§ 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 v_o for the surface velocity at the
thread of the stream, and v_m for the mean velocity of the whole cross
section, v_m/v_o has been found to have the following values:--
v_m/v_o
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 v_m/v_o. Bazin found from his experiments the
empirical expression
v_m = v_o - 25.4 [root](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 v_o the surface velocity, both measured
at the thread of the stream. Then v_o² = 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 3½ to 4½ 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 [Omega]1, [Omega]2 ... then the total discharge
is
Q = k([Omega]1v1 + [Omega]2v2 + ... ).
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.
[Illustration: FIG. 148.]
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. 148 represent a portion of the
river, which should be straight and free from obstructions. 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. Observers 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 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, 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 the 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 D. Thus all the data are determined for plotting the path CD
of the float and determining its velocity. By dropping in a series of
floats, a number of surface velocities can be determined. When all
these have been plotted, the river can be divided into convenient
compartments. The observations belonging to each compartment are then
averaged, and the mean velocity and discharge calculated. It is
obvious that, as the surface velocity is greatly altered by wind,
experiments of this kind should be made in very calm weather.
The ratio of the surface velocity to the mean velocity in the same
vertical can be ascertained from the formulae for the vertical
velocity curve already given (§ 101). Exner, in _Erbkam's Zeitschrift_
for 1875, gave the following convenient formula. Let v be the mean and
V the surface velocity in any given vertical longitudinal section, the
depth of which is h
v/V = (1 + 0.1478 [root]h)/(1 + 0.2216 [root]h).
If vertical velocity rods are used instead of common floats, the mean
velocity is directly determined for the vertical section in which the
rod floats. No formula of reduction is then necessary. The observed
velocity has simply to be multiplied by the area of the compartment to
which it belongs.
§ 148. _Mean Velocity of the Stream from a Series of Mid Depth
Velocities._--In the gaugings of the Mississippi it was found that the
mid depth velocity differed by only a very small quantity from the
mean velocity in the vertical section, and it was uninfluenced by
wind. If therefore a series of mid depth velocities are determined by
double floats or by a current meter, they may be taken to be the mean
velocities of the compartments in which they occur, and no formula of
reduction is necessary. If floats are used, the method is precisely
the same as that described in the last paragraph for surface floats.
The paths of the double floats are observed and plotted, and the mean
taken of those corresponding to each of the compartments into which
the river is divided. The discharge is the sum of the products of the
observed mean mid depth velocities and the areas of the compartments.
§ 149. _P. P. Boileau's Process for Gauging Streams._--Let U be the
mean velocity at a given section of a stream, V the maximum velocity,
or that of the principal filament, which is generally a little below
the surface, W and w the greatest and least velocities at the surface.
The distance of the principal filament from the surface is generally
less than one-fourth of the depth of the stream; W is a little less
than V; and U lies between W and w. As the surface velocities change
continuously from the centre towards the sides there are at the
surface two filaments having a velocity equal to U. The determination
of the position of these filaments, which Boileau terms the gauging
filaments, cannot be effected entirely by theory. But, for sections of
a stream in which there are no abrupt changes of depth, their position
can be very approximately assigned. Let [Delta] and l be the
horizontal distances of the surface filament, having the velocity W,
from the gauging filament, which has the velocity U, and from the bank
on one side. Then
[Delta]/l = c^4 [root]{(W + 2w)/7(W - w)},
c being a numerical constant. From gaugings by Humphreys and Abbot,
Bazin and Baumgarten, the values c = 0.919, 0.922 and 0.925 are
obtained. Boileau adopts as a mean value 0.922. Hence, if W and w are
determined by float gauging or otherwise, [Delta] can be found, and
then a single velocity observation at [Delta] ft. from the filament of
maximum velocity gives, without need of any reduction, the mean
velocity of the stream. More conveniently W, w, and U can be measured
from a horizontal surface velocity curve, obtained from a series of
float observations.
§ 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.
[Illustration: FIG. 149.]
§ 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.
[Illustration: FIG. 150.]
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 [Omega]0 be the area of the cross section of the
river, [Omega]1, [Omega]2 ... 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
½x([Omega]0 + [Omega]1), ½x([Omega]1 + [Omega]2), and so on.