean pressures on each are the same. (b) The component of the weight W of the mass in the direction of motion, acting at its centre of gravity g. The weight of the mass aa'bb' is GS2dl, and the component of the weight in the direction of motion is G52dl X the cosine of the angle between Wg and ab, that is, GSldl cos abc =GSZdl be/ab = GS2idl. (c) There is the friction of the stream on the sides and bottom of the channel. This is proportional to the area xdl of rubbing surface and to a function of the velocity which may be written f(v) ; f(v) being the friction per sq. ft. at a velocity v. Hence the friction is xdl f(v). Equating the sum of the forces to zero,
Gtti dl  xdl f (v) =o,
f (v) /G = Sti/x =mi. (I) But it has been already shown (§ 66) that f(v) = jGv2/2g, :. i v2/2 g = mi.
This may be put in the form
v = 1/ (2gR) Il (mi) = c J (mi) ; (2a) where c is a coefficient depending on the roughness and form of the channel.
The coefficient of friction varies greatly with the degree of roughness of the channel sides, and somewhat also with the velocity. It must also be made to depend on the absolute dimensions of the section, to eliminate the error of neglecting the variations of velocity in the cross section A common mean value assumed for is 0.00757
.. The range of values will be discussed presently.
It is often convenient to estimate the fall of the stream in feet per mile, instead of in feet per foot. If f is the fall in feet per mile. f=5280 i.
Putting this and the above value of in (2a), we get the very simple and longknown approximate formula for the mean velocity of a stream
(2)
The flow down the
§ 99. Ganguillet and Mutter's Modified Darcy Formula.Starting from the general expression v=cjmi, Ganguillet and Kutter examined the variations of c for a wider variety of cases than those discussed by Darcy and Bazin. Darcy and Bazin's experiments v=fLz1 (2mf). (3) were confined to channels of moderate section, and to a limited
stream per second, or discharge of the stream, variation of slope. Ganguillet and Kutter brcugnt into the dis
cussion two very distinct and important additional series of results. is Q=S2v=S2cll (mi). (4) The gaugings of the Mississippi by A. A. Humphreys and H. L.
§ 97. Coefficient of Friction for Oben Channels.'Tarious ex . Abbot afford data of discharge for the case of a stream of exceo€onpressions have been proposed iur cne coefficient of friction for ally large section anti or very low slope. cm the otner nand, their channels as :,,r pipes. Weisbach, giving attention chiefly to the own measurements of the flow in the regulated channels of some
Swiss torrents gave data for cases in which the inclination and roughness of the channels were exceptionally great. Darcy and Bazin's experiments alone were conclusive as to the dependence of the coefficient c on the dimensions of the channel and on its roughness of surface. Plotting values of c for channels of different inclination appeared to indicate that it also depended on the slope of the stream. Taking the Mississippi data only, they found
c=256 for an inclination of 0.0034 per thousand,
=154 „ 0.02
so that for very low inclinations no constant value of c independent of the slope would furnish good values of the discharge. in small rivers, on the other hand, the values of c vary little with the slope. As regards the influence of roughness of the sides of the channel a different law holds. For very small channels differences of roughness have a great influence on the discharge, but for very large channels different degrees of roughness have but little influence, and for indefinitely large channels the influence of different degrees of roughness must be assumed to vanish. The coefficients given by Darcy and Bazin are different for each of the classes of channels of different roughness, even when the dimensions of the channel are infinite. But, as it is much more'probable that the influence of the nature of the sides diminishes indefinitely as the channel is larger, this must be regarded as a defect in their formula.
Comparing their own measurements in torrential streams in Switzerland with those of Darcy and Bazin, Ganguillet and Kutter found that the four classes of coefficients proposed by Darcy and Bazin were insufficient to cover all cases. Some of the Swiss streams gave results which showed that the roughness of the' bed was markedly greater than in any of the channels tried by the French engineers. It was necessary therefore in adopting the plan of arranging the different channels in classes of approximately similar roughness to increase the number of classes. Especially an additional class was required for channels obstructed by detritus.
