§ 150. Direct Determination of the Mean Velocity by a Current Meter or Darcy Gauge.—The only method of determining the mean velocity at a cross section of a stream which involves no assumption of the ratio of the mean velocity to other quantities is this—a plank bridge is fixed across the stream near its surface. From this, velocities are observed at a sufficient number of points in the cross section of the stream, evenly distributed over its area. The mean of these is the true mean velocity of the stream. In Darcy and Bazin’s experiments on small streams, the velocity was thus observed at 36 points in the cross section.
When the stream is too large to fix a bridge across it, the observations may be taken from a boat, or from a couple of boats with a gangway between them, anchored successively at a series of points across the width of the stream. The position of the boat for each series of observations is fixed by angular observations to a base line on shore.
| Fig. 149. |
| Fig. 150. |
§ 151. A. R. Harlacher’s Graphic Method of determining the Discharge from a Series of Current Meter Observations.—Let ABC (fig. 149) be the cross section of a river at which a complete series of current meter observations have been taken. Let I., II., III., ... be the verticals at different points of which the velocities were measured. Suppose the depths at I., II., III., ... (fig. 149), set off as vertical ordinates in fig. 150, and on these vertical ordinates suppose the velocities set off horizontally at their proper depths. Thus, if v is the measured velocity at the depth h from the surface in fig. 149, on vertical marked III., then at III. in fig. 150 take cd = h and ac = v. Then d is a point in the vertical velocity curve for the vertical III., and, all the velocities for that ordinate being similarly set off, the curve can be drawn. Suppose all the vertical velocity curves I.... V. (fig. 150), thus drawn. On each of these figures draw verticals corresponding to velocities of x, 2x, 3x ... ft. per second. Then for instance cd at III. (fig. 150) is the depth at which a velocity of 2x ft. per second existed on the vertical III. in fig. 149 and if cd is set off at III. in fig. 149 it gives a point in a curve passing through points of the section where the velocity was 2x ft. per second. Set off on each of the verticals in fig. 149 all the depths thus found in the corresponding diagram in fig. 150. Curves drawn through the corresponding points on the verticals are curves of equal velocity.
The discharge of the stream per second may be regarded as a solid having the cross section of the river (fig. 149) as a base, and cross sections normal to the plane of fig. 149 given by the diagrams in fig. 150. The curves of equal velocity may therefore be considered as contour lines of the solid whose volume is the discharge of the stream per second. Let Ω0 be the area of the cross section of the river, Ω1, Ω2 ... the areas contained by the successive curves of equal velocity, or, if these cut the surface of the stream, by the curves and that surface. Let x be the difference of velocity for which the successive curves are drawn, assumed above for simplicity at 1 ft. per second. Then the volume of the successive layers of the solid body whose volume represents the discharge, limited by successive planes passing through the contour curves, will be
1⁄2 x (Ω0 + Ω1), 1⁄2 x (Ω1 + Ω2), and so on.
Consequently the discharge is
Q = x {1⁄2 (Ω0 + Ωn) + Ω1 = Ω2 + ... + Ωn−1}.
The areas Ω0, Ω1 ... are easily ascertained by means of the polar planimeter. A slight difficulty arises in the part of the solid lying above the last contour curve. This will have generally a height which is not exactly x, and a form more rounded than the other layers and less like a conical frustum. The volume of this may be estimated separately, and taken to be the area of its base (the area Ωn) multiplied by 1⁄3 to 1⁄2 its height.