§ 136. _Surface Floats_ are convenient for determining the surface
velocities of a stream, though their use is difficult near the banks.
The floats may be small balls of wood, of wax or of hollow metal, so
loaded as to float nearly flush with the water surface. To render
them visible they may have a vertical painted stem. In experiments on
the Seine, cork balls 1(3/4) in. diameter were used, loaded to float
flush with the water, and provided with a stem. In A. J. C.
Cunningham's observations at Roorkee, the floats were thin circular
disks of English deal, 3 in. diameter and ¼ in. thick. For
observations near the banks, floats 1 in. diameter and 1/8 in. thick
were used. To render them visible a tuft of cotton wool was used
loosely fixed in a hole at the centre.
The velocity is obtained by allowing the float to be carried down, and
noting the time of passage over a measured length of the stream. If v
is the velocity of any float, t the time of passing over a length l,
then v = l/t. To mark out distinctly the length of stream over which
the floats pass, two ropes may be stretched across the stream at a
distance apart, which varies usually from 50 to 250 ft., according to
the size and rapidity of the river. In the Roorkee experiments a
length of run of 50 ft. was found best for the central two-fifths of
the width, and 25 ft. for the remainder, except very close to the
banks, where the run was made 12½ ft. only. The longer the run the
less is the proportionate error of the time observations, but on the
other hand the greater the deviation of the floats from a straight
course parallel to the axis of the stream. To mark the precise
position at which the floats cross the ropes, Cunningham used short
white rope pendants, hanging so as nearly to touch the surface of the
water. In this case the streams were 80 to 180 ft. in width. In wider
streams the use of ropes to mark the length of run is impossible, and
recourse must be had to box sextants or theodolites to mark the path
of the floats.
[Illustration: FIG. 136.]
Let AB (fig. 136) be a measured base line strictly parallel to the
thread of the stream, and AA1, BB1 lines at right angles to AB marked
out by ranging rods at A1 and B1. Suppose observers stationed at A and
B with sextants or theodolites, and let CD be the path of any float
down stream. As the float approaches AA1, the observer at B keeps it
on the cross wire of his instrument. The observer at A observes the
instant of the float reaching the line AA1, and signals to B who then
reads off the angle ABC. Similarly, as the float approaches BB1, the
observer at A keeps it in sight, and when signalled to by B reads the
angle BAD. The data so obtained are sufficient for plotting the path
of the float and determining the distances AC, BD.
The time taken by the float in passing over the measured distance may
be observed by a chronograph, started as the float passes the upper
rope or line, and stopped when it passes the lower. In Cunningham's
observations two chronometers were sometimes used, the time of passing
one end of the run being noted on one, and that of passing the other
end of the run being noted on the other. The chronometers were
compared immediately before the observations. In other cases a single
chronometer was used placed midway of the run. The moment of the
floats passing the ends of the run was signalled to a time-keeper at
the chronometer by shouting. It was found quite possible to count the
chronometer beats to the nearest half second, and in some cases to the
nearest quarter second.
[Illustration: FIG. 137.]
§ 137. _Sub-surface Floats._--The velocity at different depths below
the surface of a stream may be obtained by sub-surface floats, used
precisely in the same way as surface floats. The most usual
arrangement is to have a large float, of slightly greater density than
water, connected with a small and very light surface float. The motion
of the combined arrangement is not sensibly different from that of the
large float, and the small surface float enables an observer to note
the path and velocity of the sub-surface float. The instrument is,
however, not free from objection. If the large submerged float is made
of very nearly the same density as water, then it is liable to be
thrown upwards by very slight eddies in the water, and it does not
maintain its position at the depth at which it is intended to float.
On the other hand, if the large float is made sensibly heavier than
water, the indicating or surface float must be made rather large, and
then it to some extent influences the motion of the submerged float.
Fig. 137 shows one form of sub-surface float. It consists of a couple
of tin plates bent at a right angle and soldered together at the
angle. This is connected with a wooden ball at the surface by a very
thin wire or cord. As the tin alone makes a heavy submerged float, it
is better to attach to the tin float some pieces of wood to diminish
its weight in water. Fig. 138 shows the form of submerged float used
by Cunningham. It consists of a hollow metal ball connected to a
slice of cork, which serves as the surface float.
[Illustration: FIG. 138.]
[Illustration: FIG. 139.]