To obtain a new expression for the coefficient in the formula
v= d (2g/i')v' (mi) =cl/ (mi),
Ganguillet and Kutter proceeded in a purely empirical way. They found that an expression of the form
c = a/ (i +6' nz)
could be made to fit the experiments somewhat better than Darcy's expression. Inverting this, we get
I /c = I /a +13/alb nz,
an equation to a straight line having IN m for abscissa, 1/c for ordinate, and inclined to the axis of abscissae at an angle the tangent of which is No..
Plotting the experimental values of I/c and I/gym, the points so found indicated a curved rather than a straight line, so that 13 must depend on a. After much comparison the following form was arrived at
c = (A+I/n)/(i +AnN m),
where n is a coefficient depending only on the roughness oo the sides of the channel, and A and t are new coefficients, the value of which remains to be determined. From what has been already stated, the coefficient c depends on the inclination of the stream, decreasing as the slope i increases.
Let A = a+p/i.
Then c=(a+l/n+p/i)l[I+(a+pii)nl~zn},
the form of the expression for c ultimately adopted by Ganguillet and Kutter.
For the constants a, 1, p Ganguillet and Kutter obtain the values 23, I and 0.00155 for metrical measures, or 41.6, I.81I and 0.00281 for English feet. The coefficient of roughness n is found to vary from o•oo8 to 0.050 for either metrical or English measures.
The most practically useful values of the coefficient of roughness n are given in the following table:
Nature of Sides of Channel. Coefficient of Roughness n.
Wellplaned timber 0.009
Cement plaster o•ot0
Plaster of cement with onethird sand o•oI
Unplaned planks 0.012
Ashlar and brickwork 0.013
Canvas on frames 0.015
Rubble masonry 0.017
Canals in very firm gravel 0.020 Rivers and canals in perfect order, free from stones 0.025 or weeds
Rivers and canals in moderately good order, not 0,030 quite free from stones and weeds
Rivers and canals in bad order, with weeds and 0.035 detritus .
Torrential streams encumbered with detritus . 0.050
Ganguillet and Kutter's formula is so cumbrous that it is difficult to use without the aid of tables.
Lowis D'A. Jackson published complete and extensive tables for facilitating the use of the Ganguillet and Kutter formula (Canaland Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form:
M =n(41.6+0•00281/i) ;
v=(.l m/n){(M 1.811)/(M+1,1m)}I (mi).
The following table gives a selection of values of M, taken from Jackson's tables:
i= Values of M for n=
0.010 0.012 0.015 0.017 0.020 0.025 0.030
•0000I 3.2260 3.8712 4.8390 5'4842 64520 8.0650 9.6780
•00002 I.82I0 2.1852 2.7315 3.0957 3.6420 4.5525 5.4630
•00004 1.1185 1.3422 1.6777 1.9014 2.2370 2.7962 3.3555
•00006 0.8843 1.0612 1.3264 '•5033 1.7686 2.2107 2.6529
•00008 0.7672 0.9206 I.15o8 1.3042 1.5344 1.9180 2.3016
'000I0 0.6970 0.8364 1.0455 1.1849 1.3940 1.7425 2.0910
'00025 0.5284 0.6341 0.7926 o•8983 1.0568 1.3210 1.5852
.00050 0.4722 0.5666 0.7083 0.8027 0.9444 1.1805 1.4166
•00075 0'4535 0.5442 0.6802 0.7709 0.9070 1'1337 1.3605
•OOI00 0.4441 0.5329 0.6661 0.7550 o'8882 I.1IO2 1.3323
•00200 0.4300 0.5160 0.6450 0.7310 0.8600 1.0750 1.2900
•00300 0'4254 0.5105 0.6381 0.7232 0.8508 1.0635 1.2762
A difficulty in the use of this formula is the selection of the coefficient of roughness. The difficulty is one which no theory will overcome, because no absolute measure of the roughness of stream beds is possible. For channels lined with timber or masonry the difficulty is not so great. The constants in that case are few and sufficiently defined. But in the case of ordinary canals and rivers the case is different, the coefficients having a much greater range. For artificial canals in rammed earth or gravel n varies from o ot63 to 0.0301. For natural channels or rivers n varies from 0.020 to 0.035.
