Under this head three general cases are to be considered: 1, that of liquids issuing from orifices; 2, their flow through tubes, or in streams; and, 3, the effects of the momentum and impact of liquids. The principles governing the action of the two last will be introduced, 2, under that portion of the work relating to “piping,” and, 3, under the sections pertaining to the jet pump. Throughout the different portions of the book will the three cases be still further elucidated.

Now, as to the laws governing the escape of liquids under pressure through an opening, it may be understood that when the liquid escapes from a vessel, owing to the excess of the internal pressure, the volume which escapes depends on the section of the orifice and the velocity with which the liquid molecules move at the moment of their escape from it.

This velocity depends upon the density of the liquid, the excess of pressure at the opening, and the friction of the liquid, both at the opening and against the walls. When the aperture is made in a very thin wall of a large vessel, so as to remove, as much as possible, the causes tending to modify the motion of the escaping fluid, the laws of the escape are comprised in the following theorem, discovered by Torricelli, in 1643, as a consequence of the law of the fall of bodies discovered by Galileo: “Liquid molecules, flowing from an orifice, have the same velocity, as if they fell freely in vacuo from a height equal to the vertical distance from the surface to the center of the orifice.”

Deductions from the above:—1, the velocity depends on the depth of the orifice from the surface, and is independent of the density of the liquid. Water and mercury in vacuo would fall from the same height in the same time; and so escaping from an orifice at the same depth, below the surface, would pass out with equal velocity; but mercury, being 13·5 times as heavy as water, the pressure exerted at the aperture of a vessel filled with mercury, will be 13·5 times as great as the pressure exerted at the aperture of a vessel filled with water; 2, the velocity of liquids is as the square roots of the depths of the orifices below the surfaces of the liquids.

Thus stating, the velocity of a liquid escaping from an orifice one foot below the surface to be one; from a similar orifice four feet below the surface, it will be two, and at nine feet three, at sixteen feet four, and so on.

From comparative experiments made by a great number of observers, it is learned that the actual flow is only about two-thirds of the theoretical flow. See note.

The form and constitution of liquid veins have been studied and found to be:

1.—That the fluid issuing vertically from an orifice made in a plane and thin horizontal wall, is always composed of two distinct parts, Fig. 97, the portion nearest the orifice is calm and transparent, like a rod of glass, gradually decreasing in diameter. The lower part, on the contrary, is always agitated, and takes an irregular form, in which are regularly distributed elongated swellings, called ventres, whose maximum diameter is greater than that of the orifice.

2.—In the lower part of the vein, the liquid is not continuous; for if we employ an opaque liquid, as mercury, we can see through the vein, Fig. 98. The apparent continuity in a vein of water is owing to the fact that the globules which constitute it succeed each other at a distance inappreciable to the eye.

The time it takes by a vessel to empty itself is to the time required, when it is kept constantly full, to discharge the same quantity of water, as 2 to 1, and the spaces described by the surface in its descent in a column of equal size throughout, are as the odd numbers, 9, 7, 5, 3, 1. Thus these spaces measure equal times. Since liquids are not perfectly mobile, and their exit at an orifice must be retarded by cohesion and friction, the results thus far given are much modified in practice.

When a liquid flows through an orifice in a vessel, eddies are formed about the sides of the orifice, preventing the escape of a jet equivalent to its full size; and owing to these, and to acceleration of velocity, if the jet be downward, it rapidly contracts in its diameter. At a distance outside about equal to diameter of the opening, it is contracted to ²⁄₃ or ⁵⁄₇ its original area; and this part has been called the “contracted vein.” It has been shown, that below this the stream still contracts, though less rapidly.

These swellings separate more widely as they descend with increased rapidity; but falling through great heights, the whole may finally be dissipated in a mist.

If an orifice in a vessel looks downward, and the column of liquid over it be short, this will simply drop out by its own weight, starting at a velocity of o. But if a considerable depth of liquid be above, its gravity produces a corresponding pressure on its base, or on that liquid which is near it; so that, if a plug be removed from an orifice in or close to the base, the liquid starts at once into rapid motion.

Each particle of a jet A issuing from the side of a vessel moves horizontally with the velocity above mentioned, but it is at once drawn downward by the force of gravity in the same manner as a bullet fired from a gun, with its axis horizontal. It is well known that the bullet describes a parabola with a vertical axis, the vertex being the muzzle of the gun. Now, since each particle of the jet moves in the same curve, this jet C takes the parabolic form. In every parabola there is a certain point called the focus, and the distance from the vertex to the focus fixes the magnitude of a parabola in much the same manner as the distance from the center to the circumference fixes the magnitude of a circle.

Now it can be proved that the focus B is as much below as the surface of the water is above the orifice. Accordingly, if water issues through orifices which are small in comparison with the contents of the vessel, the jets from orifices at different depths below the surface take different forms, as shown at D. If these curves are traced on paper held behind the jet, then, knowing the horizontal distance and the vertical height, it is easy to demonstrate that the jet forms a parabola.

Quantity of Efflux.—If we suppose the bottom of a vessel containing water to be thin, and the orifice to be a small circle whose area is A (see Fig. 100) where A B represents an orifice in the bottom of a vessel.

Every particle above A B tries to pass out of the vessel, at once and in so doing exerts a pressure on those nearest. Those that issue near A and B exert pressures in the directions M M and N N; those near the center of the orifice in the direction R Q, those in the intermediate parts in the directions P Q, P Q. In consequence, the water within the space P Q P is unable to escape, and that which does escape, instead of assuming a cylindrical form, at first contracts, and takes the form of a truncated cone.

It is found that the escaping jet continues to contract until at a distance from the orifice about equal to the diameter of the orifice; this part of the jet is called the vena contracta or contracted vein, as explained on a previous page.

Influence of tubes on the quantity of efflux.—The result before given has reference to an aperture in a thin wall. If a cylindrical or conical efflux tube is fitted to the aperture, the amount of the flow is considerably increased. A short tube, whose length is from two to three times its diameter, has been found to increase the actual efflux per second to about 82 per cent. of the theoretical. In this case the water on entering the tube forms a contracted vein, Fig. 101. just as it would do on issuing freely into the air; but afterwards it expands, and, in consequence of the adhesion of the water to the interior surface of the tube, has, on leaving the tube, a section greater than that of the contracted vein. The contraction of the jet within the tube causes a partial vacuum shown in black in the figure.

Now, if an aperture is made in the tube, near the point of greatest contraction, and is carefully fitted with a vertical tube, the lower end of which dips into water, Fig. 101, it is found that water rises in the vertical tube, thereby proving conclusively the formation of a partial vacuum.

If the nozzle has the form of a conic frustum whose larger end is at the aperture, the efflux in a second may be raised to 92 per cent., provided the dimensions are properly chosen. If the smaller end of a frustum of a cone of suitable dimensions be fitted to the orifice, the efflux may be still further increased, which will fall very little short of the theoretical amount.

Velocities of streams.—The velocity of streams varies greatly. The slower flow of rivers has a velocity of less than three feet per second, and the more rapid, as much as six feet per second, which gives respectively about two and four miles per hour. The velocities vary in different parts of the same transverse section of a stream, for the air upon the surface of the water, as well also as the solid bottom of the stream, has a certain effect in retarding the current. The velocity is found to be greatest in the middle, where the water is deepest, Fig. 102, somewhere in m, below the surface; then it decreases with the depth towards the sides, being least at a and b.

Appearance of the surface during a discharge.—A vessel containing a liquid, discharging itself through an orifice, does not always preserve a horizontal surface. When the vein issues from an orifice in the bottom of a vessel, and the level of the liquid is near the orifice, the liquid forms a whirlpool, Fig. 103. If the liquid has a rotary movement, the funnel is formed sooner; if the orifice is at the side of the vessel, there is a depression of the surface upon that side, above the orifice, Fig. 104. These movements depend upon the form of the vessel, the height of the liquid in it, and the dimensions and form of the orifice.

In order to verify many of the laws of hydraulics in an accurate manner, it is necessary to maintain a uniform pressure on the escaping liquid, thereby obtaining a constant velocity at the orifice. This may be done in various ways, as by allowing the water to flow into the vessel in a little larger quantity than can escape from the orifice, the excess being discharged over the upper edge of the vessel; also by means of the syphon.

By suspending solid particles, such as charred paper, pulverized in the water, we render the currents that are formed visible. These solid particles arrange themselves, in curved lines, towards and into the orifice, as a center of attraction, Fig. 105. The particles in immediate contact with the orifice, not moving so easily as those within, must cause contraction; so, also, we can see that gravity in accelerating the velocity, must cause continual decrease in the section of the jet.

Upward jets of water.—As the velocity of a liquid escaping from an orifice is the same as that which a body acquires, falling from a height equal to the distance from the level of the liquid to the orifice, a jet of water escaping from a horizontal opening upwards, should theoretically reach the level of the liquid in the vessel. But this never takes place, Fig. 106, because of—1st, the friction in the conducting tubes destroying the velocity. 2nd, the resistance of the air. 3rd, the returning water falling upon that which is rising. The height of the jet is increased by having the orifices very small, in comparison with the conducting tube; piercing them in a very thin wall, and inclining the jet a little, thus avoiding the effect of the returning water.

Height of the jet.—If a jet issuing from an orifice in a vertical direction has the same velocity as a body would have which fell from the surface of the liquid to that orifice, the jet ought to rise to the level of the liquid. It does not, however, reach this; for the particles which fall hinder it. But by inclining the jet at a small angle with the vertical it reaches about ⁹⁄₁₀ of the theoretical height, the difference being due to friction and to the resistance of the air.

The quantities of water which issue from orifices of different areas are very nearly proportional to the size of the orifice, provided the level remains constant, and this is true irrespective of the form of the opening which may be round, square, or any other shape.

Escape of liquids through short tubes.—We often place in an orifice, to increase the flow, a short tube (called an adjutage) either cylindrical or conical. If the vein pass through the tube without adhering to it, the flow is not modified; if the vein adhere (the liquid wetting the interior walls) the contracted part is dilated, and the flow is increased.

In the last case, and with a cylindrical adjutage, its length not being more than four times its diameter, the flow is augmented about one-third. Conical pipes, converging towards the exterior, increase the flow still more than the preceding, the flow and velocity of the vein varying with the angle.

Escape of liquids through long tubes.—When a liquid passes through a long straight tube, the flow soon diminishes greatly in velocity, because of the friction which takes place between the liquid particles and the walls. If there be any bends or curves in the tube, it is still further diminished by the same cause. The discharge is then much less than it would be from an orifice in a thin wall, and therefore the tube is generally inclined; the liquid then passes down this inclined plane, or it is forced through by pressure, applied at the opposite end.

Direction of the jet from lateral orifices.—From the principle of the equal transmission of pressure, water issues from an orifice in the side of a vessel with the same velocity as from an aperture in the bottom of a vessel at the same depth.

MEASUREMENT OF WATER PRESSURE.

In reference to the table on the next page, it may be well to say that it has two uses; by it when the “head” is known the pressure can be ascertained to a fraction, thus, Ex. 1, If the head is 140 feet, then the pressure is 60·64 pounds per square inch. Again, Ex. 2, If the pressure is 15·16 per square inch, then the head is 35 feet.

PRESSURE OF WATER.

The pressure of water in pounds per square inch for every foot in height to 300 feet; and then by intervals, to 1000 feet head· By this table, from the pounds pressure per square inch, the feet head is readily obtained; and vice versa.