§ 138. _Twin Floats._--Suppose two equal and similar floats (fig. 139)
connected by a wire. Let one float be a little lighter and the other a
little heavier than water. Then the velocity of the combined floats
will be the mean of the surface velocity and the velocity at the depth
at which the heavier float swims, which is determined by the length of
the connecting wire. Thus if v_s is the surface velocity and v_d the
velocity at the depth to which the lower float is sunk, the velocity
of the combined floats will be
v = ½(v_s + v_d).
Consequently, if v is observed, and v_s determined by an experiment
with a single float,
v_d = 2v - v_s
According to Cunningham, the twin float gives better results than the
sub-surface float.
[Illustration: FIG. 140.]
§ 139. _Velocity Rods._--Another form of float is shown in fig. 140.
This consists of a cylindrical rod loaded at the lower end so as to
float nearly vertical in water. A wooden rod, with a metal cap at the
bottom in which shot can be placed, answers better than anything else,
and sometimes the wooden rod is made in lengths, which can be screwed
together so as to suit streams of different depths. A tuft of cotton
wool at the top serves to make the float more easily visible. Such a
rod, so adjusted in length that it sinks nearly to the bed of the
stream, gives directly the mean velocity of the whole vertical section
in which it floats.
§ 140. _Revy's Current Meter._--No instrument has been so much used in
directly determining the velocity of a stream at a given point as the
screw current meter. Of this there are a dozen varieties at least. As
an example of the instrument in its simplest form, Revy's meter may be
selected. This is an ordinary screw meter of a larger size than usual,
more carefully made, and with its details carefully studied (figs.
141, 142). It was designed after experience in gauging the great South
American rivers. The screw, which is actuated by the water, is 6 in.
in diameter, and is of the type of the Griffiths screw used in ships.
The hollow spherical boss serves to make the weight of the screw
sensibly equal to its displacement, so that friction is much reduced.
On the axis aa of the screw is a worm which drives the counter. This
consists of two worm wheels g and h fixed on a common axis. The worm
wheels are carried on a frame attached to the pin l. By means of a
string attached to l they can be pulled into gear with the worm, or
dropped out of gear and stopped at any instant. A nut m can be screwed
up, if necessary, to keep the counter permanently in gear. The worm is
two-threaded, and the worm wheel g has 200 teeth. Consequently it
makes one rotation for 100 rotations of the screw, and the number of
rotations up to 100 is marked by the passage of the graduations on its
edge in front of a fixed index. The second worm wheel has 196 teeth,
and its edge is divided into 49 divisions. Hence it falls behind the
first wheel one division for a complete rotation of the latter. The
number of hundreds of rotations of the screw are therefore shown by
the number of divisions on h passed over by an index fixed to g. One
difficulty in the use of the ordinary screw meter is that particles of
grit, getting into the working parts, very sensibly alter the
friction, and therefore the speed of the meter. Revy obviates this by
enclosing the counter in a brass box with a glass face. This box is
filled with pure water, which ensures a constant coefficient of
friction for the rubbing parts, and prevents any mud or grit finding
its way in. In order that the meter may place itself with the axis
parallel to the current, it is pivoted on a vertical axis and directed
by a large vane shown in fig. 142. To give the vane more
directing power the vertical axis is nearer the screw than in ordinary
meters, and the vane is larger. A second horizontal vane is attached
by the screws x, x, the object of which is to allow the meter to rest
on the ground without the motion of the screw being interfered with.
The string or wire for starting and stopping the meter is carried
through the centre of the vertical axis, so that the strain on it may
not tend to pull the meter oblique to the current. The pitch of the
screw is about 9 in. The screws at x serve for filling the meter with
water. The whole apparatus is fixed to a rod (fig. 142), of a length
proportionate to the depth, or for very great depths it is fixed to a
weighted bar lowered by ropes, a plan invented by Revy. The instrument
is generally used thus. The reading of the counter is noted, and it is
put out of gear. The meter is then lowered into the water to the
required position from a platform between two boats, or better from a
temporary bridge. Then the counter is put into gear for one, two or
five minutes. Lastly, the instrument is raised and the counter again
read. The velocity is deduced from the number of rotations in unit
time by the formulae given below. For surface velocities the counter
may be kept permanently in gear, the screw being started and stopped
by hand.
[Illustration: FIG. 141.]
[Illustration: FIG. 142.]
§ 141. _The Harlacher Current Meter._--In this the ordinary counting
apparatus is abandoned. A worm drives a worm wheel, which makes an
electrical contact once for each 100 rotations of the worm. This
contact gives a signal above water. With this arrangement, a series of
velocity observations can be made, without removing the instrument
from the water, and a number of practical difficulties attending the
accurate starting and stopping of the ordinary counter are entirely
got rid of. Fig. 143 shows the meter. The worm wheel z makes one
rotation for 100 of the screw. A pin moving the lever x makes the
electrical contact. The wires b, c are led through a gas pipe B; this
also serves to adjust the meter to any required position on the wooden
rod dd. The rudder or vane is shown at WH. The galvanic current acts
on the electromagnet m, which is fixed in a small metal box containing
also the battery. The magnet exposes and withdraws a coloured disk at
an opening in the cover of the box.
§ 142. _Amsler Laffon Current Meter._--A very convenient and accurate
current meter is constructed by Amsler Laffon of Schaffhausen. This
can be used on a rod, and put into and out of gear by a ratchet. The
peculiarity in this case is that there is a double ratchet, so that
one pull on the string puts the counter into gear and a second puts it
out of gear. The string may be slack during the action of the meter,
and there is less uncertainty than when the counter has to be held in
gear. For deep streams the meter A is suspended by a wire with a heavy
lenticular weight below (fig. 144). The wire is payed out from a small
winch D, with an index showing the depth of the meter, and passes over
a pulley B. The meter is in gimbals and is directed by a conical
rudder which keeps it facing the stream with its axis horizontal.
There is an electric circuit from a battery C through the meter, and a
contact is made closing the circuit every 100 revolutions. The moment
the circuit closes a bell rings. By a subsidiary arrangement, when the
foot of the instrument, 0.3 metres below the axis of the meter,
touches the ground the circuit is also closed and the bell rings. It
is easy to distinguish the continuous ring when the ground is reached
from the short ring when the counter signals. A convenient winch for
the wire is so graduated that if set when the axis of the meter is at
the water surface it indicates at any moment the depth of the meter
below the surface. Fig. 144 shows the meter as used on a boat. It is a
very convenient instrument for obtaining the velocity at different
depths and can also be used as a sounding instrument.
[Illustration: FIG. 143.]
§ 143. _Determination of the Coefficients of the Current
Meter._--Suppose a series of observations has been made by towing the
meter in still water at different speeds, and that it is required to
ascertain from these the constants of the meter. If v is the velocity
of the water and n the observed number of rotations per second, let
v = [alpha] + [beta]n (1)
where [alpha] and [beta] are constants. Now let the meter be towed
over a measured distance L, and let N be the revolutions of the meter
and t the time of transit. Then the speed of the meter relatively to
the water is L/t = v feet per second, and the number of revolutions
per second is N/t = n. Suppose m observations have been made in this
way, furnishing corresponding values of v and n, the speed in each
trial being as uniform as possible,
[Sigma]n = n1 + n2 + ...
[Sigma]v = v1 + v2 + ...
[Sigma]nv = n1v1 + n2v2 + ...
[Sigma]n² = n1² + n2² + ...
[[Sigma]n]² = [n1 + n2 + ...]²
Then for the determination of the constants [alpha] and [beta] in (1),
by the method of least squares--
[Sigma]n²[Sigma]v - [Sigma]n[Sigma]nv
[alpha] = -------------------------------------,
m[Sigma]n² - [[Sigma]n]²
m[Sigma]nv - [Sigma]v[Sigma]n
[beta] = -----------------------------.
m[Sigma]n² - [[Sigma]n]²
[Illustration: FIG. 144.]
In a few cases the constants for screw current meters have been
determined by towing them in R. E. Froude's experimental tank in which
the resistance of ship models is ascertained. In that case the data
are found with exceptional accuracy.
§ 144. Darcy Gauge or modified Pitot Tube.--A very old instrument for
measuring velocities, invented by Henri Pitot in 1730 (_Histoire de
l'Académie des Sciences_, 1732, p. 376), consisted simply of a
vertical glass tube with a right-angled bend, placed so that its mouth
was normal to the direction of flow (fig. 145).
[Illustration: FIG. 145.]
The impact of the stream on the mouth of the tube balances a column in
the tube, the height of which is approximately h = v²/2g, where v is
the velocity at the depth x. Placed with its mouth parallel to the
stream the water inside the tube is nearly at the same level as the
surface of the stream, and turned with the mouth down stream, the
fluid sinks a depth h´ = v²/2g nearly, though the tube in that case
interferes with the free flow of the liquid and somewhat modifies the
result. Pitot expanded the mouth of the tube so as to form a funnel or
bell mouth. In that case he found by experiment
h = 1.5v²/2g.
But there is more disturbance of the stream. Darcy preferred to make
the mouth of the tube very small to avoid interference with the
stream and to check oscillations of the water column. Let the
difference of level of a pair of tubes A and B (fig. 145) be taken to
be h = kv²/2g, then k may be taken to be a corrective coefficient
whose value in well-shaped instruments is very nearly unity. By
placing his instrument in front of a boat towed through water Darcy
found k = 1.034; by placing the instrument in a stream the velocity of
which had been ascertained by floats, he found k = 1.006; by readings
taken in different parts of the section of a canal in which a known
volume of water was flowing, he found k = 0.993. He believed the first
value to be too high in consequence of the disturbance caused by the
boat. The mean of the other two values is almost exactly unity
(_Recherches hydrauliques_, Darcy and Bazin, 1865, p. 63). W. B.
Gregory used somewhat differently formed Pitot tubes for which the k =
1 (_Am. Soc. Mech. Eng._, 1903, 25). T. E. Stanton used a Pitot tube
in determining the velocity of an air current, and for his instrument
he found k = 1.030 to k = 1.032 ("On the Resistance of Plane Surfaces
in a Current of Air," _Proc. Inst. Civ. Eng._, 1904, 156).
One objection to the Pitot tube in its original form was the great
difficulty and inconvenience of reading the height h in the immediate
neighbourhood of the stream surface. This is obviated in the Darcy
gauge, which can be removed from the stream to be read.
Fig. 146 shows a Darcy gauge. It consists of two Pitot tubes having
their mouths at right angles. In the instrument shown, the two tubes,
formed of copper in the lower part, are united into one for strength,
and the mouths of the tubes open vertically and horizontally. The
upper part of the tubes is of glass, and they are provided with a
brass scale and two verniers b, b. The whole instrument is supported
on a vertical rod or small pile AA, the fixing at B permitting the
instrument to be adjusted to any height on the rod, and at the same
time allowing free rotation, so that it can be held parallel to the
current. At c is a two-way cock, which can be opened or closed by
cords. If this is shut, the instrument can be lifted out of the stream
for reading. The glass tubes are connected at top by a brass fixing,
with a stop cock a, and a flexible tube and mouthpiece m. The use of
this is as follows. If the velocity is required at a point near the
surface of the stream, one at least of the water columns would be
below the level at which it could be read. It would be in the copper
part of the instrument. Suppose then a little air is sucked out by the
tube m, and the cock a closed, the two columns will be forced up an
amount corresponding to the difference between atmospheric pressure
and that in the tubes. But the difference of level will remain
unaltered.
When the velocities to be measured are not very small, this instrument
is an admirable one. It requires observation only of a single linear
quantity, and does not require any time observation. The law
connecting the velocity and the observed height is a rational one, and
it is not absolutely necessary to make any experiments on the
coefficient of the instrument. If we take v = k[root](2gh), then it
appears from Darcy's experiments that for a well-formed instrument k
does not sensibly differ from unity. It gives the velocity at a
definite point in the stream. The chief difficulty arises from the
fact that at any given point in a stream the velocity is not
absolutely constant, but varies a little from moment to moment. Darcy
in some of his experiments took several readings, and deduced the
velocity from the mean of the highest and lowest.
§ 145. _Perrodil Hydrodynamometer._--This consists of a frame abcd
(fig. 147) placed vertically in the stream, and of a height not less
than the stream's depth. The two vertical members of this frame are
connected by cross bars, and united above water by a circular bar,
situated in the vertical plane and carrying a horizontal graduated
circle ef. This whole system is movable round its axis, being
suspended on a pivot at g connected with the fixed support mn. Other
horizontal arms serve as guides. The central vertical rod gr forms a
torsion rod, being fixed at r to the frame abcd, and, passing freely
upwards through the guides, it carries a horizontal 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.
[Illustration: FIG. 146.]
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.
[Illustration: FIG. 147.]
Let r be the radius of the torsion rod, l its length from the needle
over ef to r, and [alpha] the observed torsion angle. Then the moment
of the couple due to the molecular forces in the torsion rod is