In Jackson's opinion even Kutter's numerous classes of channels seem inadequately graduated, and he proposes for artificial canals the following classification:
•
detritus .
Ganguillet and Kutter's formula has been considerably used partly from its adoption in calculating tables for irrigation work in India. But it is an empirical formula of an unsatisfactory form. Some engineers apparently have assumed that because it is complicated it must be more accurate than simpler formulae. Comparison with the results of gaugings shows that this is not the case. The term involving the slope was introduced to secure agreement with some early experiments on the Mississippi, and there is strong reason for doubting the accuracy of these results.
§ Too. Bazin's New Formula.Bazin subsequently reexamined all the trustworthy gaugings of flow in channels and proposed a modification of the original Darcy formula which appears to be more satisfactory than any hitherto suggested (Etude d'une nouvelle formule, Paris, 1898). He points out that Darcy's original formula, which is of the form mi/v2=a+13/m, does not agree with experiments on channels as well as with experiments on pipes. It is an objection to it that if m increases indefinitely the limit towards which mi/v2 tends is different for different values of the roughness. It would seem that if the dimensions of a canal are indefinitely increased the variation of resistance due to differing roughness should vanish. This objection is met if it is assumed that 1 (mi/v2) = a+$Jd m, so that if a is a constant mi/v2 tends to the limit a when m increases. A very careful discussion of the results of gaugings shows that they can be expressed more satisfactorily by this new formula than by Ganguillet and Kutter's. Putting the equation in the form 'v2/2g= mi, i'=0•002594(I+y/Jm), where y has the following values:
I. Very smooth sides, cement, planed plank, y = 0.109
II. Smooth sides, planks, brickwork . . . . 0.290
IV. Sides of very smooth earth, or pitching . . 1.539
V. Canals in earth in ordinary condition . . . 2.353
VI. Canals in earth exceptionally rough . . . 3.168
§ Tot. The Vertical Velocity Curve.If at each point along 'a vertical representing the depth of a stream, the velocity at that point is plotted horizontally, the curve obtained is the vertical velocity curve and it has been shown by many observations that it approximates to a parabola with horizontal axis. The vertex of the parabola is at the level of the greatest velocity. Thus in fig. 104 OA is the vertical at which 'velocities are observed ; v, is the surface; v~ the maximum and vd the bottom velocity. B C D is the vertical velocity curve which corresponds with a parabola having its vertex at C. The mean velocity at the vertical is
vm = 3I2vz+vd+ (dz/ d) (v0 vd)) •
The Horizontal Velocity Curve.Similarly if at each point along a horizontal representing the width of the stream the velocities are
I. Canals in very firm gravel, in perfect order n=0.02
I I. Canals in earth, above the average in order n=0'0225
I V. Canals in earth, below the average in order n= 0.0275
V. Canals in earth, in rather bad order, partially n= 0.03
overgrown with weeds and obstructed by
plotted, a curve is obtained called the horizontal velocity curve. In streams of symmetrical section this is a curve symmetrical about the centre line of the stream. The velocity varies little near the centre of the stream, but very rapidly near the banks. In un
symmetrical symmetrical sections the greatest
velocity is at the point where the
stream is deepest, and the general
form of the horizontal velocity curve  is roughly similar to the section of the stream.
§ 102. Curves or Contours of Equal Velocity.—If velocities are observed at a number of points at different widths and depths in a stream, it is possible to draw curves on the cross section through points at which the velocity is the same. These represent contours of a solid, the volume of which is the discharge of the stream per second. Fig. io5 shows
the vertical and horizontal velocity curves and the contours of equal velocity in a rectangular channel, from one of Bazin's gaugings.
§ 103. Experimental Observations on the Vertical Velocity Curve.—A preliminary difficulty arises in observing the velocity at a given point in a stream because the velocity rapidly varies, the motion not being strictly steady. If an average of several velocities at the same point is taken, or the average velocity for a sensible period of time, this average is found to be constant. It may be inferred that
a
though the velocity at a point fluctuates about a mean value, the fluctuations being due to eddying motions superposed on the general motion of the stream, yet these fluctuations produce effects which disappear in the mean of a series of observations and, in calculating the volume of flow, may be disregarded.
