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 |
Various formulae, either empirical or based on some theory of the vertical and horizontal velocity curves, have been proposed for determining the ratio vm/v0. Bazin found from his experiments the empirical expression
vm = v0 − 25.4 √ (mi);
where m is the hydraulic mean depth and i the slope of the stream.
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. Cunningham (Roorkee Prof. Papers, iv. 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 conditions 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 v0 the surface velocity, both measured at the thread of the stream. Then v02 = cz; where c is a constant which for the Solani aqueduct had the values 1.9 to 2, the depths being 6 to 10 ft., and the velocities 31⁄2 to 41⁄2 ft. 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.
§ 147. Mean Velocity determined by observing a Series 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 compartment and the discharge can be calculated. The sum of the partial discharges will be the total discharge of the stream. When 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 the 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 can be ascertained. The mean velocity in each compartment is 0.85 to 0.91 of the surface velocity in that compartment. Putting k for this ratio, and v1, v2 ... for the observed velocities, in compartments of area Ω1, Ω2 ... then the total discharge is
Q = k (Ω1v1 + Ω2v2 + ... ).