plotted, a curve is obtained called the horizontal velocity curve. In streams of symmetrical section this is a curve symmetrical about the centre line of the stream. The velocity varies little near the centre of the stream, but very rapidly near the banks. In unsymmetrical sections the greatest velocity is at the point where the stream is deepest, and the general form of the horizontal velocity curve is roughly similar to the section of the stream.
§ 102. Curves or Contours of Equal Velocity.—If velocities are observed at a number of points at different widths and depths in a stream, it is possible to draw curves on the cross section through points at which the velocity is the same. These represent contours of a solid, the volume of which is the discharge of the stream per second. Fig. 105 shows the vertical and horizontal velocity curves and the contours of equal velocity in a rectangular channel, from one of Bazin’s gaugings.
§ 103. Experimental Observations on the Vertical Velocity Curve.—A preliminary difficulty arises in observing the velocity at a given point in a stream because the velocity rapidly varies, the motion not being strictly steady. If an average of several velocities at the same point is taken, or the average velocity for a sensible period of time, this average is found to be constant. It may be inferred that though the velocity at a point fluctuates about a mean value, the fluctuations being due to eddying motions superposed on the general motion of the stream, yet these fluctuations produce effects which disappear in the mean of a series of observations and, in calculating the volume of flow, may be disregarded.
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| Fig. 105. |
In the next place it is found that in most of the best observations on the velocity in streams, the greatest velocity at any vertical is found not at the surface but at some distance below it. In various river gaugings the depth dz at the centre of the stream has been found to vary from 0 to 0.3d.
§ 104. Influence of the Wind.—In the experiments on the Mississippi the vertical velocity curve in calm weather was found to agree fairly with a parabola, the greatest velocity being at 310ths of the depth of the stream from the surface. With a wind blowing down stream the surface velocity is increased, and the axis of the parabola approaches the surface. On the contrary, with a wind blowing up stream the surface velocity is diminished, and the axis of the parabola is lowered, sometimes to half the depth of the stream. The American observers drew from their observations the conclusion that there was an energetic retarding action at the surface of a stream like that due to the bottom and sides. If there were such a retarding action the position of the filament of maximum velocity below the surface would be explained.
It is not difficult to understand that a wind acting on surface ripples or waves should accelerate or retard the surface motion of the stream, and the Mississippi results may be accepted so far as showing that the surface velocity of a stream is variable when the mean velocity of the stream is constant. Hence observations of surface velocity by floats or otherwise should only be made in very calm weather. But it is very difficult to suppose that, in still air, there is a resistance at the free surface of the stream at all analogous to that at the sides and bottom. Further, in very careful experiments, P. P. Boileau found the maximum velocity, though raised a little above its position for calm weather, still at a considerable distance below the surface, even when the wind was blowing down stream with a velocity greater than that of the stream, and when the action of the air must have been an accelerating and not a retarding action. A much more probable explanation of the diminution of the velocity at and near the free surface is that portions of water, with a diminished velocity from retardation by the sides or bottom, are thrown off in eddying masses and mingle with the rest of the stream. These eddying masses modify the velocity in all parts of the stream, but have their greatest influence at the free surface. Reaching the free surface they spread out and remain there, mingling with the water at that level and diminishing the velocity which would otherwise be found there.
Influence of the Wind on the Depth at which the Maximum Velocity is found.—In the gaugings of the Mississippi the vertical velocity curve was found to agree well with a parabola having a horizontal axis at some distance below the water surface, the ordinate of the parabola at the axis being the maximum velocity of the section. During the gaugings the force of the wind was registered on a scale ranging from 0 for a calm to 10 for a hurricane. Arranging the velocity curves in three sets—(1) with the wind blowing up stream, (2) with the wind blowing down stream, (3) calm or wind blowing across stream—it was found that an upstream wind lowered, and a down-stream wind raised, the axis of the parabolic velocity curve. In calm weather the axis was at 310ths of the total depth from the surface for all conditions of the stream.
Let h′ be the depth of the axis of the parabola, m the hydraulic mean depth, f the number expressing the force of the wind, which may range from +10 to −10, positive if the wind is up stream, negative if it is down stream. Then Humphreys and Abbot find their results agree with the expression
Fig. 106 shows the parabolic velocity curves according to the American observers for calm weather, and for an up- or down-stream wind of a force represented by 4.
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| Fig. 106. |
It is impossible at present to give a theoretical rule for the vertical velocity curve, but in very many gaugings it has been found that a parabola with horizontal axis fits the observed results fairly well. The mean velocity on any vertical in a stream varies from 0.85 to 0.92 of the surface velocity at that vertical, and on the average if v0 is the surface and vm the mean velocity at a vertical vm = 67v0, a result useful in float gauging. On any vertical there is a point at which the velocity is equal to the mean velocity, and if this point were known it would be useful in gauging. Humphreys and Abbot in the Mississippi found the mean velocity at 0.66 of the depth; G. H. L. Hagen and H. Heinemann at 0.56 to 0.58 of the depth. The mean of observations by various observers gave the mean velocity at from 0.587 to 0.62 of the depth, the average of all being almost exactly 0.6 of the depth. The mid-depth velocity is therefore nearly equal to, but a little greater than, the mean velocity on a vertical. If vmd is the mid-depth velocity, then on the average vm = 0.98vmd.
§ 105. Mean Velocity on a Vertical from Two Velocity Observations.—A. J. C. Cunningham, in gaugings on the Ganges canal, found the following useful results. Let v0 be the surface, vm the mean, and vxd the velocity at the depth xd; then
= 12 (v.211d+v.789d ).
§ 106. Ratio of Mean to Greatest Surface Velocity, for the whole Cross Section in Trapezoidal Channels.—It is often very important to be able to deduce the mean velocity, and thence the discharge, from observation of the greatest surface velocity. The simplest method of gauging small streams and channels is to observe the greatest surface velocity by floats, and thence to deduce the mean velocity. In general in streams of fairly regular section the mean velocity for the whole section varies from 0.7 to 0.85 of the greatest surface velocity. For channels not widely differing from those experimented on by Bazin, the expression obtained by him for the ratio of surface to mean velocity may be relied on as at least a good approximation to the truth. Let v0 be the greatest surface velocity, vm the mean velocity of the stream. Then, according to Bazin,
But
where c is a coefficient, the values of which have been already given in the table in § 98. Hence