In the next place it is found that in most of the best observations on the velocity in streams, the greatest velocity at any vertical is found not at the surface but at some distance below it. In various river gaugings the depth d,at the centre of the stream has been found to vary from 0 to 0.3d.
§ 104. Influence of the Wind.—In the experiments on the Mississippi the vertical velocity curve in calm weather was found to agree fairly with a parabola, the greatest velocity being at Aths of the depth of the stream from the surface. With a wind blowing down stream the surface velocity is increased, and the axis of the parabola approaches the surface. On the contrary, with a wind blowing up stream the surface velocity is diminished, and the axis of the parabola is lowered, sometimes to half the depth of the stream. The American observers drew from their observations the conclusion that there was an energetic retarding action at the surface of a stream like that due to the bottom and sides. If there were such a retarding action the position of the filament of maximum velocity below the surface would be explained.
It is not difficult to understand that a wind acting on surface ripples or waves should accelerate or retard the surface motion of the stream, and the Mississippi results may be accepted so far as showing that the surface velocity of a stream is variable when the mean velocity of the stream is constant. Hence observations of surface velocity by floats or otherwise should only be made in very calm weather. But it is very difficult to suppose that, in still air, there is a resistance at the free surface of the stream at all analogous to that at the sides and bottom. Further, in very careful experiments, P. P. Boileau found the maximum velocity, though raised a little above its position for calm weather, still at a considerable distance below the surface, even when the wind was blowing down stream with a velocity greater than that of the stream, and when the action of the air must have been an accelerating and not a retarding action. A much more probable explanation of the diminutionof the velocity at and near the free surface is that portions of water, with a diminished velocity from retardation by the sides or bottom, are thrown off in eddying masses and mingle with the rest of the stream. These eddying masses modify the velocity in all parts of the stream, but have their greatest influence at the free surface. Reaching the free surface they spread out and remain there, mingling with the water at that level and diminishing the velocity which would otherwise be found there.
Influence of the Wind on the Depth at which the Maximum Velocity is found.—In the gaugings of the Mississippi the vertical velocity curve was found to agree well with a parabola having a horizontal axis at some distance below the water surface, the ordinate of the parabola at the axis being the maximum velocity of the section. During the gaugings the force of the wind was registered on a scale ranging from o for a calm to Io for a hurricane. Arranging the velocity curves in three sets—(1) with the wind blowing up stream, (2) with the wind blowing down stream, (3) calm or wind blowing across stream—it was found that an upstream wind lowered, and a downstream wind raised, the axis of the parabolic velocity curve. In calm weather the axis was at gths of the total depth from the surface for all conditions of the stream.
Let h' be the depth of the axis of the parabola, m the hydraulic mean depth, f the number expressing the force of the wind, which may range from+lo to — Io, positive if the wind is up stream, negative if it is down stream. Then Humphreys and Abbot find their results agree with the expression
h'/m=0.317 =o•o6f.
Fig. 106 shows the parabolic velocity curves according to the American observers for calm weather, and for an up or downstream wind of a force represented by 4.
It is impossible at present to give a theoretical rule for the vertical velocity curve, but in very many gaugings it has been found that a parabola with horizontal axis fits the observed results fairly well. The mean velocity on any vertical in a stream varies from 0.85 to o•92 of the surface velocity at that vertical, and on the average if V. is the surface and v,,, the mean velocity at a vertical v,,,= Bvo, a result useful in float gauging. On any vertical there is a point at which the velocity is equal to the mean velocity, and if this point were known it would be useful in gauging. Humphreys and Abbot in the Mississippi found the mean velocity at o•66 of the depth ; G. H. L. Hagen and H. Heinemann at 0.56 to 0.58 of the depth. The mean of observations by various observers gave the mean velocity at from 0.587 to o•62 of the depth, the average of all being almost exactly o•6 of the depth. The middepth velocity is therefore nearly equal to, but a little greater than, the mean velocity on a vertical. If v,,,d is the middepth velocity, then on the average v,,, = o•98v,,,d.
