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69
HYDRAULICS


to the direction of motion is treated as sliding down the channel to a'a'b'b' without deformation. The component of the weight parallel to the channel bed balances the friction against the channel, and in estimating the friction the velocity of rubbing is taken to be the mean velocity of the stream. In actual streams, however, the velocity of rubbing on which the friction depends is not the mean AULICS 69

variation of the coefficient of friction with the velocity, proposed an expression of the form

§ =<1(l +3/U), (5)

and from 255 experiments obtained for the constants the values 0. =0-007409; li =0- 1920.

This gives the following values at different velocities 2v- 0-3 0-5 0-7 1 15 2 3 5 7 10 I5

g'= 0~01215 0-01025 0-00944 0-00883 0-00836 O°OO8I2 0-90788 0-00769 0'OO76I 0-00755 0-00750 l

velocity of the stream, and is not in any simple relation with it, for ' In using this value of § ' when 11 is not known, it is best to proceed channels of different forms. The by approximation. Q 6 theory is therefore obviously based § 98. Darcy and Bazirfs Expression for the Coejicient of Friction.-;, , I on an imperfect hypothesis. How- Darcy and BRZIHYS researches have sh0w11 that I varies very greatly Q 5 ever, by taking variable values for for different degrees of roughness of the channel bed, and that it WW qzj- /5 5 ' the coefficient of friction, the errors also varies with the dimensions of the channel. They give for I an %@ 0/, E of the ordinary formulae are to a empirical expression (similar to that for pipes) of the form %/y M' great extent neutralized, and they 5-:au +5/my (6) are/4...

may be used without leading to

practical errors. Formulae have

been obtained based on less represent 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 ao/, 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 S2 be th f

e area 0 the cross sections, X the wetted perimeter, Fic.. 102.

stricted hypotheses but at

g ... .... ........ .. . . .-.v

f =1

where rn is the hydraulic mean depth. For different kinds of channels they give the following values of the coefficient of friction 1-7 I Kind of Channel. | 0., s

cement or planed timber ...

Smooth channels, sides of ashlar, brickwork, planks .......

Rough channels, sides of rubble masonry or pitched with stone

IV. Very rough canals in earth ..... V. Torrential streams encumbered with detritus 0-00294 0-10

II.

III.

0-00373 0-23

I. Very smooth channels, sides of smooth


FIG. 103.

pq-l-qr-l-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 GS2:ll, and the component of the weight in the direction of motion is GSZdl>< the cosine of the angle between Wg and ab, that is, Gfldl cos abc=G§ ?dl bc/ab-Gftidl. (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(1'): f(v) being the friction per sq. ft. at a velocity 11. Hence the friction is -xdl f('v). Equating the sum of the forces to zero, GQi dl-Xdl f(v) =0,

f(°u)/G=§ Zi/X=mi. (1)

But it has been already shown (§ 66) thatf(v) = § 'G1J2/2g, (112/2g =mi. (2)

This may be put in the form

v = / (28/DV (mi) =v~/ ( H); (211)

where c is a coefficient depending on the roughness and form of the channel.

The coefficient of friction f 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 elirnin. t th f l ' 1 ' ' 1 " 1 e e error 0 neg ecting the xarlations of xelocity in the cross section. A c0mm0n mean value assumed for I 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. Iff is the fall in feet per mile, f= 5280 i.

Putting this and the above value of of in (20), we get the very simple and long-k110wn approximate formula for the mean velocity of a stream-

”=1l%/(2H1f)- (3)

The How down the stream per second, or discharge of the stream, is Q=§ Z'v=9cx/ (mi). (4)

§ 97. Coejicient of Friction for Oben Cliannels.-Various expressions have beeu proposed for the coemcient of friction ior| channels as for pipes. Weisbach, giving attention chiefly to the The last yalues (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, 1; =c/ (mi),

where c=/ (2g/5), 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:- Valuex ofc in v = cy/ (mi), cleflucedfrovn Darcy and Bazin's Values.

Q.; gd; om 52 -57.5 Tggn =§ - E um 'Sf .555 'gg e * E “ -se -2' -.1 =-'= e ., J*, -5 “ ~ 2 m gd 2°-3 E2 1153 'go;=>§ 3; E15 § '§ $5 = "5 £5 -iii 80° 6 '5 as £5 eil 55 § “' < §

-25 125 95 57 26 18-5 8-5 147 130 112 89 . -55 135 112 Q2 36 25-2 9-3 147 130 112 90 71 -7 139 Il 1 42 30- 9- 147 I3O II2 90 . 1-1; 141 119 87 34-9 10-0 147 130 112 QI 72 1- 143 122 94 41-2 II 147 I3O 113 92 . 2-0 144 124 98 62 46°C 12 147 130 II3 93 74 2-5 145 122 101 37 13 147 130 II3 94 . 3-0 145 12 104 0 3 14 147 130 II3 95 .. 3-5 146 127 105 73 . 15 147 130 114 96 77 4-0 146 128 IO6 76 58 16 147 130 II4 97 .. 4-5 146 128 107 78 . 17 147 130 114 97 .. 5-0 146 128 IO8 80 62 18 147 130 II4 98 .. 5-5 146 129 109 82 20 147 131 II4 98 80 6-0 147 IZQ 110 84 65 25 148 ISI 115 100 . 6-5 147 129 110 85 . 30 148 I3I IIS 102 83

7-0 147 I2Q 110 86 67 40 148 I3I 116 103 85

7-5 147 129 III 87 50 148 131 II6 104 86 L8-0 147 I3O III 88 69 oo 5348 131 ' 117 ' 108 591 § 99. Gangnillet and Kutter's Modi ed Darcy' Formula.-Startin 3

§ from the general expression z1=c/ rni, 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 were confined to channels of moderate section, and to a limited variation of slope. Ganguillet and Kutter brought into the discussion two very distinct and important additional series of results. The gaugings of the Mississippi by A. A. Humphreys and H. L. Abbot afford data of discharge for the case of a stream of exceptionally large section and OI very row slope. Un the otner hand their

li own measurements of the flow in the regulated channels of some