§ 121. The formation of a standing wave was first observed by Bidone. Into a small rectangular masonry channel, having a slope of 0.023 ft. per foot, he admitted water till it flowed uniformly with a depth of 0.2 ft. He then placed a plank across the stream which raised the level just above the obstruction to 0.95 ft. He found that the stream above the obstruction was sensibly unaffected up to a point 15 ft. from it. At that point the depth suddenly increased from o•2 ft. to o•56 ft. The velocity of the stream in the part unaffected by the obstruction was 5.54 ft. per second. Above the point where the abrupt change of depth occurred 142=5.542=30.7, and gh =32.2 X0.2 =6.44; hence u2 was>gh. Just below the abrupt change of depth u=5.54X0.2/0.56=1.97; u2=3.88; and gh= 32'2 X0-56=-18.03; hence at this point u2 I, mouth down stream, the
ii fluid sinks a depth
Ijl~ y i ! h' =v2/2g nearly, though
the tube in that case
interferes with the free
A B C flow of the liquid and
result. Pitot expanded
the mouth of the tube so as to form a funnel or bell mouth. In that case he found by experiment
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•o34 ; by placing the instrument in a stream the velocity of which had been ascertained by floats, he found k= I •oo6 ; 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 con-sequence 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 = I (Am. Soc. Mech. Eng., 1903, 25). T. E. Stanton used a Pitot tube in deter-mining the velocity of an air current, and for his instrument he found k=1•o3o to k=1.x32 (" 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 It 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 horizon-tally. 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 sup-ported 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 re-
quired 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=kJ (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 abed (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 abed, 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.
Suppose the apparatus arranged so that the disk x is at that level 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.
Let r be the radius of the torsion rod, 1 its length from the needle over ef to r, and a the observed torsion angle. Then the moment of the couple due to the molecular forces in the torsion rod is
where Et is the modulus of elasticity for torsion, and I the polar moment of inertia of the section of the rod. If the rod is of circular section, I =11-r4. Let R be the radius of the disk, and b its leverage, or the distance of its centre from the axis of the torsion rod. The moment of the pressure of the water on the disk is
Fb = kb(G/2g)irR2v2,
where G is the heaviness of water and k an experimental coefficient. Then
Et I a/l = kb (G/2g)1R2v2.
For any given instrument,
where c is a constant coefficient for the instrument.
The instrument as constructed had three disks which could be used at will. Their radii and leverages were in feet
1st disk . 0.052 0.16
2nd „ . 0.105 0.32
3rd „ . . . 0.210 o•66
For a thin circular plate, the coefficient k =1.12. In the actual instrument the torsion rod was a brass wire 0.06 in. diameter and 61 ft. long. Supposing a measured in degrees, we get by calculation
V=0.33531 a; o.115-V a; 0.042s/ a.
Very careful experiments were made with the instrument. It was fixed to a wooden turning bridge. revolving over a circular channel of 2 ft. width, and about z6 ft. 'circumferential length. An allowance was made for the slight current produced in the channel. These experiments gave for the coefficient c, in the formula v =c' a,
1st disk, c=0.3126 for velocities of 3 to 16 ft.
2nd „ 0.1177 „ „ I; to 31
3rd 0.0349 , less than 11 „
The instrument is preferable to the current meter in giving the velocity in terms of a single observed quantity, the angle of torsion, while the current meter involves the observation of two quantities, the number of rotations and the time. The current meter, except in some improved forms, must be withdrawn from the water to read the result of each experiment, and the law connecting the velocity and number of rotations of a current meter is less well-determined than that connecting the pressure on a disk and the torsion of the wire of a hydrodynamometer.
The Pitot tube, like the hydrodynamometer, does not require a time observation. But, where the velocity is a varying one, and consequently the columns of water in the Pitot tube are oscillating, there is room for doubt as to whether, at any given moment of closing the cock, the difference of level exactly measures the impulse of the stream at the moment. The Pitot tube also fails to give measurable indications of very low velocities.
End of Article: STANDING