§ 105. Mean Velocity on a Vertical from Two Velocity Observations. —A. J. C. Cunningham, in gaugings on the Ganges canal, found the following useful results. Let vo be the surface, v,,, the mean, and Vail the velocity at the depth xd; then
vm = (vu+3v2/3d)
= (v 211d+v789d) •
§ Io6. Ratio of Mean to Greatest Surface Velocity, for the whole Cross Section in Trapezoidal Channels.—It is often very important to be able to deduce the mean velocity, and thence the discharge, from observation of the greatest surface velocity. The simplest method of gauging small streams and channels is to observe the greatest surface velocity by floats, and thence to deduce the mean velocity. In general in streams of fairly regular section the mean velocity for the whole section varies from 0.7 to 0.85 of the greatest surface velocity. For channels not widely differing from those experimented on by Bazin, the expression obtained by him for the ratio of surface to mean velocity may be relied on as at least a good approximation to the truth. Let v be the greatest surface velocity, v,,, the mean velocity of the stream. Then, according to Bazin,
Vm=vo.—25.4) (mi).
But vm, = c (mi),
where c is a coefficient, the values of which have been already given in the table in § 98. Hence
vm = cvo/(c+25.4) •
'a i
Vertical Velocity Horizontal Velocity Curves Curves
e._f._:,_.•g~. „.,.. ,,....~... v k
Contours of Equal Velocity
Values of Coefficient c/(c+25.4) in the Formula vm =cvol(c+25.4) •
Hydraulic Very Smooth Rough Very Rough Channels
mean Depth Smooth Channels. Channels. Channels. encumbered
=in 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
I0•0 .. .. .. .. ..
15.0 .. .. •79 '75
20.0 .. .. •80 .76
30.0 .. .. •82 .. '77
40.0 .. .. ..
50.0 .. .. ..
'79
§ 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
l':?1 14111 i!', .I .I I ~li 11f'f lllrpl Uli v stream, forming a horse
shoe 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
FIG. I07. much more ingenious
account of the action at the bend, which he completely confirmed by experiment.
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. Io8).
For the greater part of the water flowing in curved paths, this
difference of pressure produces no tendency to transverse motion.
But the water im
mediately in contact
with the rough bot
tom 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 there
fore flows inwards towards the inner side of the bend, carrying
with it detritus which is deposited at the inner bank. Con
jointly 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 Saone, 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.
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 castiron 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. IIo)
Concrete FIG. II0.
be the bottom breadth, be 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 12= (b+nd)d = (bond)d, and the wetted perimeter x=b+2d J (n2+1).
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 I being the one most commonly adopted.
Figs. Ill, 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.
§ Ho. 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 Kr, the hydraulic mean radius is ,ur and the area of section of the waterway vr2, where K, u, and v havethe following values:
Depth of water in }  .or .10 .20 .60 .80 .85 .95 1.0
terms of radius . .00668 05 .15 .1278 .25 .30 .35 .40 .45 .50 .55 .65 .70 75 .90 .484 .500
Hydraulic mean depth 7 .0963 .1852 .269 .408 .466
in terms of radius . .00189 .0321 .0523 .io67 .1651 .1574 .294 .2142 .242 .293 .320 .343 .365 .387 11.075 .429 .449 1.371 1.470 1.571
waterway in terms off. ,, = .0598 .228 .370 .450 .614 .885 x.175 1.276
square of radius . y I .0211 .532 .709 .795 .979
§ III. EggShaped Channels or Sewers.—In sewers for discharging I could he found satisfying the foregoing conditions. To render storm water and house drainage the volume or flow is extremely ! the problem determinate, let it be remembered that, since for variable; and there is a great liability for deposits to be left when a given discharge Stoo /x, other things being the same, the the flow is small, which are not removed during the short periods I amount of excavation will be least for that channel which has when the flow is large. The sewer in consequence becomes choked. the least wetted perimeter. Let d be the depth and b the bottom width of the channel, and let the sides slope n horizontal to i vertical (fig. 114), then
S2 = (b+nd)d ;
x=b+2d l (n2+1).
Both S2 and x are to be minima. Differentiating, and equating to zero.
