needle moving over the graduated circle ef. The support g, which carries the apparatus, also receives in a tubular guide the end of the torsion rod gr and a set screw for fixing the upper end of the torsion rod when necessary. The impulse of the stream of water is received on a circular disk x, in the plane of the torsion rod and the frame abcd. To raise and lower the apparatus easily, it is not fixed directly to the rod mn, but to a tube kl sliding on mn.
Fig. 146.
 |
| Fig. 147. |
Suppose the apparatus arranged so that the disk x is at that level in the stream where the velocity is to be determined. The plane abcd is placed parallel to the direction of motion of the water. Then the disk x (acting as a rudder) will place itself parallel to the stream on the down stream side of the frame. The torsion rod will be unstrained, and the needle will be at zero on the graduated circle. If, then, the instrument is turned by pressing the needle, till the plane abcd of the disk and the zero of the graduated circle is at right angles to the stream, the torsion rod will be twisted through an angle which measures the normal impulse of the stream on the disk x. That angle will be given by the distance of the needle from zero. Observation shows that the velocity of the water at a given point is not constant. It varies between limits more or less wide. When the apparatus is nearly in its right position, the set screw at g is made to clamp the torsion spring. Then the needle is fixed, and the apparatus carrying the graduated circle oscillates. It is not, then, difficult to note the mean angle marked by the needle.
Let r be the radius of the torsion rod, l its length from the needle over ef to r, and α the observed torsion angle. Then the moment of the couple due to the molecular forces in the torsion rod is
where Et 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 = 12πr4. Let R be the radius of the disk, and b its leverage, 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,
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 = | b = | |
| 1st disk | 0.052 | 0.16 |
| 2nd ’’ | 0.105 | 0.32 |
| 3rd ’’ | 0.210 | 0.66 |
For a thin circular plate, the coefficient k = 1.12. In the actual instrument the torsion rod was a brass wire 0.06 in. diameter and 612 ft. long. Supposing α measured in degrees, we get by calculation
Very careful experiments were made with the instrument. It was fixed to a wooden turning bridge, revolving over a circular channel of 2 ft. width, and about 76 ft. 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 = c√α,
| 1st disk, | c = 0.3126 for velocities of 3 to 16 ft. |
| 2nd disk, | c = 0.1177 for velocities of 114 to 314 ft. |
| 3rd disk, | c = 0.0349 for velocities of less than 114 ft. |
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.
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 Pitot tube are oscillating, there is room for doubt as to whether, at any given moment of closing 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
§ 146. Gauging by Observation of the Maximum Surface Velocity.—The method of gauging which involves the least trouble is to determine 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 the ratio of the maximum surface to the mean velocity is extremely variable. Thus putting v0 for the surface velocity at the thread of the stream, and vm for the mean velocity of the whole cross section, vm/v0 has been found to have the following values:—
| vm/v0 | |
| De Prony, experiments on small wooden channels | 0.8164 |
| Experiments on the Seine | 0.62 |
| Destrem and De Prony, experiments on the Neva | 0.78 |
| Boileau, experiments on canals | 0.82 |
| Baumgartner, experiments on the Garonne | 0.80 |
| Brünings (mean) | 0.85 |
| Cunningham, Solani aqueduct | 0.823 |
