Page:EB1911 - Volume 14.djvu/82

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 = √ (2g / ζ) √ (mi) = c √ (mi),

Ganguillet and Kutter proceeded in a purely empirical way. They found that an expression of the form

c = α / (1 + β/√ m)

could be made to fit the experiments somewhat better than Darcy’s expression. Inverting this, we get

1/c = 1/α + β/α √ m,

an equation to a straight line having 1/√m for abscissa, 1/c for ordinate, and inclined to the axis of abscissae at an angle the tangent of which is β/α.

Plotting the experimental values of 1/c and 1/√ m, the points so found indicated a curved rather than a straight line, so that β must depend on α. After much comparison the following form was arrived at—

c = (A + l/n) / (1 + An / √ m),

where n is a coefficient depending only on the roughness of the sides of the channel, and A and l 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) / {1 + (a + p/i) n/√ m},

the form of the expression for c ultimately adopted by Ganguillet and Kutter.

For the constants a, l, p Ganguillet and Kutter obtain the values 23, 1 and 0.00155 for metrical measures, or 41.6, 1.811 and 0.00281 for English feet. The coefficient of roughness n is found to vary from 0.008 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.
Well-planed timber	0.009
Cement plaster	0.010
Plaster of cement with one-third sand	0.011
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 or weeds	0.025
Rivers and canals in moderately good order, not quite free from stones and weeds⁠${\big \}}$	0.030
Rivers and canals in bad order, with weeds and detritus	0.035
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 (Canal and Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form:—

M = n (41.6 + 0.00281/i);

v = (√ m/n) {(M + 1.811) / (M + √m)} √ (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
.00001	3.2260	3.8712	4.8390	5.4842	6.4520	8.0650	9.6780
.00002	1.8210	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	1.5033	1.7686	2.2107	2.6529
.00008	0.7672	0.9206	1.1508	1.3042	1.5344	1.9180	2.3016
.00010	0.6970	0.8364	1.0455	1.1849	1.3940	1.7425	2.0910
.00025	0.5284	0.6341	0.7926	0.8983	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
.00100	0.4441	0.5329	0.6661	0.7550	0.8882	1.1102	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 0.0163 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:—

I.	Canals in very firm gravel, in perfect order	n = 0.02
II.	Canals in earth, above the average in order	n = 0.0225
III.	Canals in earth, in fair order	n = 0.025
IV.	Canals in earth, below the average in order	n = 0.0275
V.	Canals in earth, in rather bad order, partially over- grown with weeds and obstructed by detritus.${\big \}}$	n = 0.03

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.

§ 100. Bazin’s New Formula.—Bazin subsequently re-examined 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 (Étude d’une nouvelle formule, Paris, 1898). He points out that Darcy’s original formula, which is of the form mi/v² = α + β/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 √ (mi/v²) = α + β/√ 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 ζv²/2g = mi, ζ = 0.002594 (1 + γ/√ m), where γ has the following values:—

I.	Very smooth sides, cement, planed plank, γ =	0.109
II.	Smooth sides, planks, brickwork	0.290
III.	Rubble masonry sides	0.833
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

§ 101. 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; v₀ is the surface; v_z the maximum and v_d 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

v_m = 1/3 [2v_z + v_d + (d_z/d) (v₀ − v_d)].

Fig. 104.

The Horizontal Velocity Curve.—Similarly if at each point along a horizontal representing the width of the stream the velocities are