(db/dd+n)d+b+nd =o,
db/dd +2d (n2+1) =o; eliminating db/dd,
{n— b=2 {d (n }! ) n}d.—o,
%'t/x=
(b+nd)d/{b+2dV (n2+I)}. Inserting the value of b, m=S2/x={2dV (n2+I)—nd}/
{4dl/ (n2+1) 2nd} =Id.
That is, with given side slopes, the section is least for a given discharge when the hydraulic mean depth is half the actual depth.
A simple construction gives the form of the channel which fulfils
To obtain uniform scouring action, the velocity of flow should be this condition, for it can be shown that when m = Zd the sides constant or nearly so; a complete uniformity of velocity cannot be of the channel are tangential to a semicircle drawn on the obtained with any form of section suitable for sewers, but an ap water line.
proximation to uniform velocity is obtained by making the sewers since S2/x = Id,
of oval section. Various forms of oval have been suggested, the therefore S2=lxd. (i) simplest being one in r
which the radius of the f Let ABCD be the channel (fig. 115) ; from E the centre of AD drop crown is double the radius perpendiculars EF, EG, EH on the sides.
of the invert, and the ' Let
greatest width is two 1 AB=CD=a; BC=b; EF=EH=c; and EG=d.
thirds the height. The 12= area AEB+BEC+CED,
section of such a sewer I =acEZbd.
is shown in fig. 113, the 2
numbers marked on the X_— a+b.
 figure being proportional Putting these values in (1), numbers.
§ 112. Problems on Channels 'in which the Flow is Steady and at Uniform Velocity.—The general equations given in §§ 96, 98 are
.. lC__. r i=a(l+13/m); (I)
=Stv.' (3)
Problem I.—Given the transverse section of stream and dis
charge, to find the slope. From the dimensions of the section
find 12 and m; from (i) find from (3) find v, and lastly from (2)
find i. That is, EF, EG, EH are all equal, hence a semicircle struck Problem II.—Given the transverse section and slope, to find the ~ from E with radius equal eo the depth of the stream will pass
discharge. Find v from (2), then Q from (3). through F and H and be
Problem III.—Given the discharge and slope, and either the tangential to the sides of breadth, depth, or general form of the section of the channel, to f the channel.
determine its remaining dimensions. This must generally be solved To draw the channel,
by approximations. A breadth or depth or both are chosen, and describe a semicircle on the discharge calculated. If this is greater than the given discharge, a' horizontal line with
the dimensions are reduced and the discharge recalculated. radius =depth of channel.
Since m lies generally between the limits m =d and m= Id, where i The bottom will be a d is the depth of the stream, and since, moreover, the velocity horizontal tangent of that
varies as v (m) so that an error in the value of m leads only to a much I semicircle, and the sides tangents drawn at the required side less error in the value of the velocity calculated from it, we may
proceed thus. Assume a value for m, and calculate v from it. Let yr be this first approximation to v. Then is a first approximation to S2, say S2,. With this value of S2 design the section of the channel; calculate a second value for m; calculate from it a second value of v, and from that a second value for Q. Repeat the process till the successive values of m approximately coincide.
§ 113. Problem IV. Most
Economical Form of Channel
for given Side Slopes.—Sup
pose the channel is to be
trapezoidal in section (fig. 114), and that the sides are to have a
given slope. Let the longitudinal slope of the stream be given,
and also the mean velocity. An infinite number of channels
,,
. ; ,
k 28.0xrM 148'' 0'...  y:1['.280*. 12 •
9 ^ 
.
ac+2bd=(a+Zb)d; and hence c=d.
E

B G FIG. 115.
d
...b.... FIG. 114.
slopes.
The above result may be obtained thus (fig. 116) :
x=b+2d/sin 0. (I) S2=d(b+d cot 13) ;
S2/d=b+d cot /3; (2) 
12/d2=b/d+cot (3. (3)
From (1) and (2),
x=S2/d—d cot ii +2d/sin #.
