HYDRAULICS.] HYDROMECHANICS 509 M-Wf ; where E t is the modulus of elasticity for torsion, and I the polar moment of inertia of the section of the rod. If the rod is of circular section, I = ^Tr7 -4 . Let R be the radius of the disk, and b its lever age, or the distance of its centre from the axis of the torsion rod. The moment of the pressure of the water on the disk is where G is the heaviness of water and k an experimental coefficient. Then For any given instrument, i> = cVa; where c is a constant coefficient for the instrument. The instrument as constructed had three disks which could be used at will. Their radii and leverages were in feet R= 6 = 1st disk ............... 0-052 16 2d ............... 0-105 0-32 3d ............... 0-210 0-66 For a thin circular plate, the coefficient k "12. In the actual Instrument the torsion rod was a brass wire 06 inch diameter and 6 feet long. Supposing a measured in degrees, we get by calculation for the three disks. Very careful experiments were made with the instrument. It was fixed to a wooden turning bridge, revolving over a circular channel of 2 feet width, and about 76 feet circumferential length. An allowance was made for the slight current produced in the channel. These experiments gave for the coefficient c, in the formula v = cVo, 1st disk, c = 0-3126 for velocities of 3 to 16 feet 2d 0-1177 ,, li to 3i 3d ,, 0-0349,, ,, less than li or values little different from the values calculated from the torsion. The instrument is preferable, to the current meter in giving the velocity in terms of a single observed quantity, the angle of torsion, while the current meter involves the observation of two quantities, the number of rotations and the time. The current meter, except in some improved forms, must be withdrawn from the water to read the result of each experiment, and the law connecting the velocity and number of rotations of a current meter is less well-determined than that connecting the pressure on a disk and the torsion of the wire of a hydrodynamometer. At very low velocities the current meter fails altogether. The Pitot tube, like the hydrodynamometer, does not require a time observation. But, where the velocity is a varying one, and consequently the columns of water in the Fit ot tube are oscillating, there is room for doubt as to whether, at any given moment of clos ing the cock, the difference of level exactly measures the impulse of the stream at the moment. The Pitot tube also fails to give measurable indications of very low velocities. PROCESSES FOR GAUGING STREAMS. 133. Gauging by Observation of tJie Maximum Surface Velocity. The method of gauging which involves the least trouble is to deter mine the surface velocity at the thread of the stream, and to deduce from it the mean velocity of the whole cross section. The maximum surface velocity may be determined by floats or by a current meter. Unfortunately, however, the ratio of the maximum surface to the mean velocity is extremely variable. Thus putting r for the sur face velocity at the thread of the stream, and r m for the mean velocity of the whole cross section, has been found to have the following values : De Prony, experiments on small wooden channels, 8164 Experiments on the Seine, ... ... ... 62 Destrem and De Prony, experiments on the Neva, 078 Boileau, experiments on canals, ... ... ... 82 Baumgartner, experiments on the Garonne, ... 80 Briinings (mean), 85 Cunningham, Solani aqueduct, ... . ... 823 Various formulae, either empirical or based on some theory of the vertical and horizontal velocity curves, have been proposed for determining the ratio . Bazin found from his experiments the empirical expression where m is the hydraulic mean depth and ithe slope of the stream. In article 101, it has already been shown how from this formula the ratio can be obtained for different kinds of channels. v o In the case of irrigation canals and rivers, it is often important to determine the discharge either daily or at other intervals of time, while the depth and consequently the mean velocity is varying. Captain Cunningham, R, E. (RoorkceProf. Papers, vol. iv. p. 47), has shown that, for a given part of such a stream, where the bed is regular and of permanent section, a simple formula may be found for the variation of the central surface velocity with the depth. When once the constants of this formula have been determined by measuring the central surface velocity and depth, in different con ditions of the stream, the surface velocity can be obtained by simply observing the depth of the stream, and from this the mean velocity and discharge can be calculated. Let z be the depth of the stream, and r the surface velocity, both measured at the thread, of the stream. Then -; where c is a constant which for the Solani aqueduct had the values 1-9 to 2, the depths being 6 to 10 feet, and the velocities 3J to 4 feet. Without any assumption of a formula, however, the surface velocities, or still better the mean velocities, for different conditions of the stream may be plotted on a diagram in which the abscissae are depths and the ordinates velocities. The continuous curve through points so found would then always give the velocity for any observed depth of the stream, without the need of making any new float or current meter observations. 134. Mean Velocity determined l>y observing a Scries of Surface Velocities. The ratio of the mean velocity to the surface velocity in one longitudinal section is better ascertained than the ratio of the central surface velocity to the mean velocity of the whole cross section. Suppose the river divided into a number of compartments by equidistant longitudinal planes, and the surface velocity observed in each compartment. From this the mean velocity in each com partment and the discharge can be calculated. The sum of the partial discharges will be the total discharge of the stream. W T hen wires or ropes can be stretched across the stream, the compartments can be marked out by tags attached to them. Suppose two such ropes stretched across the stream, and floats dropped in above tho upper rope. By observing within which compartment the path of the float lies, and noting the time of transit between the ropes, the surface velocity in each compartment e;m be ascertained. The mean velocity in each compartment is 85 to 91 of the surface velocity in that compartment. Putting k for this ratio, and TJ, v. 2 . . . for the observed velocities, in compartments of area fi lt S2 2 . . . then the total discharge is Q = (Vi + i V ! i + . . . ) . If several floats are allowed to pass over each compartment, the mean of all those corresponding to one compartment is to be taken as the surface velocity of that compartment. This method is very applicable in the case of large streams or rivers too wide to stretch a rope across. The paths of the floats are then ascertained in this way. Let fig. 149 represent a portion of the river, which should be straight and free from obstruc tions. Suppose a base line AB measured parallel to the thread of the stream, and let the mean " cross section of the stream be ascertained either by sounding the terminal cross sections AE, BF, or by sounding a series of equidistant cross sections. The cross sections are taken at right angles to the base line. Obser vers are placed at A and B with theodolites or box sextants. The floats are dropped in from a boat above AE, and picked up by another boat below BF. An observer with a chronograph or -nj_ I i L L..____r watch notes the time in which each float passes from AE to BF. The method of proceeding is this. The observer A sets his theodolite in the direction AE, Fi g- 149 - and gives a signal to drop a float. B keeps his instrument on the float as it comes down. At the moment the float arrives at C in the line AE, the observer at A calls out. B clamps his instrument and reads off the angle ABC, and the time observer begins to note tho time of transit. B now points his instrument in the direction BF, and A keeps the float on the cross wire of his instrument. At the moment the float arrives at D in the line BF, the observer B calls out, A clamps his instrument and reads off the angle BAD, and the time observer notes the time of transit from C to IX Thus all c
1 7 1
1
i J
i... r . n i
1
