The streams of fluid deviated in front of the plate, supposed for definiteness to be moving through the fluid, receive from it forward momentum. Portions of this forward moving water are thrown off laterally at the edges of the plate, and diffused through the surrounding fluid, instead of falling to their original position behind the plate. Other portions of comparatively still water are dragged into motion to fill the space left behind the plate; and there is thus a pressure less than hydrostatic pressure at the back of the plate. The whole resistance to the motion of the plate is the sum of the excess of pressure in front and deficiency of pressure behind. This resistance is independent of any friction or viscosity in the fluid, and is due simply to its inertia resisting a sudden change of direction at the edge of the plate.
Experiments made by a whirling machine, in which the plate is fixed on a long arm and moved circularly, gave the following values of the coefficient f. The method is not free from objection, as the centrifugal force causes a flow outwards across the plate.
| Approximate Area of Plate in sq. ft. | Values of f. | ||
| Borda. | Hutton. | Thibault. | |
| 0.13 | 1.39 | 1.24 | .. |
| 0.25 | 1.49 | 1.43 | 1.525 |
| 0.63 | 1.64 | .. | .. |
| 1.11 | .. | .. | 1.784 |
There is a steady increase of resistance with the size of the plate, in part or wholly due to centrifugal action.
P. L. G. Dubuat (1734–1809) made experiments on a plane 1 ft. square, moved in a straight line in water at 3 to 612 ft. per second. Calling m the coefficient of excess of pressure in front, and n the coefficient of deficiency of pressure behind, so that f = m + n, he found the following values:—
The pressures were measured by pressure columns. Experiments by A. J. Morin (1795–1880), G. Piobert (1793–1871) and I. Didion (1798–1878) on plates of 0.3 to 2.7 sq. ft. area, drawn vertically through water, gave f = 2.18; but the experiments were made in a reservoir of comparatively small depth. For similar plates moved through air they found f = 1.36, a result more in accordance with those which precede.
For a fixed plane in a moving current of water E. Mariotte found f = 1.25. Dubuat, in experiments in a current of water like those mentioned above, obtained the values m = 1.186; n = 0.670; f = 1.856. Thibault exposed to wind pressure planes of 1.17 and 2.5 sq. ft. area, and found f to vary from 1.568 to 2.125, the mean value being f = 1.834, a result agreeing well with Dubuat.
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| Fig. 169. |
§ 167. Stanton’s Experiments on the Pressure of Air on Surfaces.—At the National Physical Laboratory, London, T. E. Stanton carried out a series of experiments on the distribution of pressure on surfaces in a current of air passing through an air trunk. These were on a small scale but with exceptionally accurate means of measurement. These experiments differ from those already given in that the plane is small relatively to the cross section of the current (Proc. Inst. Civ. Eng. clvi., 1904). Fig. 169 shows the distribution of pressure on a square plate. ab is the plate in vertical section. acb the distribution of pressure on the windward and adb that on the leeward side of the central section. Similarly aeb is the distribution of pressure on the windward and afb on the leeward side of a diagonal section. The intensity of pressure at the centre of the plate on the windward side was in all cases p = Gv 2/2g ℔ per sq. ft., where G is the weight of a cubic foot of air and v the velocity of the current in ft. per sec. On the leeward side the negative pressure is uniform except near the edges, and its value depends on the form of the plate. For a circular plate the pressure on the leeward side was 0.48 Gv 2/2g and for a rectangular plate 0.66 Gv 2/2g. For circular or square plates the resultant pressure on the plate was P = 0.00126 v 2 ℔ per sq. ft. where v is the velocity of the current in ft. per sec. On a long narrow rectangular plate the resultant pressure was nearly 60% greater than on a circular plate. In later tests on larger planes in free air, Stanton found resistances 18% greater than those observed with small planes in the air trunk.
§ 168. Case when the Direction of Motion is oblique to the Plane.—The determination of the pressure between a fluid and surface in this case is of importance in many practical questions, for instance, in assigning the load due to wind pressure on sloping and curved roofs, and experiments have been made by Hutton, Vince, and Thibault on planes moved circularly through air and water on a whirling machine.
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| Fig. 170. |
Let AB (fig. 170) be a plane moving in the direction R making an angle φ with the plane. The resultant pressure between the fluid and the plane will be a normal pressure N. The component R of this normal pressure is the resistance to the motion of the plane and the other component L is a lateral force resisted by the guides which support the plane. Obviously
R = N sin φ;
In the case of wind pressure on a sloping roof surface, R is the horizontal and L the vertical component of the normal pressure.
In experiments with the whirling machine it is the resistance to motion, R, which is directly measured. Let P be the pressure on a plane moved normally through a fluid. Then, for the same plane inclined at an angle φ to its direction of motion, the resistance was found by Hutton to be
A simpler and more convenient expression given by Colonel Duchemin is
Consequently, the total pressure between the fluid and plane is
and the lateral force is
In 1872 some experiments were made for the Aeronautical Society on the pressure of air on oblique planes. These plates, of 1 to 2 ft. square, were balanced by ingenious mechanism designed by F. H. Wenham and Spencer Browning, in such a manner that both the pressure in the direction of the air current and the lateral force were separately measured. These planes were placed opposite a blast from a fan issuing from a wooden pipe 18 in. square. The pressure of the blast varied from 610 to 1 in. of water pressure. The following are the results given in pounds per square foot of the plane, and a comparison of the experimental results with the pressures given by Duchemin’s rule. These last values are obtained by taking P = 3.31, the observed pressure on a normal surface:—
| Angle between Plane and Direction of Blast | 15° | 20° | 60° | 90° |
| Horizontal pressure R | 0.4 | 0.61 | 2.73 | 3.31 |
| Lateral pressure L | 1.6 | 1.96 | 1.26 | .. |
| Normal pressure √ (L2 + R2) | 1.65 | 2.05 | 3.01 | 3.31 |
| Normal pressure by Duchemin’s rule | 1.605 | 2.027 | 3.276 | 3.31 |
Water Motors
In every system of machinery deriving energy from a natural waterfall there exist the following parts:—
1. A supply channel or head race, leading the water from the highest accessible level to the site of the machine. This may be an open channel of earth, masonry or wood, laid at as small a slope as is consistent with the delivery of the necessary supply of water, or it may be a closed cast or wrought-iron pipe, laid at the natural slope of the ground, and about 3 ft. below the surface. In some cases part of the head race is an open channel, part a closed pipe. The channel often starts from a small storage reservoir, constructed near the stream supplying the water motor, in which the water accumulates when the motor is not working. There are sluices or penstocks by which the supply can be cut off when necessary.
2. Leading from the motor there is a tail race, culvert, or discharge pipe delivering the water after it has done its work at the lowest convenient level.
3. A waste channel, weir, or bye-wash is placed at the origin of the head race, by which surplus water, in floods, escapes.
4. The motor itself, of one of the kinds to be described presently, which either overcomes a useful resistance directly, as in the case of a ram acting on a lift or crane chain, or indirectly by actuating transmissive machinery, as when a turbine drives the shafting, belting and gearing of a mill. With the motor is usually combined regulating machinery for adjusting the power and speed to the work done. This may be controlled in some cases by automatic governing machinery.
