We have already, therefore, two conditions to be satisfied by the
pressure curve.
[Illustration: FIG. 168.--Curves of Pressure of Jets impinging
normally on a Plane.]
Some very interesting experiments on the distribution of pressure on a
surface struck by a jet have been made by J. S. Beresford (_Prof.
Papers on Indian Engineering_, No. cccxxii.), with a view to afford
information as to the forces acting on the aprons of weirs.
Cylindrical jets ½ in. to 2 in. diameter, issuing from a vessel in
which the water level was constant, were allowed to fall vertically on
a brass plate 9 in. in diameter. A small hole in the brass plate
communicated by a flexible tube with a vertical pressure column.
Arrangements were made by which this aperture could be moved 1/20
in. at a time across the area struck by the jet. The height of the
pressure column, for each position of the aperture, gave the pressure
at that point of the area struck by the jet. When the aperture was
exactly in the axis of the jet, the pressure column was very nearly
level with the free surface in the reservoir supplying the jet; that
is, the pressure was very nearly v²/2g. As the aperture moved away
from the axis of the jet, the pressure diminished, and it became
insensibly small at a distance from the axis of the jet about equal to
the diameter of the jet. Hence, roughly, the pressure due to the jet
extends over an area about four times the area of section of the jet.
Fig. 168 shows the pressure curves obtained in three experiments with
three jets of the sizes shown, and with the free surface level in the
reservoir at the heights marked.
+------------------------------------------------------+
| Experiment 1. Jet .475 in. diameter. |
+----------------+------------------+------------------+
| Height from | Distance from | |
| Free Surface | Axis of Jet | Pressure in. |
| to Brass Plate | in inches. | inches of Water. |
| in inches. | | |
+----------------+------------------+------------------+
| 43 | 0 | 40.5 |
| " | .05 | 39.40 |
| " | .1 | 37.5-39.5 |
| " | .15 | 35 |
| " | .2 | 33.5-37 |
| " | .25 | 31 |
| " | .3 | 21-27 |
| " | .35 | 21 |
| " | .4 | 14 |
| " | .45 | 8 |
| " | .5 | 3.5 |
| " | .55 | 1 |
| " | .6 | 0.5 |
| " | .65 | 0 |
+----------------+------------------+------------------+
| Experiment 2. Jet .988 in. diameter. |
+----------------+------------------+------------------+
| 42.15 | 0 | 42 |
| " | .05 | 41.9 |
| " | .1 | 41.5-41.8 |
| " | .15 | 41 |
| " | .2 | 40.3 |
| " | .25 | 39.2 |
| " | .3 | 37.5 |
| " | .35 | 34.8 |
| " | .45 | 27 |
| 42.25 | .5 | 23 |
| " | .55 | 18.5 |
| " | .6 | 13 |
| " | .65 | 8.3 |
| " | .7 | 5 |
| " | .75 | 3 |
| " | .8 | 2.2 |
| 42.15 | .85 | 1.6 |
| " | .95 | 1 |
+----------------+------------------+------------------+
| Experiment 3. Jet 19.5 in. diameter. |
+----------------+------------------+------------------+
| 27.15 | 0 | 26.9 |
| " | .08 | 26.9 |
| " | .13 | 26.8 |
| " | .18 | 26.5-26.6 |
| " | .23 | 26.4-26.5 |
| " | .28 | 26.3-26.6 |
| 27 | .33 | 26.2 |
| " | .38 | 25.9 |
| " | .43 | 25.5 |
| " | .48 | 25 |
| " | .53 | 24.5 |
| " | .58 | 24 |
| " | .63 | 23.3 |
| " | .68 | 22.5 |
| " | .73 | 21.8 |
| " | .78 | 21 |
| " | .83 | 20.3 |
| " | .88 | 19.3 |
| " | .93 | 18 |
| " | .98 | 17 |
| 26.5 | 1.13 | 13.5 |
| " | 1.18 | 12.5 |
| " | 1.23 | 10.8 |
| " | 1.28 | 9.5 |
| " | 1.33 | 8 |
| " | 1.38 | 7 |
| " | 1.43 | 6.3 |
| " | 1.48 | 5 |
| " | 1.53 | 4.3 |
| " | 1.58 | 3.5 |
| " | 1.9 | 2 |
+----------------+------------------+------------------+
As the general form of the pressure curve has been already indicated,
it may be assumed that its equation is of the form
y = ab^(-x²).
But it has already been shown that for x = 0, y = h, hence a = h. To
determine the remaining constant, the other condition may be used,
that the solid formed by rotating the pressure curve represents the
total pressure on the plane. The volume of the solid is
_
/[oo]
V = | 2[pi]xy dx
_/0
_
/[oo]
= 2[pi]h | b^(-x²)x dx
_/0
_ _
| |[oo]
= ([pi]h/log_eb) |-b^(-x²)|
|_ _|0
= [pi]h/log_e b.
Using the condition already stated,
2[omega] [root](hh1) = [pi]h/log_e b,
log_e b = ([pi]/2[omega]) [root](h/h1).
Putting the value of b in (2) in eq. (1), and also r for the radius of
the jet at the orifice, so that [omega] = [pi]r², the equation to the
pressure curve is
h x²
y = h[epsilon]^(-½) [root]-- --.
h1 r²
§ 166. _Resistance of a Plane moving through a Fluid, or Pressure of a
Current on a Plane._--When a thin plate moves through the air, or
through an indefinitely large mass of still water, in a direction
normal to its surface, there is an excess of pressure on the anterior
face and a diminution of pressure on the posterior face. Let v be the
relative velocity of the plate and fluid, [Omega] the area of the
plate, G the density of the fluid, h the height due to the velocity,
then the total resistance is expressed by the equation
R = fG[Omega]v²/2g pounds = fG[Omega]h;
where f is a coefficient having about the value 1.3 for a plate moving
in still fluid, and 1.8 for a current impinging on a fixed plane,
whether the fluid is air or water. The difference in the value of the
coefficient in the two cases is perhaps due to errors of experiment.
There is a similar resistance to motion in the case of all bodies of "
_unfair_ " form, that is, in which the surfaces over which the water
slides are not of gradual and continuous curvature.
The stress between the fluid and plate arises chiefly in this way.
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 | Values of f. |
| Area of Plate +------+-------+---------+
| in sq. ft. |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 6½ 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:--
m = 1; n = 0.433; f = 1.433.
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.
[Illustration: 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 = Gv²/2g lb. 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 Gv²/2g
and for a rectangular plate 0.66 Gv²/2g. For circular or square plates
the resultant pressure on the plate was P = 0.00126 v² lb. 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.
[Illustration: FIG. 170.]
Let AB (fig. 170) be a plane moving in the direction R making an angle
[phi] 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 [phi];