SECTION VIII. FLOW OF WATER IN UNIFORM CHANNELS.-MEAN VELO- TRAIN. -- EFFECTS OF FRICTION.-FORMULE FOR CALCULAT- THREE USEFUL PROBLEMS. OF In rivers the velocity is a maximum along the central line of the surface, or, more correctly, over the deepest part of the channel; and it decreases thence to the sides and bottom: but when backwater arises from any obstruction, either a submerged weir, Fig. 22, or a contracted channel, Fig. 23, the velocity in the channel approaching the obstruction is a maximum at the depth of the backwater, below the surface, and it decreases thence to the surface, sides, and bottom. When water flows in a pipe of any length, the velocity at the centre is greatest, and it decreases thence to the sides or circumference of the pipe. If the pipe be supposed divided into two portions in the direction of its length, the lower portion or channel will be analogous to a small river or stream, in which the velocity is greatest at the central line of the surface, and the upper portion will be simply the lower reversed. pipe flowing full may, therefore, be looked upon as a double stream, and it will soon appear that the formulæ for the discharge from each kind are all but identical, A though a pipe may discharge full at all inclinations, while the inclinations in rivers or streams, having uniform motion, never exceed a few feet per mile. MEAN VELOCITY. It is found, by experiment, that the mean velocity is nearly independent of the depth or width of the channel, the central or maximum velocity being the same. From a number of experiments, Du Buât derived empirical formulæ equivalent to v = Vb + v 2 1 + V = v − v 1 + 1/2, No2 = (v k − 1)2, and v= (v1+1)2; in these equations v is the mean velocity, v the maximum surface velocity, and vь the velocity at the sides, or bottom, expressed in French inches. Tables calculated from these formulæ do not give correct results for measures in English inches, though they are those generally adopted. Disregarding the difference in the measures, which are as 1 to 10678, it will be found that, in the generality of channels, the mean velocity is not an arithmetical mean between the velocity at the central surface line and that at the bottom, though nearly so between the mean bottom and mean surface velocities. Dr. Young, modifying Du Buât's formula, assumes for English inches that v + vi =V, and hence v = v + 1 − (v + 4)3. This gives results very nearly the same as the other formula for v, but something less, particularly for small surface velocities. For instance, Du Buât's formula gives 5 inch for the mean velocity when the central surface velocity is 1 Philosophical Transactions, 1808, p. 487. inch, whereas Dr. Young's makes it 38 inch. For large velocities both formulæ agree very closely, disregarding the difference between the measures, which is only seven per cent. They are best suited to very small channels or pipes, but unless at mean velocities of about 3 feet per second, they are wholly inapplicable to rivers. Prony found, from Du Buât's experiments, that for (2.37187 + v 3.15312 + v measures in metres v = v, in which v is also the maximum surface velocity. This, reduced for measures in English feet, becomes For medium velocities v = '81 v. The experiments from which these formulæ were derived were made with small channels. The author has calculated the values of from that of v, equation (71A), and given the results in columns 3, 6, and 9 in TABLE VII. Ximenes, Funk, and Brünning's experiments in larger channels. give the mean velocity at the centre of the depth equal 914 v, when the central or maximum surface velocity * Francis, Lowell Experiments, p. 150, finds this formula to give 15 per cent. less than the result found by weir measurement from the formula D = 3·33 (1 — •1 n h) h, the quantity discharged being about 250 cubic feet per second, and the velocity about 3-2 feet. It appears, however, that Francis uses the mean surface velocity, and not the maximum surface velocity required by the formula: if the latter were used, the difference would be reduced to 6 per cent., or thereabouts, in equation (72). is v; but as the velocity also decreases in nearly the same ratio at the surface from the centre to the sides of the channel, we shall get the mean velocity in the whole section equal 914 x 914 v = 835 v; and hence, for large channels, in which equation v is the maximum velocity at the surface. The author has also calculated the values of v from this formula, and given the results in columns 2, 5, and 8 of TABLE VII. This table will be found to vary considerably from those calculated from Du Buât's formula in French inches, hitherto generally used in this country, and much more applicable for all practical purposes. MEAN RADIUS.-HYDRAULIC MEAN DEPTH.-BORDER.COEFFICIENT OF FRICTION. If, in the diagrams 1 and 2, Fig. 30, exhibiting the sections of cylindrical and rectangular tubes filled with flowing water, the areas be divided respectively by the perimeters A CBDA and A B D C A, the quotients are termed "the mean radii" of the tubes, diagrams 1 and 2; and the wetted perimeters in contact with the flowing water are termed "the borders." In the diagrams 3 and 4, the surface A B is not in contact with the channel, and the width of the bed and sides, taken together, A CD B, becomes "the border." "The mean radius" is equal to the area A B D C A divided by the length of the border a C D B. "The hydraulic mean depth" is the same as "the mean radius," this latter term being perhaps most applicable to pipes flowing full, as in diagrams 1 and 2; and the former to streams and rivers which have a surface line A B, diagrams 3 and 4. Throughout the following equations, the value of the mean radius," "hydraulic mean depth," or quotient, area A B D C A * is designated by the letter r, remarking border B D CA here that for cylindrical pipes flowing full, or rivers with semicircular beds, it is always equal to half the radius, or one-fourth of the diameter. Du Buât was the first to observe that the head due to the resistance of friction for water flowing in a uniform channel increased directly as the length of the channel 1, directly as the border, and inversely† as the M. Girard has conceived it necessary to introduce the coefficient of correction 17 as a multiplier to the border for finding r, to allow for the increased resistance from aquatic plants; so that, according to his reduction, See Rennie's First Report on Hydraulics as a branch of Engineering; Third Report of the British Association, p. 167; also, equation (85), p. 216. The Author has known cases in very irregular channels in which for this sort of correction area r = 4 border' In other words, where the velocity found from the common formula, from the fall per mile, required to be reduced one-half to find the actual mean velocity. Pitot had previously, in 1726, remarked that the diminution arising from friction in pipes is, cæteris paribus, inversely as the diameters. |