This will be a minimum for
dx/dd =S2/d2+cot /3—2/sin 0=0, or 12/d2 = 2 cosec. — cot /3.
or d = {S2 sin /3/(z —cos S)}. From (3) and (4),
b/d=2(1 —cos /3) /sin 1=2 tan
(4)
velocity and slope are greatest. If in a stream of tolerably uniform slope an obstruction such as a weir is built, that will cause an alteration of flow similar to that of an alteration of the slope of the bed for a greater or less distance above the weir, and the originally uniform cross section of the stream will become a varied one. In such cases it is often of much practical importance to determine the longitudinal section of the stream.
The cases now considered will be those in which the changes of velocity and cross section are gradual and not abrupt, and in which the only internal work which needs to be taken into account is that due to the friction of the stream bed, as in cases of uniform motion. Further, the motion will be supposed to be steady, the mean velocity ateach given cross section remaining constant, though it varies from section to section along the course of the stream.
Let fig. 118 represent a longitudinal section of the stream, AoA1 being the water surface, oB1 the stream bed. Let AoBo, A1BI be
Proportions of Channels of Maximum Discharge for given Area and Side Slopes. Depth of channel =d; Hydraulic mean depth =1d ; Area of section =12.
Inclination Ratio of Top width =
of Sides to Side Area of Bottom twice len*th
Horizon. Slopes. Section H. Width. of each Side
Slope.
Semicircle . . . 1.57f d2 0 2d
Semihexagon . 6o° o' 3 :5 I•732d2 1.155d 2.31od
Semisquare 9o° o' o :I 2d2 2d 2d
75° 58' 1 :4 I.812d2 1 562d 2•o62d
63° 26' I :2 I•736d2 I.236d 2.236d
53° 8' 3 :4 I.750d2 d 2.500d
45° o' I : I.828d2 0•828d 2.828d
38° 40' I1 :1 I•952d2 0•702d 3.202d
33° 42' I1 : I 2•Io6d2 o•6o6d 3.6o6d
29° 44' II :1 2.282d2 0•532d 4.032d
26° 34' 2 : I 2.472d2 0'472d 4•472d
23° 58' 2; : I 2.674d2 0•424d 4.924d
21° 48' 21 :I 2.885d2 0•385d 5.385d
19° 58' :1 3.1o4d2 0'354d 5.854d
18° 26' 3 :1 3.325d2 0•325d 6.325d
Half the top width is the length of each side slope. The wetted perimeter is the sum of the top and bottom widths
§ 114. Form of Cross Section of Channel in which the Mean Velocity is Constant with Varying Discharge.—In designing waste channels from canals, and in some other cases, it is desirable that the mean velocity should be restricted within narrow limits with very different volumes of discharge. In channels of trapezoidal form the velocity increases and diminishes with the discharge. Hence when the discharge is large there is danger of erosion, and when it is small of silting or obstruction by weeds. A theoretical form of section for which the mean velocity would be constant can be found, and, although this is not very suitable for practical purposes, it can be more or less approximated to in actual channels.
Let fig. 117 represent the cross section of the channel. From the symmetry of the section, only half the channel need be considered.
o: x b '"
Let obac be any section suitable for the minimum flow, and let it be required to find the curve beg for the upper part of the channel so that the mean velocity shall be constant. Take o as origin of coordinates, and let de, fg be two levels of the water above ob.
Let ob=b(2; de=y, fg=y+dy, od=x, of =x+dx; eg=ds.
The condition to be satisfied is that
v=c ' (mi)
should be constant, whether the waterlevel is at ob, de, or fg. Consequently
m= constant =k
for all three sections, and can be found from the section obac. Hence also
Increment of section ydx =k. Increment of perimeter_ — ds
y2dx2 = k2ds2 = k2(dx +dye) and dx = kdy/ ' (y2k2). Integrating,
x = k log. (y+ (y2 —k2)}}constant; and, since y=b,/2 when x=o,
x=klog [[y+ (y2—k2)/1zb+\1 (4b2—k2) l].
Assuming values for y, the values of x can be found and the curve drawn.
The figure has been drawn for a channel the minimum section of which is a half hexagon of 4 ft. depth. Hence k=2; b=9.2; the rapid flattening of the side slopes is remarkable.
End of Article: FTC 

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