Hydromechanics

HYDROMECHANICS Historical Introduction.

THE word Hydromechanics is derived from the Greek υδpo-∕n7χαvικ0i, meaning the mechanics of water and fluids in general. The science is divided into three branches :— Hydrostatics, which deals with the equilibrium of fluids; Hydrodynamics, which deals with the mathematical theory of the motion of fluids, neglecting the viscosity; and Hydraulics, in which the motion of water in pipes and canals is considered, and hydrodynamical questions of practical application are investigated.

The science of hydromechanics was cultivated with less success among the ancients than any other branch of mechanical philosophy. When the human mind had made considerable progress in the other departments of physical science, the doctrine of fluids had not begun to occupy the attention of philosophers; and, if we except a few propositions on the pressure and equilibrium of water, hydromechanics must be regarded as a modern science, which owes its existence and improvement to these great men who adorned the 17th and 18th centuries.

Those general principles of hydrostatics which are to this day employed as the foundation of that part of the science were first given by Archimedes in his work ∏∈pι τωv oχoυp∙∈vωιz, or De iis quoe vehuntur in humido, about 250 B.c., and were afterwards applied to experiments by Marinus Ghetaldus in his Promotus Archimedes (1603). Archimedes maintained that each particle of a fluid mass, when in equilibrium, is equally pressed in every direction; and he inquired into the conditions according to which a solid body floating in a fluid should assume and preserve a position of equilibrium. We are also indebted to him for that ingenious hydrostatic process by which the purity of the precious metals can be ascertained, and for the screw engine which goes by his name.

In the Greek school at Alexandria, which flourished under the auspices of the Ptolemies, the first attempts were made at the construction of hydraulic machinery. About 120 b.c. the fountain of compression, the siphon, and the forcing pump were invented by Ctesibius and Hero; and, though these machines operated by the pressure of the air, yet their inventors had no distinct notions of the preliminary branches of pneumatical science. The siphon is a simple instrument; but the forcing pump is a complicated and abstruse invention, which could scarcely have been expected in the infancy of hydraulics. It was probably suggested to Ctesibius by the Egyptian Wheel or Noria, which was common at that time, and which was a kind of chain pump, consisting of a number of earthen pots carried round by a wheel. In some of these machines the pots have a valve in the bottom which enables them to descend without much resistance, and diminishes greatly the load upon the wheel; and, if we suppose that this valve was introduced so early as the time of Ctesibius, it is not difficult to perceive how such a machine might have led that philosopher to the invention of the forcing pump.

Notwithstanding these inventions of the Alexandrian school, its attention does not seem to have been directed to the motion of fluids. The first attempt to investigate this subject was made by Sextus Julius Frontinus, inspector of the public fountains at Rome in the reigns of Nerva and Trajan; and we may justly suppose that his work, entitled De Aquoeductibus Urbis Pomae Commentarius, contains all the hydraulic knowledge of the ancients. After describing the nine great Roman aqueducts, to which he himself added five more, and mentioning the dates of their erection, he considers the methods which were at that time employed for ascertaining the quantity of water discharged from ajutages, and the mode of distributing the waters of an aqueduct or a fountain. He justly remarks that the flow of water from an orifice depended not only on the magnitude of the orifice itself, but also on the height of the water in the reservoir; and that a pipe employed to carry off a portion of water from an aqueduct should, as circumstances required, have a position more or less inclined to the original direction of the current. But as he was unacquainted with the true law of the velocities of running water as depending upon the depth of the orifice, we can scarcely be surprised at the want of precision which appears in his results.

It has generally been supposed that the Romans were ignorant of the art of conducting and raising water by means of pipes; but it can scarcely be doubted, from the statement of Pliny and other authors, not only that they were acquainted with the hydrostatical principle, but that they actually used leaden pipes for the purpose. Pliny asserts that water will always rise to the height of its source, and he also adds that, in order to raise water up to an eminence, leaden pipes must be employed.

Castelli and Torricelli, two of the disciples of Galileo, applied the discoveries of their master to the science of hydrodynamics. In 1628 Castelli published a small work, Della Misura dell' acque correηti, in which he gave a very satisfactory explanation of several phenomena in the motion of fluids in rivers and canals. But he committed a great paralogism in supposing the velocity of the water proportional to the depth of the orifice below the surface of the vessel. Torricelli, observing that in a jet where the water rushed through a small ajutage it rose to nearly the same height with the reservoir from which it was supplied, imagined that it ought to move with the same velocity as if it had fallen through that height by the force of gravity. And hence he deduced this beautiful and important proposition, that the velocities of fluids are as the square root of the head, allowing for the resistance of the air and the friction of the orifice. This theorem was published in 1643, at the end of his treatise De Motu gravium Projectorum. It was afterwards confirmed by the experiments of Raphael Magiotti on the quantities of water discharged from different ajutages under different pressures; and, though it is true only in small orifices, it gave a new turn to the science of hydraulics.

After the death of the celebrated Pascal, who discovered the pressure of the atmosphere, a treatise on the equilibrium of fluids (Sur l ’ Hquilibre des Liqueurs') was found among his manuscripts, and was given to the public in 1663. In the hands of Pascal hydrostatics assumed the dignity of a science. The laws of the equilibrium of fluids were demonstrated in the most perspicuous and simple manner, and amply confirmed by experiments. The discovery of Torricelli, it may be supposed, would have incited Pascal to the study of hydraulics. But as he has not treated this subject in the work mentioned, it was probably composed before that discovery had been made public.

The theorem of Torricelli was employed by many succeeding writers, but particularly by the celebrated Mariotte, whose labours in this department of physics deserve to be recorded. His Traité du Mouvement des Eaux, which was published after his death in the year 1686, is founded on a great variety of well-conducted experiments on the motion of fluids, performed at Versailles and Chantilly. In the discussion of some points he has committed considerable mistakes. Others he has treated very superficially, and in none of his experiments does he seem to have attended to the diminution of efflux arising from the contraction of the fluid vein, when the orifice is merely a perforation in a thin plate; but he appears to have been the first who attempted to ascribe the discrepancy between theory and experiment to the retardation of the water’s velocity arising from friction. His contemporary Guglielmini, who was inspector of the rivers and canals in the Milanese, had ascribed this diminution of velocity in rivers to transverse motions arising from inequalities in their bottom. But as Mariotte observed similar obstructions even in glass pipes, where no transverse currents could exist, the cause assigned by Guglielmini seemed destitute of foundation. The French philosopher, therefore, regarded these obstructions as the effects of friction. He supposes that the filaments of water which graze along the sides of the pipe lose a portion of their velocity; that the contiguous filaments, having on this account a greater velocity, rub upon the former, and suffer a diminution of their celerity; and that the other filaments are affected with similar retardations proportional to their distance from the axis of the pipe. In this, way the medium velocity of the current may be diminished, and consequently the quantity of water discharged in a given time must, from the effects of friction, be considerably less than that which is computed from theory. Guglielmini was the first who attended to the motion of water in rivers and open canals (La Misura dell ’ acque cοrrenti). Embracing the theorem of Torricelli, which had been confirmed by repeated experiments, Guglielmini concluded that each particle in the perpendicular section of a current has a tendency to move with the same velocity as if it issued from an orifice at the same depth from the surface. The consequences deducible from this theory of running waters are in every respect repugnant to experience, and it is really surprising that it should have been so hastily adopted by succeeding writers. Guglielmini himself was sufficiently sensible that his parabolic theory was contrary to fact, and endeavoured to reconcile them by supposing the motion of rivers to be obstructed by transverse currents arising from irregularities in their bed. The solution of this difficulty, as given by Mariotte, was more satisfactory, and was afterwards adopted by Guglielmini, who maintained also that the viscosity of water had a considerable share in retarding its motion.

The effects of friction and viscosity in diminishing the velocity of running water were noticed in the Principia of Sir Isaac Newton, who threw much light upon several branches of hydromechanics. At a time when the Cartesian system of vortices universally prevailed, this great man found it necessary to investigate that absurd hypothesis, and in the course of his investigations he showed thμt the velocity of any stratum of the vortex is an arithmetical mean between the velocities of the strata which enclose it; and from this it evidently follows that the velocity of a filament of water moving in a pipe is an arithmetical mean between the velocities of the filaments which surround it. Taking advantage of these results, it was afterwards shown by Pitot that the retardations arising from friction are inversely as the diameters of the pipes in which the fluid moves. The attention of Newton was also directed to the discharge of water from orifices in the bottom of vessels. He supposed a cylindrical vessel full of water to be perforated in its bottom with a small hole by which the water escaped, and the vessel to be supplied with water in such a manner that it always remained full at the same height. He then supposed this cylindrical column of water to be divided into two parts,—the first, which he calls the “cataract,” being an hyperboloid generated by the revolution of an hyperbola of the fifth degree around the axis of the cylinder which should pass through the orifice, and the second the remainder of the water in the cylindrical vessel. He considered the horizontal strata of this hyperboloid as always in motion, while the remainder of the water was in a state of rest, and imagined that there was a kind of cataract in the middle of the fluid. When the results of this theory were compared with the quantity of water actually discharged, Newton concluded that the velocity with which the water issued from the orifice was equal to that which a falling body would receive by descending through half the height of water in the reservoir. This conclusion, however, is absolutely irreconcilable with the known fact that jets of water rise nearly to the same height as their reservoirs, and Newton seems to have been aware of this objection. In the second edition of his Principia accordingly, which appeared in 1714, he reconsidered his theory. He had discovered a contraction in the vein of fluid (vena contracta) which issued from the orifice, and found that, at the distance of about a diameter of the aperture, the section of the vein was contracted in the subduplicate ratio of two to one. He regarded, therefore, the section of the contracted vein as the true orifice from which the discharge of water ought to be deduced, and the velocity of the effluent water as due to the whole height of water in the reservoir; and by this means his theory became more conformable to the results of experience. This theory, however, is still liable to serious objections, The formation of a cataract is by no means agreeable to the laws of hydrostatics; for when a vessel is emptied by the efflux of water through an orifice in its bottom, all the particles of the fluid direct themselves toward this orifice, aud therefore no part of it can be considered as in a state of repose.

The subject of the oscillation of waves, one of the most difficult in the science of hydrodynamics, was first investigated by Newton. In the forty-fourth proposition of the second book of his Principia, he has furnished us with a method of ascertaining the velocity of the waves of the sea, by observing the time in which they rise and fall. If the two vertical branches of a siphon, which communicate by means of a horizontal branch, are filled with a fluid of known density, the two fluid columns, when in a state of rest, will be in equilibrium and their surfaces horizontal. But if the one column is raised above the level of the other, and left to itself, it will descend below that level, and raise the other column above it, and, after a few oscillations, they will return to a state of repose. Newton occupied himself in determining the duration of these oscillations, or the length of a pendulum isochronous to their duration; and he found, by a simple process of reasoning, that, allowing for the effects of friction, the length of a synchronous pendulum is equal to one-half of the length of the siphon, that is, of the two vertical branches and the horizontal one, and hence he deduced the isochronism of these oscillations. From this Newton concluded that the velocity of waves formed on the surface of water, either by the wind or by a body thrown into it, was in the subduplicate ratio of their size. When their velocity, therefore, is measured, which can be easily done, the size of the waves will be determined by means of a pendulum which oscillates in the time that a wave takes to rise and fall.

Such was the state of hydrodynamics in 1738, when Daniel Bernoulli published his Hydrodynamica, give de Viribus et Motibus Fluidorum Commentarii. The gerra of Daniel Bernoulli’s theory was first published in his memoir entitled Theoria Nova de Motu Aquarum per Canales quocunque fluentes, which he had communicated to the Academy of St Petersburg as early as 1726. His theory of the motion of fluids was founded on two suppositions, which appeared to him conformable to experience. He supposed that the surface of a fluid, contained in a vessel which is emptying itself by an orifice, remains always horizontal; and, if the fluid mass is conceived to be divided into an infinite number of horizontal strata of the same bulk, that these strata remain contiguous to each other, and that all their points descend vertically, with velocities inversely proportional to their breadth, or to the horizontal sections of the reservoir. In order to determine the motion of each stratum, he employed the principle of the conservatio vilium vivarum, and obtained very elegant solutions. In the opinion of the Abbé Bossut, his work was one of the finest productions of mathematical genius. The uncertainty of the principle employed by Daniel Bernoulli, which has never been demonstrated in a general manner, deprived his results of that confidence which they would otherwise have deserved, and rendered it desirable to have a theory more certain, and depending solely on the fundamental laws of mechanics. Maclaurin and John Bernoulli, who were of this opinion, resolved the problem by more direct methods, the one in his Fluxions, published in 1742, and the other in his Hydraulica nunc primum detecta, et demonstrata directe ex fundamentis pure mechanicis, which forms the fourth volume of his works. The method employed by Maclaurin has been thought not sufficiently rigorous; and that of John Bernoulli is, in the opinion of Lagrange, defective in perspicuity and precision.

The theory of Daniel Bernoulli was opposed also by the celebrated D’Alembert. When generalizing James Bernoulli’s theory of pendulums he discovered a principle of dynamics so simple and general that it reduced the laws of the motions of bodies to that of their equilibrium. He applied this principle to the motion of fluids, and gave a specimen of its application at the end of his Dynamics in 1743. It was more fully developed in his Traité des Fluides, which was published in 1744, where he has resolved, in the most simple and elegant manner, all the problems which relate to the equilibrium and motion of fluids. He makes use of the very same suppositions as Daniel Bernoulli, though his calculus is established in a very different manner. He considers, at every instant, the actual motion of a stratum as composed of a motion which it had in the preceding instant and of a motion which it has lost. The laws of equilibrium between the motions lost furnish him with equations which represent the motion of the fluid, Although the science of hydrodynamics had then made considerable progress, yet it was chiefly founded on hypothesis. It remained a desideratum to express by equations the motion of a particle of the fluid in any assigned direction. These equations were found by D’Alembert from two principles,—that a rectangular canal, taken in a mass of fluid in equilibrium, is itself in equilibrium; and that a portion of the fluid, in passing from one place to another, preserves the same volume when the fluid is incompressible, or dilates itself according to a given law when the fluid is elastic. His very ingenious method was published in 1752, in his Essai sur la Resistance des Fluides. It was brought to perfection in his Opuscules Mathématiques, and was adopted by Euler.

Before the time of D’Alembert, it was the great object of philosophers to submit the motion of fluids to general formulae, independent of all hypothesis. Their attempts, however, were altogether fruitless; for the method of fluxions, which produced such important changes in the physical sciences, was but a feeble auxiliary in the science of hydraulics. For the resolution of the questions concerning the motion of fluids, we are indebted to the method of partial differences, a new calculus, with which Euler enriched the sciences. This great discovery was first applied to the motion of water by D’Alembert, and enabled both him and Euler to represent the theory of fluids in formulae restricted by no particular hypothesis.

The most successful labourer in the science of hydrodynamics was the Chevalier Dubuat, engineer in ordinary to the king of France. Following in the steps of the Abbé Bossut {Nouvelles expériences sur la resistance des fluides, 1777), he prosecuted the inquiries of that philosopher with uncommon ingenuity; and in the year 1786 he published, in two volumes, his Principes dj Hydraulique, which contains a satisfactory theory of the motion of fluids, founded solely upon experiments. Dubuat considered that if water were a perfect fluid, and the channels in which it flowed infinitely smooth, its motion would be continually accelerated, like that of bodies descending in an inclined plane. But as the motion of rivers is not continually accelerated, and soon arrives at a state of uniformity, it is evident that the viscosity of the water, and the friction of the channel in which it descends, must equal the accelerating force. Dubuat therefore assumes it as a proposition of fundamental importance that, when water flows in any channel or bed, the accelerating force which obliges it to move is equal to the sum of all the resistances which it meets with, whether they arise from its own viscosity or from the friction of its bed. This principle was employed by Dubuat in the first edition of his work, which appeared in 1779, but the theory contained in that edition was founded on the experiments of others. He soon saw, however, that a theory so new, and leading to results so different from the ordinary theory, should be founded on new experiments more direct than the former, and he was employed in the performance of these from 1780 to 1783. The experiments of Bossut having been made only on pipes of a moderate declivity, Dubuat found it necessary to supply this defect. He used declivities of every kind, and made his experiments upon channels from a line and a half in diameter to seven or eight square toises.

The theory of running water was greatly advanced by the researches of Prony. From a collection of the best experiments by Couplet, Bossut, and Dubuat he selected eighty-two (fifty-one on the velocity of water in conduit pipes, and thirty-one on its velocity in open canals); and, discussing these on physical and mechanical principles, he succeeded in drawing up general formulae, which afford a simple expression for the velocity of running water.

Ey tel wein of Berlin published, in 1801, a valuable compendium of Hydraulics, entitled Handbuch der Mechanik und der Hydraulik, which contains an account of many new and valuable experiments made by himself. He investigates the subject of the discharge of water by compound pipes, the motions of jets, and their impulses against, plane and oblique surfaces; and he shows theoretically that a water wheel will have its effect a maximum when its circumference moves with half the velocity of the stream.

A series of interesting hydraulic experiments was made at Rome in 1809 by Mallet and Vici. They found that a pipe whose gauge was five ounces French measure (or 0Ό3059 French kilolitres) furnished one-seventh more water than five pipes of one ounce, an effect arising from the velocity being diminished by friction in the ratio of the perimeters of the orifices as compared with their areas.

Hachette, in the year 1816, presented to the National Institute a memoir containing the results of experiments which he had made on the spouting of fluids, and the discharge of vessels. The objects he had in view were to measure the contracted part of a fluid vein, to examine the phenomena attendant on additional tubes, and to investigate and describe the figure of the fluid vein, and the results which take place when different forms of orifices are employed. Hachette showed in the second part of his memoir that greater or lesser volumes of water will be discharged in the same time through tubes of different figures, the apertures in all having the same dimensions. He also gave several remarkable results respecting other fluids issuing out of orifices into air or a vacuum.

Several very interesting experiments on the propagation of waves have been made by the brothers Weber and by Bidone. Mr John Scott Russell performed a number of experiments on waves, which are described in the Edinburgh Transactions, vol. xiv., and in the British Association Report for 1837. The mathematical theory has been worked out by Green, Stokes, Rankine, and other mathematicians, but still offers an interesting field for the investigator. Stokes’s Report of the British Association for 1846 on Recent Researches in Hydrodynamics gives an account of the subject as it existed at that date.

In 1826 Bidone, besides his experiments on waves, made a series on the velocity of running water at the hydraulic establishment of the university of Turin, and he published an account of them in 1829. After giving a description of his apparatus and method of experimenting, he gives the figures obtained from fluid veins, sections of which were taken at different distances from the orifice.

In the year 1827 Poncelet published a Mémoire sur les Roues Hydrauliques à Aubes Courbes, containing his experiments on the undershot wheel with curved palettes, which he had invented in the year 1824. The best undershot previous to the introduction of the Poncelet wheel never developed more than 0 - 25 of the work of the water, whereas this utilized 0 ’ 60 of that work, which is nearly equivalent to the maximum effect of the breast wheel. The principle on which the Poncelet wheel acts, and that which makes it utilize so much of the work of the water, is that the water is received by the curved floats without any shock, and is discharged finally with a small velocity. This undershot wheel is much used in France.

Previous to the year 1827, the wheels required in the mills and manufactories of Germany and France were generally those which worked with the axis horizontal, or the tub and spoon wheels with the axis vertical; but in that year a young mechanician named Fourneyron introduced a wheel working with the axis vertical, yet wholly different from the latter kind. Fourneyron showed that in existing wheels with a vertical axis the water left the wheel with considerable velocity in the direction of the motion of the wheel, and thus carried away and wasted much of the energy of the fall. By the introduction of a series of fixed guide blades, which gave the water initially a backward velocity of rotation, the water left the wheel with a much smaller velocity of discharge. He thus invented the first complete turbine, a kind of water motor which has largely superseded the more cumbrous water wheels previously in use. Shortly after the invention was made public, Fourneyron was awarded the prize of 6000 francs which was offered by the Society for the Encouragement of National Industry.

The most extensive experiments on the discharge of water: from orifices are those made under the direction of the ’ French Government by Poncelet and Lesbros (Experiences i Hydrauliques, Paris, 1851). Boileau (Traite de la mesure des eaux courantes) has discussed these results and added experiments of his own. Bornemann has re-examined all these results with great care, and has been able to express in formulae the variation of the coefficients of discharge in different conditions (Civilingenieur, 1880), Very valuable experiments leading to a modification of the usual formula for the discharge over weirs were made by Mr J. B. Francis (Lowell Hydraulic Experiments, Boston, Mass., 1855). Wiesbach also has made many experimental investigations of the discharge of fluids.

The friction of water investigated originally by Coulomb at slow speeds has been measured for higher speeds by Mr W. Froude, whose researches have very great value in the theory of ship resistance (Report of Brit. Assoc., 1869).

The flow of air and steam from orifices has been measured by many experimenters from Young to Saint Venant. Mr Napier in some interesting experiments first pointed out that when the ratio of the pressures on the two sides of an orifice exceeded a certain limit the measured discharge was very different from that calculated by the accepted formulae. Since then numerous experiments have been made, and the theory of the flow of elastic fluids has been discussed in numerous memoirs. The valuable investigations of Fliegner (Civilingenieur, 1878) deserve special mention.

A most valuable investigation of the flow of water in pipes and channels has been carried out with exceptional accuracy and on a very large scale by the late M. Darcy, and continued by his successor the late M. Bazin, at the expense of the French Government (Recherches Hydrauliques, Paris, 1866). The measurement of the flow in rivers has been extensively carried out, especially by German engineers. Harlacher’s Beitrage zur Hydrographie des Königreiches Böhmen contains exceedingly valuable measure ments of this kind, and a comparison of the experimental results with all the formulae of flow which have been proposed. Messrs Humphreys and Abbott’s gaugings of the Mississippi for the United States Government, Mr Gordon’s gaugings of the Irrawaddi, and Captain Cunningham’s experiments on the Ganges Canal may be referred to as having materially advanced hydraulic science.

The first adequate theory of turbines is that of Poncelet in the Comptes Rendus de VAcademie de Paris, 1838. Redtenbacher’s Theorie und Bau der Turbinen und Fen- tilatoren (Mannheim, 1844) is the first complete treatise on the subject. Girard’s turbine, which was of an entirely new type, was discussed in Le Genie industrielle, 1856-1857, and lately by Fink (Civilingenieur, 1880). Important experiments on turbines were made by Francis (Lowell Hydraulic Experiments, Boston, Mass., 1855).

General Principles.

Hydromechanics is the science of the equilibrium and motion of fluids, both elastic and non elastic. A fluid is defined to be a substance which yields continually to the slightest tangential stress; and, consequently, when the fluid has come to rest, the stress across any surface in the fluid must be normal to the surface.

The stress considered in hydromechanics is always a pressure, as fluids are in general capable of sustaining only a slight tension without disruption (see Capillary Action). The intensity of the pressure is measured, as in the subject of elasticity, by the number of units of force per unit of area.

If an area a be pushed by P units of force, then the mean p pressure p over the· area α is —; or, if p be variable, then p at any point p = It —, a being any small area enclosing the point at which p is required, and P the number of units <>f force with which α is pushed.

The pressure across any surface being normal to the surface, it follows from the general equations of internal stress (see Elasticity) that at a point the pressure is the same in all directions about the point. This may be proved independently by considering the equilibrium of a tetrahedron ABCD. Let p, p' be the pressures on the faces ABC, BCD, and resolve the forces parallel to the edge AD; the face ABC will be pushed by a force p × area ABC, and the face BCD will be pushed by a force p' × area BCD; and the projections of the areas ABC, BCD on a plane perpendicular to AD being equal, it follows that p = p'.

If forces act throughout the fluid so that the pressure varies from one point to another, we must suppose the tetrahedron taken indefinitely small, and then the impressed forces, depending on the volume, may be neglected in comparison with the forces acting on the faces, which depend on the surface of the tetrahedron.

When fluids such as exist in nature are in motion, the stresses across any surfaces are no longer normal, but tangential stresses are called into play, the intensities of which depend upon the relative motion of the parts of the fluid; those tangential stresses are said to be due to the viscosity of the fluid.

The difference between a solid and a fluid is that in a solid the tangential stress must exceed a certain amount before permanent shearing takes place, otherwise the stress being removed the solid regains its shape; in a fluid the slightest tangential stress produces a permanent deformation, and if continued long enough will cause a complete change of form, however great the viscosity of the fluid may be. But little progress has been made in the theory of the motion of viscous fluids, those cases which have been worked out mostly falling under the head of the practical subject of hydraulics. In the theoretical subject of hydrodynamics the motion of the so-called perfect fluids only is considered,—fluids in which no viscosity is supposed to exist, and in which therefore the pressure at a point is the same in all directions.

Fluids are divided into two classes, incompressible fluids called liquids, and compressible fluids called gases. The so-called incompressible fluids are in reality compressible, but the compressibility being small is neglected in ordinary problems.

The compression of a substance is defined to be the ratio of the diminution of the volume to the original volume; and the compressibility is defined to be the ratio of the compression to the pressure producing it. The elasticity or resilience of volume is the ratio of the pressure to the compression produced, and is therefore the reciprocal of the compressibility. Fluids, from the definition, possess only elasticity of volume; an elastic solid possesses in addition an elasticity of figure, called also the rigidity.

A table of the elasticities and compressibilities of liquids is given in Everett’s Units and Physical Constants, expressed in C. G. S. units; for instance, at 8° C. the elasticity of water is 2Ό8 × 10 10 , and the compressibility per atmosphere is 4*81 × 10 -5 . The elasticity of water is also proved to exist, and can be determined from the velocity of sound in water; for instance, in water of density unity, the velocity is the square root of the elasticity, and there fore 144,000 centimetres per second, which agrees closely with the velocity determined by experiment.

Compressible fluids or gases are assumed to obey the two gaseous laws.

The first gaseous law, discovered by Boyle, and generally called Boyle’s law, asserts that the pressure of a given quantity of a gas at a given temperature varies inversely as the volume, or directly as the density.

The density is defined to be the number of units of mass in the unit of volume, and the specific volume is the volume of the unit of mass. Hence Boyle’s law asserts that the pressure of a gas at a given temperature varies as the density, or inversely as the specific volume.

Dalton generalized Boyle’s law for a mixture of gases by enunciating the law that the pressure of a mixture of gases is the sum of the separate pressures each gas would have if it existed alone in the containing vessel.

If we suppose all the gases the same, we are led to Boyle’s law, since, if a gramme of air in a centimetre cube produces a certain pressure, then two grammes will produce double the pressure, three grammes treble the pressure, and so on; hence the pressure varies as the density. This method of exhibiting Boyle’s law is due to Rankine (Maxwell, Heat, p. 27).

The second gaseous law was discovered by Charles, but it generally goes by Gay-Lussac’s name. It asserts that under constant pressure the volume of a given quantity of every gas increases by about of its volume at 0° C. for a rise of temperature of 1° C. Therefore, if ν be the volume at τ° C. and v 0 the volume at 0° C., then v _ 1 τ 273 + τ v 0 + 273 273 '

If we reckon temperature from - 273° C., and put 273 + τ=√, then t is called the absolute temperature, and -273° C. is called the absolute zero; and the second gaseous law asserts that at a constant pressure the volume of a given quantity of gas is proportional to the absolute temperature.

Combined with Boyle’s law, this leads to the result that ρv <×t, = Ri suppose, where p is the pressure, ν the specific volume, and t the absolute temperature; also, if p denote the density, ρ = —, and p =Rp∕.

Todetermine the numerical value of ^ = R, in C.G.S. units, sup- . t pose at 80° C. a centimetre cube of air is Ό01 of a gramme, the height of the barometer being 76 centimetres, and the numerical value of g to be 981; then

2? = 981 × 1342 × 76 = 10 β very nearly, t = 273 + 80 = 353,

- ¾-

If the temperature is kept constant, then, an increment of pressure dp producing a compression - —, the elasticity of volume dp dp ≈ -vf- = p -, =p; dυ r dp and therefore at a constant temperature the elasticity is equal to the pressure.

Professor Maxwell {Heat, p. 171) has proved from first principles that the ratio of the elasticity of volume when no heat escapes is to the elasticity at constant temperature as the specific heat at constant pressure to the specific heat at constant volume.

Consequently if γ denotes the ratio of the specific heat at constant pressure to the specific heat at constant volume, and if the gas be compressed and no heat allowed to escape, then

-<⅛-∙w.

dv dp ι d υ n or — + γ-=0, ρ ν or pv y = constant.

Consequently, if p, p' be the pressures, v, v' the specific volumes, p, p the densities, and t, t' the absolute temperatures at two different states of a gas, when no heat has been allowed to escape, then iruH'Hi·U

. pv t since ⅜- 7 =-τ 7 -.

pν t

The value of γ for air is about 1*4; for instance, the velocity of sound is = sj y ; and, for dry air at τ 0 C. ,

£=7'8376 ×10√ 1 + ’ );

P ∖ 273/ and therefore the velocity of sound in centimetres per second

= 33240√(l + = 33240+ 60τ,∙ nearly.

We have defined the density of a substance to be the number of units of mass in the unit of volume; for instance, with English units the density is the number of pounds in a cubic foot, and the density of water would be 62 - 5, a cubic foot of water being 1000 oz. But the densities of substances are generally tabulated relatively to water, and the old-fashioned name for the density relative to water was specific gravity. With French units of the centimetre and the gramme, a centimetre cube of water being a gramme, the density of water is unity.

PART I.—HYDROSTATICS.

Hydrostatics is the science of the equilibrium of fluids. When a fluid has come to rest there can be no tangential stress, and consequently the stress across any surface is normal to the surface, and therefore the same in all directions about a point. We shall begin with a few elementary propositions about the equilibrium of liquids like water under gravity, and afterwards proceed to the consideration of the equilibrium of any fluid under any forces.

Prop. I. The pressure is the same at all points in a horizontal plane of a liquid at rest under gravity.

. For, taking any two points in the same horizontal plane, and joining them, and describing about the joining line as axis an indefinitely thin cylinder, then, since the weight and the pressures on the sides of the cylinder are normal to the axis of the cylinder, resolving parallel to the axis of the cylinder, the pressures on the ends must be equal for equilibrum, and must therefore be of equal intensity.

Corollary.— It follows that the free surface must be a horizontal plane, supposing the pressure uniform over it, and gravity to act in parallel vertical lines.

That the free surface is a plane is verified experimentally by the fact that objects are seen by reflexion at the surface undisturbed as in a plane mirror; and that it is a horizontal plane is verified by the fact that a plumb line and its image by reflexion at the surface always appear in the same straight line.

Prop. II. The pressure at any point of a liquid at rest under gravity is proportional to the depth.

For, let P (fig. 1) be any point at a depth AP==z in the liquid, and about ÀP as axis describe a cylinder, of sectional area α suppose. Then, considering the equilibrium of the liquid in this cylinder, and resolving vertically, the pressure, p suppose, at P acting over the area a must balance the weight of the liquid in the cylinder, neglecting the atmospheric pressure, and therefore pa=ρza , or p = pz , p denoting the weight of the liquid per unit of volume.

In practical hydrostatics the gravitation measure of forces is employed, and p the pressure is generally estimated in ft> per square inch, and then, an inch being the unit of length, z is given in inches, and p is the number of pounds to a cubic inch of the liquid considered.

Thus with water, taking a cubic foot of water to be 62∙4Ib,

62 ∙4

^=⅛=0∙036× a .

1 ∕ Zo

If, however, z be measured in feet and p in lb per square inch, j> = ^⅛=0∙433 ×z.

144

Consequently the head of each foot of water produces a pressure of 0∙433 lb to the square inch.

With the French metric system of units, a centimetre cube of water is one gramme, and therefore with the centimetre as the unit of length and gramme as unit of weight, using gravitation units of force, ρ=l, and 79 = 2.

Even if the point C considered (fig. 2) should not be vertically below the actual free surface, it still follows that the pressure at C is proportional to the depth below the free surface. For the pressure at C is equal to the pressure at D in the same horizontal plane, and this is proportional to the depth DE, that is, to the depth CF.

If the atmospheric pressure w be taken into account, then p=zr + pz, and generally, at different depths z and z', p-p'≈p{z-ziγ t or the difference of the pressures at any two points of a liquid at rest under gravity is proportional to the difference of the depths.

Corollary.— It follows from this that a liquid rises to the same level in a series of communicating vessels (fig. 3), since the fluid must have one horizontal plane as the free surface.

If liquids of different densities p and p' be poured into the two branches of a bent tube, the heights of the free surfaces above the plane of separation will be inversely as the densities. For the pressure at the level of the common surface being constant, and in one case due to a height h of liquid of density p and in the other case to a height h' of liquid of density p, therefore ph = p'h' , or h: h!=p': ρ .

The barometer (fig. 4) is in reality an instrument of this character, for a column of mercury, of density σ and height h suppose, supports a column of air, which is of density p suppose, and, if homogeneous, would reach to a height H. Hence the pressure at the level of the common surface p = σh = pH , estimated in gravitation measure.

If the tube AB should not be exactly vertical, then h must be taken to denote the vertical distance between the level of the mercury at P and Q, which becomes difficult to estimate at sea when the ship is rolling.

When Λis observed the temperature must also be observed; for at the same pressure the coefficient of linear expansion of h is the coefficient of cubical expansion of mercury. For, if the density of mercury were halved, the height h would be doubled, and so on in any proportion.

H is called the pressure height (Everett, Units and Physical Constants'). For dry air at 0° C.,

H = 7 ’ 988 × 10 5 centimetres.

Prop. III. To find the pressure on the bottom of a vessel of any form containing liquid, the bottom of the vessel being supposed a horizontal plane.

The pressure at any point of the base being the same and due to the depth below the free surface, whatever be the shape of the sides of the vessel, as in fig. 5, it follows that the pressure on the base AB is equal to the weight of the liquid contained in an imaginary cylinder, traced out by vertical lines round the base AB reaching to the surface.

In order to illustrate this, let there be four vessels A, B, C, D (fig. 6), having bottoms all of the same area, and closed by plates E, F, G, H, of the same weight. Let the plates also be kept in their places by means of strings passing over pulleys and supporting equal weights w,w , ,w",w.'" The weights will measure the vertical forces acting on the plates, i.e., the bases of the vessels. It will be found that water must be poured into each vessel to the same height to cause the plates to descend. The same will be the case whatever be the shapes of the vessels, and the extreme cases of a very large base acted upon by the pressure of water in a very contracted vessel, and of a very small area kept closed by a very small weight when the vessel is very much enlarged above, constitute what used to be called the hydrostatic paradox. Another kind of hydrostatic paradox is the hydraulic press, where water, pumped in with small exertion by a forcing pump of which the plunger is of small diameter, causes a second plunger of very much greater diameter to rise and produce a very great pressure.

Prop. IV. {Archimedes's Principle).— The resultant pressure of a fluid on a body immersed in it acts vertically upwards through the centre of gravity, and is equal to the weight of the displaced liquid.

For, suppose the body removed and its place filled up with fluid as it would be at rest, and imagine for clearness this fluid solidified. The fluid pressures which act upon this solidified fluid are the same as before, and since the fluid is in equilibrium under its own weight and the fluid pressures, the resultant of the fluid pressures must°be a single vertical force equal to the weight of the displaced liquid, and acting upwards through the centre of gravity of the displaced liquid.

Corollary.— If a body float at rest in fluid, the weight of the body must be equal to the weight of the displaced fluid, and the C.G.s of the body and displaced fluid must be in the same vertical line. For instance, the weight of water displaced by a ship is equal to the weight of the whole ship, masts, rigging and all. When a balloon is at rest in the air, the weight of the balloon is equal to the weight of air displaced.

Archimedes’s principle is employed to determine the densities of bodies; for, if w be the weight of a body weighed in a balance in air (strictly speaking in vacuo), and if w ’ be its apparent weight when immersed in water, then w - w ’ , the resultant upward pressure of the water, is equal to the weight of water displaced, and the density relative to water is therefore w w-w ’ '

In very accurate weighings the weight of air displaced must be taken into account, the real weight of a body being its apparent weight in air plus the weight of air displaced. Thus, if in air a pound of lead and a pound of feathers balance in a pair of scales, if placed under a receiver and the air exhausted the feathers would preponderate.

'1 he densities of liquids and solids are also determined by hydrometers, for a description of which consult the article Hydrometer.

The numerical measure of the density of a substance being the quotient of the number of units of mass (or weight) by the number of units of volume, it follows that in a mixture of fluids the density of the mixture is the quotient of the number of units of mass in the component fluids by the volume of the mixture.

The volume of the mixture will in general be the sum of the volumes of the component fluids, except in some cases where the fluids combine chemically with each other. If equal volumes of the component fluids be taken, the density of the mixture will therefore be the arithmetical mean of the densities of the component fluids; but if equal weights be taken, the density of the mixture will be the harmonical mean of the densities of the component fluids, no change of volume being supposed to take place.

Prop. V. To find the resultant vertical and horizontal pressures on one side of a portion of a surface immersed in fluid at rest under gravity.

The resultant vertical pressure is the weight of the superincumbent fluid contained by vertical lines drawn round the bounding curve of the surface to the free surface. If, however, the free surface does not extend over the surface, we must suppose it made to do so by filling up the empty space with fluid. The line of action of the resultant vertical pressure passes through the centre of gravity of this superincumbent fluid.

The resultant horizontal pressure in any direction is equal to the resultant pressure on the plane area traced out on a plane perpendicular to the given horizontal direction by horizontal lines drawn through the bounding line of the surface in the given direction; this plane area is called the orthogonal projection of the surface on a plane perpendicular to the given direction. For, resolving in the given direction the weight of the enclosed liquid and the pressures on the cylindrical surface traced out by the horizontal lines acting in a direction at right angles, the horizontal components of the pressures on the ends balance, which proves the proposition. The line of action of this horizontal pressure passes through the centre of pressure of the plane area {vide “ Centre of Pressure ”).

If a plane area be immersed in homogeneous liquid at rest under gravity, the resultant force acting on one side of the area will be the product of the area and the pressure at the centre of gravity of the area

For, dA denoting an element of the area, and · its depth. the resultant pressure —ffrdk — ρHh which proves the proposition.

General Equations of Equilibrium of any Fluid at rest under any Forces.

If we take any arbitrary origin 0, and three rectangular axes of reference Ox, Oy, Oz, then, if p be the pressure, ρ the density, and X, Y, Z the components of the impressed force per unit of mass at the point xyz, the equilibrium of the fluid in any closed surface S requires, resolving parallel to the axis of x, ≈JJJiVAdxdydz, the integrations extending respectively over the surface and through the volume of the space S, and l, m, n denoting the direction-cosines of the outward drawn normal at the surface element dS.

But by Green’s transformation fflpd^jjj^dxdydz, and therefore ⅛~∙ dxd V dz ”, 1 ∕ZZ* ιXdxdydz, leading to the differential relation g- p x.

ax

Sirniiariy, ⅛ - ,Y , ⅛ - p Z .

The three equations of equilibrium obtained by taking moments about the axis will be found to be satisfied identically.

Hence the space variation of the pressure in any direction is equal to the resolved force per unit of volume in that direction. The resultant force is therefore in the direction of the greatest space variation of the pressure, that is, normal to the surface of equal pressure; and the lines of force must therefore be capable of being cut orthogonally by a system of surfaces, which will be the surface? of equal pressure.

If we neglect changes of temperature, then, the density at any point being a function of the pressure, it follows that surfaces of equal pressure are also surfaces of equal density, and therefore — ⅛, t p (LX

JL ⅛ J_ ⅛ are the partial differential coefficients with respect p dy p dz to x,y,z of some function P of x,y,z such that υ≈J'⅛L ∖ and therefore also X,Y,Z must be the partial differential coefficients of some function - V, called the potential, such that the force in any direction is rate of diminution of V in that direction, and the conditions of equilibrium — ⅛* = X, &c., are equivalent to

^ + "-o,⅛ + ≤-o,≤ + ^.o i dx dx dy dy dz dz or P + V = constant.

If the temperature be variable, then the surfaces of equal pressure and density need not be coincident; but, since the pressure is a function of the density and temperature only, it follows that the surfaces of equal temperature and of equal density must intersect in curves which lie on surfaces of equal pressure.

As an example of the use of the general equations, take the simplest case of a homogeneous liquid at rest under gravity; then, the axis of z being directed vertically downwards, the equations become l⅛ = o JL⅛ = o — —=7

p dx ’ p dy ’ p dz if, and therefore p=τv + gpz, w being the pressure at the level of the origin.

We are here employing a new unit of force—the absolute unit— which is defined as the force which causes unit acceleration in the unit of mass. With the same units of length, time, and mass, the gravitation unit of force is g times the absolute unit of force; for instance, in the equations p = pz, and p=gpz, the 7? in the first equation is measured in gravitation units of force per unit of area, and in the second equation in absolute units of force per unit of area.

Again, suppose the density to vary as any power of the depth, and put

ρ=^ n ; then ^-=gp≈gμz n ; (IZ and p = w+-^ τ 2" +1 .

r n + 1

If the fluid be elastic, and the temperature uniform, p≈kg t and

1 dp _ k dp _ p dz°° p dz & ’ and integrating, 1 og⅛=f-; V p = ∙πek; a k P°-k e ·

Consequently, as we go up in the air, if the temperature be uniform, as the heights increase in arithmetical progression, the pressures and densities diminish in geometrical progression.

If H denote the pressure height, then

7>≈⅛p = 7pH, k and therefore H=—; V and if p υ p. 1 denote the pressures at levels z 1 , z a , then g ι-⅞ P±~ β h , Pa or z 1 -z 2 =H log , Pa

= H × 2'30261og 10 .

Pa

For dry air at 0° C., taking H = 26,000 feet, z 1 -z 2 =60,000 log 10 7h-nearly.

Pa

--(Maxwell, Heat, chap, χiv.)

The Figure of the Earth.

Suppose a fluid mass arranged under the gravitation of its parts in concentric spherical strata, the density increasing towards the centre for stability of equilibrium, and p being the density at a distance k from the centre. If we suppose this mass to be rotating without relative motion of its parts about an axis, and to be slightly disturbed in consequence of the rotation from the spherical arrangement of the strata, we can gain an idea of the figure of the earth on the hypothesis of original fluidity.

If ω be the angular velocity, we must suppose a disturbing function, whose potential is ⅛ω 2 r 2 sin 2 0, added to the gravitation potential, θ being the angular distance from the axis of revolution and of figure. Denoting the zonal surface harmonic of the second degree ⅜μ 2 - ⅜, where μ = cos θ, by Q 2 , the disturbing function may be written

⅜α> 2 r 2 (l-Q 2 ) 5 and we shall assume in consequence that the disturbance of each stratum from the spherical form is also a zonal harmonic of the second degree, so that when disturbed the equation of a stratum will bo r=⅛(l-ieQ 2 ) 5 and e, which is the ratio of the difference of the equatorial and polar axes to the mean axis, is called the ellipticity of the stratum.

The gravitation potential of a homogeneous spheroid of density p, and bounded by r=⅛(l-f e Q 2 ), is the same as that of a homogeneous sphere of radius k and density p, and of a distribution of matter on the sphere of radius k i of surface density - ∣⅛∈Q 2 ρ, neglecting ∈ 2 .

Therefore, for an internal point the potential is

2πp⅛ 2 - 11rpr 2 - ⅛τrper 2 Q 2 ; and for an external point the potential is

4 * 3 8 ⅛

⅛*P--⅛*P^Qa∙

Therefore, for the shell of density p, enclosed by the stratum r=*(l-∣eQ s )

and the consecutive stratum, the potential <ZU^

= 4π pkdk — for an internal point, and

= 4πp-^- -

for an external point; and, therefore, for any point in the interior of the whole mass on the stratum r=⅛(l -geQ 2 ),

∕*κ ∕-K J, τ⅛τ n Γh d V-4,f p m-Λ"' 2 Q≈∕ f⅛ rft +τ->'∙'⅜∕ ∣ ⅛p ’ ∙'>< a ∙ M denoting the mass enclosed by the stratum of mean radius k, and K being the mean radius of the exterior stratum. Neglecting < 2 , ∕- κ ∕- κ de

U = 4π∕ pkdk-⅛pdC 2 ⅛J p^flk *J k ∙J k

+⅛(1 + ∣.Q.)-.v⅝∕⅛(^∙

The equation of equilibrium is, since the force in any direction is the rate of increase of the gravitation potential,

∕"⅛ _ U - ⅜w 2 r 2 (l - Q 2 ) = constant; J P and, supposing surfaces of equal pressure to be also surfaces of equal density, we must have to our order of approximation

U + ⅜ω 2 ⅛ 2 (l - Q 2 ) = constant, over a surface of equal density, OŸ, equating to zero the coefficient of Q 2 ,

Dividing by k 2 , and differentiating with respect to k, ω 2 disappears, and we obtain fc3 ⅛F 5 - 3;fc2Me - 4πpfrr ' i + 5

and, differentiating again with respect to k, ⅛⅝ ga 4,y g - 6*Me - 4πfc 8 ⅛ = 0 , dk 2 dk m j⅛M∙-(' β + f⅛i⅛')M<-0 ι dk 2 ∖ M dkj a differential equation of the second order to determine Me, and therefore e, provided we know what function p and therefore M is of⅛.

Properly speaking, from the elasticities of the substances of the various strata we should know the relation between the pressure and the density, and then from the conditions of equilibrium of the strata when undisturbed and in this spherical shape we could determine what function p is of k,- the pressure and density at a stratum with our approximations being the same in the undisturbed and disturbed states; and therefore

1 a⅛> M

p dk = & '

Laplace, on the assumption that the cubical elasticity is double the pressure, and therefore the pressure proportional to the square of the density, has integrated the differential equation for the ellipticity,—his assumption amounting to putting

4τr⅛ 2 dp 2 , - iτ - -, r r = -q i , a constant.

M dk ’

More generally, to make the equation for the ellipticity integrable, we may put 4πk i dp _ ∕ k ∖ n

M dk~ ∖ a J ’ where a and n are constants, the negative sign being taken because ⅛ is negative for stability. This assumption reduces the differential OJC equations for p, M, and Me to equations reducible to Bessel’s <lifferential equation, and therefore p, M, and e can be expressed by Bessel’s functions.

Laplace’s assumption amounts to putting τι=2, and then the Bessel’s functions which occur are of the order ⅜, f, and ⅞.

Then p==<r s * n .^ , where σ is the density at the centre of the qk earth, and therefore

M = ^7U(sin qk - qk cos qk),

∕ 3 ∖ 3

. _5ω 2 sin a - a cos a ∖ q 2 k 2 ∕ 8 ^ n qk C0S and * - 8ιrσ∙ 2 „ 1 , 1 ' sin qk-qk cos qk ’

- s sin 2 α—⅞ sin a cos a * a a a i a where a=<∕K, and the value of g ’ must be determined from the condition that the mean density of the earth is twice the density at the surface.

Centre of Pressure.

When a plane area is exposed to fluid pressure on one side, the resultant force experienced by the area is a single force perpendicular to the area, the sum of all the separate pressures, and acts through a definite point called the centre of pressure.

If p be the pressure at the point xy, the axes being taken in the plane of the area, then the resultant force B≈ffpdxdy, and, if x, ÿ denote the coordinates of the centre of pressure, _ ff^pdxdg _ ffypdxdy X — , V — ffpdxdy ffpdxdy '

The centre of pressure is therefore the C. G. of the plane area, supposed a lamina of variable density p. If ρ is uniform, the centre of pressure is obviously the C.G. of the area.

For a homogeneous liquid at rest under gravity, p is proportional to the depth below the surface, thatis, proportional to the perpendicular distances from the line of intersection of the plane of the area with the free surface of the liquid. If the equation of this line be x cos a + y sin α-^=0 , «... . >⅛- x cos a - y sin a)dxdy then s "^7λ »

JJ ∖ P-x cos a- y sin a)dxdy - ∕∕ylp- x cos a - y sin a)dxdy ffp-x cos a- y sin a)dxdy

If the origin be taken at the centre of gravity of the area, and if the axes be the principal axes of the area, then ffdxdy= A, the area, ffxdxdy - 0, ffydxdy - 0, ffxydxdy ≈ 0 and ffypdxdy - Aα 2 , ffx l dxdy =Aδ 2 ;

« and ⅛ being the semi-axes of the momental ellipse of the area.

Therefore r

α 2 J 2 .

x= cos α, y≈ sin a; P p and therefore the centre of pressure is the antipole of the line of intersection with the free surface, with respect to the momental ellipse.

The centre of pressure of a rectangular area, with a side in the free surface, is at ∣ of the depth of the lower side; of a triangle with vertex in the free surface and base horizontal is ∣ of the depth of the base; of a triangle with base in the surface is ⅜ the depth of the vertex.

Metacentre. y Ns have found from Archimedes’s principle the conditions of equilibrium of a floating body, and we must now examine whether the equilibrium is stable or unstable.

Let ACB (fig. 7) represent the cross section of a floating body, like a ship, and let G be the C.G. of the body, and II that of the liquid displaced, supposed homogeneous. Let the body be turned through a small angle in the plane of the paper, whose circular measure is θ, so that the volume of liquid displaced remains the same. Then, if W denote the weight of the body, and therefore also of the liquid displaced, the resultant force due to the liquid on the body in the displaced position is a vertical force W acting vertically upwards through H', the new C. G. of the displaced liquid.

In order that the volume displaced may remain unaltered, it is necessary that the line of intersection of the two planes of flotation AB and DE should pass through the C.G. of the area of the curve of flotation. For dA denoting an element of the area A of flotation, and x its distance from the line of intersection of the planes AB and DE, the element of volume traced out by dA when the body is displaced being θxdA, we must have fθxdA = 0, or fxdA≈Q∙, which proves that the line of intersection of the planes of AB and DE passes through the C.G. of the area of flotation.

The force W acting upwards through H' is therefore equivalent to an equal force W acting upwards through H, and a couple, due to the moment of the weight of AOD upwards and BOE downwards, the moment of which is therefore, in gravitation units, pf9x i dA = ρθAk 2 , k denoting the radius of gyration of the area A about the line whose projection on the plane is 0.

Since HH' is the arc of a curve, such that the tangent at H is parallel to AB, therefore HH' is ultimately a straight line perpendicular to GH, and

W . HH ’ = pβA⅛ 2 , or if V be the volume of liquid displaced.

If H'M be drawn vertically upwards to meet HG in M, then M is ultimately the centre of curvature of the locus of H in the body, and is called the metacentre, and θ V

If M lies above G, the fluid pressure tends to restore the body to its position of equilibrium, and the equilibrium is therefore stable; but if M lies below G the equilibrium is unstable.

Generally we see that, if planes be drawn cutting off constant volumes from a solid, the principal radii of curvature of the locus ∖ k 2 of H, the centre of gravity of the volume cut off, will be and Ak 2 -f-, where V is the volume cut off, A the area of the cutting plane intercepted by the surfaces, and k 1 , k 2 the principal radii of gyration of this area about the centre of gravity of the area. The vertical through H' will intersect HG only when the plane of displacement is a plane of symmetry, that is, if it is perpendicular to k 1 or k, 1 .

Generally for finite displacements in a plane the locus of M will be the evolute of the curve which is the intersection of the plane with the locus of H.

The surface the locus of H is called the surface of buoyancy, and the surface which is the envelope of the planes of flotation is called the surface of flotation.

If r 1 , r, i be the principal radii of curvature of the surface of buoyancy, then r =N c L =Iι. r -A⅛-½ 1 V V ’ 2 V “V *

If R 1 , R, 3 be the principal radii of gyration of the surface of flotation, Mons. E. Ledert has proved that p dl 1 Υdr 1 dl 2 , Vrfr 2 1 1= dv -ri+ czv ’ Rä_ dv“ri+ rfV ‘

For small oscillations, if we suppose the liquid pressure the same as if the liquid were at rest, the body oscillates as if the surface of buoyancy moved upon a horizontal plane.

Next, suppose a body completely immersed in heterogeneous liquid, which must for equilibrium be arranged in horizontal strata of equal density, and suppose p=∕(z) the density of the liquid at any depth z ; let G be the centre of gravity of the body and H that of the liquid displaced (fig. 8). When the body is in equilibrium, we must have G and H in the same vertical line, and fffpdxdydx-C®, the weight of the liquid displaced, using gravitation units of force again.

Suppose the axes of coordinates fixed in the body, and take GM as the axis of z ; suppose the body turned through a small angle θ about the axis Oy, and let H' be the centre of gravity of the displaced liquid in the new position. The density of the liquid displaced at a point P, whose coordinates are x, y, z, is now

∕(NP)=j(z cos θ + x sin θ) =f(z + xθ) ≈f(z') + θxf(z), neglecting 0 2 ; and, to the same order of approximation, mr JTf x {∙A≈) + ftr ∕(z )} dxdydz j w a fffiH , ^dxdydz

~ 0 w ’

since fffxflf)dxdydz=Q,

H lying in the axis of z.

If the vertical through H' meet Oz in Μ, M is called the metacentre, and is the centre of curvature of the locus of H, and ττι , HH" fffx t f ’ (f)dxdydz J kki ^ dz θ W W ’

if A⅛ 2 denote the moment of inertia of a horizontal plane section of the body at a depth z, about the line of intersection with the plane of yz.

We have here supposed that H ’ lies in the plane of xz ; but thia will only be true for two directions of displacement.

In general for any displacement, if x, ÿ, z be the coordinates of H', - _ fff ’ M 2 ) + dxdydz fffxyf(z)dxdydz y W β W~

and therefore ÿ= 0, only when the axis of y is a principal axis of the body, supposed of density ⅛ .

CLZ

When p is discontinuous, as in the case of the body floating in homogeneous fluid, then the integral ι ∕A⅛ 2 dp will have a term Ak i p, due to the discontinuity at the surface, and the rest of the integral will vanish, because ⅛ = 0.

dz

For a body floating wholly immersed in two liquids, the upper of uniform density p and the lower of uniform density p, Tension of Flexible Surfaces exposed to Pressure.

In hydrostatics it is usual to determine the circumferential and longitudinal tension produced in a thin circular cylinder, due to uniform internal pressure, and also to determine the tension of a spherical surface, like a soap-bubble, due to the excess of the internal pressure over the atmospheric pressure.

Let r be the internal radius of the cylinder, e the thickness (supposed small), p the internal pressure, and Z,i'the circumferential and longitudinal tension per unit of area caused by the pressure p.

If we suppose the cylinder divided into two halves by a diametral plane, and consider the equilibrium of unit length of either half under the resultant of the fluid pressure over the half-cylinder and the tensions at the ends of the diameters, the resultant of the tensions must balance the resultant of the fluid pressures, which is the resultant pressure on the diametral plane, since the resultant of the uniform pressure on a closed surface is zero. Therefore 2te*=2pr; t r or —=— .

p e

To determine the longitudinal tension t , , consider that the resultant pressure on the end of the cylinder, which is jo.πr 2 , is balanced by the resultant of the tensions round a circumferential seam, which is t ’ e.2πr ; and therefore 2πt ’ er = πpr*; t ’ i r .

or — = ⅜~ > p ∙ i e and therefore t ’ =⅛t.

Thus in a boiler, half an inch thick, and 3 feet in diameter, a pressure of 150 lb to the square inch makes Z = 5400, i ’ =2700. .

For a sphere of internal radius r, and small thickness e, supposing it divided by a diametral plane, then the resultant tension round the circumference, te. 2ιrr, must balance the resultant fluid pressure p.- ∣ rr' 2 , supposing ρ the excess of the internal over the external pressure; and therefore

2πfer = πyr 2 ; i 1 r or — = ⅛

p e

In the experiment with the Magdeburg hemispheres, where two hemispheres were joined by an air-tight joint and the air say half exhausted, then, with a pound and inch as units, - p=7'5, supposing 15 the atmospheric pressure; and if the diameter of the hemispheres be 3 feet, then r = 18; and the force required to separate the hemispheres would be η × 18 2 × 7’5=7634 ft>.

The tension of flexible surfaces is considered more fully in the article Capillary Action.

PART II.—HYDRODYNAMICS.

In considering the motion of fluids we shall suppose them non-viscous, so that whatever be the state of motion, the stress across any section is normal to the section, and therefore the stress is a pressure and the same in all directions about a poiut, as in Hydrostatics.

Two methods are employed in hydrodynamics, called the Eulerian and Lagrangian, although both are due to Euler; in the Eulerian we fix our attention on particular points of space, and observe the changes of pressure, density, and velocity which take place there, and in the Lagrangian we follow a particle of fluid and observe its changes. The first may be called the statistical and the second the historical method, according to Prof. J. C. Maxwell. The Eulerian method is generally employed except where the fluid has a moving boundary.

The Eulerian Form of the Equations of Motion.

The first equation to be established is the equation of continuity, which expresses the fact that the increase of matter inside a fixed surface is due to the flow of fluid across the surface into the interior, supposing there are no hypothetical sources or sinks in the interior of the surface.

Lemma.— The quantity of fluid, estimated in units of mass, which flows across a plane area in a given time is equal to the product of the area, the density, the time, and the resolved part of the velocity perpendicular to the area.

For if q be the velocity, the quantity of fluid which flows across the area A in the time t will form an oblique cylinder of length qt, with its generating lines in the direction of motion (fig. 9). If θ denote the angle between the normal to the area and the velocity, the mass of the cylinder — Aptq cos θ, which is therefore the flux across the area A in the time t.

Generally, if S denote any fixed surface, M the mass of the fluid inside it, and θ the angle which the normal drawn outwards at any point of the surface makes with the velocity q at that point, then — == rate of increase of quantity of fluid inside the surface per unit of time = flux across the surface per unit of time = -/ P<l cos 0 dS; or ~zr+ fa cos βdS = 0. ... . . (1), toi fc ∕

the integral equation of continuity.

In the Eulerian equations of motion u, v, w are taken to denote the components of the velocity q parallel to the coordinate axes at the point xyz at the time t; u, v, w are therefore functions of x, y, z, t the independent variables, and d is used to denote partial differentiation with respect to these four independent variables.

To transfer the integral equation into the differential equation of continuity, we require Green’s transformation, namely, + ⅛ + + +; or, individually, ⅛ d8 > J7f-JJ)rn v dS, Jff^dxdydz≈ffnζd^; where the integrations extend respectively through the volume and over the surface of a closed space S; km,n denote the directioncosines of the outward drawn normal at the surface element dS, an 4 ξ> 7> C are continuous functions of x, y, z.

The integral equation of continuity may now be written fff⅛d*!*ιl y +Jβ jf u + mpυ + ηpw)dS — 0, which by Green’s transformation becomes leading to the differential equation of continuity

⅛ ⅛ ⅛ ⅛J, dt dx dy dz '

It is customary to establish the differential equation of continuity immediately by considering the fluid which enters and leaves an infinitesimal parallelepiped, whose edges are dx, dy, dz, in the time dt, but this requires us to suppose in succession each of the elements dx, dy, dz, though infinitesimal, to be infinite compared with the other two, and with the infinitesimal element of time dt ; this violation of the principles of the differential calculus is avoided by establishing the equation in its integral form first.

We shall establish the equations of motion in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the forces acting throughout the space S and by the pressures acting at the surface S.

Taking the fixed direction parallel to the axis of x, the rate of increase of momentum in that direction per unit of time, due to the fluid which crosses the surface, is ffCpC l + mpuv + npuw)d^>, which by Green’s transformation

J(⅞÷⅞÷⅞κ

and, adding this to the rate of increase of momentum per unit of time of the fluid inside the surface

J z '⅛' ω ≈*-

we obtain, as the total rate of increase of momentum per unit of time of the fluid which fills the space S, JJJ ∖ dt dx dy dz ) in the direction of the axis of x.

The rate of generation of momentum in this direction by the forces of components X,Y,Z per unit of mass in the interior is fffpXdxdydz, and by the pressures at the surface is

√⅛ , which by Green’s transformation is equal to

-fff i ^^∙

and therefore ΓΓΓ( d ^ + dp j* + d Pff + ⅛ ∖ dxdydz JJJ ∖ dt dx dy dz J leading to the differential equation of motion dpu dρu 2 ι dpuv . dpuw v dp /ox

Λ-+Ί& + -¾- + ⅛- ,,χ -⅛ · < 3 > i with two similar equations in y and z.

These equations may be slightly simplified; for dpu dpu 2 dpuv dpuw dt dx dy dz ( du du dιι du ∖ = Pl -77 ÷ w-r- + v τ^ + w-υ- I ∖ dt dx dy dz J + v( dp ι' dpu i dpυ i dpw ∖ ∖ dt " r dx dy dz J ’ which reduces to the first line, the second line vanishing in consequence of the equation of continuity; and therefore the equations of motion may be written du du du du yr 1 dp ... — +u—+v— +w-≈X-- . . . (4); dt dx dy dz ρ dx with the two similar equations dv dv dv dυ , τ 1 dp

- π - +u 3 - + v- r + w- r = Y ≠ . . . (5), dt dx dy dz p dy dw dw dw dw r7 1 dp ro .

- 77 + tt- 5 -+v-τ- + w-,- = Z √- . . . (6).

dt dx dy dz p dz As a rule these equations are established immediately by determining the component accelerations of the fluid particle which is at xyz at the instant of time t considered, and saying that these accelerations reversed, combined with the impressed forces per unit of mass, will form a system in equilibrium according to D’Alembert’s principle.

To determine the component accelerations, suppose F any function of x, y, z, t, and determine the rate of change of F per unit of time

DF

for a moving particle; denoting this change by —— , we have (Lt

DF 1 F(¾ + ¾δl, y + vbt, g÷wδ<, l + δi)-F(ar,y,z,i) dt St dF dF dF dF =τn + u τr + υ τr + w zr > dt dx dy dz so that

D d d d d - r . = λ + u Υ~ + υ ι~ + w j~∙ dt dt dx dy dz

~ is called particle differentiation, because we follow the rate of 1 i x·L , · 1 ., 1 ., . . , . dF dF change of the particle as it leaves the point xyz ; but — , —, cot ctx

— i IZ are the rates of change of F at the time t at the point dy ’ dz xyz, fixed in space. Consequently the component accelerations parallel to the axes of coordinates of a particle of fluid are du du du du

-j 7 + u- r - + v- r - + w τ , dt dx dy dz dv dv dv dv dt dx dy dz dw dw dw dw dt dx dy dz leading to the equations of motion last established.

If F(z,y,z,i) = 0 be the equation of a surface containing always the same particles of fluid, it follows from the preceding that

∞-o.

(Lb dF dF dF dF n or — + w— + v-- + w— ≡0 (7).

dt dx dy dz

This is called the differential equation of the bounding surface, as particles of fluid once in the bounding surface always remain in it.

To integrate the equations of motion (4), (5), and (6), suppose the impressed forces due to a potential V, such that the force in any direction is the rate of diminution of V in that direction, then v= _£V 7=-^Σ · dx ’ dy ’ dz ’ and putting dw d¾ 9t du dw 9 dv du oz dy dz ξ , dz dx = y ’ dx dy~ the equations may be written

*- 2<+2w , + ®.0 (8),

⅜-^ + 2 < + ^-< ’ <’)

⅞- 2 ""> + 2 't + ⅛-o .... (10), where R = f⅛T + V + —⅞ ,2 , J P 2

and ⅛ 2 = u 2 + v 3 + w 2 (Lamb, Motion of Fluids, Appendix D; also Proc. London Math. Society, vol. ix.).

A stream line is defined to be the actual path of a particle, and a line of flow to be a line such that the tangent at every point is in the direction of the velocity at the point; the stream lines and lines of flow are coincident only when the motion is steady; and when the motion is irrotational, the lines of flow are orthogonal to the surfaces obtained by equating the velocity function to a constant.

A vortex line is defined to be a line whose tangent at any point is in the direction of the resultant ω of the component angular velocities ξ, ∙η, ζ at that point; and ω is called the spin (Clifford, Kinematic). ξ,η,ζ are called the components of molecular rotation (or spin) at xyz, for a reason to be explained afterwards; and when they vanish the motion is said to be irrotational, and a function φ exists, called the velocity function, such that dφ dφ dφ u=-f-, v≈~r-, w=-f-', dx dy dz and, generally, the velocity in any direction is then the space variation of φ.

When the motion is irrotational, equations (8), (9), and (10) become

∕⅜ + ⅛-o,⅛ + ⅛-o ⅛ + ffi. 0 . dxdt dx dydt dy ’ dzdt dz ’ and therefore ⅛ + R = H, dt or ∕⅛ + v + L 2 + ⅛= h , J ρ 2 i dt a constant throughout the fluid, which may, however, be a function of the time.

If, however, the motion be steady, that is, if the velocity at any point of space does not change with the time, then

—=o —=o dt υ ’ dt υ ’ di υ ’ and the equations become

≤⅛-2<+2wη = 0, dx

⅞^-2wξ + 2<=0, dy ~~2uη + 2νξ^0∙ dz qλ+ I ιαt ,,dR . dR r dB so that u— p v—- + wj -— =0:

dx dy dz i dR dR .dB λ

⅛ + * r ⅜ + ⅛~° 5 and therefore the surface R, = constant, contains both stream lines and vortex lines; and therefore

V + ⅜j 2 = constant (11)

along a stream line, and along a vortex line; and if the motion is irrotational, the constant is the same for all the space filled with the fluid; for then dR dR _ dR

da: ’ ^⅜"° ’ dΓ

Taking the axis of x for an instant in the direction of the normal to the surface R=constant, then w=0 and ξ=0, and (8), (9), and (10), if the motion is steady, reduce to =2tζ - 2wτj = 2g ’ ω sin Ö, where θ is the angle between the stream and the vortex line.

It is sometimes convenient to use moving exes of coordinates in Hydrodynamics, and the equations of motion then become du , .du . . du

+ (w + 2/«3 - + \ V + z <∙>l - ~ff f , . du ~ r 1 dp + (w + XUa-yaiO -j-=Λ , v 2 i, v dz p dx’ with two similar equations; w 1 , ω 2 , ω 3 denoting the component angular velocities of the moving axes, and u, v, w the components of the velocity of the fluid in space at the point xyz at the time t parallel to the axes.

For if g denote the component velocity of the particle xyz at the time t in a direction fixed in space whose direction-cosines are l, m, n, then q=lu + mv + nw; and in the infinitesimal element of time dt the coordinates of the particle will have become x + (u + yω 3 - zω 2 )dt , y + (v + za> 1 -Xa> 3 )dt , z + (w + Xa> 2 - yω 1 )dt;

., . Do dl dm dn so that -57= "J7 “ + -JT ® + w dt dt dt dt fdu . .du , . du du ∖ + l ^+(u + yω 3 -zω 2 )-^ c + (v + zωι-xω 3 ) -^+{w + Xa> 2 -yω 1 )^^ ( dv , ,dν , .dv . ∖ dv) + m )dt + ^ u + yω3 ~ zω ^dx + ^ ν+2 " 1 ~ xω ' i ' dy + ( ' W + Xa>,> ~ ya>1 ’dz) {dw , . dw , . dw . , , x dw) + n ∖ dt + + y “ 3 ~ ZW3) dx + f ' v+zωι ~ xω ^ dj j + ^ w+zω ^~ ywι >Tz ∖ '

But, since l, m, n are the direction-cosines of a line fixed in space, dl dm 7 dn 7 -^ t = mω 3 -nω 2 , -^ = nw 1 -lω 3 , -^- t ≈lω 2 -mω 1 ∙,

. Do , ( du , .du .du ’ · di * i -fι-' o ∙>>s + wω 2 + {u + yωs→<*jjfa+W + zω 1 -Xω 3 )⅛ l I χdu ∖ + {w + Xω 2 -yω 1 )-^ I

( dv , .dv . .dt> + m i dt~ wω1 + uω3 + ^ u + yω3 ~ zω ^dx + ^ θ + zw1 ~ xω3 'dy + (w + xω 2 -yω 1 )^ ( ( dw , x dw . , λj χ dw + n ]^0(- uω ∙2 + νω ι + ( u + ll ω 3 ^ zw ν^dj c + + 2 " 1 - dÿ

∕ flιv ) + {w+χω 2 -yuO-^ I

= z(χ _ J_ ⅛) + τn(γ - - + n(z - - ∖ p dx) ∖ P dy) ∖ p dzJ for all values of l, m, n, leading to the equations of motion.

As an example of the use of moving axes in hydrodynamics, consider the liquid filling the ellipsoidal case x a , 2∕ a , z a 1 α> + π + π= 1 '

and first suppose the liquid to be frozen, and to have component angular velocities ξ, η, ζ about the axes, then u= -yζ+zη, v≈-zζ+xζ, w= -xη + yξ.

If the liquid now be suddenly melted, and additional component angular velocities ∩ 1 , ∩ 2 , ∩ 3 communicated to the ellipsoid about the axes, then (vide infra) c 2 - α 2 a 2 - b 2 u^-yζ + z η + 2 ∩ 8 z + ^⅛2 ω 33∕ » α 2 _ J2 ⅛ 2 - C 2 v≈-zξ + xζ+ +jrj√ 2 ι z ’

J2 q ∣ jL u>≈-χv+ y^^Αy+^ i ^> and if U, V, W denote the component velocities of the liquid relative to the axes,

ττ 1 2α 2 2α 2 o U =m + 2/W3 _^ 2 = __o 32/ _ __ω 2 ζ , 2b 2 2b 2

2c 2 2c 2

W = w + zω 2 - y ω ^^-^ e ---ii 1 y, since ®i = Oi + f, ω 2 = Ω 2 + η, ω 3 = Ω 3 +< l , w 1 , ω 2 , ω 3 being the component angular velocities of the axes.

We see that u ⅛÷ v ⅜÷"'⅛- 0 ∙

so that a liquid particle always remains on a similar ellipsoid.

The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, are i∣+⅛ + ~ t , +w + u→V→Wy=0 (1), P dx at i i dx dy dz ' h

√ È + B ^ ⅛-^ι + ^3 + u ⅞ +V^+W^ = 0 (2 )> P a y dt i dx dy dz '”

1 dP , ∕‰ , d w dw ^ τ dw - nτ dw λ ∕λ∖ j„ ÷ '-' z + r j f uω% + Vω 1 + U^5—t^V-j—hW-j- = 0 (3): P dz dt l dx dy dz '” " tere u * 1 A⅛∙

b -* m ∕'⅛> c -* m Z⅛⅛- and P 2 =(α 2 + λ)(δ 2 +λ )(c ≡ + λ ).

With the above values of u, v, w, U, V, W, the hydrodynamical equations are of the form

1 dp

7 ⅛+ aλi +<∞+⅜+^≡o, 1 dP τ> p dy + B ll +hx + ßy+f z =o,

1 dp 7 ^ a +cz +s×+fy+yz=o.

The component accelerations in space of the liquid particle at xyz parallel to the axes are therefore ax + hy + gz, hx + βy+fz, gx+fy + yz∙, and by the dynamical equations the rates of change of angular momentum about the coordinate axes are zero, and therefore 2m{(gx+fy+yz)y-(hχ + βy + f z ) e ∣ = 0; 0r ∕2w(y 2 -z 2 ) = 0, 0r Λδ 2 -c 2 ) = 0j and therefore y≈ θ .

and similarly g and h vanish.

Therefore the hydrodynamical equations become

1 dp , l p dx + ( A + a ) x ’~ 0 >

1 ⅛ τ , 7 ⅜ + < b+ ^≈-°> 7^+(C+y)z-Q i where ^ 4c 2 ( c 2 -α 2 ) 2 _ ∕c 2 -α 2 ∖ 2

(c 2 + α a ) 2 i * a ∖c 2 + α 2iia 1, )

^ 4⅛ 2 (α 2 -⅞ 2 ) ∕α 2 -¾ 2 y (a 2 + b 2 ) 2 " 3 ∖ a 2 + lf i3+ζ J ’

4 a 2 (a 2 -⅛ 2 ) ∕a 2 -⅛ 2 γ

fi (a 2 + b 2 ) 2 ω ≡ ∖ a 2 +VP 3 ~ ζ ) w 2 -⅛ P 2 - c2 ω,+Λ 2 (δ 2 +c 2 ) 2 ° 1 " ∖δ 2 77 2 ° 1 + i J ’ W 2 -c 2 ) ⅛- c ≡ y y (δ 2 +c 2 ) 2 Ω1 ∖⅛ 2 +7 2 1 ^ ξ )

4a 2 (c 2 -a 2 ) fc 2 -a 2 o t ∖ 2 π , v c (c 2 + a 2 ) 2 ° 2 (^7^ Ω2 + ”)·

Therefore, integrating,

^~+ ⅜ ∣ (A + a)aj 2 + (B +j8)y 2 + (C + γ)2 2 I = constant; and therefore the surfaces of equal pressure are the similar and co-axial quadrics

(A + a)x 2 + (B + β)y 2 + (C + γ)z 2 =constant.

If we can make α, β, y constant, and (A + α)α 2 =(B + β)b 2 =C + γ)c 2 , the surfaces of equal pressure are similar to the external case, which can therefore be removed without affecting the motion.

This is the case when the axis of revolution is a principal axis; and, supposing it the axis of z, then

Ω 1 = 0, Ω 2 = 0, ξ = 0, η = 0.

If in addition we put Ω 3 =0, or ω 3 =ζ, we obtain the solution of the particular case considered by Jacobi, of a liquid ellipsoid of three unequal axes, rotating about its least axis in relative equilibrium; or, putting a=b, we obtain Maclaurin’s solution of the equilibrium of a rotating spheroid (Cam. Phil. Soc. Proc., iii.).

Equation (11) is called Bernoulli’s equation, and for homogeneous liquids under gravity is a very useful principle in hydraulics; the equation may be established from first principles by considering the energy which enters and leaves a certain portion of a tube of flow. (Lamb, Motion of Fluids, p. 23).

If homogeneous liquid be drawn off from a vessel, so large that the motion of the free surface may be neglected, then Bernoulli’s equation becomes, P being the atmospheric pressure and h the height of the free surface, — + gz+ iq 2 = -+gh; P P and in particular, for a jet issuing into the atmosphere, where - p = Γ ’, ⅛ 2 =g(h-z)∙, or the velocity is due to the depth below the free surface. This is Torricelli’s theorem (Be Motu gravium Projectorum, 1643).

If we suppose fluid to escape according to this law from a large closed vessel in which the pressure is p where the motion is insensible, and neglect the variations of velocity due to variations of level, p being sufficiently great, then

If A be the sectional area of the jet (at the vena contracta), the quantity of fluid which escapes per unit of time is A w ≈A√{2 p ( 2 >-P)}; the momentum per unit of time is

Ap⅛ 2 =2A(^-P); and the energy per unit of time is ⅛A p < z 3 = A( 7 >-P)* λ ∕- ∣ ∙

Suppose, for instance, two equal pipes leading one from the steam space and the other from the water space of a steam boiler at a pressure p, and suppose Torricelli’s theorem to hold for the rate of efflux of the'steam and water, then, if σ denote the density of steam, and p the density of water,

∩. The Velocity of steam jet _ ∕ p

The velocity of water jet ∖ ' σ ’ The quantity of steam jet _ /7

The quantity of water jet ^v p ’

,0, The momentum of steam jet _ j ‘

The momentum of water jet

The energy of steam jet _ ∕f_

( ' The energy of water jet σ

For instance, with steam at 8 atmospheres, or 120 lb to the square inch,

= 15 nearly.

—(Rankine, Steam Engine, appendix).

These principles assumed enable us to give a general explanation of the working of Giffard’s injector. For, if the steam jet and water jet be directed at each other, with a small interval between, the superior energy and equal momentum of the steam jet will overcome the waterjet, and the steam will flow back into the boiler. But the steam jet, without losing its momentum, is capable of being mixed with water to such an extent as to become a condensed water jet, moving with the velocity of the water jet, and still entering the boiler, a valve preventing the reversal of the motion. Consequently, the amount of water carried into the boiler per unit of time will theoretically be at most the difference between the quantities which would escape by the water and the steam jets, and therefore

=A√(^-P)(√⅜-√2σ); and the efficiency of the injector, that is, the ratio of the water pumped in to the quantity of steam used, will be

√÷-

the efficiency of a pump being -£- .

(Γ

With C.G.S. units, and a pressure of 8 atmospheres, for instance,

^>-P = 7×10 6 very nearly, a ∕-£- = 15, and p = 1. z v (Γ

Therefore, if the diameter of the nozzles of the injector bed centimetres, the delivery in grammes per second

= 1 πd 2 √14 · 10 3 (l - √⅛)

= 2180d 2 ; and since 1 gallon is 4541 cubic centimetres, the delivery in gallons per minute .

2180 × 60 z72

4541

= 28 ’ 78d 2 nearly.

The Lagrangian Form of the Equations.

Here the independent variables which define a particle are the time t, and a, b, c, the initial values of the coordinates x, y, z of a particle of fluid (or else functions of the initial coordinates, but it is best to consider a, b, c as the initial coordinates themselves).

Here x, y, z do not refer to a fixed point in space, but are the variable coordinates of a fluid particle, and are functions of a, b, c, t, the independent variables; and consequently dx du dz . U = -77 , v≈~ir > w ≈~3ι > dt dt ’ dt and the component accelerations of the fluid particle are du dv dw dt ’ dt ’ dt

Consequently the equations of motion, assuming the existence of the potential V, and putting P = ∕^y > an <i P + V = Q , ar dQ d⅛ dQ d-o dQ .

dx dt ~ ’ dy dt ’ l dz dt ’ or midtiplying by -⅛L, , , and adding, da da da dQ du da; ι dυ dy dw dz _ da dt da dt da dt da ·’·()> with two similar equations dQ d⅛ da; d¾ dy ι dw dz db + dt db + dt db + dt ^db~ 0 · · (2) ’ dQ du dx dυ dy dw dz _ dc^ + ^dΓ ^dc + ^dt ^dc + ^dΓ dc~ 0 · * '

Since the elementary parallelepiped whose edges were initially da, db, de, becomes strained into a parallepiped of volume

⅞⅛‰δdc d(α,δ,c)

therefore the equation of continuity is d(a;,y,z) p d(a,b,f) p ° ’ or, if the fluid be a homogeneous liquid, d(x,yf) i d{a,b,c)

When a, b, c are not the coordinates of a point actually occupied by the fluid particle, this equation of continuity must be replaced by d ∣ χ f(x,y,z) I =0 dt (f α(α,δ,c) )

Cauchy's Integrals of Lagrange's Equations.

Eliminating Q by differentiation between (2) and (3)

d⅜ dx d 2 u dx^^v dy d⅛ dy ^ dl·w dz d'⅛ dz _ θ

dtdb de dtdc db + dtdb de dtdc db ^ t ^ dtdb de dtdc db ’ and integrating with respect to t, du dx du dx ^dv dy dυ dy ^dw dz dw dz _dw 0 dv 0 db de de do db de de db db de de db db de ’ m 0 , ⅞> ιvo being the initial values of u, v, w, and a, b, c the initial values of x, y, z. Now du _ du dx + du dy du dz da ~ dx da dy da r dz da ’ and therefore idw dυ ∖ d(y,z) ∕du dw ∖ d(z,x) ∕dv du ∖ d(x,y) dw 0 dv 9 . ∖ dy ~ dz Jd(jb,c') ∖ dz dx)d{b,c) + ∖ dx dyjd(b,c)~ db de ’ or putting dw__dy__ 9 d¾ _ n dυ du _ 9 > dy dz ξ ’ dz dx y ’ dx dy ’ flM , d M , f d{x,y) r . a(j>,c) ^ r η d(j>,c) ^ r i d(δ,c) ~ ξ ° ’

with two similar equations t d(y,z) d(z,x) > d(g,y)

*d(c,α) 1, d(c,α) M(c,α) 1,0 ’

t d(y,¾) d(z,⅞) .d(¾,y)

⅛δ) + v d(a,b) + i >d(a,b) 510 ’ τ v dx dx . dx Therefore j ⅛=⅛ +η °dδ + ζ °~df ’ τ t dy ↑ „ dy ∖ r dy jη ~t°da +η °db + ζ °dc , . dz dz . dz ^~ ξo da + y °db + ζ °dc ’ where τ , d(≈,y,z) ∙.

wtiere j ~d(α,δ,c) ’

or, since J = ⅛, therefore P

J.⅛Io. d Ξ + 2lo, d Ξ + A ( 4 ∖ p p 0 da pQ db p 0 de

A = Jθ- dy + H9 dy , + ^ dy (5)

p p 0 da p 0 db p 0 de , '

_L = Jo. d L + jHo ⅛ + A d f, (6).

P Po da Po d l ’ Po dc

Consequently if ξ,η,ζ are ever zero they are always zero, and then dφ dφ dφ u=-f-, v ≈-fdx dy dz and a velocity function φ exists.

For instance, if motion be generated from rest in a non-viscous fluid under forces due to a potential, a velocity function always exists, and the discovery of this velocity function for different cases is one of the chief problems to be solved in hydrodynamics.

A good example of the use of the Lagrangian equations of motion is given by the state of wave motion in deep water invented by Rankine; he puts x = a + ce c sin Ç ωt + , y=β+ce c cos + ^⅛) J and therefore the coordinates of a particle are given in terms of t and « and β.

But a and β are not the initial coordinates of a particle; for putting t =0, then the coordinates are . a a=a + ce c sin — , c a b≈β + ce c cos — .

c

Therefore ⅜y) - 2 ∣

^(α,j8)

a ∏d ⅛y.ι- e -4.

<Z(a,j3)

therefore ’ = 1 5

d(a,b) and the equation of continuity is satisfied.

Cauchy’s integrals reduce to the single equation p⅛f.,y) f ς d(α,δ) ς ° ’ or f=√ 0 , where 2C,.⅛-⅛, da db and therefore 2fi,⅛⅛>-⅛⅛-⅛≤, b ¼α,3) d{a,β) d(β,a) dx ~~ i

Now « = —— = cue c cos I a>t+- ] , dt ∖ cj β ~- cue c sin ( ut + —); dt ∖ c J _ß_ _ß_ therefore u 0 =cue c cos — , v 0 ≈ -cue c sin —; c c and 2^‰α,e~⅛; d(ι α,j8)

_£

therefore C= Co~ ≡∙ « = ——— ·

- 27 2 sinh B 1-e c c and the motion cannot therefore have been generated from rest by natural forces; the fluid must have been created with the proper amount of spin at every point.

, _ß f .

We have - c ^ c e sin (ut + — ∣ , dt ∖ c ∕

, -A ∕ x . -^- = cu 2 e c cos (" i + “4 J

lid therefore the dynamical equations (1) and (2) become - cu 2 e c sin + ^c - )°°θ ’

- cw 2 e c cos ( ut + — ) + cw 2 e c -- 0; dβ ∖ c J and therefore the integral of these two equations is

_A ∕ ∖ -2Ê.

Q +∙ c 2 u 2 e c cos (ut 4- ∖ - ⅜ c 2 ω 2 e « = Η , a constant.

Now Q= ∕⅛ + V -T--gy ι J P P β ≈y -gβ - gee c cos (ut +-?-); and therefore ß β ^-gβ + c(cu 2 - g)e e cos (ut ⅛ ~) - I c 2 u 2 e~ 2 c =H.

A free surface is possible if cu 2 =g, n — 2 —

and then — ≈gβ + A c 2 u 2 e f ÷ Η, P and the pressure at a particle is constant.

The wave length λ-2n∙c; and the velocity of propagation = λ ∕⅛.

The surfaces of equal pressure are trochoids, obtained by rolling a circle of radius c on the under side of a line at a depth β - c, the distance of the carried point from the centre being ce~ e . Irrotational Motion.

If liquid originally at rest be contained in a singly-connected space, then forces due to a singly-valued function V are not capable of setting up any motion in the liquid, and any motion must be due to the motion of the bounding surface.

For, φ denoting the velocity function, by Green’s theorem the kinetic energy ι -*Λ)W) ’ ∣*

and therefore, if ⅛ = 0, then T = 0, and therefore an (W(⅛h or ⅛ = 0 ‰0 ⅛ = 0 dx ">dy 0, √7° ∙

If we suppose the actual motion at any instant to have been instantaneously generated from rest by the application of proper impulses at the bounding surface, then, since no natural forces can act impulsively throughout the liquid, the equations of impulse are

1 dw 1 dτr 1 d·a p dx p dy p dz ’ w denoting the impulsive pressure at any point of the liquid; and therefore, if φ denote the velocity function, we can put

≠-- 7 ∙

Since the work done by an impulse is the product of the impulse into half the sum of the initial and final velocities, we see how it is that the kinetic energy of the liquid

-4∕⅛-

4≠S-

Also the kinetic energy acquired thus due to the velocity function φ will be less than the kinetic energy of any other motion, wholly or partially rotational, but satisfying the equation of continuity, and the condition at the boundary that the normal velocity of the liquid is the normal velocity of the boundary.

For, if u l , v 1 , w 1 be the velocities at any point of this new motion, and T 1 the whole kinetic energy,

T 1 - T = A p βj ∖ u ι ~ w 2 + v 1 2 - r 2 + w 1 2 - w 2 )dxdydA = * f J'(f | (!<1 ^^ + ~ + (W1 ~ I dxdydz

4-, pjjj' | m ( m i - m ) + v (p ι ~ ®) + w(w 1 - w} dxdydz.

But JJJ~ u{u 1 - w) + v(v 1 - v) + w(w 1 - w) dxdydz ~fff ; È <"' - ,, > + ⅞ <' 1 - 1 '> + E <”' - "'> ∣ λ **

I - w ) + 7λi (1 'ι - r ) + 7l (w ι - w- ’)

-fff φ ∣E fe -"> + ⅛<*∙∙-'> + E<" ’--" ’ > i"≠≈

= 0.

Then T 1 - T=⅜ pJJJ' | - w) 2 + (v 1 - v) 2 + (w 1 - w} 2 dxdydz, a positive quantity; and therefore T 1 is always greater than T, a theorem due to Sir W. Thomson. If, however, φ be multiply-valued, and the space occupied by the liquid multiply-connected, we can have circulation existingin the different circuits of the space even when the bounding surface is at rest, and the motion may still be differentially irrotational, and any motion of the bounding surface will not affect these circulations. For instance, we may have φ = ~ tan -1 — , and the liquid circulating in any ring-shaped J ∣ ιr ∙⅞7

surface, whose axis of figure is the axis of z.

To find the kinetic energy of a liquid in a multiply-connected space, the motion being differentially irrotational, but circulations existing in the circuits, the space occupied by the liquid must be rendered acyclic by barriers, which may be supposed to be membranes, moving with the velocity of the liquid; and then, if k be the cyclic constant of the value of φ in any circuit, we must suppose the value of φ on one side of the membrane to exceed the value of φ on the other side by k, so that the integral^Ç> ÉÉ dS over the membrane must be replaced by hjj~ dS ; so that to the term ⅞ p <∕>^ over the outside surface must be added a number of terms of the form ipk^/^^-dS, to express the energy due to the circulation in the circuits; and the condition of continuity shows that^^^dS over one of these membranes which render a circuit acyclic is independent of the form of the membrane.

On Flow, Circulation, and Vortex Motion.

The line integral of the tangential velo ity∕*^u^ + + ds

-or f{udx + υdy + wdz), from one point to another of a curve, is called the flow along the curve from the initial to the final point; and, if the curve be closed, the line integral round the curve is called the circulation in the curve.

If a velocity function φ exist, then the flow =∕d<∕> = <∕> 2 - <∕> 1 , where φ l and φ, i are the initial and final values of φ ; and therefore the flow is independent of the curve for all mutually reconcilable curves; and the circulation in any closed curve, capable of being reduced to a point without leaving space for which φ is single-valued, is zero.

If through every point of a small closed curve the vortex lines be drawn, a tube is obtained and the fluid contained is called a vortex filament.

By analogy with the spin of a rigid body the component spin of the fluid in any plane at any point is defined as the circulation round any infinitesimal area in the plane enclosing the point divided ‘by twice the area. For in a rigid body, rotating about the axis of X with angular velocity ξ suppose, the circulation round a curve in tthe plane of yz is

∕√∙¾-⅛)*

= ω times twice the area of the curve.

Now if, in the fluid at the point xyz, we take the circulation round the elementary area dydz, it is equal to + w+^wly^dz - (v + c ~^dz - wdz ∖ dy dzj and therefore the component spin in the plane yz is ~ > which we have denoted by ξ. Similarly the component spins in , , , j 1 (du dw ∖ j 1 (dm du ∖ * the planes of zx and xy are ( _ ) = η and —( ) = f r 2 ∖ dz dx) 2 ∖ dx dy ) respectively.

Since the circulation round any triangular area is the sum of the circulations round the projections of the area on the coordinate planes, the composition of the component spins ξ, η, ζ is according to the vector law. Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through a certain line, the resultant axis of spin of ξ, η, £ at that point of the fluid. Consequently the circulation round any closed curve drawn on the surface of a vortex filament is zero; and therefore, if at any two points of a vortex filament we draw the cross sections ABC, A'B'C', joined by the line AA', then, since the flow in AA' in the complete circuit ABCAA'B'C'A'A is taken in opposite directions, the resultant flow in AA' vanishes, and therefore the circulations in ABC, A'B'C', estimated in the same direction, are equal. This is expressed by saying that, at all points of a vortex filament, ωα is constant, where a is the sectional area of the filament, and ω the spin (Clifford, Kinematic, Book iii. ).

So far the theorems about vortex motion are kinematical; but, .introducing Euler’s equation of motion

Dιt l n = o Dw l dQ

dt dx , dt dy ’ dt dz ’ where Q = /*— + V, J P 1 D

.then (udx + vdy + wdz)

Ddae ^Ddy Odz ι Dw , Dt>, Dw , - u ~dΓ + ' ’ ~dΓ +w ~di + S dx+ di d > + -lι d2

, , , dQ , = udu + vdv + wdιv - -j-dx - -j~dy --^-dz ax ay az

= ⅜c⅛ 2 -dQ; and therefore, by integration round a closed curve, — ∕"udx + vdy + wdz^) = 0; dtj and therefore the circulation in any circuit composed of the same fluid particles is constant, and, if the motion is differentially irrotational, is zero round all reconcilable paths.

The circulation round any small plane curve passing through the axis of spin at any point being always zero, it follows conversely that a vortex filament Is always composed of the same fluid particles; and, since the circulation round any cross section is constant for different times, it follows from the previous kinematical propositions that aω is constant for all the time, and the same at all points of a vortex filament.

Professor Clifford {Proc. London Mathematical Society, vol. ix,)has given a simple quaternion proof of the theorem—To determine the velocity at any point of a fluid, when the spin is given.

If σ denote the velocity and ω the spin at any point, then

2ω = Vv<r; also, if k denote the cubical expansion, k= - Sv<r .

Hence the quaternion q or-⅛ + 2ω is simply v<>∙; consequently the problem to be solved is to determine σ from the equation

2=V<r, q being given.

Operating by v, vg ’ = V 2 σ; therefore σ is the potential of V?; and therefore ∕*Vq ∣ , dv b σ *~V t J D α ⅛ ’

where σ a means the value of σ at the point a, dv b means an element of volume at the point b, and D α i, the distance between the points a, b.

Returning to Euler’s equations of motion, du du du du cZQ .

dt dx dy dz dx and eliminating Q,

Dξ f dw r(d u q . dt dx r, dx dx * ∖ dx dy dz J ’ and, since by the equation of continuity

1 Dp ι dω d¾ d w _(y p didadydz ’ therefore

D ( ξ ∖ ξ du η dυ ζ dw . dt ∖ ρ ) p dx p dx p dx ’ and similarly

D ( y\_ ξ du η dv ζ dw dt ∖ P J~ P dy + p dy p dy ’

D ( ζ ∖ _ ξ du^r^dv^ζ dw dt ∖ p J p dzp dz p dz

These equations, first given by Professor Stokes for homogeneous liquid, were generalized for any fluid by Professor Nanson, Messenger of Mathematics, 1873. They may also be obtained immediately by the differentiation of Cauchy’s integrals (4), (5), and (6), given above.

Plane Vortex Motion.

When a series of straight vertical vortices (called columnar vortices by Sir W. Thomson) are present in homogeneous liquid, bounded by two horizontal planes, we can determine the motion of any vortex by supposing it due to the remaining vortices.

A single vortex will remain at rest, and cause a velocity at any point perpendicular to the plane through the point and the vortex inversely as the distance from the vortex.

If m denote the strength of the vortex, i.e., the circulation in any circuit enclosing the vortex once, then the velocity at a distance r from the vortex will be —, and the current function ψ will be 2πr

— log r, and the velocity function φ will be θ, where θ is the 2π da angle between any fixed plane and the plane through the vortex and the point.

The surface of equal pressure under gravity will be of the form the axis of the vortex being the axis bf z.

When there are more than one vortex present, each vortex moving with the velocity due to the other vortices will describe the curve whose equation is

Σ — log r = constant,

Z7Γ

where m is the strength of one of the remaining vortices, and r the distance between it and the vortex whose motion is considered; this equation may also be written

∏ r m = constant.

When the liquid is bounded by a vertical cylindrical surface, the motion of a vortex may be determined as due to a series of vortices considered as images of the original vortex, and so arranged as to make the velocity across the boundary zero at every point of the boundary.

When the boundaries are nlane surfaces, the images are the optical images by reflexion of tne original vortex, considered as positive or negative, according as formed by an even or odd number of reflexions.

Thus the curve described by a vortex inside the angle bounded by the planes θ= ±dL is the Cotes’s spiral r cos nff=C, and inside the space bounded by the planes x≈0, x=a, y=0, y = b is cot 2 am K -y, k^ + cot 2 am ^K ’-y , k' ) = constant,

, K cι where = r . = - r

I√ b (Quarterly Journal of Mathematics, vol. xv.).

A single vortex of strength m in a circular cylinder of radius a at a distance c from the centre will move with the velocity due to an image of strength - m at a distance — from the centre, and c therefore describe a circle of radius c with velocity m a _m c 2ιr α 2 2ιr ’ α 2 - c 2 ’

c c A 2

and therefore in the periodic time — (α 2 - c 2 ).

m

A single circular vortex in infinite fluid will move with a certain velocity in the direction of its axis (“Vortex Motion,” Trans. BS.E., 1869; “Vortex Motion,” Helmholtz, Grelle, 1858); and, if another equal circular vortex be projected coaxially after the first, the motion of the first must be compounded with that due to the second. Consequently the first vortex will dilate and move slower till the second vortex passes through it, when it will contract and move faster till it passes through the second, and so on. This can be verified experimentally with smoke rings projected from the same circular hole, or with half vortex rings, formed on the surface of water by drawing a semi-circular blade a short distance through the water.

The motion of a vortex ring projected perpendicularly against a plane boundary will be determined by compounding it with the motion due to an equal and opposite vortex ring, its optical image in the wall. The vortex ring will therefore spread out and move more slowly in the direction of its axis as it approaches the wall; at the same time the molecular rotation, being inversely proportional to the cross section of the vortex, will be seen to increase.

Plane Motion of Liquids. hen the velocity of the fluid is always parallel to a fixed plane, we take this plane as the plane of xy, and then w=0, and u and v are functions of x and y, and the stream lines are plane curves.

Considering only the cases where the fluid is incompressible, the equation-of continuity becomes du dυ dx + dy~ ’ and therefore a function ψ exists, called the stream function, such that u=-⅛, tr .⅛j dy dx and ψ = constant is the equation of a line of flow.

The spin at any υoint

1 f dν duy i ~ 2 ∖ dx dy)

=J√⅛ .≤Ψ∖

2 ∖ dx* + df)∙

If the motion is irrotational, then £—0, and a velocity function φ exists such that w = ⅛= -⅛ dx dy ’ t ,-⅛- + ⅛ i dy dx therefore ψ and φ are ccnjugate functions of x and y, and

ψ + f≠=∕(z + iy).

Β ϊ »’signing particular values to this function, Helmholtz and Kirchhoff have discovered the solution of various problems of discontinuous plane liquid motion, an account of which is riven in Lamb s Motion of Fluids.

The kinetic energy of the liquid bounded by two planes perpendicular to the axis of z at unit distance is

--i·#!®'·®» -∣Λi) ’ ∙(⅛) ’ H

i d ^ p ∕ φ ^, d8 =^ p ∕^ d3 > when ⅛ ⅛ are the rates of change of φ and ψ in the direction dn dn o r τ of the outward drawn normal to the bounding curve at the element ds.

Since ⅜≡⅜ dφ^~dψ dn ds ' dn ds ’ therefore T = - -⅛- p fφdφ = -L p ∕ψdφ. à y 2 J

We can interchange φ and ψ, and make φ the stream function and ψ the velocity function; thus from any given irrotational motion in two dimensions another may be derived by turning the velocity through a right angle without altering its magnitude.

For instance, if the axis of z be a line source of delivery m, then, since the flow across any cylinder of radius r is m, the velocity must be; and therefore

2πr Φ = ^ log τ, and ψ=‰

Z7Γ Z7Γ

where Ö is the angle made by a plane through the axis of z and the point with a fixed, plane.

If the values of φ and ψ be interchanged, we obtain a vortex round the axis of z, of strength m. Plane Motion in a Liquid due to the Motion of Rigid Cylinders perpendicular to their Axes.

Suppose a rigid cylindrical surface moving in the direction of the axis of x with velocity V, and other fixed rigid cylindrical surfaces to be present in the liquid, which is supposed for simplicity to be bounded also by two fixed planes perpendicular to the axis of z at unit distance from each other, the generating lines of the cylinders being supposed parallel to the axis of z ; then at all points of the boundary of the moving surface normal velocity of fluid = velocity of boundary normal to itself

_ γ ⅜ .

ds ’ and therefore ψ= -Vy + constant; and at all points of the fixed surfaces ^⅛ = 0, and therefore ψ = constant.

ds

We must therefore discover a function ψ which satisfies the equation

⅛ + ⅛=n dx 2 df 2 ’ and is equal to a constant round a fixed boundary, and equal to -Vy +constant round a moving boundary, moving with velocity V in the direction of the axis of x ; and φ, the conjugate function, can then easily be written down.

Ex. 1. The moving cylinder a circular cylinder of radius a, and the fixed cylinder a circular cylinder of radius b, both having the axis of z as axis. Then

δ 2 r ψ= - Vα sin θ a

--v√i√≤. i ∙)⅛β, b 2 -a 2 ∖ r ∕ and therefore

Φ≈-Vv⅛-(- + r)cos θ. b 2 -a 2 ∖ r )

If φ ’ denote the velocity function of liquid filling the cylinder r≈a, φ ’ =Vr cos 0, and therefore, when r=a, ψ ¾ 2 + α 2 φ ’ ~ b' 2 - d 2

In determining the kinetic energy of the liquid intermediate to the cylinders, ^^^ = 0 when r≈b ; and when r = α, 5 an ^

therefore the kinetic energy of the liquid intermediate to the cylinders is + of the kinetic energy of the liquid filling the cylinder b^ 2 - a' 2 r=a. Consequently, if the cylinder r=α be moved, the inertia to be overcome will be its own inertia, together with the inertia of a mass of a liquid ^+ α times the volume of the cylinder; this is b- - α 2 called the effective inertia of the cylinder.

In particular, if Z> = ∞, the effective inertia is the mass of the cylinder, increased by a mass of liquid of equal volume with the cylinder; and then

ψ= - V— sin 0, φ= - V — cos θ, r r so that

. , √ Vα 2 τ r z{x + τy) Ex, 2. The moving cylinder an elliptic cylinder, and the fixed cylinder a confocal elliptic cylinder.

Using elliptic coordinates ξ, η, such that c cosh ∙η, c sinh η are the semi-axes of the confocal ellipse, c cos ξ, c sin ξ of the confocal hyberbola passing through a point, 2c being the distance between the foci; thenae = c coshη cos ξ, y = c sinh ∙η sin ξ; and if η = α is the equation of the moving ellipse, η = β of the fixed, ellipse, then ψ=-Vcsinh≈ si " 11 ^~^sinξ y sinh (β - a) satisfies the conditions that

(i) ≤≠ + ⅛ = 0, W dξ 2 dy 2 ’

(ii) ψ = - Vc sinh a sin ξ= - Ny, when η=-α,

(iii) ψ = 0, when η = β.

Therefore the conjugate function φ = Nc sinh a cos ξ; r sinh ∙, β -a) so that

≠ + i <'- v ' , c ° s +

If φ' denote the velocity function of the liquid filling the elliptic cylinder η = a, then

<j⅛'=Va3=Vc cosh η cos ξ; and round the ellipse η = a, φ tanh a

ø' = tanh {β-d) ’ while — is the same for each, and vanishes when η = β; therein dn fore the kinetic energy of the liquid between ∙η = a and η=β is —^ ap ^ 1 — of the kinetic energy of the liquid inside ∙η = α, which is tanh (0 - α)

⅜πpV 2 C 2 sinh a cosh a.

Hence the mass of the cylinder η = a must be increased by —tanh a — times the mass of an equal volume of liquid to give tanh(0-α) 1 . .

the effective inertia for motion in the direction of the major axis, the space between the cylinder η = α and a fixed cylinder η = β being filled with liquid.

Similarly for motion parallel to the minor axis, ψ∙=Vc cosh α s ^ n ^ 1 ~ r, ∖ cos ξ, ’ sinh {β - a) s φ = Vc cosh a ~ sin ξ

sinh (β- a ) {Quarterly Journal of Mathematics, vol. xvi.).

Ex. 3. When the moving and fixed cylinders are any two circular cylinders, not co-axial, the limiting points are taken as the foci of reference; and, supposing 2c the distance between them, and ξ, η the dipolar system of coordinates, we have x ≈ c sinh η y-c sin ⅞ , cosh η - cos ξ cosh η - cos ξ * and then *~T 1 ° g ⅞~c) 2 +’

ξ=tan -1 --tan -1 —; x~c x+c so that ξ=constant is the equation of a circle passing through the two limiting points, and η=. constant is the equation of an orthogonal circle.

If η~a be thè moving cylinder, moving in the direction of the axis of X (the line of centres) with velocity V, and if η = β be the fixed cylinder, we must make

ψ= - Vc _ s l≡l 1 α 4. constant, cosh a - cos ξ when η -= a ; ψ = 0 when η = β ; and ¾ + ⅛ = 0 in the intervening αξ 2 dη i space.

Now, expanding, anh α i + 2 τi -∞ _ na cog nξ , cosh a - cos ξ and therefore n=∞ sinh n(ιι - β) 4, ^^ 2τ ⅛^~0^^3) c0snξ, and φ - 2Ve'Se-"- ∞> h *⅛~W 8 j 11 ,, s , sinh n(α - β)

Similarly for a velocity V of the cylinder η = α perpendicular to the line of centres, the cylinder η=β being fixed,

ψ = 2Vc ’ ∑e→ l * sinh ¾(η - 0) gin nξ y sinh n(a-β) φ -= 2VcV e~ na cos nξ . sinh n(α - 0)

Next, suppose a rigid cylindιical surface to be rotating about the axis of e with angular velocity ω ; we must have - ⅛=velocity of boundary normal to itself dx dy = - a>X — - ωl∕ - r , ds j ds and therefore ψ=⅛ w(z 2 + y 2 ) + constant, at all points of the moving boundary, and ψ = constant, at all points of a fixed cylindrical boundary. . 1

Ex. 4. Take the two elliptic cylinders of Ex. 2, and suppose the cylinder η = a to be rotating with angular velocity ω, and the cylinder η = β to be fixed; since

αj 2 + y 2 =⅛c 2 (cosh 2η + cos 2ξ), if we put ψ = ∣ fttc ≡⅛⅛2(0-η) cos 2ξ y i sinh 2(0-a)

then (i) when y = a , ψ=∣ωc 2 cos 2ξ

= ⅜ω(x 2 + y 2 ) + constant;

(ii) when η = 0 , ψ = 0;

< ui > S + S^ o i and therefore ψ satisfies the required conditions.

and from the value of the kinetic energy of the intermediate liquid the instantaneous value of the effective moment of inertia can be inferred.

If the cylinder η = 0 be also rotating with angular velocity α>, the cylinders will remain confocal, and the values of ψ and φ will not change; then

ψ = l ωc 2 sinh 2⅛ - ≈) +sinh 2(0- n) cos 2ξ γ 4 sinh 2(0-a)

= |a , c2 cosh(2n- a - _g) 2ξ 4 cosh(0-α) 5 and φ = ia,c 2 ≡h(2^-α→) siπ 2ξ .

cosh {β-a)

To find the kinetic energy of the liquid, since ∕4>-∕*S λ therefore T = ∣p ∕~φ dξ; and when η = a, and η = 0, φd½ = ^ ω 2 β 4 ⅛ an j 1 _ α ) s i n 2 2 ξ.

dη

∕-2n∙

therefore T= ∣pw 2 c 4 tanh (0 - α) ∕ sin 2 2ξ<7f

~ ’ o = lπρω 2 C i tanh (0 - a); and, if k denote the effective radius of gyration of the liquid,

T = ∣τrpω 2 c 2 F(sinh 0 cosh Ø-sinh a cosh α); therefore k 2 = ⅛c 2 tfmh√0a) —

sinh 0 cosh ø - sinh a cosh a i 4 a⅞j-<⅛⅞

- ‘ c {aa 1 - bb 1 )(a 1 b 1 - ab) ’ where a 1 , b 1 are the semi-axes of the ellipse η = β, and a, b of the ellipse ∙η = a. Ex. 5. Suppose a sector, bounded by r=αand 0=±α, rotating about the axis with angular velocity ω; we must put ψ.J^^29 + "∑ a, irγ n+ '⅛e° a yn + T^, cos 2a n=0 2n+l∖^ a ∕ ' 2α

which satisfies the conditions v 2 ψ=0, and ψ = ⅜ ωr 2 when θ = ± a; in order that ψ = ⅛ωr 2 when r=a, we must have n ≡∞ ιτθ „ ∕, cos 2θ ∖

∑ 0 A 2n+1 cos(2π + l)^ = ⅜ ωα 2 ^ 1 --2^ y ).

and therefore, by Fourier’s theorem,

A = 2 ( -D n+1 I 1 2 1 1

2n÷l ( (2π+l)ιr - 4<x (2n + l)ιr^'(27i + l)π + 4α 1

When all the cylinders present rotate, as if rigidly connected, about the axis of z with angular velocity ω at any instant, then 4 =i<v(x 2 + y 2 ) + constant round the boundary of every cylinder; and if we put χ=ψ - ⅜<⅛>(x 2 + t∕ 2 ), then χ is the stream function of the relative motion, relative to the cylinders, and satisfies the conditions + ⅛* = - 2» at every point of the liquid, and χ = con- ctx> ccy stant round the boundaries.

Since χ involves « as a factor, which is a function of t only, it follows that -V = constant is the equation of a stream line of the relative motion, and any alteration in ω does not affect the shape of the relative stream lines, the liquid being frictionless, and the motion generated from rest.

Ex. 6. Put a? y 2

α δ+ 0 2 . χ =- ft , τ - τ , tf + T 2 then ⅛ + ⅛.- 2u .

cte 2 dy 2 ’ and the relative stream lines are similar ellipses.

Then ψ = χ + J. ω (ar 8 +y≡)

and therefore φ=ω ——— xυ ,

α 2 + ⅛ 2 for the motion between two similar elliptic cylinders, rotating with angular velocity ω.

The velocity of any liquid particle is of what it would be α j + o-

if rigidly connected to the cylinders; hence the effective radius of gyration of the liquid is —L of the radius of gyration of the Cl> ÷ O^ homogenous rigid body occupying the space.

Ex. 7. Put X ^ ⅛⅛i " ^ 2 ) + a = 2 + y 2 - ai } { √ 2 (* 2 - y 2 ) - & - f + b 2 }; then v 2 χ = - 2«, and χ = 0, when

√2(z 2 - y 2 ) + a? + y 2 - α 2 ≈ 0, √2(z 2 - 2 ∕ 2 )-z 2 -y 2 + ⅛ 2 = 0 , which may therefore be taken as boundaries of the liquid. This problem is due to Mr Ferrers.

Again put x ” ⅛⅛≡ ~ y) ~ 4 { 2l ^ x + y} ~ 4 5

then v 2 χ = - 2ω, and the hyperbolas 2x(x-y)-a 2 =0, 2y( x + y)-b 2 =Q may be taken as boundaries, but these hyperbolas are only the previous ones turned through a quarter of a right angle.

Ex. 8. When the liquid fills a rectangular cylinder bounded bv x ** ±®, and y≈ ±δ the conditions drf + dy 2 ~ ~ 2ft " and u- -^=0 when x≈±a, v ~ ^ = θ when t∕-=±δ, are satisfied by putting

<⅛-.⅛-⅛ tan -ι ta (κ ⅞∙*)

g.¾-⅛ hm ->⅛lιr.).

ax dx i π ø χ ∖ ,

Λ∙cn(κ⅞,Λ∙)

where

K' b (Quarterly Journal of Mathematics, vol. xv.).

. fi i=∞ cosh (2ι + l)^ cos (2i + l)-

In fact, if ψ = «-^a 2 ∑ ( - l) i — —

1t ' i = 0 (2i+l) 3 cosh (2ι + l)-

2λ

i=< β ∞sh(2i + l)^ cos (2i + l)^

+ «4 * 2 Σ( - l) i - 2 *, π t = 0 (2ι + l) 3 cosh (2i + l)-

2b

U>en (l)⅛ + ⅛.o i dx 2 dy 2

(2) 4 = ⅛ ω (δ 2- 3∕ 2 ) whenae=±a;

(3) 4 = ⅛w(α2 -∞ 2 ) when y = ±δ; and therefore ψ satisfies the required conditions, and is therefore the value of 4 required.

Ex. 9. Consider liquid filling the interior of a cylinder, whose cross section is an equilateral triangle of altitude h, and let α, β, y denote the perpendicular distances of a point in the interior from the sides. If we put

Z = 2.⅛, tw S÷⅛=- 2 "-

and χ is the stream function of the relative motion, supposing the cylinder rotating with angular velocity w.

Therefore the cubic α3γ = constant is the equation of the path of a liquid particle relative to the cylinder, when it is moved in any manner; and also for the cylinder bounded by aβy=c 1 and aβy = c 2 .

We have supposed the liquid motion to have been generated from rest by the motion of the moving cylinders, but we might also have supposed the liquid to have been of infinite extent, and streaming past the cylinders as fixed obstacles; in that case, the stream function of the relative motion χ = ψ + Vτ∕, and χ satisfies the relations

⅛ + ⅛~n dx 2 'dy 2 ’ and χ = constant, the equation of a stream line, and therefore also of a boundary; also at infinity

⅛-o, ⅛-v.

dx dy

For instance, if in liquid, moving with velocity-V parallel to the axis of x, the fixed circular cylinder r=α be introduced, then χ= - V ysin Θ + Vy = γ( r ~) sin θ.

If the elliptic.cylinder η = a be introduced, then, since β≈∞,

V. tr-sinh V ~ cosh n . t y-Vc sinh α ——i r -' sin ξ

sinh a — cosh a

= Vc sinh η sin ξ - Vc sinh a e~ η+a sin ξ

== Vce a sinh (ιj - a) sin ξ

≈ V(a + δ) sinh (η - a) sin ξ.

If the axis of z be horizontal, and the liquid supposed of infinite extent, and originally at rest, then a circular cylinder of density σ, projected in any manner perpendicular to its length, will describe a parabola with vertical acceleration tL∑Jl g. <r + ρ

If, however, previously to projection, a vortex exist in the liquid, co-axial with the cylinder and of strength m, then any motion of the cylinder will not affect the circulation of the liquid round the cylinder due to the vortex, and inequalities of pressure round the cylinder will arise from the vortex motion.

Lord Rayleigh has shown (Messenger of Mathematics, vol. vii. ) that, if no forces act, the cylinder will describe a circle in the same direction as the circulation of the vortex in the periodic time — * r ^*^P , where the circulation of the vortex is 2ir« 2 w, a being the ω ρ radius of the cylinder.

If the axis of the cylinder be horizontal, and the influence of the boundaries of the liquid neglected, then the cylinder will describe a trochoid, and for a particular velocity of projection can be made to describe a horizontal straight line (Messenger of Mathematics, vol. ix. p. 113).

On the Motion of a Solid through a Liquid.

If we take an origin 0, and axes 0x, 0y, Qz fixed in the body, then, if u, v, w, p, q, r denote the component linear and angular velocities of the body at any instant, the velocity function φ = uψ 1 + vψ s + wψ 3 + pχ 1 + qχ 2 + rχ 3 , where the ψ ’ s and χ ’ s are functions of x, y, z, depending only upon the shape of the body.

To determine ψ 1 , we may suppose the velocity u only to exist, and thus ψ 1 must satisfy the conditions—

(i) v 3 ≠ 1 = 0;

(ii) ⅛ = l the cosine of the angle between the normal to ' dn the surface and the axis of x, at the surface of the moving body; (iff) ⅛ = o, over a fixed surface.

' ' dn

Similarly for ψ 2 and ψ 3 .

To determine χ 1 , we may suppose the velocity p only to exist, and and then, l, m, n being the direction-cosines of the normal to the surface, χ 1 satisfies the conditions—

(i) V 2 χ 1 = 0j

(ii) ⅛ = uy - mz at the surface of the moving body; ' dn

(iii) ⅛l = 0 at a fixed surface. ' dn

Similarly for χ 2 and χ 3 .

For a cavity filled with liquid in the interior of a moving body, since the liquid moves as if solid when the moving body has a motion of translation only, therefore ψ 1 =a5, ψ 2 = y , ψ 3 =z,

The only cases practically solved are those where the bounding surfaces are similar or confocal surfaces of the second degree.

z^∙2 qΛ Ex. 1. Consider the space between the ellipsoid + 4--^= 1

and a similar and similarly situated ellipsoid rigidly connected with

Ä2 __ z»2

it; then obviously χ 1 = - 5 —÷yz, as for plane motion; and therefore δ 2 + c 2

δ 2 - c 2 c 2 - α 2 α 2 - b 2 φ = ux + vy + wz + p yz + g - q -- 2 z χ + r xy.

The liquid filling this space will behave therefore like a body of ^2 ^»2 ^>2 ^2 ^2

equal mass, and of principal radii of gyration 2 , f a +a z > α 2 + ¾2 of the radii of gyration if the liquid were solidified.

Ex. 2. Consider the liquid filling the space between the ellipsoids ⅛+⅛ + ⅛-ι . . . . (1)

α 2 b 2 c i

-a ⅛+^+⅞- 1 ∙ · · · < 2 >∙

the ellipsoids being confocal, such that

α 1 2 =a 2 + λ 1 , δ 1 2 =δ 2 + λ 1 , e 1 2 = c 2 + λ 1 . p . a ∕ r °° „ P* dκ p ∕∙°° dκ

Λ (a 2 + λ)P ’ A (⅛ 2 + λ)P ’ Λ (<≈ 2 + λ)P

where P 2 —(a 2 + λ)(δ 2 + λ)(c 2 + λ), and ⅛ + δ⅛ + ⅛= 1 (8) ’

so that α 2 + λ, δ 2 + λ, c 2 + λ are the squares of the semi-axis of the confocal ellipsoid passing through xyz. Then

9

A+B + C-f; also, if p be the length of the perpendicular from the centre on the tangent plane to (3), p 2 = {a 2 + λ)Z 2 + (δ 2 +λ)wι 2 + (c 2 + λ)n 2 , and therefore = 2ρ - 4~ ·

dp dn

Suppose the ellipsoid (1) moving with velocity u, and the ellipsoid (2) fixed, then ψ 1 can be made to satisfv the required conditions by putting

ψ 1 =Mz∙ + NAaj, where M and N are constants.

For v 2 ψ 1 =0,

«nd ⅛.4' ti - i ⅛A

dn dn dn dn - m ' +na '-(⅛ n -(m + NA-2^)z = ⅛-N(B + C)j∙ l.

Consequently, when λ=0, we must have

M-N(B 0 + C 0 )-l, and when λ = λ 1 ,

M-N(B 1 + C 1 ) = 0, where A o , B o , C o are the values of A, B, C, when λ=0, and A 1 , B 1 , C 1 when λ=λ 1 .

Hence y _ 1

B o + C o - B 1 - C 1 >

B 1 + C 1 m ^^ B 0 + C 0 -B j -C√

. , A + B 1 + C 1 n , ,nd *-M⅞-⅝⅛ , ∙

Similarly

. _ A 1 + B + C 1 ≠ 2 ^ c 0 + a 0 -c i -a∕ , 1 A 1 + B 1 + C v

* 3 ~ - A 0 +B 0 -A 1 -B 1 ·

If the inner ellipsoid had been fixed and the outer moved, we should have had i _ A + B o + C o ψl ~ B o + C o - Bi - Ci ’

A o + B + C o

* 2 C 0 + A 0 -C 1 -A 1 y

A o + B o + C

* 3 A o + B o -A 1 -B 1

Next suppose the outer ellipsoid fixed, and the inner to have the angular velocity p ; then χ 1 can be made to satisfy the required conditions by putting

χ 1 -Myz + N(B-C>, where M and N are constants.

For then V 2 χ 1 = θ, and

⅛=jM + κ (B-θ!(⅛ +i ⅛) + N(≤^>.

. I M + N(B-C) I (⅛⅛ + ⅛) -N(s ⅛-⅛)⅛ -[jM + S(B-C)j(s ⅛ i+ 4-)

-N(A + B+C)(j⅛ χ -⅛ χ )>, which when λ=0 must

=, ψ -≡=f,-a 2 =(⅜4) w .

and when λ = λ 1 must=0.

Therefore M and N must be determined from the equations

I M + N(B 0 - C o ) I ^-^- + -^-^-N(A o + B o + tj o)^∙⅛∙ - ∙^τ)

1

^c 2 ^δ 2 ^ ’

and

I M + N(B i - Cl) I (⅛ + ⅛)-N(A1 + B1 + C1)(^-⅛) = O.

Similarly χ 2 and χ 3 can be determined, and also χ 1 , χ 2 , and χ 3 when the inner ellipsoid is fixed and the outer moved with given angular velocities.

When the outer ellipsoid is indefinitely great, then A 1 , B i , C i are zero, as also is Μ. Then

JL-JL

n c2 62 ,

(B 0 - C0)^^j2~ (A 0 + Bo÷θo)^ j2

and χι=N(B - C)yz.

To find the effective inertia of the inner ellipsoid, when the outer ellipsoid is fixed, and first for motion parallel to the axis of x; whenλ=λι, 4r i ^=θ > but when λ = 0, = l, and the ψ i for the 1 dn dn liquid in the interspace is - the ψ 1 for the liquid filling the inner ellipsoid; and hence, since the kinetic energy = ⅛ρ∕φ t Υ~dS, it follows that the kinetic energy of the liquid in the interspace is A ° * θ 1 of the liquid filling the interior B o + Cq — — C 1 .

ellipsoid for motion parallel to the axis of x, and therefore the effective inertia parallel to the axis of x is a , A o + B 1 + C 1 .

i *'*% 0 + C 0 -B 1 -C 1 ’ with similar expressions for the effective inertia parallel to the axes of y and z.

If the outer ellipsoid be indefinitely large, then the effective inertia of the liquid parallel to the axis of x (since A 1 = B 1 ∙≈ C 1 = 0) ∙=⅜*pa⅛Φr ’

B o + G o which, in the case of the sphere, is half the effective inertia of the liquid inside the sphere, since in the sphere A 0 = B 0 =C 0 ∙

For a rotation about the axis of x of the inner ellipsoid, it follows in the same way that the effective inertia of the liquid in the interspace is to the effective inertia of the liquid filling the inner ellipsoid in the ratio of the χ ’ s of the two motions, which, supposing the outer ellipsoid indefinitely large,

(⅜-⅛)⅞- c °>

(⅛+⅛)<B,-C s )-(A,+ B.+⅛)(-i--^) 0 '^ e * (B 0 - C 0 )(⅞ 2 + c 2 ).

’( B o - C 0 )(6 2 + c 2 ) + ( A o + B o + C 0 )(δ 2 - c 2 ) ’ and therefore the effective moment of inertia of the liquid about the axis of x A iro gb c (B 0 -C 0 X⅞ 2 -c 2 ) 2 τ5 B o - C 0 )(⅛ 2 + c 2 ) + ( A o + B o + C o )(0 2 - c 2 ) ’

with similar expressions for the effective moment of inertia about the other axes.

Ex. 3. In the case of two spheres and the liquid between, the χ's are all zero, and, if the spheres be instantaneously concentric, α 3 ( α 1 s ∙c )

* 1 ~α 3 -^0∕ j ar+ 2(x 2 + 2∕ 2 + z 2 )l I

a 3 - a 1 3 ∖ 2r 2 )

supposing a the radius of the moving, and α 1 that of the fixed sphere.

This is a particular case of the confocal ellipsoids, when α=δ = c. For then

, _ A + 2A 1

* 1 2(A 0 -A 1 ) ’

ami A=∕- — = I tt 1 -.≡⅛ · j κ (α 2 + λ)* (α s + λ)1 3r 3

Therefore

J +-

ψl r, ⅝ 3 »" ∕ ’- + a⅛∖

* 2 (l-2 3 ) “'-“A 2 ' j λ

When the spheres are not concentric, expressions for the effective inertias have been obtained by the method of images by Mr W. Μ. Hicks {Philosophical Transactions, 1880).

The image of a source at P of strength μ outside a sphere is a source inside the sphere of strength T f C at a distance from the centre, a being the radius of the sphere, and a line sink reaching from the image to the centre of line strength - A; this α combination will be found to produce no flow across the surface of the sphere.

Again for a source P of strength μ inside the sphere, the images will be a source of strength at the inverse point of P, that is, at a distance θp from the centre, and a line sink - ~ thence to infinity.

In order that there should be no flow across the spherical boundary, another sink of equal strength must exist inside the sphere, and the infinite parts of the line sinks will then cancel.

The determination of the ψ ’ s andχ ’ s is a kinematical problem, as yet solved only for the cases we have mentioned, and the discovery of the solution of fresh problems is at present engaging the attention of mathematicians.

But supposing them determined for the motion of a body through uquιd, then T, the kinetic energy of the body and the liquid, will be a. quadratic function of u, v 9 w 9 p, q, τ ; so that we may put

2T - c 11 u 2 + c 22 √ 1 + c 33 w 2 + c 44 p 2 + c m9 2 + c ββ r 2

+ 2⅛,vw + .... + 2c sβ qτ + .... + 2c u υp + . . . .

In all twenty-one terms; and, in order to determine the c’s, we may suppose all the velocities except one or two to vanish, and then we see that c u -M + p∕∕*Zψ 1 dS,. . . .

where M is the mass of the body, c 44=A + p∕∕χ 1 (ny-77<z)dS, . . . .

where A is the moment of inertia of the body about the axis of x ; these are obtained by supposing all to vanish except u or p.

If we suppose all to vanish except v and w, we find Φ = ψ 2 ν + ψ 3 w, and 2T - ρjjφ t ⅛dS + M(r 2 + w 2 )

=c∙^ 2 + c 33 w 2 + p vw^ψ 2 ⅛ + ≠⅛) rf8 »

Similarly the other coefficients may be determined (Kirchhoff^ Vorlesungen uber Mathematische Physik, p. 240).

In particular cases of symmetry, the coefficients of the products of u, v, w, p, q, r can be made to vanish by a proper choice of axes; and in the case of the ellipsoid, the only case for which the coefficients have as yet been determined, c 11 - M + 4π ρ abc .A⅞ n 0 ‘ v 0

, - a 1 1 - f -Λ∙ (B 0 -C 0 )(0 2 - c 2 ) 2 44 Λ B o - C 0 )(δ 2 + c 2 ) + (A o + B β + C 0 )(δ 2 - c 2 ) ’

while c 12 , . . . vanish, the origin 0 being at the centre of the ellipsoid, and the axes of the ellipsoid its principal axes.

In the case of a sphere of mean density <r, projected in infinite liquid of density ρ, and subject to gravity, the sphere will describe a parabola, with vertical acceleration ~~ p g. σ + ⅛p

Having expressed T now as a quadratic function of u, v,w,p, q, r> the coefficients being functions of the shape but independent ol the position and orientation of the body, the Hamiltonian equations of motion lead to the equations <f(<ff ∖ _ ≤E γ dt ∖ du J r dv Jdw* z d fdlΓ ∖ dT dT dT dT τ dt ∖ dp J dq ^dr dv dw

For if P demote the resultant linear impulse in the direction, fixed in space, whose direction-cosines are l, rn, n, then p ΛT dΥ d r V du dv dw and, differentiating with respect to the time, since ^=mr-nq, ^≈np-rl, ~^lq^rnp, dt dt dt dP = l I -≤∕≤∏ dT dT)

’ ' dt ( dv ∖ du) r dv + ^dw}

. β , j dfdT ∖ cΓΓ ι dT)

( dt ∖ dv ) ?dw + r du )

, m ( d∕<ΓΓ∖ dT ι 7T )

( dt ∖ dwj ^du + P dv I

= ZX + τnY + nZ, for all values of l, m, n.

Again, taking a fixed origin, and supposing G the impulsive couple about a straight line through the origin fixed in space whose direction-cosines are l, m, n; p j fdT ι dΥ d r Γ ∖

G = 41 J- + y— - z·y J

∖ dp s, dw dv )

, ∕dT dT dT∖

∖ dq du dw J .HcΓT dT dT ∖ ∖ dr dv s, du ) where x, y, z are the coordinates of the centre of the body.

Differentiating with respect to the time, and supposing afterwards that the centre of the body and the fixed origin are coincident, then,since^=z4 ⅜ = w > butx=0, y=0, z = 0, ’ dt dt di dG A d ∕<fΓ∖ <ZT ι d r Y cZT ι dT 1

dt “ ( dt ∖ dp) r dq dr W dv +ν dw )

( d∕dT∖ dΥ dT dΥ T)

+rn H ·y -P~r + r τ- ~ u τ~ + w τr~ ( ( dt ∖ dq) dr dp dw du ) ( dfdΥ ∖ dΥ dT dT dlι ∖

+ n j - π -5- ]-Q-γ + Pj— v j~ + u ττ^ ( ( dt ∖ dr j dp dq du do )

= ZL + wιM + nN , for all values of l, m, n.

If no external forces act, then three integrals of the equations of motion are

(1) T = constant; ( 2) ^y + ^y + /«y =consUnt .

∖ duj ∖ dυj ∖ dw J + + " "constant; ' du dp dυ dq dw dr expressing the fact· that the energy is constant, and also the force and couple constituents of the resultant impulse.

For a body like an ellipsoid, using single suffixes,

T = ⅛ (c 1 u 2 + c 3 v 2 + c 3 w 2 + c 4 p 2 + c 5 q 2 + c β r 2 ); and the integration of the equations of motion under no forces leads to hyperelliptic and double θ functions (Weber, Mathematische Annalen, vol. xiv. ).

The equations of motion become [table]

Multiplying the equations by u, v, w, p, q, r, and adding, du do dw dn dq dr n c ∙'⅛ + c >⅛ + c ∙*⅛ +c ⅛ + c ∙⅛ + ν di- 0 or ⅜ (c 1 u 2 + c 2 v 2 + c 3 w 2 + c i p 2 + c- a q 2 + c β r 2 ) = T, a constant . . (7 ).

Multiply (1) by c 1 u, (2) by c 2 v, and (3) by c 3 w, and add; then 2 du »do 2 dw n and c 1 2 u 2 + c 2 2 υ 2 + c 3 2 w 2 =Ε 2 (8),

F being a constant, the resultant linear impulse of the motion.

Again, multiplying the equations (1 to 6) by c 4 p, c i q, c 6 r, c 1 u, c 2 ν, c 3 w, adding and integrating, c 1 c 4 up + c. 2 c 6 vq + c 3 e 6 wr=G, a constant . . . (9).

Equations (4), (5), (6) show that the body is acted upon by component couples about the principal axes (c 2 -c 3 )νw, (c 3 -c 1 )wu, (c 1 ~c 2 )uυ, the principal moments of inertia being supposed to be c 4> <⅛ c β∙

If the body be of revolution, c 1 = c 2 and c 4 = c 5 , the motion can be expressed by elliptic functions. For equation (6) shows that r is constant, and equation (3) becomes

¾ 2 (⅞) 2 = c ι 2 (^ - ^) 2 - C 1 2 {(u 2 + v 2 ')(p 2 + q 2 ) - (up + t 7 ) 2 }

= ~(F 2 - c 3 2 w 2 } ∖ ρι - F 2 - C ti r 2 - -½-(c 1 - ¾> 2 } - ^-W , ) 2 . . (10),

4 * Cj ' C 4 a biquadratic function of w, and therefore w is an elliptic function of t, the time.

Put 14=3 cos/ ® = -3 sin/; then from (1) and (2)

c ι s2 ^ *= c 1 (ur - nv) = c 1 rs 2 - c 3 (up + νq)w or i ff ≈ r _ c 3 l H>+W w . dt r c 1 u 2 +ν 2 ’ a rational function of w ; and therefore f is expressed in terms of the time by elliptic integrals of the third kind.

Again, put p = σ cos g , q≈ - σ sin g ; then from (4) and (5) =c l {ρl-ρq} = (e 4 - c β )σ 3 r + (e 1 - <¾)(t¾J + νq)w; or ⅛-(l-A), + ½Σ⅛ ≡⅛, dt ∖ c i J c i p 2 + q 2 a rational function of w ; and therefore g is expressed in terms of the time by elliptic integrals of the third kind.

In a state of steady motion, w is constant, and -£ = ⅛; also dt dt uρ+vq=sσ ; and therefore c β y ⅞ σ i c ι~ c s g .

c 4 w c 1 s c i σ ’ and we must therefore have

4>4-⅛-(q-⅞)-⅝

w 2 c 1 c β 2 for the roots of this quadratic in s: σ to be real.

If we employ the Lagrangian coordinates x, y, z, θ, φ, ψ, and take OZ in the direction of the resultant linear impulse F, then

(fig. 10) the eye being supposed at 0 the centre of the sphere, c 1 u= component momentum in direction OA= - F sin θ cosφ , c 1 v= ,, ,, OB = F sin θ sin φ , c λ w= ,, „ OC = F cos 0; and therefore equation (10) gives cos θ as an elliptic function of l. Since p = sin φθ - sin θ cos ≠ψ, q = cos φθ + sin θ sin φφ, equation (9) becomes c i sin 2 θψ + c β r cos 0=G ... . . (11); dψ G - c β r cos θ dt c 4 sin 2 0

= 1_ G-¾r 1 l 1 G + ¾r 1.

2 c 4 1 - cos 0 2 c 4 1 + cos 0 ’

and therefore ψ will consist of elliptic integrals of the third kind. Equations (4), (5), (6). show that the body is acted upon at every instant by a couple whose axis is OE, of magnitude = -⅛(c 1 -c 3 )w 2 tan 0 , c ι c 4 being supposed to be the equatorial moment of inertia of the body.

If OT be the direction of motion of 0, then OT lies in the plane ZOC, and tan COT = tan 0. c ι

We may determine the steady motion from elementary reasoning; for if OG be the axis of the resultant angular momentum (also lying in the plane ZOC) making an angle β with OC, and if μ be the constant value of ψ, then Gμ sin (0 - £) = impressed couple .⅛ (c 1 -c 3 )w s tan 0.

But G cos β = c a r, G sin 0= ~c i p; and ρ= -μ sin 0; therefore tan sin 0; c 6 r and if 7 he the angle made by the axis of instantaneous rotation with OC, tan 7=— sin 0.

Therefore Gμ sin (θ-β')≈ c β rμ ain (∂ cos β

-=c β rμ(sin 6-cos θ tan β) = c 6 rμ sin θ - c i μ 2 sin θ cos θ

= ⅛ (c 1 - c 3 )uf tan θ; and, dropping the factor sin θ, „ c 9 , , w 2 C i COS θμ 2 ~ c 6 rμ + ^(c 1 - ¾) —— = 0, a quadratic equation in μ, the condition for steady motion.

The least admissible value of r in order that the roots should be real is given by c β 2 r 2 = 4 ⅛ (c 1 -c 3 )c i w 2 , c ι or r 2 →½(<⅛-<⅛)-⅞w 2 ∙

¾ Cq

In an oblate solid of revolution c 1 - c 3 is negative, and the roots of the quadratic in μ are always real for all values of r. In a prolate solid c 1 - c 3 is positive, and a certain spin r is required to keep the motion stable.

An interesting application is to determine the proper amount of rifling of a gun. The following table has been calculated, from the formulae given below, by Captain J. P. Cundill, R.A., and the results appear to agree very fairly with what is observed in practice.

Table calculated for Stability of Rotation of Projectiles. [table]

Suppose the rifling at the muzzle makes one turn in n calibres, and 2α is the calibre and β the angle of the rifling; then tan 3=-- = -≈2 ∕ I ⅛(cr - c )-⅛ j .

n w ∖∕ j c 1 1 * c β 2 )

If W = weight of shot, and w = weight of air displaced, then c 1 -W + wα, c 3 ≡W + wy, c 4 = W⅛ 1 2 + w⅛i 2 a', c u = W⅛ 2 , where k l , k are the radii of gyration of the shot about an equatorial •diameter and the axis, and k[ of the air displaced, supposed rigidified, about an equatorial axis; and then a, y, a will be certain quantities depending only upon the external shape of the projectile, supposing the surrounding medium frictionless and incompressible.

When, as in practice, the fraction ~ is so small that its square may be neglected, tan ⅛ = ξ = 4⅜ 1 -⅞) ⅛ n 2 c 1 ' 1 3 ’ cf 1

W + w ' υ w ⅛ 1 ≡ + 2 √

1 4- w '

WU 1 w⅛ 1 ' 2 Λ

4 W (a y, k 4 w ∖ 1 + W k 2 a ) 1+ w c

. w . . α 2 ⅛ 1 2 = 4 W^

+ higher powers of-U which are neglected.

The only body for which α, y, and α' have as yet been determined is the ellipsoid; and in the case of a prolate spheroid of semi-axes a and c,

A C a A + C ’ 7 2A ’

e , = (C - A)(c 2 - q 2 ) 2 j (C - A)(c 2 - α 2 ) + (2A + C)(c 2 + a 2 ) I (c 2 + a 2 )

, ∕∙∞ dx where A = ∕ ——-

½> (a 2 +λ) 2 (c 2 + λ)l

£ 1 1 c + √(c 2 -a 2 ) a 2 (c 2 - a 2 ) 2 (c 2 - a 2 )i 1 ° gf c - √(c 2 - a 2 ) ’

c -Γ co d ^

Jo (a 2 + λ)(c 2 + λ)i ’

1 . c + √(c 2 - a 2 ) 2

^ (c 2 -a 2 )⅛ ge c-√(c 2 -a 2 ) “c(c 2 -a 2 ) 5

2

and therefore 2 A + C=- 5 - ∙

a 2 c Wave Motion in Liquids.

First consider plane waves propagated in the direction of the axis of x in liquid of depth h, the undisturbed surface being taken as the plane of xy and the axis of z drawn vertically upwards.

The equation of continuity, supposing a velocity function φ to exist, being d 2 φ d 2 φ dx 2 + dz 2 ’ we must first seek a solution of this equation, involving a periodic

9 9 7γ V term of the form sin (mx-nt), where m = — , n≈~f~ , λ being λ À

the wave length and V the velocity of propagation of the waves.

If we put Φ=fz) sin {mx-nt), then the solution of which, under the condition that — = o when dz

2= -h, is fz^) = A cosh τn(z + Λ), and therefore φ = A cosh m{z + h) sin {mx-nt).

We must now endeavour to make the free surface a surface of equal pressure, and in order to do this we must suppose A small enough for its square to be neglected; and therefore the square of the velocity is to be neglected too.

The dynamical equation then becomes — + qz +^ = II, a constant; p dt and at the surface where z≈0,-^ j = 0, and-- may be put = ^ , dt dt ∙dz therefore g~ + c ~ = 0, when z = 0.

dz dt

Therefore Amg sinh mh — n 2 A cosh τnΛ = 0 , or ri 2 =mg tanh mh , λ— XT2 tlh a. 1 2ττ7t or V 3 ="- tanh .

2ιr λ

If the depth h be very great compared with the wave length λ, then neglecting the square of A , ft

V 2 =^ .

2ιr

If the depth h be very small compared with the wave length λ, then, neglecting the square of -- ,

V 2 = <7Λ∙

Next consider the more general case of wave motion propagated in the direction of the axis of x at the common surface 2=0 of two liquids, the lower of density p and bounded below by the fixed plane z = - h, and the upper of density ρ , and bounded by the fixed plane z=lι ’ , and suppose U and U' the mean velocities of currents in the liquids making ang es α and a with the axis of x ; suppose in addition there is a surface tension T at the common surface of the liquids.

Denoting the velocity functions by φ and ψ , φ =U cos a . ae + U sin a . y + A cosh m(z + h) cos (mx-nt) φ' = U' cos α . x + U' sin a! . y + M cosh m(z-h ’ ) cos {mx-nt); then φ and φ' satisfy the equations of continuity, and the conditions that — = 0 when z≈ -h, and ⅛-==0 when z = ⅛'.

dz dz

Supposing the equation of the moving surface of separation to be z = b sin {mx - nt), then the direction of motion of each liquid, relative to the moving surface of separation, must be a tangent to the surface, and therefore, when s=0, dφ dφ' dz _ . dx dx or, neglecting A 2 and M 2 ,

A sinh mh - A' sinh mh' ,

U cosα-V - U'cosa'-V “ ’ dividing out by the common factor m cos Cnx - nt), and therefore φ = U cos a. X +U sin a. y + (U cos a - V)δ cosh ÷ -cos (mx -nt), j ' ' sιn h m f l φ'= U ’ cos a!.x + U'sin a!. y - (U'cos« - V^}cos(rn¾ - nt). Ύ j x sinh mh!

The dynamical equations are j , + ^+,⅛+⅜ W 1 -H, p'+0p'≡+p'⅛ + ⅛ l>⅛ ’-≡'i (M∕ and at the surface of separation, where z=0, we must have p —ρ' = - T = m 2 Tδ sin {mx - nt) dx i

= H - H' - g(p - p')b sin (mx - nt) - p(V cos a - V) coth mh . nb sin {mx - nt)

- p'(U ’ cos a' - V) coth mh'. nb sin (mx - nt)

- ⅛ p{U cos a - (U cos a - V) coth mh . mb sin (rnx - ni)J 2

- ⅜ pU 2 sin 2 a - ⅜ p (U cos a - V) 2 m 2 b 2 cos 2 (mx - nt)

+ ⅛p'{U' cos α' + (U' cos a! - V') coth mh'. mb sin Crvx-nt)} i

+ ⅜ p'U' 2 sin 2 a' + ⅜ p'(U' cos a - V) 2 τn⅞ 2 cos 2 {mx - nt); and neglecting b 2 and equating to zero the coefficient of sin {mx - nt), τn 2 T +g(p - p') - (U cos a - V)(mU cos a - n)p coth mh

- (U' Cos a! - V)(mU' cos α' - n)p' coth mh' = 0, which, since — ≈ V, reduces to m m 2 Υ + g{p - p') - τn(U cos α - V) 2 ρ coth mh

- τn(U' cos a' - V) 2 ρ' coth mh'≈ 0, or, since m = — , λ ’

⅛ T = - j (U cos a-V) 2 ρ coth ⅛.(U'cosa'- V) 2 p'coth^' j λ λ ( λ λ )

-S f (p-p') = O∙

If U = 0, U' = 0, p'=0, we find vs ∕j* ⅛T∖ ⅛A

∖2π Kp } λ

as at first, if T=0.

A discussion of the different cases that can arise is given by Lord Rayleigh in his papers on the “Instability of Jets” published in the Proceedings of the Royal Society and of the London Mathematical Society ; also in a paper by Sir W. Thomson in the Phil. Mag., 1871.

In the last-mentioned paper an interesting application of the above equations is made to determine the ripples produced by wind blowing over the surface of still water.

Put U = 0, Λ=oo , 7√=oo; then w 2 T + g(p - p ’ ) - rnV 2 p - τn(U' - V) a p'=0.

If W be the velocity of propagation of waves of the same length with no wind, then rn 2 T + g(g> - p') - mW 2 (p + p') = 0; or W 2 =-; 'm p + p p + p' the minimum value of which for different values of m is given by W 2 ≈2√(yT) ^~* ’ '>, p+p' and then m i = T( p - p ') .

But V 2 p + (W-V) 2 p'=W 2 (p + p)

and therefore V ⅛ . ∕ j W 2 - ,⅛U" ⅛; p + p' 3z V ( (p + p'√ )

giving the velocities of propagation of waves with and against the wind.

The least value of U' 2 is less than + times the least value PP of W 2 , and is therefore 2√(^T)e±∕√(p-p') . PP

If the wind be blowing with a velocity greater than this minimum value of U', the surface of the water as a plane level surface becomes unstable, and ripples are produced.

With C. G. S. units, <j = 981, T = 81, p = l, p'='0012759, and then the minimum value of U ’ is about 664, equivalent to about 14‘8 miles an hour. This velocity is of course much greater than what is required to ruffle the surface of water in reality, the discrepancy being due to the viscosity of the air.

In the case of standing waves in a circular tank, cylindrical coordinates r, θ, z being used, where aj=r cos θ, y≈r sin θ, the equation of continuity becomes d 2 φ 1 dφ 1 cPφ cPφ _θ dr 2 + r dr + r 2 db 2 ^ dz 2 ~

If the liquid be of depth h, we must put φ = φ 1 cosh k(z + h) cos 2πnt, where n is the number of oscillations per second, and then ⅛ + ± ⅛ + 2 ⅛ + J⅜ 1 -o. dr 2 r dr r 2 dθ 2

If we put <J> 1 = ψ cos mθ, then

≤≠ + l ⅛' + Λ-2.^ 2λ l ψ = 0, dr 2 r dr + ∖ r 2 ∕ Bessel’s differential equation; and therefore ψ = AJ m der), and φ = AJ„(&?·) cos mθ cosh k{z + h) cos 2ιτ)ii; and k must be determined from the condition that ^ = 0, when r=a; dr or Jk(⅛α) = 0.

At the free surface

⅜ ^0 = 0 , y dz + dt 2 or gk sinh kh - 4π⅛ 2 cosh kh≈G, or n 2 ∙= -⅛ tanh kh. 4π-

For circular waves, τn = 0, and the roots of Jθ dca)=0, are *α-=3∙832, 7'016, 10’173, 13∙323

(Rayleigh, Sound, p. 274).

When the tank is limited by the radial plane 6 = 0, then the slowest oscillation corresponds to m= ⅛, and then

J i (fcr)=≡⅛, _ √⅛Γ

and Ji(fcα)= z ^∕-^-^cos ka - ’ gives tan ka=ka, 1cci∕ and therefore — = 1 ∙4303

7Γ (Rayleigh, Sound, p. 279).

When the tank is bounded by the radial planes β = 0, β=-∣ ιr, the slowest oscillation corresponds to wι=∣, and then τ zτ ∖ 1 /sin kr , ∖

V r >-√κ(τ^- cosfc ∙)∙ and the equation J'(⅛α)=≡0 leads to taniα -≡^∙

For the discussion of the free oscillations of an ocean of uniform depth, covering a central nucleus, under the gravitation of the parts, and the surface tension at this free surface, consult Lamb’s. Motion of Fluids, p. 196, and Lord Rayleigh’s papers in the Proceedings of the London Mathematical Society.

The propagation of plane waves of longitudinal displacement in air, and the notes produced by open and closed pipes, have been considered under the heading Acoustics.

When the air is limited by special surfaces, the problem of its vibrations is worked out by Lord Rayleigh in the Proceedings of the-London Mathematical Society, 1872.

A list of references to the memoirs and treatises on the subject will be found at the end of Lamb’s Motion of Fluids. (A. G. G.) PART HL—HYDRAULICS.

I. THE DATA OF HYDRAULICS.

[Units.— Except where other units are given, the units throughout this article are feet, pounds, pounds per sq. ft., feet per second.]

1. Properties of Fluids.— The fluids to which the laws of practical hydraulics relate are substances the parts of which possess very great mobility, or which offer a very small resistance to distortion independently of inertia. Under the general heading Hydromechanics a fluid is defined to be a substance which yields continually to the slightest tangential stress, and hence in a fluid at rest there can be no tangential stress. But, further, in fluids such as water, air, steam, Ac., to which the present division of the article relates, the tangential stresses that are called into action between contiguous portions during distortion or change of figure are always small compared with the weight, inertia, pressure, &c., which produce the visible motions it is the object of hydraulics to estimate. On the other hand, while a fluid passes easily from one form to another, it opposes considerable resistance to change of volume.

It is easily deduced from the absence or smallness of the tangential stress that contiguous portions of fluid act on each other with a pressure which is exactly or very nearly normal to the interface which separates them. The stress must be a pressure, not a tension, or the parts would separate. Further, at any point in a fluid the pressure in all directions must be the same; or, in other words, the pressure on any small element of surface is independent of the orientation of the surface.

2. Fluids are divided into liquids, or incompressible fluids, and gases, or compressible fluids. Very great changes of pressure change the volume of liquids only by a small amount, and if the pressure on them is reduced to zero they do not sensibly dilate. In gases or compressible fluids the volume alters sensibly for small changes of pressure, and if the pressure is indefinitely diminished they dilate without limit.

In ordinary Hydraulics, liquids are treated as absolutely incompressible. In dealing with gases the changes of volume which accompany changes of pressure must be taken into account.

3. Viscous fluids are those in which change of form under a continued stress proceeds gradually and increases indefinitely. A very viscous fluid opposes great resistance to change of form in a short time, and yet may be deformed considerably by a small stress acting for a long period. A block of pitch is more easily splintered than indented by a hammer, but under the action of the mere weight of its parts acting for a long enough time it flattens out and flows like a liquid.

All actual fluids are viscous. They oppose a resistance to the relative motion of their parts. This resistance diminishes with the velocity of the relative motion, and becomes zero in a fluid the parts of which are relatively at rest. When the relative motion of different parts of a fluid is small, the viscosity may be neglected without introducing important errors. On the other hand, where there is considerable relative motion, the viscosity may be expected to have an influence too great to be neglected.

Measurement of Viscosity. Coefficient of Viscosity. — Suppose the plane ab, fig. 11, of area ω,to move with the velocity V relatively to the surface cd and parallel to it. Let the space between be filled with liquid. The layers of liquid in contact with ab and cd adhere to them. The intermediate layers all offering an equal resistance to shearing or distortion, the rectangle of fluid abed will take the form of the parallelogram a ’ b ’ cd. Further, the resistance to the motion of ab may be expressed in the form

R=>κωV (1), where κ is a coefficient the nature of which remains to be determined.

If we suppose the liquid between tib and cd divided into layers na. shown in fig. 12, it will be clear that the stress R acts, at each dividing face, forwards in the direction of motion if we consider the upper layer, backwards if we consider the lower layer. Now suppose the original thickness of the layer T increased to wT; if the bounding plane in its new position has the velocity nV, the shearing at each dividing face will be exactly the same as before, and the resistance must therefore be the same. Hence, R = √ω(nV) . . (2).

But equations (1) and (2) may both be expressed in one equation if κ and κ' are replaced by a constant varying inversely as the thickness of the layer. Putting κ = ^,, κ'≈P-, T ni p V

or, for an indefinitely thin layer, κ -'"⅛ ' 3 '∙

an expression first proposed by Navier. The coefficient μ is termed the coefficient of viscosity.

According to Maxwell, the value of μ for air at θ o Fahr, in pounds, when the velocities are expressed in feet per second, is μ = 0 ∙000 000 025 6(461° + 0); that is, the coefficient of viscosity is proportional to the absolute temperature and independent of the pressure.

The value of μ for water at 77° Fahr, is, according to Helmholtz and Piotrowski, j a = 0∙000 001 91, the units being the same as before. For water μ decreases rapidly with increase of temperature.

4. When a fluid flows in a very regular manner, as for instance when it flows in a capillary tube, the velocities vary gradually at any moment from one point of the fluid to a neighbouring point. The layer adjacent to the sides-of the tube adheres to it and is at rest. The layers more interior than this slide on each other. But the resistance developed by these regular movements is very small. If in large pipes and open channels there were a similar regularity of movement, the neighbouring filaments would acquire, especially near the sides, very great relative velocities. Boussinesq has shown that the central filament in a semicircular canal of 1 metre radius, and inclined at a slope of only 0Ό001, would have a velocity of 187 metres per second, the layer next the boundary remaining at rest. But before such a difference of velocity can arise, the motion of the fluid becomes much more complicated. Volumes of fluid are detached continually from the boundaries, and, revolving, form eddies traversing the fluid in all directions, and sliding with finite relative velocities against those surrounding them. These slidings develop resistances, incomparably greater than the viscous resistance due to movements varying continuously from point to point. The movements which produce the phenomena commonly ascribed to fluid friction must be regarded as rapidly or even suddenly varying from one point to another. The internal resistances to the motion of the fluid do not depend on the general velocities of translation at different points of the fluid (or what M. Boussinesq terms the mean local velocities), but rather on the intensity at each point of the eddying agitation. The problems of hydraulics are therefore much more complicated than problems in which a regular motion of the fluid is assumed, hindered by the viscosity of the fluid.

Relation of Pressure, Density, and Temperature of Liquids.

5. Density of Water.— Water at ordinary temperature and pressure contains 62’4 lb per cubic foot, or 1000 kilogrammes per cubic metre. The density or weight per unit of volume will be designated by G. River and spring water is not sensibly denser than pure water, being at most l-100000th heavier. Sea-water may be taken at 64 lb per cubic foot.

6. Compressibility of Liquids.— The most accurate experiments show that liquids are sensibly compressed by very great pressures, and that up to a pressure of 65 atmospheres, or about 1000 lb per square inch, the compression is proportional to the pressure. The chief results of experiment are given in the following table. Let V 1 be the volume of a liquid in cubic feet under a pressure p 1 lb per square foot, and V 2 its volume under a pressure ρ v Then the cubical compression is

V¾~ Vi

V 1 ’

and the ratio of the increase of pressure p< i - ρ l to the cubical compression is sensibly constant. That is, ^ 2 is constant.

'2 ~ ’ ι

This constant is termed the elasticity of volume, and is denoted by k (Thomson). With the notation of the differential calculus, ⅜ , V⅛- _dff y dN'

V

Elasticity of Volume of Liquids. [table]

According to the experiments of Grassi, the compressibility of water diminishes as the temperature increases, while that of ether, alcohol, and chloroform is increased.

7. Change of Volume and Density of Water with Change of Temperature.— Although the change of volume of water with change of temperature is so small that it may generally be neglected in ordinary hydraulic calculations, yet it should be noted that there is a change of volume which should be allowed for in very exact calculations. The values of ρ in the following short table, which gives data enough for hydraulic purposes, are taken from Professor .Everett’s System of Units. Density of Water at Different Temperatures. [table]

The weight per cubic foot has been calculated from the values of ρ, on the assumption that a cubic foot of water at 39 - 2° Fahr, is 62 - 425 lb. For ordinary calculations in hydraulics, the density of water (which will in future be designated by the symbol G) will be taken at 62’4 lb per cubic foot, which is its density at 53° Fahr. It may be noted also that ice at 32° Fahr, contains 57 2 lb per cubic foot. The values of ρ* are the densities in grammes per cubic centimetre.

8. Pressure Column. Free Surface Level.— Suppose a small vertical pipe introduced into a liquid at any point P (tig. 13). Then the liquid will rise in the pipe to a level OO, such that the pressure due to the column in the pipe exactly balances the pressure on its mouth. If the fluid is in motion the mouth of the pipe must be supposed accurately parallel to the direction of motion, or the impact of the liquid at the mouth of the pipe will have an influence on the height of the column. If this condition is complied with, the height h of the column is a measure of the pressure at the point P. Let ω be the area of section of the pipe, h the height of the pressure column, p the intensity of pressure at P; then pω = Chu> lb, that is, h is the height due to the pressure at p. The level OG will be termed the free surface level corresponding to the pressure at P.

Relation of Pressure, Temperature, and Density of Gases.

9. Relation ofPressure, Volume, Temperature, and Density in Compressible Fluids.— Certain problems on the flow of air and steam are so similar to those relating to the flow of water that they are conveniently treated together. It is necessary, therefore, to state as briefly as possible the properties of compressible fluids so far as knowledge of them is requisite in the solution of these problems. Air may be taken as a type of these fluids, and the numerical data here given will relate to air.

Relation of Pressure and Volume at Constant Temperature.— At constant temperature the product of the pressure y?and volume V of a given quantity of air is a constant (Boyle’s law).

Let;? 0 be mean atmospheric pressure (2116’8 lb per square foot), V # the volume of 1 lb of air at 32° Fahr, under the pressure p 0 . Then p 0 V 0 = 26214 (1).

If G o is the weight per cubic foot of air in the same conditions, 0 -⅛-≡i-≡ 5 < 2 >∙

For any other pressure p, at which the volume of 1 lb is V and the weight per cubic foot is G, the temperature being 32° Fahr.,

2>V-⅜-26214 1 orQ-⅛ <3)∙

Change ofPressure or Volume by Change of Temperature.—Let p 0 , V o , G o , as before be the pressure, the volume of a pound in cubic feet, and the weight of a cubic foot in pounds, at 32° Fahr. Let;?, V, G be the same quantities at a temperature t (measured strictly by the air thermometer, the degrees of which difler a little from those of a mercurial thermometer). Then, by experiment,

^^∙ v ∙l≡2≈^ v s · < 4 >- where τ, τ 0 are the temperatures t and 32° reckoned from the abso lute zero, which is -460’6° Fahr.; JL=P<l. L (4a);

G G o τ 0

G = ^-.ToG o (5).

Po τ

If p 0 ≈ 2116’8, G 0 =O8075, τ 0 ≡ 460’6 + 32 = 492'6, then g- = 53'2τ (M∙

II. KINEMATICS OF FLUIDS.

10. Moving fluids as commonly observed are conveniently classified thus :—

(1) Streams are moving masses of indefinite length, completely or incompletely bounded laterally by solid boundaries. When the solid boundaries are complete, the flow is said to take place in a pipe. When the solid boundary is incomplete and leaves the upper surface of the fluid free, it is termed a stream bed or channel or canal.

(2) A stream bounded laterally by differently moving fluid of the same kind is termed a current.

(3) A jet is a stream bounded by fluid of a different kind.

(4) An eddy, vortex, or whirlpool is a mass of fluid the particles of which are moving circularly or spirally.

(5) In a stream we may often regard the particles as flowing along definite paths in space. A chain of particles following each other along such a constant path may be termed a fluid filament or elementary stream.

11. Steady and Unsteady, Uniform and Varying, Motion.— There are two quite distinct ways of treating hydrodynamical questions. We may either fix attention on a given mass of fluid and consider its changes of position and energy under the action of the stresses to which it is subjected, or we may have regard to a given fixed portion of space and consider the volume and energy of the fluid entering and leaving that space.

If, in following a given path ab (fig. 14), a mass of water a has a constant velocity, the motion is said to be uniform. The kinetic energy of the mass a remains unchanged. If the velocity varies from point to point of the path, the motion is called varying motion. If at a given point a in space, the particles of water always arrive with the same velocity and in the same direction, during any given time, then the motion is termed steady motion. On the contrary, if at the point a the velocity or direction varies from moment to moment the motion is termed unsteady. Steady motion is sometimes termed permanent motion. A river which excavates its own bed is in unsteady motion so long as the slope and form of the bed is changing. It, however, tends always towards a condition in which the bed ceases to change, and it is then said to have reached a condition of permanent regime. No river probably is in absolutely permanent regime, except perhaps in rocky channels. In other cases the bed is scoured more or less during the rise of a flood, and silted again during the subsidence of the flood. But while many streams of a torrential character change the condition of their bed often and to a large extent, in others the changes are comparatively small and not easily observed.

As a stream approaches a condition which would be strictly defined as one of steady motion, its regime becomes permanent. Hence steady motion and permanent regime are sometimes used as meaning the same thing. The one, however, is a definite term applicable to the motion of the water, the other a less definite term applicable in strictness only to the condition of the stream bed.

12. Theoretical Notions on the Motion of Water.— The actual motion of the particles of water is in most cases very complex. To simplify hydrodynamic problems, simpler modes of motion are assumed, and the results of theory so obtained are compared experimentally with the actual motions.

Motion in Plane Layers. — The simplest kind of motion in a stream is one in which the particles initially situated in any plane cross section of the stream continue to be found in plane cross sections during the subsequent motion. Thus, if the particles in a thin plane layer ab (fig. 15) are found again in a thin plane layer a'b' after any interval of time, the motion is said to be motion in plane layers. In such motion the internal work in deforming the layer may usually be disregarded, and the resistance to the motion is confined to the circumference.

. Laminar Motion.— In the case of streams having solid boundaries, it is observed that the central parts move faster than the lateral parts. To take account of these differences of velocity, the stream may be conceived to be divided into thin laminae, having cross Sections somewhat similar to the solid boundary of the stream, and sliding on each other. The different laminae can then be treated as having differing velocities according to any law either observed or deduced from their mutual friction. A much closer approximation to the real motion of ordinary streams is thus obtained.

Stream Line Motion.— In the preceding hypothesis, all the particles in each lamina have the same velocity at any given cross section. of the stream. If this assumption is abandoned, the cross section of the stream must be supposed divided into indefinitely small areas, each representing the section of a fluid filament. Then these filaments may have any law of variation of velocity assigned to them. If the motion is steady motion these fluid filaments (or as they are then termed stream lines) will have fixed positions in space.

Periodic Unsteady Motion.— In ordinary streams with rough boundaries, it is observed that at any given point the velocity varies from moment to moment in magnitude and direction, but that the average velocity for a sensible period (say for 5 or 10 minutes) varies very little either in magnitude or velocity. It has hence been conceived that the variations of direction and magnitude of the velocity are periodic, and that, if for each point of the stream the mean velocity and direction of motion were substituted for the actual more or less varying motions, the motion of the stream might be treated as steady stream line or steady laminar motion.

13. Volume of Flow.— Let A (fig. 16) be any ideal plane surface, of area «, in a stream, normal to the direction of motion, and let V be the velocity of the fluid. Then the volume flowing through the surface A in unit time is

Q = ωV (1).

Thus, if the motion is rectilinear, all the particles at any instant in the surface A will be found after one second in a similar surface A', at a distance V, and as each particle is followed by a continuous thread of other particles, the volume of flow is the right prism AA' having a base ω and length V.

If the direction of motion makes an angle θ with the normal to the surface, the volume of flow is represented by an oblique prism AA' (fig. 17), and in that case

Q = ωV cos fl .

If the velocity varies at different points of the surface, let the surface be divided into very small portions, for each of which the velocity may be regarded as constant. If du is the area and ν, or V cos θ, the normal velocity for this element of the surface, the volume of flow is

Q = fvdw, or fv cos 0 dω , as the case may be.

14. Principle of Continuity. —If we consider any completely bounded fixed space in a moving liquid initially and finally filled continuously with liquid, the inflow must be equal to the outflow. Expressing the inflow with à positive and the outflow with a negative sign, and estimating the volume of flow Q for all the boundaries,

2Q = 0.

In general the space will remain filled with fluid if the pressure at every point remains positive. There will be a break of continuity, if at any point the pressure becomes negative, indicating that the stress at that point is tensile. In the case of ordinary water this statement requires modification. Water contains a variable amount of air in solution, often about one-twentieth of its volume. This air is disengaged and breaks the continuity of the liquid, if the pressure falls below a point corresponding to its tension. It is for this reason that pumps will not draw water to the full height due to atmospheric pressure.

Application of the Principle of Continuity to the case of a Stream. —If A 1 , A o are the areas of two normal cross sections of a stream, and V 1 , V 2 ' are the velocities of the stream at those sections, then, from the principle of continuity, V 1 A 1 = V 2 A 2 ;

Σ1 = Δ? (2);

V 2 A 1 ' h that is, the normal velocities are inversely as the areas of the cross sections. This is true of the mean velocities, if at each section the velocity of the stream varies. In a river of varying slope the velocity varies with the slope. It is easy therefore to see that in parts of large cross section the slope is smaller than in parts of small cross section.

If we conceive a space in a liquid bounded by normal sections at A 1 , A 2 and between A 1 , A 2 by stream lines (fig. 18), then, as there is no flow across the stream lines,

V 1- A 2 V 2 A 1 ’ as in a stream with rigid boundaries.

In the case of compressible fluids the variation of volume due to the difference of pressure at the two sections must be taken into account. If the motion is steady the weight of fluid between two cross sections of a stream must remain constant. Hence the weight flowing in must be the same as the weight flowing out. Let p v ρ 2 be the pressures, υ 1 , v 2 the velocities, G 1 , G 2 the weight per cubic foot of fluid, at cross sections of a stream of areas A 1 , A 2 . The volumes of inflow and outflow are

Ap> 1 and A 2 v 2 , and, if the weights of these are the same, G 1 A 1 v 1 = G 2 A 2 ¾; and hence, from (5α) § 9, if the temperature is constant,

Λ a Λ=7⅞ a 2 v 2 (3 )∙

III. PHENOMENA OF THE DISCHARGE OF LIQUIDS FROM ORIFICES AS ASCERTAINABLE BY EXPERIMENTS.

15. When a liquid issues vertically from a small orifice, it forms a jet which rises nearly to the level of the free surface of the liquid in the vessel from which it flows. The difference of level h r (fig. 19) is so small that it may be at once suspected to be due either to air resistance on the surface of the jet or to the viscosity of the liquid or to friction against the sides of the orifice. Neglecting for the moment this small quantity, we may infer, from the elevation of the jet, that each molecule on leaving the orifice possessed the velocity required to lift it against gravity to the height h. From ordinary dynamics, the relation between the velocity and height of projection is given by the equation

As this velocity is nearly reached in the flow from well-formed orifices, it is sometimes called the theoretical velocity of discharge. This relation was first obtained by Torricelli.

If the orifice is of a suitable conoidal form, the water issues in filaments normal to the plane of the orifice. Let ω be the area of the orifice, then the discharge per second must be, from eq. (1),

Q = art’=» √ 2gh nearly (2).

This is often termed the theoretical discharge.

Use of the term Head in Hydraulics.— The term head is an old millwright’s term, and meant primarily the height through which a mass of water descended in actuating a hydraulic machine. Since the water in fig. 19 descends through a height A to the orifice, we may say there are h feet of head above the orifice. Still more generally any mass of liquid h feet above a horizontal plane may be said to have h feet of elevation head relatively to that datum plane. Further, since the pressure p at the orifice which produces outflow is connected with h by the relation = h, the quantity ∙^- may be termed the pressure head at the orifice. Lastly, the velocity v is *y 2 z y 2 connected with h by the relation g- = Λ, s °that may be termed the head due to the velocity v.

16. Coefficients of Velocity and Resistance.— As the actual velocity of discharge differs from V 2gh by a small quantity, let the actual velocity = v α =c,√2y⅛ . (3), where c β is a coefficient to be determined by experiment, called the coefficient of velocity. This coefficient is found to be tolerably constant for different heads with well-formed simple orifices, and it very often has the value 0 97.

The difference between the velocity of discharge and the velocity due to the head may be reckoned in another way. The total height A causing outflow consists of two parts,—one part h e expended in producing the velocity of outflow, another h r in overcoming the resistances due to viscosity and friction. Let hr Cjhe , where c r is a coefficient determined by experiment, and called the coefficient of resistance of the orifice. It is tolerably constant for different heads with well-formed orifices. Then v a ≈ ∖ ∕2gh t ≈ χ∕2g (4)∙

The relation between c v and c r for any orifice is easily found :— v α =c,√2^⅛= χ ∕2ff-A- ·

V 1 +c r

*-.√S < 5 >∙

c r =⅛-l (5α).

C, 2

Thus if c, = 0'97, then c r =0O628. That is, for such an orifice about 6} per cent, of the head is expended in overcoming frictional resistances to flow.

Coefficient of Contraction—Sharp-edged Orifices in Plane Surfaces. —When a jet issues from an aperture in a vessel, it may either spring clear from the inner edge of the orifice as at a or b (fig. 20), or it may adhere to the sides of the orifice as at c. The former condition will be found if the orifice is bevelled outwards as at a, so as to be sharp edged, and it will also occur generally for a prismatic aperture like b, provided the thickness of the vessel round the aperture is less than the diameter of the jet. But if the thickness is greater the condition shown at c will occur.

When the discharge takes place as at a or b, the section of the jet is smaller than the section of the orifice. This is due to the formation of the jet from filaments converging to the orifice in all directions inside the vessel. The inertia of the filaments opposes sudden change of direction of motion at the edge of the orifice, and the convergence continues for a distance of about half the diameter of the orifice beyond it. Let ω be the area of the orifice, and c e ω the area of the jet at the point where convergence ceases; then c c is a coefficient to be determined experimentally for each kind of orifice, called the coefficient of contraction. When the orifice is a sharp-edged orifice in a plane surface, the value of c c is on the average 0'64, or the section of the jet is very nearly five-eighths ot the area of the orifice.

Coefficient of Discharge.— In applying the general formula Q = ωυ to a stream, it is assumed that the filaments have a common velocity v normal to the section ω. But if the jet contracts, it is at the contracted section of the jet that the direction of motion is normal to a transverse section of the jet. Hence the actual discharge when contraction occurs is

Q β — c,υ × c c ω — ca<d√¾A, or simply, if c≡<⅛ c n Q a =c< ∣ Cd2gh, where c is called the coefficient of discharge. Thus for a sharp-edged plane orifice c=0 - 97 × 0∙64 = 0 ’ 62.

17. Experimental determination, of c v , c c , and c.— The coefficient of contraction c e is directly determined by measuring the dimensions of the jet. For this purpose fixed screws of fine pitch (fig. 21) are convenient. These are set to touch the jet, and then the distance between them can be measured at leisure.

The coefficient of velocity is determined directly by measuring the parabolic path of a horizontal jet.

Let OX, OY (fig. 22) be horizontal and vertical axes, the origin being at the orifice. Let h be the head, and x, y the coordinates of a point A on the parabolic path of the jet. If v a is the velocity at the orifice, and t the time in which a particle moves from 0 to A, then m . . .. t χ=v a ,t∙, y=⅛t*.

Eliminating t, v α = . Ae.

V 2y

Then c,--⅛=- ∕z ∖ l2gh V 4y⅛ ’

If the jet is not initially horizontal, let OB (fig. 23) be any horizontal datum line, and let the vertical distances OC, AD, BE be measured, the point A being taken conveniently midway between O and B. Then y 1 = OC-AD, and y a = OC-BE.

Let a be the inclination of the jet at C to the horizontal, so that ν a cos a is its horizontal and v a sin a its vertical velocity at that point. If t is the time in which a particle moves from C to D, then

4 = v a cos a t, Λ y 1 = t> α sin at - .

Eliminating t, y 1 =4 ta ∏ < χ - r7⅛ 1+ tan2 ®)· Δ ov a

Similarly, y i ≈x tan a- ~ i 0- + tan 2 α).

Hence tan a -fej⅛, x v = . ∕9 , {≈ 2 + (^i-3∕ 2 ) 2 }

“V 4(2y 1 -y s ) ’

c , ∕*+'(ιy 1 -y,)*

∖ l2gh V *h(2y 1 -y a ) ’ where for h is to be put the depth of C below the free water surface.

The coefficient of discharge is determined independently, by measuring the discharge in a gauging tank for a given time. Then, if Q is the measured volume discharged in one second, e—¾=.

o >^f2gh

18. Coefficients for Bellmouths and Bellmouthed Orifices.— If an orifice is furnished with a mouthpiece exactly of the form of the contracted vein, then the whole of the contraction occurs within the mouthpiece, and if the area of the orifice is measured at the smaller end, c c must be put = l. It is often desirable to bellmouth the ends of pipes, to avoid the loss of head which occurs if this is not done; and such a bellmouth may also have the form of the contracted jet. Fig. 24 shows the proportions of such a bellmouth or bellmouthed orifice, which approximates to the form of the contracted jet sufficiently for any practical purpose.

For such an orifice Weisbach has found the following values of the coefficients with different heads. [table]

As there is no contraction after the jet issues from the orifice, c β =l, c = c,; and therefore

Q = c r ω√¾Λ = ω ∖ ∕⅛lrj~r ·

▼ 1 "Γ Cf

19. Coefficients for Sharp-edged or virtually Sharp-edged Orifices. —The coefficient of velocity for sharp-edged orifices of different areas and under different heads is not very accurately determined. Its mean value is about 0 - 96.

The coefficient of contraction is also dependent on circumstances the relative influence of which is not so perfectly known as is desirable. Its mean value for well-placed orifices in a plane surface is 0 ’ 64. For conditions similar in other respects, the contraction is less (that is, the area of the stream is greater) the smaller the orifice and the less the head. If the surface surrounding the orifice is not plane, the coefficient is greater for a surface convex to the interior of the reservoir and less for a concave surface. The thickening of the edges of the orifice modifies the contraction in a slight degree, and if a border or rim is placed round part of the edge of the orifice, and projects inwards or outwards, the coefficient is very considerably altered, and the contraction is then termed incomplete. If the orifice is placed in a contracted part of the vessel so that the water approaches the orifice with considerable velocity, the coefficient is increased, and the contraction is said to be imperfect.

The coefficient of discharge has been determined for sharp-edged orifices under a great variety of conditions. Its mean value, taking the values of c v and c e given above, is 0 , 62.

For circular orifices, sharp-edged and with complete and perfect contraction, Weisbach found the following values :—

Coefficients of Discharge for Sharp-edged Circular Orifices. [table]

The following table, compiled by Mr Fanning {Treatise on Water Supply Engineering), gives values for rectangular orifices in vertical plane surfaces, the head being measured, not immediately over the orifice, where the surface is depressed, but to the still-water surface at some distance from the orifice. The values were obtained by graphic interpolation, all the most reliable experiments being plotted and curves drawn so as to average the discrepancies.

Coefficients of Discharge for Rectangular Orifices, Sharp-edged, in Vertical Plane Surfaces. [table]

20. Orifices with Edges of Sensible Thickness.— When the edges of the orifice are not bevelled outwards, but have a sensible thickness, the coefficient of discharge is somewhat altered. The following table gives values of the coefficient of discharge for the arrangements of the orifice shown in vertical section at P, Q, R (fig. 25). The plan of all the orifices is shown at S. The planks forming the orifice and sluice were each 2 inches thick, and the orifices were all 24 inches wide. The heads were measured immediately over the orifice. The formula above becomes, in this case,

Q-c⅛(H-⅛)

Table of Coefficients of Discharge for Rectangular Vertical Orifices in Fig. 25. [table]

21. Partially Suppressed Contraction.— Since the contraction of the jet is due to the convergence towards the orifice of the issuing streams, it will be diminished if for any portion of the edge of the orifice the convergence is prevented. Thus, if an internal rim or border is applied to part of the edge of the orifice (fig. 26), the convergence for so much of the edge is suppressed. For such cases Bidone found the following empirical formulae applicable :—

For rectangular orifices, c c = 0∙62^1 + 0∙152y)l and for circular orifices, c e ≡O∙62(l + O∙128^) 5 where n is the length of the edge of the orifice over which the border extends, and p is the whole length of edge or perimeter of the orifice. The following are the values of c e , when the border extends over ∣, ⅜, or ∣ of the whole perimeter :— [table]

For larger values of — the formulae are not applicable. Bornemann P has shown, however, that these formulae for suppressed contraction are not reliable.

22. Imperfect Contraction — If the sides of the vessel approach near to the edge of the orifice, they interfere with the convergence of the streams to which the contraction is due, and the contraction is then modified. It is generally stated that the influence of the sides begins to be felt if their distance from the edge of the orifice is less than 2 , 7 times the corresponding width of the orifice. The coefficients of contraction for this case are imperfectly known.

23. Orifices Furnished with Channels of Discharge.— These external borders to an orifice also modify the contraction.

The following coefficients of discharge were obtained with openings 8 inches wide, and small in proportion to the channel of approach (fig. 27, A, B, C).

[table]

24. Inversion of the Jet.— When a jet issues from a horizontal orifice, or is of small size compared with the head, it presents no marked peculiarity of form. But if the orifice is in a vertical surface, and if its dimensions are not small compared with the head, it undergoes a series of singular changes of form after leaving the orifice. These were first investigated by Bidone; subsequently Magnus measured jets from different orifices; and lately Lord Rayleigh (Proc. Roy. Soc.,xxix. 71) has investigated them anew.

Fig. 28 shows some remarkable forms, the upper row giving the shape of the orifices, and the others sections of the jef. The jet first contracts as described above, in consequence of the convergence of the fluid streams within the vessel, retaining, however, a form similar to that of the orifice. Afterwards it expands into sheets in planes perpendicular to the sides of the orifice. Thus the jet from a triangular orifice expands into three sheets, in planes bisecting at right angles the three sides of the triangle. Generally a jet from an orifice, in the form of a regular polygon of n sides, forms n sheets in planes perpendicular to the sides of the polygon.

Bidone explains this by reference to the simpler case of meeting streams. If two equal streams having the same axis, but moving in opposite directions, meet, they spread out into a thin disk nor mal to the common axis of the streams. If the directions of two streams intersect obliquely they spread into a symmetrical sheet perpendicular to the plane of the streams. Now those portions of a jet which proceed from different portions of an orifice are conceived to behave in some degree like independent meeting streams.

Let α 1 , α a (fig. 29) be two points in an orifice at depths h 1 , h 2 from the free surface. The filaments issuing atα 1 , a i will have the different velocities f2gh λ and ∖ j2gh i , Consequently they will tend to describe parabolic paths a x cb l and α a cδ a of different horizontal range, and intersecting in the point c. But since two filaments cannot simultaneously flow through the same point, they must exercise mutual pressure, and will be deflected out of the paths they tend to describe. It is this mutual pressure which’ causes the expansion of the jet into sheets.

Lord Rayleigh has pointed out that, when the orifices are small and the head is not great, the expansion of the sheets in directions perpendicular to the direction of flow reaches a limit. Sections taken at greater distance from the orifice show a contraction of the sheets until a compact form is reached similar to that at the first contraction. This is shown in the elevation of the jet c. Beyond this point, if the jet retains its coherence, sheets are thrown out again, but in directions bisecting the angles between the previous sheets. Lord Rayleigh accepts an explanation of this contraction first suggested by Buff, namely, that it is due to surface tension or capillarity. The fluid is enclosed in an envelope of constant tension, and the recurrent form of the jet is due to vibrations of the fluid column, about a circular figure of equilibrium, superposed on the general progressive motion. Since the phase of vibration depends on the time elapsed, it is always the same at the same point in space, and thus the motion is steady and the boundary of the jet is a fixed surface.

In so far as the vibrations may be considered isochronous, the distance between consecutive corresponding points of the recurrent figure, or, as it may be termed, the wave length of the figure, is directly proportional to the velocity of the jet, that is, to the square root of the head of water. For low he ids the measurements confirm this law. For higher heads there is an increase of the wave lengths in a higher ratio than the velocity of the jet. This points to a departure from isochronous vibration, the nature of which is investigated in Lord Rayleigh’s paper.

IV. THEORY OF THE STEADY MOTION OF FLUIDS.

25. The general equation of the steady motion of a fluid given under Hydrodynamics furnishes immediately three results as to the distribution of pressure in a stream which may here be assumed.

(a. ) If the motion is rectilinear and uniform, the variation of pressure is the same as in a fluid at rest. In a stream flowing in an open channel, for instance, when the effect of eddies produced by the roughness of the sides is neglected, the pressure at each point is simply the hydrostatic pressure due to the depth below the free surface.

(δ. ) If the velocity of the fluid is very small, the distribution of pressure is approximately the same as in a fluid at rest.

(c.) If the fluid molecules take precisely the accelerations which they would have if independent and submitted only to the external forces, the pressure is uniform. Thus in a jet falling freely in the air the pressure throughout any cross section is uniform and equal to the atmospheric pressure.

(cl.) In any bounded plane section traversed normally by streams which are rectilinear for a certain distance on either side of the section, the distribution of pressure is the same as in a fluid at rest.

Distribution of Energy in Incompressible Fluids.

26. Application of the Principle of the Conservation of Energy to Cases of Stream Line Motion.— The external and internal work done on a mass is equal to the change of kinetic energy produced. In many hydraulic questions this principle is difficult to apply, because from the complicated nature of the motion produced it is difficult to estimate the total kinetic energy generated, and because in some cases the internal work done in overcoming frictional or viscous resistances cannot be ascertained; but in the case of stream line motion it furnishes a simple and important result known as Bernouilli’s theorem.

Let AB (fig. 30) be any one elementary stream, in a steadily moving fluid mass. Then, from the steadiness of the motion, AB is a fixed path in space through which a stream of fluid is constantly flowing. Let 00 be the free surface and XX any horizontal datum line. Let ω be the area of a normal cross section, ν the velocity, p the intensity of pressure, and z the elevation above XX, of the elementary stream AB at A, and w 1 , p- i , r 1 , z 1 the same quantities at B. Suppose that in a short time t the mass of fluid initially occupying AB comes to A'B'. Then AA', BB' are equal to vt, υ 1 t, and the volumes of fluid A A', BB' are the equal inflow and outflow — Qi = uvt = ω l υ l t, in the given time. If we suppose the filament AB surrounded by other filaments moving with not very different velocities, the frictional or viscous resistance on its surface will be small enough to be neglected, and if the fluid is incompressible no internal work is done in change of volume. Then the work done by external forces will be equal to the kinetic energy produced in the time considered.

The normal pressures on the surface of the mass (excluding the ends A, B) are at each point normal to the direction of motion, and do no work. Hence the only external forces to be reckoned are gravity and the pressures on the ends of the stream.

The work of gravity when AB falls to A' B ’ is the same as that of transferring A A'to BB'; that is, GQi (z-z 1 ). The work of the pressures on the ends, reckoning that at B negative, because it is opposite to the direction of motion, is (pω×vt) - (p l ω- i ×vf) = Qt (ρ ~ 7>ι)∙ The change of kinetic energy in the time t is the difference of the kinetic energy originally possessed by AA' and that finally acquired by BB', for in the intermediate part A'B there is no change of kinetic energy, in consequence of the steadiness of the motion. But the mass of A A' and BB' is —Qi, and the change of kinetic energy is therefore — Qi( ∏- -L-). Equating this to the 9 ∖ 2 2 j work done on the mass AB,

GQt(z - z 1 ) + Qi(p -p 1 ) = y Q<(⅞ - ⅜) ·

Dividing by GQi and reaιτanging the terms, ⅛÷⅜ + ≈-⅛ ’ ÷⅜÷ 2 ∙ < 1 >'

or, as A and B are any two points,

~ + + constant = H .... (2).

2ø G I *⅛

Now Lis the head due to the velocity v, -7L is the head equivalent to the pressure, and z is the elevation above the datum (see § 15). Hence the terms on the left are the total head due to velocity, pressure, and elevation at a given cross section of the filament. z is easily seen to be the work in foot-pounds which would be done by 1 tt> of fluid falling to the datum line, and similarly -∣L and Lare the quantities of work which would be done by 1 lb of fluid due to the pressure p and velocity v. The expression on the left of the equation is, therefore, the total energy of the stream at the section considered, per lb of fluid, estimated with reference to the datum line XX. Hence we see that in stream line motion, under the restrictions named above, the total energy per lb of fluid is uniformly distributed along the stream line. If the free surface of the fluid 00 is taken as the datum, and - h, - h 1 are the depths of A and B measured down from the free surface, the equation takes the form

⅞÷⅜->-⅛ ’ ÷⅛- a * < 8 >∙

or generally

L- + T r - h = constant (3α).

2<∕ G

27. Second Form of the Theorem of Bernoulli.— Suppose at the two sections A, B (fig. 31) of an elementary stream small vertical pipes are introduced, which may be termed pressure columns.

(§ 8), having their lower ends accurately parallel to the direction of flow. In such tubes the water will rise to heights corresponding to the pressures at A and B. Hence

⅛-2-, audy-fi-Vr Lr

Consequently the tops of the pressure columns A' and B' will be at total heights b + c≈L r + z and b' + c'=yh + z 1 above the datum

G G

line XX. The difference of level of the pressure column tops, or the fall of free surface level between A and B, is therefore

Vτ

and this by equation (1), § 26, is

_ν 1 β -·υ 8

2ø ‘ . . ∙J

That is, the fall of free surface level between two sections is equal to the difference of the heights due to the velocities at the sections. The line A'B' is sometimes called the line of hydraulic gradient, though this term is also used in cases where friction needs to be taken into account. It is the line the height of which above datum is the sum of the elevation and pressure head at that point, and it falls below a horizontal line A"B" drawn at H feet above XX by the quantities a∞- and b=ff when friction is absent.

2y 2g ’

28. Illustrations of the Theorem of Bernoulli.— In a lecture to the mechanical section of the British Association in 1875, the late Mr W. Froude gave some experimental illustrations of the principle of Bernoulli. Mr Froude remarked that it was a common but erroneous impression that a fluid exercises in a contracting pipe A (fig. 32) an excess of pressure against the entire converging surface whichitmeets, and that, conversely, as it enters an enlargement B, a relief of pressure is experienced by the entire diverging surface of the pipe. Further it is commonly assumed that when passing through a contraction C, there is in the narrow neck an excess of pressure due to the squeezing together of the liquid at that point. These impressions are in no respect correct; the pressure is smaller as the section of the pipe is smaller and conversely.

taken into the reckoning; if the motion is in a horizontal plane, the terms expressing variation of elevation of the filament will disappear. tive stream lines, at present assumed to be in a vertical plane, and PQ a normal to these lines making an angle φ with the vertical. Let P, Q be two particles moving along these lines at a distance PQ = ds, and let z be the height of Q above the horizontal plane with reference to which the energy is measured, v its velocity, and p its pressure. Then, if H is the total energy at Q per unit of weight of fluid, κ - i+ ⅜÷⅞∙

Differentiating, we get

<ZH = tZz + ⅛ + - (1)

g g for the increment of energy between Q and P. But dz=PQ cos φ≈ds cos φ;

.∙. <ZH = ⅛ + - + cfccos≠ (la), g g where the last term disappears if the motion is in a horizontal plane.

Now imagine a small cylinder of section ω described round PQ as an axis. This will be in equilibrium under the action of its centrifugal force, its weight and the pressure on its ends. But its volume is ωds and its weight Ga>ds. Hence, taking the components of the forces parallel to PQ—

< ∣ >dp ÷--- — — ωds - Gω cos φ ds. g P where p is the radius of curvature of the stream line at Q. Consequently, introducing these values in (1),

-L + ⅞[ ifa . . . (2).

gp g g ( p <⅛ >

Now it is already known that if, through any particle A, lines be drawn through B and C two particles near to A, such that AB and AC are at right angles at the instant considered, then the mean angular velocity of these lines is the same in whatever direction they are drawn, and is equal to the angular velocity with which a small cylindrical element described round A would rotate if supposed suddenly solidified. This mean angular velocity may be conveniently called the molecular rotation, and will be denoted by . In the present case — is the angular velocity of the tangent at Q, and p as is the angular velocity, reckoned in the same direction, of a line perpendicular to the tangent through P and Q. The sum of these is, therefore, twice the molecular rotation, and dH = 2-‰ f is (3).

g

Now vds is constant, being the flow in an elementary stream of breadth unity, and thickness ds. Therefore the difference of energy between two consecutive elementary streams is proportional to the molecular rotation at any point of either.

Currents.

30. Rectilinear Current. — Suppose the motion is in parallel straight stream lines (fig. 38) in a vertical plane. Then p is infinite, and from eq. (2), § 29,

<ZH = ⅛

g

Comparing this with (1) we see that λ ÷⅞- 0 i . ∙. z + T- = constant (4);

G

or the pressure varies hydrostatically as in a fluid at rest. For two stream lines in a horizontal plane, z is constant, and therefore p is constant.

Radiating Current.— Suppose water flowing radially between horizontal parallel planes, at a distance apart = δ. Conceive two cylindrical sections of the current at radii r 1 and r 2 , where the velocities are r 1 and r⅛, and the pressures p 1 and m. Since the .flow across each cylindrical section of the cur'rent is the same,

Q = 2πr 1 bv 1 = 2τrr 2 δ ι> 2 r 1 ι> 1 = r 2 v 2 rι=¾ (5).

r 2 v ι

The velocity would be infinite at radius 0, if the current could be conceived to extend to the axis. Now, if the motion is steady, H-A 4 .⅛l a .⅛ + ⅛

G ' 2g G 2<∕ ’

-Α+Ί 5 ^1 8 .

G r 3 2 2g ’ P*ΣTi = v BJ 1 _ . . . . (6);

G 2g ∖ l r a >J

2⅛- = H-- — .... (6α).

G " r 2 2g k 7

Hence the pressure increases from the interior outwards, in a way indicated by the pressure columns in fig. 39, the curve through the free surfaces of the pressure columns being, in a radial section, the quasi-hyperbola of the form xι∕ 2 =c s . This curve is asymptotic to a horizontal line, H feet above the line from which the pressures are measured, and to the axis of the current.

Free Circular Vortex.—A free circular vortex is a revolving mass of water, in which the stream lines are concentric circles, and in which the total head for each stream line is the same. Hence, if by any slow radial motion portions of the water strayed from one stream line to another, they would take freely the velocities proper to their new positions under the action of the existing fluid pressures only.

For such a current, the motion being horizontal, we have for all the circular elementary streams

H = L- +—-constant;

G 2g

.∙.dH-⅜ + ⅛0 (7).

g g

Consider two stream lines at radii r and r + dr (fig. 39). Then in (2), § 29, p = r and ds=dr, gr g dv dr V r ’

® α 1 (8), r precisely as in a radiating current; and hence the distribution of pressure is the same, and formulae 6, 6α are applicable to this case. ,¾

Free Spiral Vortex.— As in a radiating and circular current the equations of motion are the same, they will also apply to a vortex in which the motion is compounded of these motions in any proportions, provided the radial component of the motion varies inversely as the radius as in a radial current, and the tangential component varies inversely as the radius as in a free vortex. Then the whole velocity at any point will be inversely proportional to the radius of the point, and the fluid will describe stream lines having a constant inclination to the radius drawn to the axis of the current. That is, the stream lines will be logarithmic spirals. When water is delivered from the circumference of a centrifugal pump or turbine into a chamber, it forms a free vortex of this kind. The water flows spirally outwards, its velocity diminishing and its pressure increasing according to the law stated above, and the head along each spiral stream line is constant.

31. Forced Vortex.— If the law of motion in a rotating current is diff erent from that in a free vortex, some force must be applied to cause the variation of velocity. The simplest case is that of a rotating current in which all the particles have equal angular velocity, as for instance when they are driven round by radiating paddles revolving uniformly. Then in equation (2), § 29, considering two circular stream lines of radii r and r + dr (fig. 40), we have p=r,ds=dr. If the angular velocity is α, then v=«r and dv=adr. Hence dH = ⅛r + ^=‰.

9 9 9

Comparing this with (1), § 29, and putting dz=Q, because the motion is horizontal, dp a*rdr 2cfrj r

G g 9 ’

⅛-⅛fr,

G g

= + constant (9).

G 2g

Let Pι> r ι> ν ι be the pressure, radius, and velocity of one cylindrical section, p 2 , r 2 , v 2 those of another; then p l ct 2 r 1 8 jt⅛ <⅜ 8 r a 8 .

G " 2g ~ G 2g’

· · * · (10) · M r ΔlJ Δ(J

That is, the pressure increases from within outwards in a curve which in radial sections is a parabola, and surfaces of equal pressure are paraboloids of revolution (fig. 40).

Dissipation of Head in Shock.

32. Relation of Pressure and Velocity in a Stream in Steady Motion when the Changes of Section of the Stream arc Abrupt.— non a stream changes section abruptly, rotating eddies are lormed which dissipate energy. The energy absorbed in producing rotation is at once abstracted from that effective in causing the flow, and sooner or later it is wasted by frictional resistances due to the rapid relative motion of the eddying parts of the fluid. In such cases the work thus expended internally in the fluid is too important to be neglected, and the energy thus lost is commonly termed energy lost in shock. Suppose fig. 41 to represent a stream having such an abrupt change of section. Let AB, CD be normal sections at points where ordinary streamline motion has not been dis turbed and where it has been re-established. Let ω,p, υ be the area of section, pressure, and velocity at AB, and c∣> 1 , p i , r l corresponding quantities at CD. Then if no work were expended internally, and. assuming the stream horizontal, we should have

2L + Li = ⅛ + ∏l (11

G + 2g G + 2g

But if work is expended in producing irregular eddying motion, the head at the section CD will be diminished.

Suppose the mass ABCD comes in a short time t to A'B'C'D'. The resultant force parallel to the axis of the stream is Po> +P 0 (ω ι~ ω ) ~7 ’ ι ω ι> where p 0 is put for the unknown pressure on the annular space be. tween AB and EF. The impulse of that force is

{?« +P 0 (ωι - «) -J¾∙ι} t .∙

The horizontal change of momentum in the same time is the difference of the momenta of CDC'D' and ABA'B', because the amount of momentum between A'B' and CD remains unchanged if the motion is steady. The volume of ABA'B' or CDC'D', being the inflow and outflow in the time t, is tyt≈wvt≈ωp)ιt, and the momen-

G G

tum of these masses is — Q l vt and —Qv 1 tf. The change of momen- J p turn is therefore — Qi(τ 1 - v). Equating this to the impulse,

Q

{ρ a > +2 ’ o(ω ι - ω)-p i u 1 }t=- — Qi(v 1 -v) .

9

Assume that p 0 =p, the pressure at AB extending unchanged through the portions of fluid in contact with AE, BF which lie out of the path of the stream. Then (since Q = ω 1 v 1 )

G <P~Pι)^-vfP 1 -v)∙, P Pi v 1 (v 1 -v)

G G - g (2) ’

G + 2g G + 2< z + 2g

This differs from the expression (1), § 26, obtained for cases where no sensible internal work is done, by the last term on the right. That is, ⅛—has to be added to the total head at CD, which is 2g β 2 ⅛- + Ll— , to make it equal to the total head at AB, or -- y ~ v λ l is G τ 2g * 2g the head lost in shock at the abrupt change of section. But V - v 1 is the relative velocity of the two parts of the stream. Hence, . when an abrupt change of section occurs, the head due to the relative velocity is lost in shock, or —^ ιl ^- foot-pounds of energy is wasted for each pound of fluid. Experiment verifies this result, so that the assumption that p 0 ≈p appears to be admissible.

If there is no shock, Pi TL. v i∑ v ι

G G 2g

If there is shock, p l p vfPι ~ C

G “G g

Hence the pressure head at CD in the second case is less than in the former by the quantity

(f→ ’ ι) 8 2g or, putting u i y 1 ≈ωv, by the quantity

⅞0-i i ) ’ «>·

V. THEORY OF THE DISCHARGE FROM ORIFICES AND MOUTHPIECES.

33. Minimum Coefficient of Contraction. Re-entrant Mouthpiece of Borda. — In one special case the coefficient of contraction can be determined, theoretically, and, as it is the case where the convergence of the streams· approaching the orifice takes place through the greatest possible angle, the coefficient thus determined is the minimum coefficient.

Let fig. 42 represent a vessel with vertical sides, 00 being the free water surface, at λvhich the pressure is p a . Suppose the liquid issues by a horizontal mouthpiece, which is re-entrant and of the greatest length which permits the jet to spring clear from the inner end of the orifice, without adhering to its sides. With such an orifice the velocity near the points CD is negligible, and the pressure at those points may be taken equal to the hydrostatic pressure due to the depth from the free surface. Let ∩ be the area of the mouthpiece AB, ω that of the contracted jet aa. Suppose that in a short time t, the mass 00αα comes to the position 0Ό' a'a' ; the impulse of the horizontal external forces acting on the mass during that time is equal to the horizontal change of momentum.

The pressure on the side OC of the mass will be balanced by the pressure on the opposite side 0E, and so for all other portions of the vertical surfaces of the mass, excepting the portion EF opposite the mouthpiece and the surface AααB of the jet. On EF the pressure is simply the hydrostatic pressure due to the depth, that is, (p α + GA)∩. On the surface and section AααB of the jet, the horizontal resultant of the pressure is equal to the atmospheric pressure p a acting on the vertical projection AB of the jet; that is, the resultant pressure is- ρ a Ω. Hence the resultant horizontal force for the whole mass 00aa is (p a + Gh')Ω-p a Ω≈GhΩ. Its impulse in the time t is GhΩh Since the motion is steady there is no change of momentum between 0'0' and aa. The change of horizontal momentum is, therefore, the difference of the horizontal momentum lost in the space 000'0' and gained in the space aaa'a'. In the former space there is no horizontal momentum.

The volume of the space aaa!a' is ωvt ; the mass of liquid in that • G . C 1 space is — wvt ; its momentum is — ωv i t. Equating impulse to momentum gained, GhΩt - — ωv i t; 9 ω gh ~Ω v 2 '

But J s = 2gh, and —=c β ; Ω

• w - 1 -c ·

• · -Ω~2~ c ' ’ a result confirmed by experiment with mouthpieces of this kind. A similar theoretical investigation is not possible for orifices in plane surfaces, because the velocity along the sides of the vessel in the neighbourhood of the orifice is not so small that it can be neglected. The resultant horizontal pressure is therefore greater than GhΩ, and the contraction is less. The experimental values of the coefficient of discharge for a re-entrant mouthpiece are 0'5149 (Borda), 0 ’ 5547 (Bidone), 0 ’ 5324 (Weisbach), values which differ little from the theoretical value, 0 - 5, given above.

°∙^ Fil aments issuing in a Jet.— A jet is composed of nuιd filaments or elementary streams, which start into motion at some point in the interior of the vessel from which the fluid is discharged, and gradually acquire the velocity of the jet. Let Mm, ⅛∙ be such a filament, the point M being taken where the velocity is insensibly small, and m at the most contracted section of the jet, where the filaments have become parallel and exercise uniform mutual pressure. Take the free surface AB for datum line, and let p v v ι> ^ι> b e the pressure, velocity, and depth below datum at M; p, V, A, the corresponding quantities at m. Then § 26, eq. (3), <÷¾-*>-⅞÷⅜- a < 1 >∙

But at M, since the velocity is insensible, the pressure is the hydrostatic pressure due to the depth; that is, v 1 = 0, p l =p a + Gh 1 . At m, P = Pa> the atmospheric pressure round the jet. Hence, inserting these values,

0÷⅜÷⅛-⅛-⅞÷⅜-M

<-- a ∙ < 2 > i or υ== V2<jA=8O25V)Γ . . (2α).

That is, neglecting the viscosity of the fluid, the velocity of filaments at the contracted section of the jet is simply the velocity due to the difference of level of the free surface in the reservoir and the orifice. If the orifice is small in dimensions compared with h, the filaments will all have nearly the same velocity, and if h is measured to the centre of the orifice, the equation above gives the mean velocity of the jet.

Case of a Submerged Orifice. — Lettheorifice discharge below the level of the tail water. Then using the notation shown in fig. 44, we have at M, τ 1 = 0, p i = GA 1 +p a ; at m, p = Gh i +p a . Inserting these values in (3), § 26,

θ+½ι+^- -A i +h 3 -h i +^ l ;

-^--⅞-⅛ s =A (3), *9 where h is the difference of level of the head and tail water, and may be termed the effective head producing flow.

Case where the Pressures are different on the Free Surface and at the Orifice.— Let the fluid flow from a vessel in which the pressure is p 0 into a vessel in which the pressure is p, fig. 45. The pressure p 0 will produce the same effect as a layer of fluid of thickness ff added to the G head water; and the pressure p will produce the same effect as a layer of thickness T-

G added to the tail water. Hence the effective difference of level, or effective head producing flow, will be , Po P .

Λ-Λ o + ---, and the velocity of discharge will be t - χ∕ 2 i'i 4 . +a <Γ i < 4 >∙

We may express this result by saying that differences of pressure at the free surface and at the orifice are to be reckoned as part of the effective head.

Hence in all cases thus far treated the velocity of the jet is the velocity due to the effective head, and the discharge, allowing for contraction of the jet, is

Q = c w v = c<vV2g⅛ (5), where ω is the area of the orifice, cω the area of the contracted section of the jet, and h the effective head measured to the centre of the orifice. If h and ω are taken in feet, Q is in cubic feet per second.

It is obvious, however, that this formula assumes that all the filaments have sensibly the same velocity. That will be true for horizontal orifices, and very approximately true in other cases, if the dimensions of the orifice are not large compared with the head h. In large orifices in say a vertical surface, the value of h is different for different filaments, and then the velocity of different filaments is not sensibly the same.

Simple Orifices—Head Constant.

35. Large Rectangular Jets from Orifices' in Vertical Plane Surfaces.— Let an orifice in a vertical plane surface be so formed that it produces a jet having a rectangular contracted section with vertical and horizontal sides. Let b (fig. 46) be the breadth of the jet, h 1 and ⅛ 2 the depths below the free surface of its upper and lower surfaces. Consider a lamina of the jet between the depths h and h + dh. Its normal section is bclh, and the velocity of discharge f f2gh. The discharge per second in this lamina is therefore b ’ f2gh dh, and that of the whole jet is therefore ∕∙fr 2 .

Q = ∕ bf2ghdh

∙ z Λ 1

= ⅜⅛√2y j h, 2 i -⅛ 1 3 ∣ (6), where the first factor on the right is a coefficient depending on the form of the orifice.

Now an orifice producing a rectangular jet must itself be very approximately rectangular. Let B be the breadth, H 1 , H„ the depths to the upper and lower edges of the orifice. Put

⅛U⅛=c (7).

B(H a i ∙- H∕)

Then the discharge, in terms of the dimensions of the orifice, instead of those of the jet, is

Q = ⅜cB√¾i(H a 3 - H 1 ’) (8), the formula commonly given for the discharge of rectangular orifices. The coefficient c is not, however, simply the coefficient of contraction, the value of which is

δ(⅛ a -¼) B(H a -H 1 ) ’ and not that given in (7). It cannot be assumed, therefore, that c in equation (8) is constant, and in fact it is found to vary for different B B

values of — and —- , and must be ascertained experimentally.

I±2 Γl 1 Relation between the Expressions (5) and (8).—For a rectangular orifice the area of the orifice is ω = B∫H 2 - H 1 ), and the depth measured to its centre is ⅜ (H a + H 1 ). Putting these values in (5),

Q 1 =cB(H a - H 1 )√y(H s + H 1 ).

From (8) the discharge is

Q i -3cB√⅜(H s *-H 1 i ).

Hence, for the same value of c in the two cases, Q, _ » H a ι - H 1 3

QΓ s (H a -H 1 )√⅛(H a +H 1 ) ’

ι .5ι ∕5χ

= 0 9427 ⅛ 2 -V¾ i9 ∖

1 -⅛√(1 ÷¾)

If H 1 varies from 0 to oo , 51 varies from 0 to 1. The following table gives values of the two estimates of the discharge for different values of ⅛ :—

H 2 [table]

Hence it is obvious that, except for very small values of 51, the • i L

simpler equation (5) gives values sensibly identical with those of (8). When 51 < 0 5 it is better to use equation (8) with values of

H 2 c determined experimentally for the particular proportions of orifiee which are in question.

36. Large Jets having a Circular Section from Orifices in a Vertical Plane Surface.— Let fig. 47 represent the section of the jet;

OO being the free surface level in the reservoir. The discharge through the horizontal strip aabb, of breadth aa=b, between the h 1 + y and depths h 1 + y + dy, is dQ, = b∙f{2fih 1 + y}}dy.

The whole discharge of the jet is

Q=∕' d 0√{20(Λ 1 + y)}<fy .

Jo

But b=d sin φ ; y=⅜<Z(l - cos ≠); dy=⅛d sin φ dφ. Let d 2h 1 + d ’ then

Q = -y- ∖ J I + -7θ ∖ ff sin 2 ≠V1 - e cos φdφ .

From eq. (5), putting ω≈-^-d 2 , h=h 1 -±~, c = l when d is the dia- 4 Δ meter of the jet and not that of the orifice,

⅛-U i √M⅛ + i)b

Q- = — ∕^ sin 2 ≠χ∕{l - e cos φ}dφ.

Ql Tr∕ 0

For 7⅛ = °o, e = 0 and S- = l;

Qi and for h 1 ≈ 0 , e=l and 0- = 0 ’ 96.

Qi

So that in this case also the difference between the simple formula (5) and the formula above, in which the variation of head at different parts of the orifice is taken into account, is very small.

Notches and Weirs.

37. Notches, Weirs, and Byewashes.— A notch is an orifice extending up to the free surface level in the reservoir from which the discharge takes place. A weir is a structure over which the water flows, the discharge being in the same conditions as for a notch. The formula of discharge for an orifice of this kind is ordinarily deduced by putting H l = 0 in the formula for the corresponding orifice, obtained as in the preceding section. Thus for a rectangular notch, put H 1 = 0 in (8). Then

Q = f cB√27 H ? . . . . . . (11), where H is put for the depth to the crest of the weir or the bottom of the notch. Fig. 48 shows the mode in which the discharge occurs in the case of a rectangular notch or weir with a level crest. As the free surface level falls very sensibly near the notch, the head H should be measured at some distance back from the notch, at a point where the velocity of the water is very small.

Since the area of the notch opening is BH, the above formula is of the form

Q = c×BH×½√¾H, where k is a factor depending on the form of the notch and expressing the ratio of the mean velocity of discharge to the velocity due to the depth H.

38. Francis’s Formula for Rectangular Notches.— The jet discharged through a rectangular notch has a section smaller than BH, (a) because of the fall of the water surface from the point where H is measured towards the weir, (δ) in consequence of the crest contraction, (c) in consequence of the end contractions. It may be pointed out that while the diminution of the section of the jet due to the surface fall and to the crest contraction is proportional to the length of the weir, the end contractions have nearly the same effect whether the weir is wide or narrow.

Mr Francis’s experiments showed that a perfect end contraction, when the heads varied from 3 to 24 inches, and the length of the weir was not less than three times the head, diminished the effective length of the weir by an amount approximately equal to one-tenth of the head. Hence, if l is the length of the notch or weir, and H the head measured behind the weir where the water is nearly still, then the width of the jet passing through the notch would be Z-0 - 2H, allowing for two end contractions. In a weir divided by posts there may be more than two end contractions. Hence, generally, the width of the jet is l- O’L·iH, where n is the number of end contractions of the stream. The contractions due to the fall of surface and to the crest contraction are proportional to the width of the jet. Hence, if cH is the thickness of the stream over the weir, measured at the contracted section, the section of the jet will be c(Z - 0 lτιH)H and (§ 37) the mean velocity will be ∣V ’ 2gH. Consequently the discharge will be given by an equation of the form

Q = g c (Z-0∙lnH)H√2^H -5∙35c(Z-0∙lnH)H*.

This is Francis’s formula, in which the coefficient of discharge c is much more nearly constant for different values of l and h than in the ordinary formula. Francis found for c the mean value 0 , 622, the weiι ’ being sharp-edged.

39. Triangular Notch (fig. 49).—Consider a lamina issuing between the depths h and h + dh. Its area, negle ctin g contraction, will be bdh, and the velocity at that depth is √2<∕⅛. Hence the discharge for this lamina is b V 2gh dh.

But

B H

b H-Ä ’

5 - b ≡γ∙

Hence discharge of lamina and total discharge of notch

= Q = B√2ξ^" ¾⅛d∕ t = τ ⅛B√⅞Hi, or, introducing a coefficient to allow for contraction,

Q = ⅛ cB λ √ z 2<∕H^ .

When a notch is used to gauge a stream of varying flow, the ratio T> y*- varies if the notch is rectangular, but is constant if the notch is triangular. This led Professor James Thomson to suspect that the coefficient of discharge, c, would be much more constant with different values of H in a triangular than in a rectangular notch, and this has been experimentally shown to be the case. Hence a triangular notch is more suitable for accurate gaugings than a rectangular notch. For a sharp-edged triangular notch Professor J. Thomson found c = 0 ’ 617. It will be seen, as in § 37, that since ⅜BH is the area of section of the stream through the notch, the formula is again of the form

Q = c× ⅜BH ×⅛√2yH, where ⅛=⅛ is the ratio of the mean velocity in the notch to the velocity at the depth H. It may easily be shown that for all notches the discharge can be expressed in this form.

40. Weir with a Broad Sloping Crest.— Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a move· ment sensibly rectilinear and uniform (fig. 50). Let the inner edge be so rounded as to prevent a crest contraction. Consider a filament aa', the point a being so far back from the weir that the velocity of approach.is negligible. Let 00 be the surface level in the reservoir, and let α be at a height h" below 00, and h' above a'. Let h be the distance from 00 to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at a is GN; at a' it is Gz. If v be the velocity at a', — ≈h'+h"-z≈h-e;

Q = δe V 2g{h - e) .

Theory does not furnish a value for e, but Q = 0 for e = 0 and for e=h. Q has therefore a maximum for a value of e between 0 and Ä, obtained by equating to zero. This gives e≈ lh, and, inserting de this value,

Q = 0∙385δ⅛√i≠, as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately applicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonry sluice openings than the ordinary weir formula for sharp-edged weirs. It should be remembered, however, that the friction on the sides and crest of the weir has been neglected, and that this tends to reduce a little the discharge. The formula is equivalent to the ordinary weil· formula with c = 0 ’ 577.

Coefficients for the Discharge over Weirs, derived from the Experiments of Mr Blackwell. When more than one experiment was made with the same head, and the results were pretty uniform, the resulting coefficients are marked with an (*). The effect of the converging wing-boards is very strongly marked. [table]

43. Separating Weirs.— Many towns derive their water supply from streams in high moorland districts, in which the flow is extremely variable. The water is collected in large storage reservoirs, from which an uniform supply can be sent to the town. In such cases it is desirable to separate the coloured water which comes down the streams in high floods from the purer water of ordinary flow. The latter is sent into the reservoirs; the former is allowed to flow away down the original stream channel, or is stored in separate reservoirs and used as compensation water. To accomplish the separation of the flood and ordinary water, advantage is taken of the different horizontal range of the parabolic path of the water falling over a weir, as the depth on the weir and, consequently, the velocity change. Fig.

54 shows one of these separating weirs in the form in which they were first introduced on the Manchester Waterworks; fig

55 a more modern weir of the same kind designed by Mr Binnie for the Bradford Waterworks. When the quantity of water coming down the stream is not excessive, it drops over tne weir into a transverse channel leading to the reservoirs. In flood, the water springs over the mouth of this channel and is led into a waste channel.

It may be assumed, probably with accuracy enough for practical purposes, that the particles describe the parabolas due to the mean velocity of the water passing over the weir, that is, to a velocity

∣√2gΛ, where h is the head above the crest of the weir.

Let cb=x be the width of the orifice and ac=y the difference of level of its edges (fig. 56). Then, if a particle passes from a to b in t seconds,

∞=⅛^ 2 > y=∣√2gAi;

·· ∙→⅞.

which gives the width x for any given difference of level y and head Jι, which the jet will just pass over the orifice. Set off ad verti-callyand equal to ⅛g on any scale; af horizontally and equal to I ’ fgh . Divide af, fe into an equal number of equal parts. Join a with the divisions on ef. The intersections of these lines with verticals from the divisions on af give the parabolic path of the jet.

Mouthpieces—Head Constant.

44. Cylindrical Mouthpieces.— When water issues from a short cylindrical pipe or mouthpiece of a length at least equal to 1⅜ times its smallest transverse dimension, thestream, after contraction within the mouthpiece, expands to fill it and issues full bore, or without »contraction, at the point of discharge. The discharge is found to be about one-third greater than that from a simple orifice of the same size. On the other hand, the energy of the fluid per unit of weight is less than that of the stream from a simple orifice with the same head, because part of the energy is wasted in eddies produced at the point where the stream expands to fill the mouthpiece, the action being something like that which occurs at an abrupt change of section.

Let fig. 57 represent a vessel discharging through a cylindrical mouthpiece at the depth h from the free surface, and let the axis of the jet XX be taken as the datum with reference to which the head is estimated. Let ∩ be the area of the mouthpiece, ω the area of the stream at the contracted section EF. Let V, ρ be the velocity and pressure at EF, and v γ ,p- i the same quantities at GH. If the discharge is into the air, p 1 is equal to the atmospheric pressure p a .

The total head of any filament which goes to form the jet, taken at a point where its velocity is sensibly zero, is h + ¾L; at EF the

G

total head is L- -f-; at GH it is + .

Between EF and GH there is a loss of head due to abrupt change of velocity, which from eq. (3), § 32, may have the value (v - t> 1 ) a ⅛ '

Adding this head lost to the head at GH, before equating it to the heads at EF and at the point where the filaments start into motion,—

h 4 ,P^ - νi 4.P V l 4. Pl √ v " i, l) 2 a+ ^G ⅞ + G ¾ + G +- 2Γ^'

But ωv=nv- l , and ω = c c ∩, if c e is the coefficient of contraction within the mouthpiece. Hence ω C, Supposing the discharge into the air, so thaty> 1 =p α , i ÷⅜-¾ s ÷⅜÷¾ ’ a-√ i i⅛K-) ’ H'

* 1 ^ ∕ /I τ* d⅛ ii m '∙

√ l+ (⅜- 1 )

where the first term on the right is evidently the coefficient of velocity for the cylindrical mouthpiece in terms of the coefficient of contraction at EF. Let c c = 0∙64, the value for simple orifices, then the coefficient of velocity is

<⅛ . 1 - — 0-S7 .... (2).

√*÷(⅜->y

The actual value of c t found by experiment is 0 ’ 82, which does not differ more from the theoretical value than might be expected if the friction of the mouthpiece is allowed for. Hence, for mouthpieces of this kind, and for the section at GH, c, = 0∙82 e c =l - 00 c = 0∙82,

Q=0∙82θ√2gΛ

It is easy to see from the equations that the pressure p at EF is less than atmospheric pressure. Eliminating v 1 , we get

= JΛ nearly (3); or P=Pa - ∣GΛ lb per sq. ft.

If a pipe connected with a reservoir on a lower level is introduced into the mouthpiece at the part where the contraction is formed (fig. 58), the water will rise in this pipe to a height

KL= t h nearly.

G

If the distance X is less than this, the water from the lower reservoir will be forced continuously into the jet by the atmospheric pressure, and discharged with it. This is the crudest form of a kind of pump known as the jet pump.

45. Convergent Mouthpieces.— With convergent mouthpieces there is a contraction within the mouthpiece causing a loss of head, and a diminution of the velocity of discharge, as with cylindrical mouthpieces. There is also a second contraction of the stream outside the mouthpiece. Hence the discharge is given by an equation •of the form

Q = <⅛c e ∩√2gΛ (4), where ∩ is the area of the external end of the mouthpiece, and c e ∩ the section of the contracted jet beyond the mouthpiece.

Convergent Mouthpieces {Castcl , s Experiments').—Smallest diameter of orifice =0’05085 feet. Length of mouthpiece = 2 - 6 diameters. [table]

The maximum coefficient of discharge is that for α mouthpiece with a convergence of 13’ 24'.

The values of c, and c e must here be determined by experiment. The above table gives values sufficient for practical purposes. Since the contraction beyond the mouthpiece increases with the convergence, or, what is the same thing, c e diminishes, and on the other hand the loss of energy diminishes, so that c β increases with the convergence, there is an angle for which the product e c c v , and consequently the discharge, is a maximum.

46. Divergent Conoidal Mouthpiece.— Suppose a mouthpiece so designed that there is no abrupt change in the section or velocity of the stream passing through it. It may have a form at the inner end approximately the same as that of a simple contracted vein, and may then enlarge gradually, as shown in fig. 59. Suppose that at EF it becomes cylindrical, so that the jet may be taken to be of the diameter EF. Let ω, v, p be the section, velocity, and pressure at CD, and ∩, v 1 , p 1 the same quantities at EF, p a being as usual the atmospheric pressure, or pressure on the free surface AB. Then, since there is no loss of energy, except the small frictional resistance of the surface of the mouthpiece, + G 2gr + G 2y + G ’

If the jet discharges into the air, ρ 1 =p a ; and i⅛ 1>1 = √2ifΛ; or, if a coefficient is introduced to allow for friction.

v 1 ≈c 9 ∖ Z t igh ; where c e is about 0’97 if the mouthpiece is smooth and well-formed.

Q = ∩t> 1 = c v n, ∖ jιgh.

Hence the discharge depends on the area of the stream at EF, and not at all on that at CD, and the latter may be made as small as we please without affecting the amount of water discharged.

There is, however, a limit to this. As the velocity at CD is greater than at EF the pressure is less, and therefore less than atmospheric pressure, if the discharge is into the air. If CD is so contracted that p = 0, the continuity of flow is impossible. In fact the stream disengages itself from the mouthpiece for some value of p greater than 0 (fig. 60).

From the equations, P P« ( v i -vι a ∖ G G"k ⅛ T

Let — =m. Then ω v≈vpn ;

⅛-⅞-¾ ’ '" 2 - 1 >

( r whence we find that will become zero or negative if

G

∕, , R a λ =- ∕- g = ∕ι + ^∙ ω = ∖ ' h V + GV

or, putting ^£ = 34 feet, if

∆s- ∕ 7t + 34 ω ~ ∖ ∕ h

In practice there will be an interruption of the full bore flow with a less ratio of —, because of the disengagement of air from the water.

ω

But, supposing this does not occur, the maximum discharge of a mouthpiece of this kind is q-^∕¾(λ+⅞); that is, the discharge is the same as for a well-bellmouthed mouthpiece of area a>, and without the expanding part, discharging into a vacuum.

47. Jet Bump.— A divergent mouthpiece may be arranged to act as a pump, as shown in fig. 61. The water which supplies the energy required for pumping enters at A. The water to be pumped enters at B. The streams combine at DD where the velocity is greatest and the pressure least. Beyond DD the stream enlarges in section, and its pressure increases, till it is sufficient to balance the head due to the height of the lift, and the water flows away by the discharge pipe C.

Fig. 62 shows the whole arrangement in a diagrammatic way. A is the reservoir which supplies the water that effects the pump ing; B is the reservoir of water to be pumped; C is the reservoir into which the water is pumped.

Discharge with Varying Head.

48. Flou) from a Vessel when the Effective Head varies with the Time. —Various useful problems arise relating to the time of emptying and filling vessels, reservoirs, lock chambers, &c., where the flow is dependent on a head which increases or diminishes during the operation. The simplest of these problems is the case of filling or emptying a vessel of constant horizontal section.

Time of Emptying or Filling a Vertical-sided Lock Chamber.— Suppose the lock chamber, which has a water surface of ∩ square feet, is emptied through a sluice in the tail gates, of area w, placed below the tail-water level. Then the effective head producing flow through the sluice is the difference of level in the chamber and tail bay. Let H (fig. 63) be the initial difference of level, h the difference of level after t seconds. Let - dh be the fall of level in the chamber during an interval dt. Then in the time dt the volume in the chamber is altered by the amount - Ωdh, and the outflow from the sluice in the same time is cωV 2gh dt. Hence the differential equation connecting h and t is c»V 2gh dt + Ωh = 0.

For the time t, during which the initial head H diminishes to any other value h, - n f li d ^≈f t dt Cω ∖ f2gJH fh f = Û1 /2H_ ∕2Aj CO) ( V g ∖ g )

For the whole time of emptying, during which h diminishes from H to 0, T _ ∩ ∕2H , co> ∖ g

Comparing this with the equation for flow under a constant head, it will be seen that the time is double that required for the discharge of an equal volume under a constant head.

The time of filling the lock through a sluice in the head gates is exactly the same, if the sluice is below the tail-water level. But if the sluice is above the tail-water level, then the head is constant till the level of the sluice is reached, and afterwards it diminishes with the time.

Practical Use of Orifices in Gauging Water.

49. If the water to be measured is passed through a known orifice under an arrangement by which the constancy of the head is ensured, the amount which passes in a given time can be ascertained by the formulae already given. It will obviously be best to make the orifices of the forms for which the coefficients are most accurately determined; hence sharp-edged orifices or notches are most commonly used.

Water Inch.— For measuring small quantities of water circular sharp-edged orifices have been used. The discharge from a circular orifice one French inch in diameter, with a head of one line above the top edge, was termed by the older hydraulic writers a waterinch. A common estimate of its value was 14 pints per minute, or 677 English cubic feet in 24 hours. An experiment by Bossutgave 634 cubic feet in 24 hours (see Navier’s edition of Belidor's Arch. Hydr., p. 212). .1

Weisbach points out that measurements of this kind would be made more accurately with a greater head over the orifice, and he proposes that the head should be equal to the diameter of the orifice. Several equal orifices may be used for larger discharges.

50. Bin Ferrules or Measuring Cocks.— To give a tolerably definite supply of water to houses, without the expense of a meter, a ferrule with an orifice of a definite size, or a cock, is introduced in the servicepipe. If the head in the water main is constant, then a definite quantity of water would be delivered in a given time. The arrangement is not a very satisfactory one, and acts chiefly as a check on extravagant use of water. It is interesting here chiefly as an example of regulation of discharge by means of an orifice. Fig. 64 shows a cock of this kind used at Zurich. It consists of three cocks, the middle one having the orifice of the predetermined size in a small circular plate, protected by wire gauze from stoppage by impurities in the water. The cock on the right hand can be used by the consumer for emptying the pipes. The one on the left and the measuring cock are connected by a key which can be locked by a padlock, which is under the control of the water company.

51. Measurement of the Flow in Streams.— To determine the quantity of water flowing off the ground in small streams, which is available for water supply or for obtaining water power, small temporary weirs are often used. These may be formed of planks supported by piles and puddled to prevent leakage. The measurement of the head may be made by a thin-edged scale at a short distance behind the weir, where the water surface has not begun to slope down to the weir and where the velocity of approach is not high. The measurements are conveniently made from a short pile driven into the bed of the river, accurately level with the crest of the weir (fig. 65). Then if at any moment the head is h, the discharge is, for a rectangular notch of breadth b,

Q = § e b h ∖ j2gh where c = 0'62; or, better, the formula in § 38 may be used.

Gauging weirs are most commonly in the form of rectangular notches; and care should be taken that the crest is accurately horizontal, and that the weir is normal to the direction of flow of the stream. If the planks are thick, they should be bevelled (fig. 66), and then the edge may be protected by a metal plate about 1 ⅛th inch thick to secure the requisite accuracy of form and sharpness of edge. In permanent gauging weirs a cast steel plate is sometimes used to form the edge of the weir crest. The weir should be large enough to discharge the maximum volume flowing in the stream, and at the same time it is desirable that the minimum head should not be too small (say half a foot) to decrease the effects of errors of measurement. The section of the jet over the weir should not exceed one-fifth the section of the stream behind the weir, or the velocity of approach will need to be taken into account. A triangular notch is very suitable for measurements of this kind.

If the flow is variable, the head h must be recorded at equidistant intervals of time, say twice daily, and then for each 12 hour period the discharge must be calculated for the mean of the heads at the beginning and end of the time. As this involves a good deal of troublesome calculation, Mr Sang has proposed to use a scale so graduated as to read off the discharge in cubic feet per second. The lengths of the principal graduations of such a scale are easily calculated by putting Q≈l, 2, 3 . . . . in the ordinary formulae for notches; the intermediate graduations may be taken accurately enough by subdividing equally the distances between the principal graduations.

^The accurate measurement of the discharge of a stream by means i? Y eir * s > h 0w θ v 0r, in practice, rather more difficult than might λ 2 inferred from the simplicity of the principle of the operation. Apaιt from the difficulty of selecting a suitable coefficient of discharge, which need not be serious if the form of the weir and the nature of its crest are properly attended to, other difficulties of measurement arise. The length of the weir should be very accurately determined, and if the weir is rectangular its deviations from exactness of level should be tested. Then the agitation of the water, the ripple on its surface, and the adhesion of the water to the scale on which the head is measured, are liable to introduce errors. Upon a weir 10 feet long, with 1 foot depth of water flowing over an error of l-1000th of a foot in measuring the head, or an error of 1-100th of a foot in measuring the length of the weir, would cause an error in computing the discharge of 2 cubic feet per minute.

52. Hook Gauge.— For the determination of the surface level of water, the most accurate instrument is the hook gauge used first by Mr U. Boyden of Boston, in 1840. It consists of a fixed frame with scale and vernier. In the instrument in fig. 67 the vernier is fixed to the frame, and the scale slides vertically. The scale carries at its lower end a hook with a fine point, and the scale can be raised or lowered by a fine pitched screw. If the hook is depressed below the water surface and then raised by the screw, the moment of its reaching the water surface will be very distinctly marked, by the reflexion from a small capillary elevation of the water surface over the point of the hook. In ordinary light, differences of level of the water of ‘001 of a foot are easily detected by the hook gauge. If such a gauge is used to determine the heads at a weir, the hook should first be set accurately level with the weir crest, and a reading taken. Then the difference of the reading at the water surface and that for the weir crest will be the head at the weir.

53. Modules used in Irrigation.— In distributing water for irrigation, the charge for the water may be simply assessed on the area of the land irrigated for each consumer, a method followed in India; or a regulated quantity of water may be given to each consumer, and the charge may be made proportional to the quantity of water supplied, a method employed for a long time in Italy and other parts of Europe. To deliver a regulated quantity of water from the irrigation channel, arrangements termed modules are used. These are constructions intended to maintain a constant or approximately constant head above an orifice of fixed size, or to regulate the size of the orifice so as to give a constant discharge, notwithstanding the variation of level in the irrigating channel.

54. Italian Module.— The Italian modules are masonry constructions, consisting of a regulating chamber, to which water is admitted by an adjustable sluice from the canal. At the other end of the chamber is an orifice in a thin flagstone of fixed size. By means of the adjustable sluice a tolerably constant head above the fixed orifice is maintained, and therefore there is a nearly constant discharge of ascertainable amount through the orifice, into the channel leading to the fields which are to be irrigated.

In fig. 68 A is the adjustable sluice by which water is admitted to the regulating chamber, B is the fixed orifice through which the water is discharged. The sluice A is adjusted from time to time by the canal officers, so as to bring the level of the water in the regulating chamber to a fixed level marked on the wall of the chamber. When adjusted it is locked. Let w 1 be the area of the orifice through the sluice at A, and ω 2 that of the fixed orifice at B; let ⅛ 1 be the difference of level between the surface of the water in the canal and regulating chamber; h i the head above the centre of the discharging orifice, when the sluice has been adjusted and the flow has become steady; Q the normal discharge in cubic feet per second. Then, since the flow through the orifices at A and B is the same,

Q = c 1 <⅛> 1 V 2gh 1 = c 2 ω 2 f2gh a , where c 1 and c, 2 are the coefficients of discharge suitable for the two orifices. Hence

<⅛wi- ∕(⅜V c 2<∙>2 t y ∖hlJ '

Suppose now that in the interval between the visits of the canal officer the level of the canal rises h feet, causing the heads relatively to the orifices A and B to become h' 1 and h' a . Since the areas of the orifices are unchanged √ h 2 h' 1 c 2 ω 2 ’ and therefore h 2 h2 or the ratio of the effective heads above the orifices A and B is unaffected by the change of level of the canal. Also

+ Λj ~½ι + ä 2 + ä .

Eliminating h' 1 , we get . z (hi + h 2 + h)h% . 2 ~ hι + h a and the discharge in the altered conditions is

Q'=c 2 ω 2 √2g⅛4

∕ ( c (hi + hi ^b h)h2 I

That is, ^i≈= ∕ hι + h i +h .

Q ∖ f hi + h 2

If the orifice at B opened directly into the canal without any intermediate regulating chamber, the discharge would increase for a given change of level in the canal in exactly the same ratio. Consequently the Italian module in no way moderates the fluctuations of discharge, except so far as it affords means of easy adjustment from time to time. It has further the advantage that the - cultivator, by observing the level of the water· in the chamber, can always see whether or not he is receiving the proper quantity of water.

On each canal the orifices are of the same height, and intended to work with the same normal head, the width of the orifices being varied to suit the demand for water. The unit of discharge varies on different canals, being fixed in each case by legal arrangements. Thus on the Canal Lodi the unit of discharge or one module of water is the discharge through an orifice Γ12 feet high, 012416 feet wide, with a head of 0 ’ 32 feet above the top edge of the orifice, or - 88 feet above the centre. This corresponds to a discharge of about 0 - 6165 cubic feet per second. Two modules would be the discharge of a similar orifice of twice the width. The following table gives some examples of different units of discharge. [table]

In the most elaborate Italian modules the regulating chamber is arched over, and its dimensions are very exactly prescribed. Thus in the modules of the Naviglio Grande of Milan, shown in fig. 69, the measuring orifice is cut in a thin stone slab, and so placed that the discharge is into the air with free contraction on all sides. The adjusting sluice is placed with its sill flush with the bottom of the canal, and is provided with a rack and lever and locking arrangement. The covered regulating chamber is about 20 feet long, with a breadth 1 ‘64 feet greater than that of the discharging orifice. At precisely the normal level of the water in the regulating chamber, there is a ceiling of planks intended to still the agitation of the water. A block of stone serves to indicate the normal level of the water in the chamber. The water is discharged into an open channel 0‘655feet wider than the orifice, splayingout till it is 1’637 feet wider than the orifice, and about 18 feet in length. This apparatus was invented in the 16th century, and is still used. The greatest objection to it is the loss of level between the canal and discharging channel. Arrangements precisely similar to an Italian module are in use in England, for discharging compensation water into streams from impounding reservoirs. The fullest account of Italian modules is to be found in Colonel Baird Smith’s Italian Irrigation.

55. Spanish Module.— On the canal of Isabella II., which supplies water to Madrid, a module much more perfect in principle than the Italian module is employed. Part of the water is supplied for irrigation, and as it is very valuable its strict measurement is essential. The module (fig. 71) consists of two chambers one above the other, the upper chamber being in free communication with the irrigation canal, and the lower chamber discharging by a culvert to the fields. In the arched roof between the chambers there . is a circular sharp edged orifice in a bronze plate. Hanging in this there is a bronze plug of variable diameter suspended from a hollow brass float. If the water level in the canal lowers, the plug descends and gives an enlarged opening, and conversely. Thus a perfectly constant discharge with a varying head can be obtained, provided nr clogging or silting of the chambers prevents the free discharge of the water or the rise and fall of the float. The theory of the module is very simple. Let R (fig. 70) be the radius of the fixed opening, r the diameter of the plug at a distance h from the plane of flotation of the float, and Q the required discharge of the module. Then Q-cπ(R a -r 2 )√2gA.

Taking c=0 63, Q = 15∙88(R 9 -r s )√Λ; r= ∖ZS r2- 15∙88√⅛ l·

Choosing a value for R, successive values of r can be found for different values of h, and from these the curve of the plug can be drawn. The module shown in fig. 71 will discharge 1 cubic metre per second. The fixed opening is 0 ’ 2 metre diameter, and the greatest head above the fixed orifice is 1 metre. The use of this module involves a great sacrifice of level between the canal and the fields. The module is described in Lieutenant Scott Moncrieffs Irrigation in Southern Europe.

56. Reservoir Gauging Basins.— In obtaining the power to store the water of streams in reservoirs, it is usual to concede to riparian owners below the reservoirs a right to a regulated supply through out the year. This compensation water requires to be measured in such a way that the miIΓowners and others interested in the matter can assure themselves that they are receiving a proper quantity, and they are generally allowed a certain amount of control as to the times during which the daily supply is discharged into the stream.

f igs 72 and 73 show an arrangement designed for the Manchester Water Works. The water enters from the reservoir a chamber

A, the object of which is to still the irregular motion of the water 1 he admission is regulated by sluices at b, b, b. The water is discharged by orifices or notches at a, a, over which a tolerably constant head is mamtained by adjusting the sluices at b, b, b. At any time the mιllownβrs can see whether the discharge is given and whether the proper head is maintained over the orifices To test at any time the discharge of the orifices, a gauging basin B is provided. The water ordinarily flows over this, without entering it, on a floor of cast-iron plates. If the discharge is to be tested, the water is turned for a definite time into the gauging basin, by suddenly opening and closing a sluice at c. The volume of flow can be ascertained from the depth in the gauging chamber. A mechanical arrangement was designed for securing an absolutely constant head over the orifices at a, a. The orifices were formed in a cast-iron plate capable of sliding up and down, without sensible leakage, on the face of the wall of the chamber. The orifice plate was attached by a link to a lever, one end of which rested on the wall and the other on floats f in the chamber A. The floats rose and fell with the changes of level in the chamber, and raised and lowered the orifice plate at the same time. This mechanical arrangement was not finally adopted, careful watching of the sluices at b, b, b, being sufficient to secure a regular discharge. The arrangement is then equivalent to an Italian module, but on a large scale.

57. Professor Fleeming Jenkins Constant Flow Valve.— In the modules thus far described constant discharge is obtained by varying the area of the orifice through which the water flows. Professor F. Jenkin has contrived a valve in which a constant pressure head is obtained, so that the orifice need not be varied (Roy. Scot. Society of Arts, 1876). Fig. 74 shows a valve of this kind suitable for a

6 inch water main. The water arriving by the main C passes through an equilibrium valve D into the chamber A, and thence through a sluice 0, which can be set for any required area of opening, into the discharging main B. The object of the arrangement is to secure a constant difference of pressure between the chambers A and B, so that a constant discharge flows through the stop valve O. The equilibrium valve D is rigidly connected with a plunger P loosely fitted in a diaphragm, separating A from a chamber B 2 connected by a pipe B 1 with the discharging main B. Any increase of the difference of pressure in A and B will drive the plunger up and close the equilibrium valve, and conversely a decrease of the difference of pressure will cause the descent of the plunger and open the equilibrium valve wider. Thus a constant difference of pressure is obtained in the chambers A and B. Let ω be the area of the plunger in square feet, p the difference of pressure in the chambers A and B in pounds per square foot, w the weight of the plunger and valve. Then if at any moment pa> exceeds w the plunger will rise, and if it is less than w the plunger will descend. Apart from friction, and assuming the valve D to be strictly an equilibrium valve, since ω and w are constant, p must be constant also, and equal to - . By making w ω small and ω large, the difference of pressure required to ensure the working of the apparatus may be made very small. Valves working with a difference of pressure of ⅜ inch of water have been constructed.

58. Aρρold's Module.— This acts on the same general principle as the Spanish module, but it secures only an approximately constant discharge. On the other hand it involves no great sacrifice of level, and is not very likely to be affected by silting. It was contrived originally as an air regulator, but it has also been tried with success as a water module. It consists simply of a horizontal pipe with an enlarged chamber, in which hangs a heavy wedge-shaped pendulum. The pressure of the water on the upstream side of this pendulum keeps it in a position inclined to the vertical, and partially closing the orifice of discharge as shown by the dotted lines in fig. 75. Any increase of pressure will cause a greater inclination of the pendulum and decrease the orifice of discharge, and vice versa.

VI. STEADY FLOW OF COMPRESSIBLE FLUIDS.

59. External Work during the Expansion of Air.— If air expands without doing any external work, its temperature remains constant. This result was first experimentally demonstrated by Joule. It leads to the conclusion that, however air changes its state, the internal work done is proportional to the change of temperature. When, in expanding, air does work against an external resistance, either heat must be supplied or the temperature falls.

To fix the conditions, suppose one pound of air confined behind a piston of one square foot area (fig. 76). Let the initial pressure be p- i and the volume of the air υ 1 , and suppose this to expand to the pressure p, 2 and volume v i . If p and v are the corresponding pressure and volume at any intermediate point in the expansion, the work done on the piston during the expansion from v to v + dv is pdv, and the whole work during the expansion from v 1 to v 2 , represented by the area abed, is ∕ pdv. J l 'ι

Amongst possible cases two may be selected.

Case 1.—So much heat is supplied to the air during expansion that the temperature remains constant. Hyperbolic expansion. Then pv=p 1 v l .

Work done during expansion per pound of air f v 2 , f υ * dv ≈J pdv=p 1 v 1 ∕ — w ∕Γ 1 J v ∖ v

=^ 1 V 1 lθg e ⅛=^ 1 lθge-J- . . . (1).

Since the weight per cubic foot is the reciprocal of the volume per pound, this inay be written

P-i<>g∙≥ d“)·

½ ^^ τ 2

Then the expansion curve ab is a common hyperbola.

Case 2.—No heat is supplied to the air during expansion. Then the air loses an amount of heat equivalent to the external work done and the temperature falls. Adiabatic expansion.

In this case it can be shown that jW γ ==pp){t , where γ is the ratio of the specific heats of air at constant pressure and volume. Its value for air is l ’ 408, and for dry steam 1 ’ 135.

Work done during expansion per pound of air ∕"⅞ , „ p*-ιdv ~f, -

_ √ι φ ι γ J 1 1 )

7-l∣t√- 1 t√- 1 i τw j i i_ )

7-lU 1 V- 1 V 1 ’

_ a⅛l ι-(⅛γ- 1 I . (⅛ 7-1 (∖ v 2 J )

The value of p 1 v 1 for any given temperature can be found from the data already given.

As before, substituting the weights G 1 , G 2 per cubic foot for the volumes per pound, we get for the work of expansion

⅛∙ γ ÷∙Γ 1 √¾Γ∣ · · ∙' u w∙

60. Modification of the Theorem of Bernoulli for the Case of a Compressible Fluid. — In the application of the principle of work to a filament of compressible fluid, the internal work done by the expansion of the fluid, or absorbed in its compression, must be taken into account. Suppose, as before, that AB (fig. 77) comes to A'B' in a short time t. Let p 1 , ω 1 , v 1 , G 1 be the pressure, sectional area of stream, velocity, and weight of a cubic foot at A, and ρ 2 , ω v v 2 , G 2 the same quantities at B. Then, from the steadiness of motion, the weight of fluid passing A in any given time must be equal to the weight passing B:

G 1 wιVji = G 2 w 2 r 2 ∕ .

Let 2 1 , z 2 be the heights of the sections A and B above any given datum. Then the work of gravity on the mass AB in t seconds is G 1 ω l v 1 t(z 1 -z 2 ) = 'W (z 1 -z a )t, where W is the weight of gas passing A or B per second. As in the case of an incompressible fluid, the work of the pressures on the ends of the mass AB is Plt>>lvlt ~ P2 ω 2 v 2^ >

"(⅛-¾) w '∙

The work done by expansion of W∕ pounds of fluid between A and B is Wi f v ' t pdw. The change of kinetic energy as before is (v 2 2 - v 1 2 )i.

Hence, equating work to change of kinetic energy,

W⅛ - ¾X + ((ζ - ¾ ) W< + (.·, - vflt;

·· · · <»·

Now the work of expansion per pound of fluid has already been given. If the temperature is constant, we get (eq. lα, § 59)

Pi ι ’ 1 2 Vo v∙> i P ∖ , G 1 * 1+ g? + ¥ ^ z ≡ + g 2 + 27" g 1 1o ^g 2 ·

But at constant temperature ¾L ≈= 7⅛;

Gj G 2

·· < 2 > i or, neglecting the difference of level,

⅛Γ⅛>S i2o >∙

Similarly, if the expansion is adiabatic (eq. 2α, § 59),

. + Λ + *ι , .. ψ i⅛.<⅛ , n 1 I 1 ∕⅛∙γ≡b 13 ∖ .

'> + c 1 ¼-¾⅛ + ⅛-0i7∏p-(⅛p j∙ < 8 >> or neglecting the difference of level

It will be seen hereafter that there is a limit in the ratio ⅛ beyond which these expressions cease to be time.

61. Discharge of Air from an Orifice.— 1 The form of the equation of work for a steady stream of compressible fluid is 21+ *ι + ^=, + ⅛ + !⅛!-Λ JL j 1 -∕⅛Vτi 1 + G l + 2g 2a+ G 2 + 2g G 1 γ-lι 1 (j√ Y I’

the expansion being adiabatic, because in the flow of the streams of air through an orifice no sensible amount of heat can be communicated from outside.

Suppose the air flows from a vessel, where the pressure is p 1 and the velocity sensibly zero, through an orifice, into a space where the pressure is p 2 . Let t> a be the velocity of the jet at a point where the convergence of the streams has ceased, so that the pressure in the jet is also p 2 . As air is light, the work of gravity will be small compared with that of the pressures and expansion, so that z 1 , z 2 may be neglected. Putting these values in the equation above—

Δ-Λ√dL∆JJ 1 l r P∆ 1 r). G 1 G a + 2 0 r G 17 -l ∣ 1 VK∕! ’

2g G 1 G 2 G 17 - 1 (∖ p l ) I

»A j L,f⅛λ 1 v I

G 1 (7 -l 7 -l∖p 1 ) γ ) G 2

But . ⅛=W 2Ù.ŸW

G 1 > G.f · · G 2 G 1 ∖ pJ 1 " (¾) γ j (1);

θr ⅞ 8 , 7 ⅛ P» I .

⅛ 7-l<G 1 "G 2 ’ an equation commonly ascribed to Weisbach (Civilingenieur, 1856), though it appears to have been given earlier by St Venant and Wantzel.

It has already (§ 9, eq. 4α) been seen that

θι G o τ 0 ’

where for air p 0 - 2116’8, G o - ∙08075 and τ 0 - 492’6 .

t ⅛ G o τ o7 -i(1 (p 1 ∕ γ ∣> · · · (2); or, inserting numerical values,

⅛ ’-ιrn τι j 1 .(⅛y 29 j w .

which gives the velocity of discharge ¾ in terms of the pressure and absolute temperature, p 1 , τ 1 , in the vessel from which the air flows, and the pressure p 2 in the vessel into which it flows.

Proceeding now as for liquids, and putting ω for the area of the orifice and c for the coefficient of discharge, the volume of air discharged per second at the pressure p 2 and temperature τ 2 is

Q 2 = cwv 2 = c ω ∖∕ 2g HL- j i _ (⅛Vγ~ I

7 -1 G 1 (∖ p 1 j )

- 108ta √N ι -(⅛Γ∏ · · · <≡)∙

If the volume discharged is measured at the pressure p 1 and absolute temperature τ 1 in the vessel from which the air flows, let Q 1 be that volume; then aQi y =^Q 2 y ;

¾-~√[ 2 ≈' i ≡y⅞≠J

= 108 ’ 7cw Vτ 1 ψ (4).

The weight of air at pressure p 1 and temperature τ 1 is

G 1 = j h pounds per cubic foot. Dθ *ZTj

Hence the weight of air discharged is

W = G 1 Q 1 = cω 5 y z p fl ,-2-j, 1 G 1 ≠ J

= 2 ’ 043cwpι (5).

Weisbach has found the following values of the coefficient of discharge c [table]

62. Limit to the Application of the above Formulae.— In the formulae above it is assumed that the fluid issuing from the orifice expands from the pressure jo 1 to the pressure^, while passing from the vessel to the section of the jet considered in estimating the area ω. Hence p 2 is strictly the pressure in the jet at the plane of the external orifice in the case of mouthpieces, or at the plane of the contracted section in the case of simple orifices. Till recently it was tacitly assumed that this pressure p 2 was identical with the· general pressure external to the orifice. Mr B. D. Napier first discovered that, when the ratio ⅛ exceeded a value which does not Pi greatly differ from 0 ’ 5, this was no longer true. In that case the expansion of the fluid down to the external pressure is not completed at the time it reaches the plane of the contracted section, and the pressure there is greater than the general external pressure; or, what amounts to the same thing, the section of the jet where the expansion is completed is a section which is greater than the area c e ω of the contracted section of the jet, and may be greater than the area « of the orifice. Mr Napier made experiments with steam which showed that, so long as 0 - 5, the formulae above Pi were trustworthy, when p 2 was taken to be the general external pressure, but that, if ⅛ < 0 5, then the pressure at the contracted section was independent of the external pressure and equal to 0'5p 1 . Hence in such cases the constant value 0 - 5 should be substituted in the formulae for the ratio of the internal and external pressures T½ . Pi

It is easily deduced from Weisbach’s theory that, if the pressure external to an orifice is gradually diminished, the weight of air discharged per second increases to a maximum for a value of the ratio

⅛=p ∖ γ ÷ 1 Pi ∖7+ 1 ∕

= 0’527 for air

= 0 ‘58 for dry steam.

For a further decrease of external pressure the discharge diminishes, —a result no doubt improbable. The new view of Weisbach’s formula is that from the point where the maximum is reached, or not greatly differing from it, the pressure at the contracted section ceases to diminish.

Fliegner has shown (Civilingenieur, xx., 1874) that for air flowing from well-rounded mouthpieces there is no discontinuity of the law of flow, as Napier’s hypothesis implies, but the curve of flow bends so sharply that Napier’s rule may be taken to be a good approximation to the true law. The limiting value of the ratio 2⅛, for which Weisbach’s formula, as originally understood, ceases Pi to apply, is for air 0 , 5767; and this is the number to be substituted for 7⅛ in the formulae when ⅛ falls below that value. For later Pi Pi researches on the flow of air, reference may be made to Zeuner’s paper (Ciνiiingenieur, 1871), and Fliegner’s papers (ibid., 1877, 1878).

VΠ. FRICTION OF LIQUIDS.

63. When a stream of fluid flows over a solid surface, or conversely when a solid moves in still fluid, a resistance to the motion is generated, commonly termed fluid friction. It is due to the viscosity of the fluid, but generally the laws of fluid friction are very different from those of simple viscous resistance. It would appear that at all speeds, except the slowest, rotating eddies are formed by the roughness of the solid surface, or by abrupt changes of velocity distributed throughout the fluid; and the energy expended in producing these eddying motions is gradually lost in overcoming the viscosity of the fluid in regions more or less distant from that where they are first produced.

The laws of fluid friction are generally stated thus :—

1. The frictional resistance is independent of the pressure between the fluid and the solid against which it flows. This may be verified by a simple direct experiment. Coulomb, for instance, oscillated a disk under water, first with atmospheric pressure acting on the water surface, afterwards with the atmospheric pressure removed. No difference in the rate of decrease of the oscillations was observed. The chief proof that the friction is independent of the pressure is that no difference of resistance has been observed in water mains and in other cases, where water flows over solid surfaces under widely different pressures.

2. The frictional resistance of large surfaces is proportional to the area of the surface.

3. At low velocities of not more than 1 inch per second for water, the frictional resistance increases directly as the relative velocity of the fluid and the surface against which it flows. At velocities of ⅜ foot per second and greater velocities, the frictional resistance is more nearly proportional to the square of the relative velocity.

In many treatises on hydraulics it is stated that the frictional resistance is independent of the nature of the solid surface. The explanation of this was supposed to be that a film of fluid remained attached to the solid surface, the resistance being generated between this fluid layer and layers more distant from the surface. At extremely low velocities the solid surface does not seem to have much influence ôn the friction. In Coulomb’s experiments a metal surface covered with tallow, and oscillated in water, had exactly the same resistance as a clean metal surface, and when sand was scattered over the tallow the resistance was only very slightly increased. The earlier calculations of the resistance of water at higher velocities in iron and wood pipes and earthen channels seemed to give a similar result. These, however, were erroneous, and it is now well understood that differences of roughness of the solid surface very greatly influence the friction, at such velocities as are common in engineering practice. Darcy’s experiments, for instance, showed that in old and incrusted water mains the resistance was twice or sometimes thrice as great as in new and clean mains.

64. Ordinary Expressions for Fluid Friction at Velocities not Extremely Small.— Let/be the frictional resistance estimated in pounds per square foot of surface at a velocity of one foot per second; ω the area of the surface in square feet; and v its velocity in feet per second relatively to the water in which it is immersed. Then, in accordance with the laws stated above, the total resistance of the surface is

R=∕wt> 2 (1), where /is a quantity approximately constant for any given surface. If t .2fl∕ s G ’

R = ξG w ^- ...... (2), where ξ is, like ∕, nearly constant for a given surface, and is termed the coefficient of friction.

The following are average values of the coefficient of friction for water, obtained from experiments on large plane surfaces, moved in an indefinitely large mass of water. [table]

The distance through which the frictional resistance is overcome is v feet per second. The work expended in fluid friction is therefore given by the equation—

Work expended =∕α>r 3 foot-pounds per second 1

-S0-≠ „ „ · · · <8) ·

2g 1

The coefficient of friction and the friction per square foot of surface can be indirectly obtained from observations of the discharge of pipes and canals. In obtaining them, however, some assumptions as to the motion of the water must be made, and it will be better therefore to discuss these values in connexion with the cases to which they are related.

Many attempts have been made to express the coefficient of friction in a form applicable to low as well as high velocities. The older hydraulic writers considered the resistance termed fluid friction to be made up of two parts,—a part due directly to the distortion of the mass of water and proportional to the velocity of the water relatively to the solid surface, and another part due to kinetic energy imparted to the water striking the roughnesses of the solid surface and proportional to the square of the velocity. Hence they proposed to take

< = α + A, s v in which expression the second term is of greatest importance at very low velocities, and of comparatively little importance at velocities over about ⅜ foot per second. Values of ξ expressed in this and similar forms will be given in connexion with pipes and canals.

All these expressions must at present be regarded as merely empirical expressions serving practical purposes. The frictional resistance will be seen to vary through wider limits than these expressions allow, and to depend on circumstances of which they do not take account.

65. Coulomb's Experiments. —The first direct experiments on fluid friction were made by Coulomb, who employed a circular disk suspended by a thin brass wire and oscillated in its own plane. His experiments were chiefly made at very low velocities. When the disk is rotated to any given angle, it oscillates under the action of its inertia and the torsion of the wire. The oscillations diminish gradually in consequence of the work done in-overcoming the friction of the disk. The diminution furnishes a means of determining the friction.

Fig. 78 shows Coulomb’s apparatus. LK supports the wire and disk; aq is the brass wire, the torsion of which causes the oscilla-tiens; DS is a graduated disk serving to measure the angles through which the apparatus oscillates. To this the friction disk is rigidly attached hanging in a vessel of water. The friction disks were from 47 to 77 inches diameter, and they generally made one oscillation in from 20 to 30 seconds, through angles varying from 360° to 6°. When the velocity of the circumference of the disk was less than 6 inches per second, the resistance was sensibly proportional to the velocity.

Beaufoy ’ 3 Experiments.— Towards the end of the last century Colonel Beaufoy made an immense mass of experiments on the resistance of bodies moved through water {Naτιtical and Hydraulic Experiments, London, 1834). Of these the only ones directly bearing on surface friction were some made in 1796 and 1798. Smooth painted planks were drawn through water and the resistance measured. For two planks differing in area by 46 square feet, at a velocity of 10 feet per second, the difference of resistance, measured on the difference of area, was 0'339 lb per square foot. Also the resistance varied as the 1 '949th power of the velocity.

66. Mr Froude’s Experiments.— The most important direct experiments on fluid friction at ordinary velocities are those made by Mr Froude at Torquay. The method adopted in these experiments was to tow a board in a still water canal, the velocity and the resistance being registered by very ingenious recording arrangements. The general arrangement of the apparatus is shown in fig. 79. AA is the board the resistance of which is to be determined. B is a cutwater giving a fine entrance to the plane surfaces of the board. CC is a bar to which the board AA is attached, and which is suspended by a parallel motion from a carriage running on rails above the still water canal. G is a link by which the resistance of the board is transmitted to a spiral spring H. A bar I rigidly connects the other end of the spring to the carriage. The dotted lines K, L indicate the position of a couple of levers by which the extension of the spring is caused to move a pen M, which records the extensions on a greatly increased scale, by a line drawn on the paper cylinder N. This cylinder revolves at a speed proportionate to that of the carriage, its motion being obtained from the axle of the carriage wheels. A second pen 0, receiving jerks at every second and a quarter from a clock P, records time on the paper cylinder. The scale for the line of resistance is ascertained by stretching the spiral spring by known weights. The boards used for the experiment were τ⅞ inch thick, 19 inches deep, and from 1 to 50 feet in length, cutwater included. A lead keel counteracted the buoyancy of the board. The boards were covered with various substances, such as paint, varnish, Hay’s composition, tinfoil, &c., so as to try the effect of different degrees of roughness of surface. The results obtained by Mr Froude may be summarized as follows :—

(1) The friction per square foot of surface varies very greatly for different surfaces, being generally greater as the sensible roughness of the surface is greater. Thus, when the surface of the board was covered as mentioned below, the resistance for boards 50 feet long, at 10 feet ner second, was— [table]

(2) The power of the velocity to which the friction is proportional varies for different surfaces. Thus, with short boards 2 feet long, For tinfoil the resistance varied as -ι> 2 ' ls For other surfaces ,, ,, v 2 ' 00

With boards 50 feet long,

For varnish or tinfoil the resistance varied as t> 1 ' 8a For sand ,, n n v ≡∙oo

(3) The average resistance per square foot of surface was much greater for short than for long boards; or, what is the same thing, the resistance per square foot at the forward part of the board was greater than the friction per square foot of portions more sternward. Thus, [table]

This remarkable result is explained thus by Mr Froude “The portion of surface that goes first in the line of motion, in experiencing resistance from the water, must in turn communicate motion to the water, in the direction in which it is itself travelling. Consequently the portion of surface which succeeds the first will be rubbing, not against stationary water, but against water partially moving in its own direction, and cannot therefore experience so much resistance from it.”

67. The following table gives a general statement of Mr Froude’s results. In all the experiments in this table, the boards had a fine cutwater and a fine stern end or run, so that the resistance was entirely due to the surface. The table gives the resistances per square foot in pounds, at the standard speed of 600 feet per minute, and the power of the speed to which the friction is proportional, so that the resistance at other speeds is easily calculated. [table]

Columns A give the power of the speed to which the resistance is approximately proportional.

Columns B give the mean resistance per square foot of the whole surface of a board of the lengths stated in the table.

Columns C give the resistance in pounds of a square foot of surface at the distance sternward from the cutwater stated in the heading.

Although these experiments do not directly deal with surfaces of greater length than 50 feet, they indicate what would be the resistances of longer surfaces. For at 50 feet the decrease of resistance for an increase of length is so small that it will make no very great difference in the estimate of the friction whether we suppose it to continue to diminish at the same rate or not to diminish at all. For a varnished surface the friction at 10 feet per second; diminishes from 0'41 to 0 - 32 lb per square foot when the length is increased from 2 to 8 feet, but it only diminishes from 0∙278 to 0’250 lb per square foot for an increase from 20 feet to 50 feet.

If the decrease of friction .sternwards is due to the generation of a current accompanying the moving plane, there is not at first sight any reason why the decrease should not be greater than that shown by the experiments. The current accompanying the board might be assumed to gain in volume and velocity stemwards, till the velocity was nearly the same as that of the moving plane and the friction per square foot nearly zero. That this does not happen appears to be due to the mixing up of the current with the still water surrounding it. Part of the water in contact with the board at any point, and receiving energy of motion from it, passes afterwards to distant regions of still water, and portions of still water are fed in towards the board to take its place. In the forward part of the board more kinetic energy is given to the current than is diffused into surrounding space, and the current gains in velocity. At a greater distance back there is an approximate balance between the energy communicated to the water and that diffused. The velocity of the current accompanying the board becomes constant or nearly constant, and the friction per square foot is therefore nearly constant also.

VIII. STEADY FLOW OF WATER IN PIPES OF UNIFORM SECTION.

68. The ordinary theory of the flow of water in pipes, on which all practical formulae are based, assumes that the variation of velocity at different points of any cross section may be neglected. The water is considered as moving in plane layers, which are driven through the pipe against the frictional resistance, by the difference of pressure at or elevation of the ends of the pipe. If the motion is steady the velocity at each cross section remains the same from moment to moment, and if the cross sectional area is constant the velocity at all sections must be the same. Hence the motion is uniform. The most important resistance to the motion of the water is the surface friction of the pipe, and it is convenient to estimate this independently of some smaller resistances which will be accounted for presently.

In any portion of a uniform pipe, excluding for the present the ends of the pipe, the water enters and leaves at the same velocity. For that portion therefore the work of the external forces and of the surface friction must be equal. Let fig. 80 represent a very short portion of the pipe, of length dl, between cross sections at z and z + dz feet above any horizontal datum line xx, the pressures at the cross sections being p and p + dp lb per square foot. Further, let Q be the volume of flow or discharge of the pipe per second, ∩ the area of a normal cross section, and χ the perimeter of the pipe. The Q cubic feet, which flow through the space considered per second, weigh GQ lb, and fall through a height - dz feet. The work done by gravity is then

-GQcfe; a positive quantity if dz is negative, and vice versa. The resultant pressure parallel to the axis of the pipe is ρ - {p + dρ) = - dp lb per square foot of the cross section. The work of this pressure on the volume Q is

- Q⅜.

The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is χ dl.

The work expended in overcoming the frictional resistance per second is (see § 64, eq. 3)

-iθχ∙w⅞, or, since Q = Ωv, the negative sign being taken because the work is done against a resistance. Adding all these portions of work, and equating the result to zero, since the motion is uniform,—

- GQ<∕z - Q⅛> - (GÀQ ^-dl≈= 0 .

∩ 2ÿ Dividing by GQ,

Integrating, z + -p- + t½ ⅛ = constant (1).

69. Let A and B (fig. 81) be any two sections of the pipes for which p, z, I have the values p 1 , z 1 , l 1 , and ρ. 2 , z 2 , l 2 respectively.

Then 2 - J' j√* ν l = z -u ρ * 4- r x v I

Then 2 i+g→⅛ 2∕1=≈¾+θ- + C∩ √ 2 ’

or, if l 2 - l 1 = L, rearranging the terms,

⅞4i(→)-(→)i⅞∙ · · «

Suppose pressure columns introduced at A and B. The water will rise in those columns to the heights Lf and fl due to the G G

pressuresj> 1 and p 2 at A and B. Hence ^z 1 + ^-^-^z 2 +^-^is the quantity represented in the figure by DE, the fall of level of the pressure columns, or virtual fall of the pipe. If there were no friction in the pipe, then by Bernoulli’s equation there would be no fall of level of the pressure columns, the velocity being the same at A and B. Hence DE or h is the head lost in friction in the distance AB. The quantity = A is termed the virtual slope

At> L·

of the pipe or virtual fall per foot of length. It is sometimes termed very conveniently the relative fall. It will be denoted by the symbol i.

The quantity — which appears in many hydraulic equations is X

called the hydraulic mean radius of the pipe. It will be denoted by m.

Introducing these values, a r 2 h . tn ∖ f ⅛- ’ "T7 -m, (3> ·

For pipes of circular section, and diameter· d,

A If A

” X “ «I i ' λ v 2 d h di z .v

Then <- — . - 7 r , (4); 2g 4 L 4

, > 4L v i , λ » or i - f τ∙¾ (4o) ·

which shows that the head lost in friction is proportional to the head due to the velocity, and is found by multiplying that head by the coefficient .

70. Hydraulic Gradient ar Line of Virtual Slope.·— Join CD. Since the head lost in friction is proportional to L, any intermediate pressure column between A and B will have its free surface on the line CD, and the vertical distance between CD and the pipe at any point measures the pressure, exclusive of atmospheric pressure, in the pipe at that point. If the pipe were laid along the line CD instead of AB, the water would flow at the same velocity by gravity without any change of pressure from section to section. Hence CD is termed the virtual slope or hydraulic gradient of the pipe. It is the line of free surface level for each point of the pipe.

If an ordinary pipe, connecting reservoirs open to the air, rises at any point above the line of virtual slope, the pressure at that point is less than the atmospheric pressure transmitted through the pipe. At such a point there is a liability that air may be disengaged from the water, and the flow stopped or impeded by the accumulation of air. If the pipe rises more than 34 feet above the line of virtual slope, the pressure is negative. But as this is impossible, the continuity of the flow will be broken.

If the pipe is not straight, the line of virtual slope becomes a curved line, but since in actual pipes the vertical alterations of level are generally small, compared with the length of the pipe, distances measured along the pipe are sensibly proportional to distances measured along the horizontal projection of the pipe. Hence the line of hydraulic gradient may be taken to be a straight line without error of practical importance.

71. Case of a Uniform Pipe connecting two Reservoirs, when all the Resistances are taken into account.— Let h (fig. 82) be the difference of level of the reservoirs, and ν the velocity, in a pipe of length L and diameter d. The whole work done per second is virtually the removal of Q cubic feet of water from the surface of the upper reservoir to the surface of the lower reservoir, that is GQÄ foot-pounds.

This is expended in three ways. (1) The head — , corresponding

-.2

to an expenditure of GQU foot-pounds of work, is employed in giving '~' f energy of motion to the water. This is ultimately wasted in eddying motions in the lower reservoir. (2) A portion of head, t> s which experience shows may be expressed in the form , corréla spending to an expenditure of GQ< 0 ^ foot-pounds of work, is employed in overcoming the resistance at the entrance to the pipe. (3) As already shown the head expended in overcoming the surface friction of the pipe is <^ ^-corresponding to GQ<-^- · foot-pounds of work. Hence

, ∕, . .4LW ∖

⅞-(ι ÷⅞+ ⅛)¾. I . (6) φ - 8 ∙° 25 V( 14-⅞‰ l ∙ 1

If the pipe is bellmouthed, is about = Ό8. If the entrance to the pipe is cylindrical, £ 0 = 0'505. Hence 14√ o =ΓO8 to 1’505.

In general this is so small compared with that, for. practical (JU calculations, it may be neglected; that is, the losses of head other than the loss in surface friction are left out of the reckoning. It is only in short pipes and at high velocities that it is necessary to take account of the first two terms in the bracket, as well as the third. For instance, in pipes for the supply of turbines, v is usually limited to 2 feet per second, and the pipe is bellmouthed. Then

1 - 08∣-- = 0 Ό67 foot. In pipes for towns’ supply ν may range from

2 to 4⅛ feet per second, and then 1'5^- = 0 - l to 0’5 foot. In either case this amount of head is small compared with the whole virtual fall in the cases which most commonly occur.

When d and ν or d and h are given, the equations above are solved quite simply. When v and h are given and d is required, it is better to proceed by approximation. Find an approximate value of d by assuming a probable value for ζ as mentioned below. Then from that value of d find a corrected value for ζ and repeat the calculation.

The equation above may be put in the form

-⅛i i ⅛FM⅛ < β >=

from which it is clear that the head expended at the mouthpiece is equivalent to that of a length

(i+C)rf of the pipe. Putting 1 + ζ 0 = 1 ∙505 and ζ= 0 Ό1, the length of pipe equivalent to the mouthpiece is 37’6 d nearly. This may be added to the actual length of the pipe to allow for mouthpiece resistance in approximate calculations.

72. Coefficient of Friction for Pipes Discharging Water— From the average of a large number of experiments, the value of £ for ordinary iron pipes is f=OOO7567 (7).

But practical experience shows that no single value can be taken applicable to very different cases. The earlier hydraulicians occupied themselves chiefly with the dependence of £ on the velocity. Having regard to the difference of the law of resistance at very low and at ordinary velocities, they assumed that £ might be expressed in the form i=α + -∙

The following are the best numerical values obtained for f so expressed:— [table]

Weisbach proposed the formula

<=α + -~= ≈0∙003598 + 0 ' 00 ^ 89 .. . . . (8).

√t> √v

The following short table gives Weisbach’s values of the coefficient of friction for different velocities in feet per second :— [table]

73. Darcy's Experiments.— All previous experiments on the resistance of pipes have been superseded by the remarkable researches carried out by the late Inspector-General of the Paris Waterworks, Μ. Darcy. His experiments were carried out on a scale, under a variation of conditions, and with a degree of accuracy which leaves little to be desired, and the results obtained are of very great practical importance. These results may be stated thus :—

(1) For new and clean pipes the friction varies considerably with the nature and polish of the surface of the pipe. For clean castiron it is about 1 ⅛ times as great as for cast-iron covered with pitch.

(2) The nature of the surface has less influence when the pipes are old and in crusted with deposits, due to the action of the water. Thus old and incrusted pipes give twice as great a frictional resistance as new and clean pipes. Μ. Darcy’s coefficients were chiefly determined from experiments on new pipes. He doubles these coefficients for old and incrusted pipes, in accordance with the results of a very limited number of experiments on pipes containing incrustations and deposits.

(3) The coefficient of friction may be expressed in the form f = « + — > but in pipes which have been some time in use it is sufficiently accurate to take f=α 1 simply, where a 1 depends on the diameter of the pipe alone, but α and β on the other hand depend both on the diameter of the pipe and the nature of its surface. The following are the values of the constants.

For pipes which have been some time in use, neglecting the term depending on the velocity; f -√ 1 ÷4) <’>· [table]

These coefficients may be put in the following very simple form, without sensibly altering their value :—

For clean pipes ∙005( 1 )

∕ 1 ∖ < 9α >∙

For slightly inerusted pipes £=-Ό1ί 1 )

Darcy's Value of the Coefficient of Friction ζ for Velocities not less than 4 inches per second. [table]

These values of fare, however, only applicable when the velocity exceeds 4 inches per second. To embrace all cases Darcy proposes the expression fl+ ~ 1 - c-(→)⅛- which is a modification of Coulomb’s, including terms expressing the influence of the diameter and of the velocity. For clean pipes Darcy found these values

[table]

74. Scraping Water Mains.— The influence of the condition of the surface of a pipe on the friction is shown by various facts known to the engineers of waterworks. Pipes are very often heated and dipped in pitch, which gives them a smooth hard surface and protects them from oxidation. Such pipes are known to give a discharge larger than that calculated by the ordinary formulae. In pipes which convey certain kinds of water oxidation proceeds rapidly, and the discharge is very perceptibly and sometimes very greatly diminished. In a main laid at Torquay the discharge diminished from this cause more than 50 per cent., and the supply became insufficient for the town. Mr Appold suggested an apparatus for scraping the interior of the pipe, and this was constructed and tried under the direction of Mr Froude. It was found that by scraping the interior of the pipe the discharge was increased 56 per cent. The scraping requires to be repeated at intervals. After each scraping the discharge diminishes rather rapidly by 10 per cent, and afterwards more slowly, the diminution in a year being about 25 per cent.

Fig. 83 shows a scraper for water mains, similar to Mr Appold’s but modified in details, as constructed by the Glenfield Company, at Kilmarnock. A is a longitudinal section of the pipe, showing the scraper in place; B is an end view of the plungers, and C, D sections of the boxes placed at intervals on the main for introducing or withdrawing the scraper. The apparatus consists of two plungers, packed with leather so as to fit the main pretty closely. On the spindle of these plungers are fixed eight steel scraping blades, with curved scraping edges fitting the surface of the main. The apparatus is placed in the main by removing the cover from one of the boxes shown at C, D. The cover is then replaced, water pressure is admitted behind the plungers, and the apparatus driven through the main. At Lancaster, after twice scraping the discharge was increased 56⅜ per cent., at Oswestry 54 ⅛ per cent. The increased discharge is due to the diminution of the friction of the pipe by removing the roughnesses due to oxidation.

75. Reduction of a long Pipe of Varying Diameter to an Equivalent Pipe of Uniform Diameter. Dupuit ’ s Equation.— Water mains for the supply of towns often consist of a series of lengths, the diameter being the same for each length, but differing from length to length. In approximate calculations of the head lost in such mains, it is generally accurate enough to neglect the smaller losses of head and to have regard to the pipe friction only, and then the calculations may be facilitated by reducing the main to a main of uniform diameter, in which there would be the same loss of head. Such a uniform main will be termed an equivalent main.

In fig. 84, let A be the main of variable diameter, and B the equivalent uniform main. In the given main of variable diameter A, le' [table]

for the successive portions, and let l, d, v, and i be corresponding quantities for the equivalent uniform main B. The total loss of head in A due to friction is

Λ=AA + ⅛ +

= + z⅛ 8 l⅞ +

⅝ ⅛ t ⅝ V “ ' t and in the uniform main il-ζf ⅛ 2<∕ d

If the mains are equivalent, as defined above, rff = . 4 A + λ!½! 4 ⅞+

s 2g d 5s 2g d 1 s, 2g d i .

But, since the discharge is the same for all portions,

Xd 2 v = τ⅛⅛ 1 =τ⅛ 2 ¾=

4 4 4

d* d 2 ν i~ υ d* iν *~ ν d*

Also suppose that f may be treated as constant for all the pipes. Then

A + Éi A + d f i 'd 1 dfdf

⅛≈⅛÷

which gives the length of the equivalent uniform main which would have the same total loss of head for any given discharge.

76. Case of a Pipe of Uniform Diameter with Discharge diminishing uniformly along its Length.— In the case of a branch main the water is delivered at nearly equal distances to service pipes along the route. Such a main approximates to the case of a main of uniform diameter, with a discharge at each point diminished by an amount proportional to the distance from the origin. Let AB (fig. 85) be a main of diameter d and length L; let Q o cubic feet per second enter at A, and let q cubic feet be delivered to service pipes per foot of its length. Then at any point C, I feet from A, the discharge is Q = Q o - ql. Consider a short length dl at c. The loss of head in that length is 4dl_ Q 2 4dl 640 QW_64£ _

To

Hence the whole head lost in the length AB will be *-i⅛∕<⅛∙-^ -V⅛-j⅛'-iQΛ+wj (1); or, putting P = ?L, the total discharge through the service pipes between A and B,

*-¾gs j⅛> ’- p ‰ + ⅛ p, j <⅛

The discharge at the end B of the pipe is Q 1 = Q 0 -P. If Q ’ is put for the discharge of the pipe when the flow into the service pipes is stopped, under the same head h, it may be shown that Q 1 = Q ’ 0 ’ 55P nearly

If the pipe is so long that

Q 1 = Q 0 P-0, all the water passing into service pipes en route,

< sμ

77. Other Losses of Head in Pipes.— Most of the losses of head in pipes, other than that due to surface friction against the pipe, are due to abrupt changes in the velocity of the stream producing eddies. The kinetic energy of these is deducted from the general energy of translation, and practically wasted.

Sudden Enlargement of Section.— Suppose a pipe enlarges in section from an area ω 0 to an area ω l (fig. 86); then ^i_"o .

or, if the section is circular,

2⅛√M 9 ⅞ ∖ dj

The head lost at the abrupt change of velocity has already been shown to be the head due to the relative velocity of the two parts of the stream. Hence head lost h =(⅞→ι) 2 Z«i 1 ∖ 2 v 1 2 i {dΛ* , I ¼ 2

’· 2? ^!⅞^U⅛-i(⅛- 1 ! ⅛

°' *∙→⅛* in if 0 is put for the expression in brackets. [table]

Abrupt Contraction of Section.— When water passes from a larger to a smaller section, as in figs. 87, 88, a contraction is formed, and the contracted stream abruptly expands to fill the section of the pipe. Let ω be the section and v the velocity of the stream at bb. At aa the section will be c c ω, and the velocity — v= — , where c t is c c a> c t the coefficient of contraction. Then the head lost is i , p " ⅛ ∖ c c 1 J 2g ’ and, if c c is taken 0 - 64, h. = 0·316g (2).

The value of the coefficient of contraction for this case is, however, not well ascertained, and the result is somewhat modified by friction. For water entering a cylindrical, not bellιnouthed, pipe from a reservoir of indefinitely large size, experiment gives f) β = 0∙505- (3).

j 2g

If there is a diaphragm at the mouth of the pipe as in fig. 88, let ω 1 be the area of this orifice. Then the area of the contracted stream is c c a> l , and the head lost is

H⅛- 1 ) ’ C

ff 3 < 4 > if Λ is put for ∕ — - 1 f.

∖CcW 1 ]

Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was considerably larger than the orifice :— [table]

When a diaphragm was placed in a tube of uniform section (fig. 89), the following values were obtained, ω 1 being the area of the orifice, and ω that of the pipe :— [table]

Elbows. —Weisbach considers the loss of head at elbows (fig. 90) to be due to a contraction formed by the stream. From experiments with a pipe 1⅛ inches diameter, he found the loss of head

. . . (5) i

<,-0∙9457 sin*ξ + 2∙047 sm 4 ⅞ .

Δ Δ [table]

Hence at a right-angled elbow the whole head due to the velocity very nearly is lost.

Bends.— Weisbach traces the loss of head at curved bends to a similar cause to that at elbows, but the coefficients for bends are not veιy satisfactorily ascertained. Weisbach obtained for the loss of head at a bend in a pipe of circular section

⅛∙-< i ¾ < β > i

⅛-O∙131 + l∙847^) 5 , where d is the diameter of the pipe and p the radius of curvature of the bend. For bends with rectangular cross sections

0-0124 + 3104 (Λ)’, where s is the length of the side of the section parallel to the radius of curvature ρ. [table] [table]

Valves, Cocks, and Sluices.— These produce a contraction of the water-stream, similar to that for an abrupt diminution of section already discussed. The loss of head may be taken as before to be

>-≤ · · · · < 7 >≈

where v is the velocity in the pipe beyond the valve and a coefficient determined by experiment. The following are Weisbach’s results.

Sluice in Pipe of Rectangular Section (fig. 91). Section at sluice = u> 1 in pipe = ω. [table]

Sluice in Cylindrical Pipe (fig. 92). [table]

Cock τn a Cylindrical Pipe (fig. 93). Angle through which cock is turned = θ. [table] [table]

Throttle Valve in a Cylindrical Pipe (fig. 94). [table] [table]

78. Practical Calculations on the Flow of Water in Pipes.— In the following explanations it will be assumed that the pipe is of so great a length that only the loss of head in friction against the surface of the pipe needs to be considered. In general it is one of the four· quantities d, i, v, or Q which requires to be determined.

For since the loss of head h is given by the relation h=il, this need not be separately considered.

There are then three equations (see eq. 4, § 69, and 9α, § 73) for ’the-solution of such problems as arise :—

f-(l +⅛) < 1,ι where α=OO05 for new and =0'01 for incrusted pipes.

<⅜-f < 2 >∙

Q-iΛ (3)∙

4

Problem 1. Given the diameter of the pipe and its virtual slope, ' to find the discharge and velocity of flow. Here d and i are given, and Q and υ are required. Find £ from (1); then ν from (2); lastly Q from (3). This case presents no difficulty.

By combining equations (1 ) and (2), ν is obtained directly :— '√P√⅛√iτ≡ · · «>· _ 12d

For new pipes ∕ -f- = 56’72

∖j Za

For incrusted pipes =80’24

For pipes not less than 1, or more than 4 feet in diameter, the mean values of f are [table]

Using these values we get the very simple expressions—

v =55’31 for new pipes ) ,. >

= 39 'll^fdi for incrusted pipes )

Within the limits stated, these are accurate enough for practical purposes, especially as the precise value of the coefficient ζ cannot be known for each special case.

Problem 2. Given the diameter of a pipe and the velocity of flow, to find the virtual slope and discharge. The discharge is given by (3); the proper value of £ by (1); and the virtual slope by (2). This also presents no special difficulty.

Problem 3. Given the diameter of the pipe and the discharge, to find the virtual slope and velocity. Find v from (3); £ from (1); lastly i from (2). If we combine (1) and (2) we get i .⅛4-2√l + A i )⅞ .... (5); b 2g d ∖ 12d∕ gd and, taking the mean values of £ for pipes from 1 to 4 feet diameter, given above, the approximate formulae are

,,2 ∖

i = 0 ‘0003268 — for new pipes J

⅛ . j . ?· · · (M.

= 0Ό006536 —for incrusted pipes ∖

d J Problem 4. Given the virtual slope and the velocity, to fiud the diameter of the pipe and the discharge. The diameter is obtained from equations (2) and (1), which give the quadratic expression

<∕t 6gz

⅛√i⅞(iH)i · · m -

For practical purposes, the approximate equations

<J-⅛⅛ («“)

ÿi 12

« 2

= 0 ’ 00031 — + Ό83 for new pipes 2»

V 2

= 0’00062 — + Ό83 for incrusted pipes I are sufficiently accurate.

Problem 5. Given the virtual slope and the discharge, to find the diameter of the pipe and velocity of flow. This case, which often occurs in designing, is the one which is least easy of direct solution. From equations (2) and (3) we get—

(7).

If now the value of ζ in (1) is introduced, the equation becomes very cumbrous. Various approximate methods of meeting the difficulty may be used.

(a) Taking the mean values of < given above for pipes of 1 to 4 feet diameter we get d . v⅛ i«)

V gπ- V l·

= Q∙221Q^^-ioν new pipes

= 0 2541 for i∏crusted pipes; equations which are interesting as showing that when the value, of f is doubled the diameter of pipe for a given discharge is only increased by 13 per cent.

( b) A second method is to obtain a rough value of d by assuming ζ = a. This value is

< f -√^√- 0β819 v z 77∙∙

Then a very approximate value of £ is + 12d ’) ’

and a revised value of d, not sensibly differing from the exact value, is

(e) Equation 7 may be put in the form -√⅛÷⅛) v w Expanding the term in brackets, ∕ i ∖⅜ 1 1

(1 + 12rf∕” 1 + 60rf - 1800rf 2

Neglecting the terms after the second,

^^∕∣⅛^T + 0 016δ7 · · · < 9 ->!

5 ∕39 α and x∕ —g- = 0 - 219 for new pipes

= 0 252 for incrusted pipes.

79. Arrangement of a Pipe Network for Town's Supply.— Excluding the service pipes which directly supply the houses, the smallest branch water mains are made 3 to 4 inches in diameter. For the smallest districts supplied, these are sufficient or more than sufficient to convey the necessary supply, and in that case the only question arising is, to lay them out so that their total length is as small as possible.

Thus if there are two places of consumption A and B there is choice of any of the three arrangements shown in fig 95, the principal main being shown by the dark line, and the branch mains by the thinner ones. If, however, the supply through thè branch mains requires pipes of more than the minimum diameter, then the condition to be fulfilled is that the sum of the products of the lengths and diameters should be a minimum, because the cost of the mains when laid in place is very approximately proportional to their length and diameter.

For a main delivering water to branch mains on each side, the best position is that which makes the branch mains of equal diameter. Suppose water is to be supplied from an intermediate main along branch mains to a and b (fig. 96). Let Q α , Q⅛ be the quantities to be delivered at a and b, and i) a , I;» the virtual fall of the two branch mains. Then, according to Dupuit, the best position for the intersection M is that for which

⅛ ⅞.<⅛ a . h "l⅛Q. a ’

or, if the consumption of water occurs uniformly throughout the length of the branch mains, then

⅛= 71-

⅛ V t⅛

In this way various points MN may be determined giving the position AMN for the main, and afterwards the nearest convenient position ABC may be fixed.

In determining the consumption of water Q for any given locality, the mode of supply must be taken into account. On the intermittent system, water is supplied for a period of t seconds daily. Then the discharge per second is S2≤, where N is the number of t inhabitants, and Q j the daily supply to each in cubic feet. With a constant supply the rate of flow is variable at different periods of the day, and the maximum rate of flow may be taken at 21 times the mean rate. Hence in this case the discharge to be provided for in the mains is - 2 · The daily supply to a district NQ a Lτ X t)U X OU

is sometimes taken proportional to the area supplied, sometimes to the length of house frontage in the district.

Determination of the Diameters of Different Parts of a Water Main.

When the plan of the arrangement of mains is determined upon, and the supply to each locality and the pressure required is ascertained, it remains to determine the diameters of the pipes. Let fig. 97 show an elevation of a main ABCD . . ., R being the reservoir from which the supply is derived. Let NN be the datum line of the levelling operations, and H α , Ht . . . the heights of the main above the datum line, H r being the height of the water surface in the reservoir from the same datum. Set up next heights AA n BB 1 , . . . representing the minimum pressure height necessary for the adequate supply of each locality. Then A 1 B l Cj Dj . . . is a line which should form a lower limit to the line of virtual slope. Then if heights fia, U, l) e · · · are taken representing the actual losses of head in each length Z β , ⅞, l e . . . of the main, A 0 B 0 C 0 will be the line of virtual slope, and it will be obvious at what points such as D o and E o , the pressure is deficient, and a different choice of diameter of main is required. For any point z in the length of the main, we have

Pressure height = H r - H t - (lj α + ι>⅜ + fy t ).

Where no other circumstance limits the loss of head to be assigned to a given length of main, a consideration of the safety of the main from fracture by hydraulic shock leads to a limitation of the velocity of flow. Generally the velocity in water mains lies between 1⅛ and 4,⅛ feet per second. Occasionally the velocity in pipes reaches 10 feet per second, and in hydraulic machinery working under enormous pressures even 20 feet per second. Usually the velocity diminishes along the main as the discharge diminishes, so as to reduce somewhat the total loss of head which is liable to render the pressure insufficient at the end of the main.

Mr Fanning gives the following velocities as suitable in pipes for towns’ supply :— [table]

80. Branched Pipe connecting Reservoirs at Different Levels.—Let A, B, 0 (fig. 98) be three reservoirs connected by the arrangement of Pipes shown,-Z 1 , d y , Q l , t , 15 Z 2 , ⅛ n v 3 ∙ ι 31 d 3 , Q 3 , v 3 being the length, diameter, discharge, and velocity in the three portions of the main pipe. Suppose the dimensions and positions of the pipes known and the discharges required.

If a pressure column is introduced at X, the water will rise to a height XR, measuring the pressure at X, and αR, RZ>, Re will be the lines of virtual slope. If the free surface level at R is above b, the reservoir A supplies B and C, and if R is below b, A and B supply C. Consequently there are three cases :—

I. R above b ; Q 1 = Q 2 +Q 3 .

II. R level with b ; Q1 = Q 3 ; Q 2 =0.

III. R below b ; Q 1 + Q 2 = Q 3 ∙

To determine which case has to be dealt with in the given conditions, suppose the pipe from X to B closed by a sluice. Then there is a simple main, and the height of free surface h' at X can be determined. For this condition 7 V Z*L 3W¼.

Λβ ~ Λ ~*>2g · d 1 gιf t d 1 s , 7 , z λ⅛ 4 ⅞ 3¾CQ⅞. h ~ h '-^2g- d i gπ*dj ’ where Q' is the common discharge of the two portions of the pipe. Hence h a -h' ~l- l df hHTTΓ l^f ’ from which h' is easily obtained. If then h' is greater than hb, opening the sluice between X and B will allow flow towards B, and the case in hand is case I. If h' is less than 7⅛, opening the sluice will allow flow from B, and the case is case III. If h'=hb, the case is case II., and is already completely solved.

The true value of h must lie between h' and A⅛. Choose a new . value of h, and recalculate Q 1 , Q 2 , Q 3 . Then if

Q1=*Q a + Q3 in case I., or Q 1 + Q 2 =*Q 3 in case IΠ., the value chosen for A is too small, and a new value must be chosen If

Q1*=Q2 + Q3 in case I., or Q i + Q 2 <Q 3 in case HL, the value of h is too great.

Since the limits between which h can vary are in practical cases not very distant, it is easy to approximate to values sufficiently accurate.

IX. FLOW OF COMPRESSIBLE FLUIDS IN PIPES.

81. Flow of Air in Long Pipes. — When air flows through a long pipe, by far the greater part of the work expended is usιd in overcoming frictional resistances due to the surface of the pipe. The work expended in friction generates heat, which for the most part must be developed in and given back to the air. Some heat may be transmitted through the sides of the pipe to surrounding materials, but in experiments hitherto made the amount so conducted away appears to be very small, and if no heat is transmitted the air in the tube must remain sensibly at the same temperature during expansion. In other words, the expansion may be regarded as isothermal expansion, the heat generated by friction exactly neutralizing the cooling due to the work done. Experiments on the pneumatic tubes used for the transmission of messages, by Messrs Culley & Sabine, show that the change of temperature of the air flowing along the tube is much less than it would be in adiabatic expansion.

82. Differential Equation of the Steady Motion of Air Flowing in a Long Pipe of Uniform Section.— When air expands at a constant absolute temperature τ, the relation between the pressure p in pounds per square foot and the density or weight per cubic foot G is given by the equation

⅛-" < 1 >∙

where c = 53 - 15. Takingτ = 521, corresponding to a temperature of 60° Fahr., cτ=27690 foot-pounds (2).

The equation of continuity, which expresses the condition that in steady motion the same weight of fluid, W, must pass through each cross section of the stream in the unit of time, is

G∩u=W = constant (3), where ∩ is the section of the pipe and u the velocity of the air. Combining (1) and (3), =cτ = constant .... (3α).

Since the work done by gravity on the air during its flow through a pipe due to variations of its level is generally small compared with the work done by changes of pressure, the former may in many cases be neglected.

Consider a short length dl of the pipe limited by sections A o , A 1 at a distance dl (fig. 99). Let p, u be the pressure and velocity at A c , p + dp and u + du those at A 1 . Further, suppose that in a very short time dt the mass of air between A 0 A 1 comes to A0A( so that A 0 Aθ=wcZi and A 1 A{ = (w + √u)di. Let ∩ be the section, and m the hydraulic mean radius of the pipe, and W the weight of air flowing through the pipe per second.

From the steadiness of the motion the weight of air between the sections A o A0, and A 1 A{ is the same. That is, ~Wdt = GΩ,udt = G∩(u + du)dt .

By analogy with liquids the head lost in friction is, for the length dl (see § 69, eq. 3), dl *2g‘ m'

LetH=- .∙ Then the 2<7

head lost is ζ*dl-, m and, since Wi⅛ pounds of air flow through the pipe in the time considered, the work expended in friction is

_ CH-WrfZ dt. ^m

The change of kinetic energy in dt seconds is the difference of the kinetic energy of A 0 Aθ and A 1 A{, that is,

W dfu + du)* - u i 9 2

W

= —ududt = W <ZH dt . 9

The work of expansion when Ωudt cubic feet of air at a pressure ρ expand to Ω(u + du)dt cubic feet is ∩ pdudt.

But from (3α)

cτW w = -χ- > np du cτ^W dp^ Ωp i

And the work done by expansion is - c ff^dpdt. P

The work done by gravity on the mass between A o and A 1 is zero if the pipe is horizontal, and may in other cases be neglected without great error. The work of the pressures at the sections A 0 A 1 iy pΩudt- (p + dp)Ω(u + du)dt = - {pdu + udp)Ωdt.

But from (3α)

pu = constant, pdu + udp = 0, and the work of the pressures is zero. Adding together the quantities of work, and equating them to the change of kinetic energy, W<ffldi= --y dpdt-ζJ^h τ dldt cτ H

dH + y⅛> + ζ- rfZ=0, f÷⅞⅛÷<- 0 *

cτW βut m = Ι⅛ ’

. ττ h 2 c 2 t 2 W 2 and II = — = n rt ., - 2 , 2g 2gΩ 2 p 2

.∙. gg + 2 9*⅝ + ( fL =Q . . i , (4α> ’ H + cτW s v + i rn For tubes of uniform section m is constant; for steady motion W is constant; and for isothermal expansion τ is constant. Integrating, a j lo β H + + ⅛≡ constant . · · · i 5 ); for l -0, let H = H o , and p =p 0 ; and for l=I, let H = H 1 , and ρ =ρ 1 .

< 6α >'

where p 0 is the greater pressure and ρ 1 the less, and the flow is from A o towards A 1 .

By replacing W and H, 1 ¾Ιr + ⅛⅛ a - y ∙ ,i+ ⅛^ 0 · · · < δ >∙

Pi u o Po '' 4

Hence the initial velocity in the pipe is β , l ∖ ff cτ W-Pι v ) I ... (7).

’ i ° s λ)

When l is great, log — is comparatively small, and then Pi v√⅛⅛ i l · · · · σβ, '

a very simple and easily used expression. For pipes of circular section m=^j-, where d is the diameter:—

4

∕ i 9 rcτi ^ Po i ~Pι I (7⅛∖.

√⅛ -tf~∖ · * · · (7ά) ’ or approximately u o -(l∙1319-O∙7264∣θ χ ∕^ . · · (7c).

83. Coefficient of Friction for Air.— A discussion by Professor Unwin of the experiments by Messrs Culley & Sabine on the rate of transmission of light carriers through pneumatic tubes, in which there is steady flow of air not sensibly affected by any resistances other than surface friction, furnished the value ζ= ’ 007. The pipes were lead pipes, slightly moist, 2} inches (0 ’ 187 ft.) in diameter, and in lengths of 2000 to nearly 6000 feet.

Some experiments on the flow of air through cast-iron pipes have been made by Μ. Arson. He found the coefficient of friction to vary with the velocity and diameter of the pipe. Putting f=y + 3 (8), he obtained the following values— [table]

It is worth while to try if these numbers can be expressed in the form proposed by Darcy for water. For a velocity of 100 feet per second, and without much error for higher velocities, these numbers agree fairly with the formula f- 006 (1 + ⅛) '»>·

which only differs from Darcy’s value for water in that the second term, which is always small except for very small pipes, is larger.

Some more recent experiments on a very large scale, by Μ. Stockalper at the St Gotthard Tunnel, agree better with the value <-°∙° 028 (1+ ⅛)∙

These pipes were probably less rough than Μ. Arson’s.

When the variation of pressure is very small, it is no longer safe to neglect the variation of level of the pipe. For that case we may neglect the work done by expansion, and then

⅞-¾-⅞ l -∙⅛--C≠-—-o . . . . (io), G o Gι 2g m ' n precisely equivalent to the equation for the flow of water, z 0 and z 1 being the elevations of the two ends of the pipe above any datum, p 0 and p 1 the pressures, G o and G, the densities, and v the mean velocity in the pipe. This equation may be used for the flow of coal gas.

84. Distribution of Pressure in a Pipe in which Air is Flowing.— From equation (7α) it results that the pressure p, at l feet from that end of the pipe where the pressure is p 0 , is j, = λ √S i -⅛⅛I < ιl >> which is of the form p-^fal+b for any given pipe with given end pressures. The curve of free surface level for the pipe is, therefore, a parabola with horizontal axis. Fig. 100 shows calculated curves of pressure for two of Mr Sabine’s experiments, in one of which the pressure was greater than atmo∙ spheric pressure, and in the other less than atmospheric pressure. The observed pressures are given in brackets and the calculated pressures without brackets. The pipe was the pneumatic tube between Fenchurch Street and the Central Station, 2818 yards in length. The pressures are given in inches of mercury.

Variation of Velocity in the Pipe.— Let p 0 , u 0 be the pressure and velocity at a given section of the pipe; p, u, the pressure and velocity at any other section. From equation (3α)

u p = constant;

12

so that, for any given uniform pipe, up=u 0 p 0 ,

(12); which gives the velocity at any section in terms of the pressure, which has already been determined. Fig. 101 gives the velocity curves for the two experiments of Messrs Culley & Sabine, for which the pressure curves have already been drawn. It will be seen that the velocity increases considerably towards that end of the pipe where the pressure is least.

85. Weight of Air Flowing per Second.— The weight of air discharged per second is (equation 3α)—

w ‰ 0 y 0 cτ

From equation (7δ), for a pipe of circular section and diameter d,

=∙'6H χ∕⅛(p ° a ^- pι, )Γ * * · (18) · Approximately ∕ d i ∖i .

W—('6916p 0 - ’ 4438p 1 )^^y · · · · (13α).

86. Application to the Case of Pneumatic Tubes for the Transmission of Messages.— In Paris, Berlin, London, and other towns, it has been found cheaper to transmit messages in pneumatic tubes than to telegraph by electricity. The tubes are laid underground with easy curves; the messages are made into a roll and placed in a light felt carrier, the resistance of which in the tubes in London is only ⅞ oz. A current of air forced into the tube or drawn through it propels the carrier. In most systems the current of air is steady and continuous, and the carriers are introduced or removed without materially altering the flow of air.

Time of Transit through the Tube.— Putting t for the time of transit from 0 to l, t -fi. Λ u

From <±a} neglecting and putting τn = A , gdtiPp dl ~2ζW⅛r d l ’

From (1) and (3)

Wcτ u ~ pn ’ ffd(iγ u '~2ζWW aρ ’ Γ ρ ∖ gdtfjl· Jρχ gdCP ( r >^-υ v ∖ (14).

- 6^W 3 c 2 τ 2 (Po Pl) 1 '

But; ∙ , ∙ '^ ∣ w⅛ w ^ as, ∙

= ± -i!lL Po 3 -∙P1 3 .... (15).

6 (gcτd)i (p 0 ⅛ 2 ) i

If τ = 521°, corresponding to 60° F.,

<=∙001412⅛* -Po 3 ~Pι 3 .... (15a); d i (Po 2 -Pι 8 ) i which gives the time of transmission in terms of the initial and final pressures and the dimensions of the tube.

Mean Velocity of Transmission.— The mean velocity is J-; or, for τ = 521°,

« -0708 . ∕! ⅛ w ,~ y * 1 ⅜- I · · · < 16 >'

mean ∖ ( ζl p 0 3 -7⅛ 3 )

The following table gives some results:— [table]

Limiting Velocity in the Pipe when the Pressure at one End is diminished indefinitely. — If in the last equation there be put p 1 = 0, then u ’ =O∙7O8a∕⅛5 mean z V ζl where the velocity is independent of the pressure p 0 at the other end, a result which apparently must be absurd. Probably for long pipes, as for orifices, there is a limit to the ratio of the initial and terminal pressures for which the formula is applicable.

X. FLOW IN RIVERS AND CANALS.

87. Flow of Water in Open Canals and Rivers.— When water flows in a pipe the section at any point is determined by the form of ths boundary. When it flows in an open channel with free upper surface, the section depends on the velocity due to the dynamical conditions.

Suppose water admitted to an unfilled canal. The channel will gradually fill, the section and velocity at each point gradually changing. But if the inflow to the canal at its head is constant, the increase of cross section and diminution of velocity at each point attain after a time a limit. Thenceforward the section and velocity at each point are constant, and the motion is steady, or permanent regime is established.

If when the motion is steady the sections of the stream are all equal, the motion is uniform. By hypothesis, the inflow Ωv is constant for all sections, and ∩ is constant; therefore v must be constant also from section to section. The case is then one of uniform steady motion. In most artificial channels the form of section is constant, and the bed has a uniform slope. In that case the motion is uniform, the depth is constant, and the stream surface is parallel to the bed. If when steady motion is established the sections are unequal, the motion is steady motion with varying velocity from section to section. Ordinary rivers are in this condition, especially where the flow is modified by weirs or obstructions. Short unobstructed lengths of a river may be treated as of uniform section without great error, the mean section in the length being put for the actual sections.

In all actual streams the different fluid filaments have different velocities, those near the surface and centre moving faster than those near the bottom and sides. The ordinary formulae for the flow of streams rest on an hypothesis that this variation of velocity may be neglected, and that all the filaments may be treated as having a common velocity equal to the mean velocity of the stream. On this hypothesis, a plane layer abab (fig. 102) between sections normal to the direction of motion is treated as sliding down the channel to a'a'b ’ b' without deformation. The component of the weight parallel to the channel bed balances the friction against the channel, and in estimating the friction the velocity of rubbing is taken to be the mean velocity of the stream. In actual streams, however, the velocity of rubbing on which the friction depends is not the mean velocity of the stream, and is not in any simple relation with it, for channels of different forms. The theory is therefore obviously based on an imperfect hypothesis. However, by taking variable values for the coefficient of friction, the errors of the ordinary formulae are to a great extent neutralized, and they may be used without leading to practical errors. Formulae have been obtained based on less restricted hypotheses, but at present they are not practically so reliable, and are more complicated than the formulae obtained in the manner described above.

88. Steady Flow of Water with Uniform Velocity in Channels of Constant Section.— Let aa', bb' (fig. 103) betwo cross sections normal to the direction of motion at a distance dl. Since the mass aa'bb' moves uniformly, the external forces acting on it are in equilibrium. Let ∩ be the area of the cross sections, χ the wetted perimeter, +qr + rs, of a section. Then the quantity m =—is termed the hydraulic mean depth of the section. Let v be the mean velocity of the stream, which is taken as the common velocity of all the particles, i, the slope or fall of the stream in feet, per foot, being the ratio -^-.

The external forces acting on aa'bb' parallel to the direction of motion are three :—(a) The pressures on aa' and bb', which are equal and opposite since the sections are equal and similar, and the mean pressures on each are the same, {b) The component of the weight W of the mass in the direction of motion, acting at its centre of gravity g. The weight of the mass aa'bb' is G∩⅛ and the component of the weight in the direction of motion is GΩdl × the cosine of the angle between W# and ab, that is, GΩdl cos abc = Gcidl^ ι = Gιlidl. ab

(c) There is the friction of the stream on the sides and bottom of the channel. This is proportional to the area χdl of rubbing surface and to a function of the velocity which may be written ftp); ftp) being the friction per square foot at a velocity v. Hence the friction is -χdlftυ). Equating the sum of the forces to zero, GQidl-χdlfv)≈0, f{υ) Ω . . ~ 3 ∕ n

7≈r = -ï = (1)∙

G *

But it has been already shown (§ 63) that - ∕(v) = ^G- ,

·· <- 1 " i , · »

This may he put in the form v= t ,J^γ ∖ Zmi~c λ fmι (2a); where c is a coefficient depending ∙on the roughness and form of the shannel.

The coefficient of friction ζ varies greatly with the degree of roughness of the channel sides, and somewhat also with the velocity. It must also be made to depend on the absolute dimensions of the section, to eliminate the error of neglecting the variations of velocity in the cross section. A common mean value assumed for £ is 0’00757. The range of values will be discussed presently.

It is often convenient to estimate the fall of the stream in feet per mile, instead of in feet per foot. If f is the fall in feet per mile

∕=5280 i.

Putting this and the above value of £ in (2α), we get the very simple and long-known approximate formula for the mean velocity of a stream—

v -H√2wi∕ ’ (3).

The flow down the stream per second, or discharge of the stream, is

Q = ih> = Ωcv∞i (4).

89. Coefficient of Friction for Open Channels.— Various expressions have been proposed for the coefficient of friction for channels as for pipes. Weisbach, giving attention chiefly to the variation of the coefficient of friction with the velocity, proposed an expression of the form

H ι+ 4) <≈⅛

and from 255 experiments obtained for the constants the values α=0'007409; 3 = 0'1920.

This gives the following values at different velocities :— [table]

In using this value of ζ when v is not known, it is best to proceed by approximation. Calculate a rough value of v by taking any mean value of ζ for instance the one given in the preceding section. Then from this value of v calculate a revised value of and from this a new value of v.

90. Darcy <k Bazin’s Expression for the Coefficient of Friction.— Darcy & Bazin’s researches have shown that ζ varies very greatly for different degrees of roughness of the channel bed, and that it also varies with the dimensions of the channel. They give for f an empirical expression (similar to that for pipes) of the form

H ι + 4) w -

where m is the hydraulic mean depth. For different kinds of channels they give the following values of the coefficient of friction :— [table]

The last values (Class V. ) are not Darcy & Bazin’s, but are taken from experiments by Ganguillet & Kutter on Swiss streams.

The following table veιy much facilitates the calculation of the mean velocity and discharge of channels, when Darcy & Bazin’s value of the coefficient of friction is used. Taking the general formula for the mean velocity already given in equation (2α) above, v≈cVτni, where c= ∖∕γ- » the following table gives values of c for channels of different degrees of roughness, an<l for such values of the hydraulic mean depths as are likely to occur in practical calculations:— Values of c in v=cVmi, deduced, from Darcy <k Bazin’s Values. [table]

91. Ganguillet <fc Kutter's modified Darcy Formula.— Starting from the general expression v = Vf mi, Messrs Ganguillet & Kutter have examined the variations of c for a wider variety of cases than those discussed by Darcy & Bazin. Darcy & Bazin’s experiments are confined to channels of moderate section, and to a limited variation of slope. Ganguillet & Kutter brought into the discussion two very distinct and important additional series of results. The gaugings of the Mississippi by Messrs Humphreys & Abbot afford data of discharge for the case of a stream of exceptionally large section and of very low slope. On the other-hand, their own measurements of the flow in the regulated channels of some Swiss torrents gave data for cases in which the inclination and roughness of the channels were exceptionally great. Darcy & Bazin’s experiments alone were conclusive as to the dependence of the coefficient c on the dimensions of the channel and on its roughness of surface. Plotting values of c for channels of different inclination indicated to Ganguillet & Kutter that it also depended on the slope of the stream. Taking the Mississippi data only, they found c = 256 for an inclination of 0 ’ 0034 per thousand, = 154 ,, „ 002

so that for very low inclinations no constant value of c independent of the slope would furnish good values of the discharge, In small rivers, on the other hand, the values of c vary little with the slope. As regards the influence of roughness of the sides of the channel a different law holds. For very small channels differences of roughness have a great influence on the discharge, but for very large channels different degrees of roughness have but little influence, and for indefinitely large channels the influence of different degrees of roughness must be assumed to vanish. The coefficients given by Darcy & Bazin are different for each of the classes of channels of different roughness, even when the dimensions of the channel are infinite. But, .as it is much more probable that the influence of the nature of the sides diminishes indefinitely as the channel is larger, this must be regarded as a defect in their formula.

Comparing their own measurements in torrential streams in Switzerland with those of Darcy & Bazin, Ganguillet & Kutter found that the four classes of coefficients proposed by Darcy & Bazin were insufficient to cover all cases. Some of the Swiss streams gave results which showed that the roughness of the bed was markedly greater than in any of the channels tried by the French engineers. It was necessary therefore in adopting the plan of arranging the different channels in classes of approximately similar roughness to increase the number of classes. Especially an additional class was required for channels obstructed by detritus.

To obtain a new expression for the coefficient in the formula v = ∖ J ^ ∖ Γrni—c ∖ fini, in which Darcy & Bazin take t '⅜⅛

Ganguillet & Kutter proceeded in a purely empirical way. They found that an expression of the form a e 'UJ Vwι

could be made to fit the experiments somewhat better than Darcy’s expression. Inverting this, we get i u i l β^ ι c « a fm an equation to a straight line having —for abscissa, À for ordinate, and inclined to the axis of ahscissae at an angle the tangent of which is — .

α

Plotting the experimental values of -ι and λ y=> t hθ points so found indicated a curved rather than a straight line, so that β must depend on a. After much comparison the following form was arrived at—

A + ±

n c b+-4≡= ’ where n is a coefficient depending only on the roughness of the sides of the channel, and A and l are new coefficients, the value of which remains to be determined. From what has been already stated, the coefficient c depends on the inclination of the stream, decreasing as the slope i increases.

Let A = α + ⅜

⅛

l p a -I 1——

Then c 7 - ,

1 + β + ∣-5=

∖ l Nm the form of the expression for c ultimately adopted by Ganguillet & Kutter.

For the constants a, l, p Ganguillet & Kutter obtain the values 23, 1, and 0Ό0155 for metrical measures, or 41 - 6, 1’811, and 0'00281 for English feet. The coefficient of roughness n is found to vary from 0 Ό08 to 0 Ό50 for either metrical or English measures.

The most practically useful values of the coefficient of roughness n are given in the following table [table]

Ganguillet & Kutter’s formula is so cumbrous that it is difficult to use without the aid of tables.

Mr Lowis D’A. Jackson has published complete and extensive tables for facilitating the use of the Ganguillet & Kutter formula (Canal and Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form xr Λi∙a θ’ 00281 ∖ · M = n( 416 -1 -. I;

√wΓ∕M + l∙811∖ r -.

r ~ n +

The following table gives a selection of values of M, taken from Mr Jackson’s tables :— [table]

One principal difficulty in the use of this formula is the selection of the coefficient of roughness. The difficulty is one which no theory will overcome, because no absolute measure of the roughness of stream beds is possible. For channels lined with timber or masonry the difficulty is not so great. The constants in that case are few and sufficiently defined. But in the case of ordinary canals and rivers the case is different, the coefficients having a much greater range. For artificial canals in rammed earth or gravel n varies from 0’0163 to 0 - 0301. For natural channels or rivers n varies from 0 Ό20 to 0 035.

In Mr Jackson’s opinion even Kutter’s numerous classes of channels seem inadequately graduated, and after careful examination he pronoses for artificial canals the following classification :— [table]

92. Forms of Section of Channels.— The simplest form of section for channels is the semicircular or nearly semicircular channel (fig. 104), a form now often adopted from the facility with which it can be executed in concrete. It has the advantage that the nibbing surface is less in proportion to the area than in any other form.

Wooden channels or flumes, of which there are examples on a large scale in America, are rectangular in section, and the same form is adopted for wrought and cast-iron aqueducts. Channels built with brickwork or masonry may be also rectangular, but they are often trapezoidal, and are always so if the sides are pitched with masonry laid dry. In a trapezoidal channel, let b (fig. 105) be the bottom breadth, b 0 the top breadth, d the depth, and let the slope of the sides be n horizontal to 1 vertical. Then the area of section is Ω = (b + nd)d = (b 0 - nd)d, and the wetted perimeter χ = δ + 2dVπ s +l.

When a channel is simply excavated in earth it is always originally trapezoidal, though it becomes more or less rounded in course of time. The slope of the sides then depends on the stability of the earth, a slope of 2 to 1 being the one most commonly adopted.

Figs. 106, 107 show the form of canals excavated in earth, the former being the section of a Navigation Canal and the latter the section of an Irrigation Canal.

93. Channels of Circular Section.— The following short table facilitates calculations of the discharge with different depths of water in the channel. Let r be the radius of the channel section; then for a depth of water ≈κr, the hydraulic mean radius is μr and the area of section of the waterway vr 2 , where κ, μ, and v have the following values: — [table]

94. Egg-Shaped Channels or Sewers.— In sewers for discharging storm water and house drainage the volume of flow is extremely variable; and there is a great liability for deposits to be left when the flow is small, which are not removed during the short periods when the flow is large. The sewer in consequence becomes choked. To obtain uniform scouring action, the velocity of flow should be constant or nearly so; a complete uniformity of velocity cannot be obtained with any form of section suitable for sewers, but an approximation to uniform velocity is obtained by making the sewers of oval section. Various forms of oval have been suggested, the simplest being one in which the radius of the crown is double the radius of the invert, and the greatest width is two-thirds the height. The section, of such a sewer is shown in fig. 108, the numbers marked on the figure being proportional numbers.

95. Problems on Channels in which the Flow is Steady and at Uniform Velocity.— The general equations given in §§ 88, 90 are f-(1+ ⅛) < 1 >≡

. V 2

......... (2);

Q = Ωv (3).

Problem I. Given the transverse section of stream and discharge, to find the slope. From the dimensions of the section find ∩ and m ; from (1) find f, from (3) find v, and lastly from (2) find t.

Problem II. Given the transverse section and slope, to find the discharge. Find v from (2), then Q from (3).

Problem III. Given the discharge and slope,and either the breadth, depth, or general form of the section of the channel, to determine its remaining dimensions. This must generally be solved by approximations. A breadth or depth or both are chosen, and the discharge calculated. If this is greater than the given discharge, the dimensions are reduced and the discharge recalculated.

Since m lies generally between the limits m = d and m = ⅛d , where d is the depth_ of the stream, and since, moreover, the velocity varies as VnΓ so that an error in the ∖ F ∕ value of m leads ’''f ^"jWi ,.-y"√., ΓfD only to a much less! V ∕; ∖ if error in the value </, ∖ ∕: ∖ ∖ f of the velocity cal-: J·V ∕ ∖ '∕H culated from it, we: V. ∕ ∖ ,∕

may proceed thus.! V^ ∙√': X∙ ∕

Assume a value for —4 V ^' ^: --"^ ∖ ∕

rn, and calculate v D G C from it. Let v 1 be this first approxi- *⅛>∙ *θ"∙

nation to u Then is a first approximation to ∩, say ∩ 1 . With this vp,lue of ∩ design the section of the channel; calculate a second value for m ; calculate from it a second value of v, and from that a second value for ∩. Repeat the process till the successive values of tn approximately coincide.

96. Problem IV. Most Economical Form of Channel forgiven Side Slopes. —Suppose the channel is to be trapezoidal in section (fig. 109), and that the sides are to have a given slope. Let the longitudinal slope of the stream be given, and also the mean velocity. An infinite number of channels could be found satisfying the foregoing conditions. To render the problem determinate, let it be remembered that, since for a given discharge ∩oc ζfχ, other things being the same, the amount of excavation will be least for that channel which has the least wetted perimeter. Let d be the depth and b the bottom width of the channel, and let the sides slope n horizontal to 1 vertical (fig. 110), then Ω=(b+nd y ) d; χ = b + 2d ∖ fn 2 +1 .

Both Ω and χ are to be minima. ·. Differentiating, and equating to zero, ( ~r ∖ + n )d + b + nd =0, ∖ dd J

+ 2√n 2 Π = 0; da ’ eliminating ,

(n-2√π a + l)cf + δ + τi6f=0, b = 2(fJn 2 ->r' ∖ . - ri)d .

But

Ω _ (b + ndfl X - ⅛ + 2d√n 2 + l ’ Inserting the value of b,

Ω 2<Z^√π 2 +1 - nd d m= p= = — ·

X 4d√π 2 + l- 2nd 2

That is, with given slide slopes, the section is least fbr a given discharge when the hydraulic mean depth is half the actual depth.

A simple construction gives the form of the channel which fulfils this condition, for it can be shown that when m = the sides of the channel are tangential to a semi-circle drawn on the water line.

Since --1, X 2 therefore

. Ω = iχd (1).

Let ABCD be the channel (fig. 109); from E the centre of AD drop perpendiculars EF, EG, EH on the sides.

Let

AB = CD = a; BC = Z>; EF = EH = c;and EG=d. Ω = area AEB + BEC + CED, = ac + ⅛ bd. χ = 2a + δ.

Putting these values in (1), ac + ⅛bd≈(a+⅛b)d; and hence c— d.

That is, EF, EG, EH are all equal, hence a semicircle struck from E with radius equal to the depth of the stream will pass through F and H and be tangential to the sides of the channel.

To draw the channel, describe a semicircle on a horizontal line with radius = depth »f channel. The hot-tom will be a horizontal tangent of that semicircle, and the sices tangents drawn at the required side slopes.

The above result may be obtained thus (fig. Ill) :—

T , 2iZ ∕η ∖

X = b+- a w∙

λ sin β

Ω = <Z(δ + cZ cot β) ;

37- = i> + α! cot β (2); CL

⅜-4+ cot * (8) ·

From (1) and (2), χ÷<-+⅛∙

This will be a minimum for ⅛-⅞ + “‘»-KB-Ö· or -⅛ = 2 cosec. β - cot β (4).

d i

√Ω sin β

2-cos3'

From (3) and (4), t ¾l--∞3⅜)- 2toni3 d sin β Proportions of Channels of Maximum Discharge for given Area and Side Slopes. Depth of channel ≈ d; Hydraulic mean depth = ⅛d; Area of section =Ω . [table]

Half the top width is the length of each side slope. The wetted penmeter is the sum of the top and bottom widths.

97. Fbrm of Cross Section of Channel in which the r Mean Velocity is Constant with Varying Discharge.

In designing waste channels from canals, and in some other cases, it is desirable that the mean velocity should be restricted within narrow limits with very different volumes of discharge. In channels of trapezoidal form the velocity increases and diminishes with the discharge. Hence when the discharge is large there is danger of erosion, and when it is small of silting or obstruction by weeds. A theoretical form of section for which the mean velocity would be constant can be found, and, although this is not very suitable for practical purposes, it can be more or less approximated to in actual channels.

Let fig. 112 represent the cross section of the channel. From the symmetry of the section, only half the channel need be considered.

Let oboe be any section suitable for the minimum flow, and let it be required to find the curve beg for the upper part of the channel so that the mean velocity shall be constant. Take o as origin of coordinates, and let de, fg be two levels of the water above ob.

Letoδ=-g-j de=y,fg=y + dy, od≈x, of=x + dx; eg=ds.

The condition to be satisfied is that υ=c*fmi should be constant, whether the water level is at ob, de, or fg. Consequently m— constant =k for all three sections, and can be found from the section oboe. Hence also

Increment of section ydx Increment of perimeter ds y 2 dx 2 =k 2 ds 2 =k 2 (dx 2 + dy 2 ) ; and dx=.

Vy 2 - k 2 Integrating, X=k log e {y + √t∕ 2 -⅛ 2 ) + constant; and, since y≈⅛- when x≈0,

χ-⅞i⅛ i>+⅝⅞Eg ι.

∕ l> ∕ι> i

2 + V 4 k ’

Assuming values for y, the values of x can be found and the curve drawn.

The figure has been drawn for a channel the minimum section of which is a half hexagon of 4 feet depth. Hence ⅛ = 2; b= 9 , 2; the rapid flattening of the side slopes is remarkable.

98. Variation of Velocity in Different Parts of the Cross Section of a Uniform Stream. Vertical Velocity Curve in a Stream.— If it is assumed that the resistance to the relative sliding of the layers of water in a stream is of the nature of a viscous resistance, then the law of the distribution of velocity in a vertical longitudinal section of the stream can be determined theoretically. For simplicity, suppose the stream of uniform depth and indefinite width. Let fig. 113 show a portion of a vertical longitudinal section of the stream, and let OA, O'A' be the intersections with this of two transverse sections at a distance apart l. Let ab, cd be the traces of two planes parallel to the free surface or to the bed, and let us consider the equilibrium of a layer abed of width unity. Letθα=y, ac≈dy, and let v be the velocity of the particles comprised in abed, v being a function of y which is to be determined. Taking the components of the forces acting on abed, parallel to 00, the pressures on ac, bd, being proportional to the depth from the free surface, are equal and opposite; also, the frictions or viscous resistances on the lateral faces of the prism are zero, since in a wide stream there is no relative sliding between abed and the layers on each side. There remain only the resistances on the upper and lower surface, and the component of the weight.

The weight of the layer is GZ dy, and if i is the slope of the stream the component of the weight parallel to 00 is Gli dy. The friction or viscous resistance on the face ab is proportional to its area and to the differential coefficient — (§ 3). The resistance is, therefore, dy -kl t ^-, the negative sign being used because, if v increases with y, dy

∣L is positive, while the action of the layers above ab is a retarding dD d*v action. The resistance on the face cd is similarly kl — + kld .

The resultant of the action of the layers above and below is, therefore, kld⅛f . dy

When the motion is uniform, Glidy + kld d T- = t); d^v Gi .

dy 1 k ’ integrating,

*--¾+c i dy k v= -⅜¾ 2 + c ZZ + v o . . (1); Δ κ> an equation which gives the velocity v at any depth y . If on a vertical line OA (fig. 114), representing the depth of the stream, the values of v are set off horizontally, a parabolic curve is obtained, termed the vertical velocity curve for the section considered. The constant ν ϋ is evidently the surface velocity, being the value of v for y=0. The parabola has a horizontal axis corresponding to the position of the filament of maximum velocity. If there is no resistance at the surface of the stream like that at the bottom and sides, the maximum velocity should be at the surface, and then c=0, and the equation becomes

*-' ’ ^-⅜⅞≈' ’ < 2 >∙

Assuming this for the present, the mean velocity is

∕^ hvdy 1 G⅛

*-=∙ z -=*∙4⅜ a1 · · · < 8 >∙

The bottom velocity is, putting the depth h for y in (2); 1 ‰

^k h; and therefore

Γm=⅛(2⅞ + n) (4).

It is now understood that the motion in a stream is much more complete than the viscous theory just stated assumes. The retardation of the stream is much greater than it would be in simple motion of that kind. This has already been partly explained in the introduction to the present article. Nevertheless the viscous theory may probably be so modified as to furnish ultimately a true theory of streams.

99. Experimental Observations on the Vertical Velocity Curve.— In obtaining the vertical velocity curve from direct observations in streams, a preliminary difficulty arises from the fact that the velocity at any given depth is not constant, and hence the motion in the strict sense is not steady. The velocities taken on a given vertical section at any given moment do not form when plotted any regular curve. But if a series of observations are taken at each depth and the results averaged, the mean velocities at each depth when plotted give a regular curve, agreeing very fairly with the parabola, which might be expected from the theory above. Hence it may be inferred that the velocity at any given point fluctuates about a mean value, the fluctuations being due to irregular eddying motions superposed on the general steady motion of the stream, and having an effect which disappears in the mean of a series of observations. For certain purposes these irregular motions may be ignored, and the constant mean velocity substituted for the actual varying velocities at each point. In the next place, all the best observations show that the maximum velocity is to be found, not at the free surface of the stream, but at some distance below it. Influence of the Wind.— In the experiments on the Mississippi the vertical velocity curve in calm weather was found to agree fairly with a parabola, the greatest velocity being at 1 ⅞ths of the depth of the stream from the surface. With a wind blowing down stream the surface velocity is increased, and the axis of the parabola approaches the surface. On the contrary, with a wind blowing up stream the surface velocity is diminished, and the axis of the parabola is lowered, sometimes to half the depth of the stream. The American observers drew from their observations the conclusion that there was an energetic retarding action at the surface of a stream like that due to the bottom and sides. If there were such a retarding action the position of the filament of maximum velocity below the surface would be explained.

It is not difficult to understand that a wind acting on surface ripples should accelerate or retard the surface motion of the stream, and the Mississippi results may be accepted so far as showing that the surface velocity of a stream is variable when the mean velocity of the stream is constant. Hence observations of surface velocity by floats or otherwise should only be made in very calm weather. But it is very difficult to suppose that, in still air, there is a resistance at the free surface of the stream at all analogous to that at the sides and bottom. Further, in very careful experiments, Boileau found the maximum velocity, though raised a little above its position for calm weather, still at a considerable distance below the surface, even when the wind was blowing down stream with a velocity greater than that of the stream, and when the action of the air must have been an accelerating and not a retarding action. Professor James Thomson has given a much more probable explanation of the diminution of the velocity at and near the free surface. He points out that portions of water, with a diminished velocity from retardation by the sides or bottom, are thrown off in eddying masses and mingle with the rest of the stream. These eddying masses modify the velocity in all parts of the stream, but have their greatest influence at the free surface. Reaching the free surface they spread out and remain there, mingling with the water at that level and diminishing the velocity which would otherwise be found there.

100. Influence of the Wind on the Depth at which the Maximum Velocity is found.— In the gaugings of the Mississippi the vertical velocity curve was found to agree well with a parabola having a horizontal axis at some distance below the water surface, the ordinate of the parabola at the axis being the maximum velocity of the section. During the gaugings the force of the wind was registered on a scale ranging from 0 for a calm to 10 for a hurricane. Arranging the velocity curves in three sets—(1) with the wind blowing up stream, (2) with the wind blowing down stream, (3) calm or wind blowing across stream—it was found that an up stream wind lowered, and a down stream wind raised, the axis of the parabolic velocity curve. In calm weather the axis was at √⅛ths of the total depth from the surface for all conditions of the stream.

Let h' be the depth of the axis of the parabola, m the hydraulic mean depth,/the number expressing the force of the wind, which may range from +10 to—10, positive if the wind is up stream, negative if it is down stream. Then Messrs Humphreys and Abbot find their results agree with the expression

— = 0∙317±OO6∕. τn

Fig. 115 shows the parabolic velocity curves according to the American observers for calm weather, and for an up or down stream wind of a force represented by 4.

101. Bazin's Formules for the Variation of Velocity in a Vertica< Longitudinal Section of a Stream. — Μ. Bazin assumes that the vertical velocity curve is a parabola, and has investigated numerica’ values for the constants from his own and other experiments. Assuming the general equation already found, § 98,

1 Cb Q ?, ∖

v=ν 0 +cy--^-^y (1), v will have the maximum value V, for a value of h' of y which makes zero. That is, dy ,, ck Gi j , . h =-τrr , or c=- f -^;

Gb ’ k and the maximum velocity is v -'.+⅜T*' ’ i

·· <¾-V-l≤⅛*.

Le Ke

Inserting these values of r 0 and c in (1),

—V-g⅛-Λ')∙ i or putting , where h is the whole depth of the section, 0-V-m(1⅛*')* <2>.

where M is constant for any given stream. Let lh = χ and ~ = a, v=V-M(z-α) 2 (2a).

Then the mean velocity on the vertical is r m = ∕^ 71 -JV-M(z-α) 2 pz=-V-M(i - α + α ’).

Let v n be the velocity at nh feet from the surface, t⅛- n the velocity at an equal depth from the bottom, i(υ f , + t>ι→)-V-M(n*- n + α 3 -α+i), ≈v m - M(n s - n + ⅜) ,

Let n = ⅛, end put vι for the velocity at mid depth, then

Vi≈ι⅛ + ⅛M (3); so that the mid depth velocity differs from the mean velocity by the small quantity ⅛ M only, whatever be the position of the axis of the parabola. Messrs Humphreys and Abbot have based on this property a method of rapidly gauging rivers which will be described hereafter.

From a discussion of experiments in which the maximum velocity was at the surface, Bazin was led to take

M = 36∙3√Λi (4); and for that case the equation to the vertical velocity curve is (α=0) v = V -36∙3√½ 2 .

In the cases in which the maximum velocity was below the surface, Bazin found that the difference between the maximum velocity V and the bottom velocity r⅛ remained constant.

But, putting X= — = 1 in the equation (2α), and v b for the bottom h

V elocιty, ι⅛=V-M(l-α) 2 (5);

V - v i = M(l - a) 2 = constant, for different positions of the axis of the parabola. Let

, f N

M (l-α) 3 *

where N is a constant; then

,=V-NtoV

∖l-αy for any position of the axis of the parabola. But this must agree with the equation (4) above, for α=0; hence,

N = 36∙3√⅛i^j and the general equation for all cases becomes v=V-36∙3√Γi(^) a (6 )∙

Bazin has shown that this equation agrees well with experiments on artificial channels by himself, and on the Saone, Seine, Garonne, and Rhine. In all these the ratio ranged from 1’10 to 1 ’ 13, except in the case of the Rhine at Basel, for which the ratio was 1'17. The parameter 3*s∕Λ-⅛ ