Page:EB1911 - Volume 14.djvu/102

Rh can in no case exceed the pressure v2/2g or h measured in feet of water, or the direction of motion of the water would be reversed, and there would be reflux. Hence the maximum intensity of the pressure of the jet on the plane is h ft. of water. If the pressure curve is drawn with pressures represented by feet of water, it will touch the free water surface at the centre of the jet.



Suppose the pressure curve rotated so as to form a solid of revolution. The weight of water contained in that solid is the total pressure of the jet on the surface, which has already been determined. Let V = volume of this solid, then GV is its weight in pounds. Consequently

GV = (G/g) v1v; V = 2 √ (hh1).

We have already, therefore, two conditions to be satisfied by the pressure curve.

—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 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 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.

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−x 2.

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

V = $$\int^{\infty}_0$$ 2xydx

= 2h $$\int^{\infty}_0$$ b−x 2 xdx

= (h / loge b) $$\Big[$$ −b−x 2 $$\Big]^{\infty}_0$$

= h / loge b.

Using the condition already stated,

2 √ (hh1) = h / loge b, logundefined b = (/2) √ (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 = r&#8202;2, the equation to the pressure curve is

y = h−1/2 √ (h / h 1) (x2 / r&#8202;2).

§ 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, 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 = f&#8202;Gv2 / 2g pounds = f&#8202;Gh;

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.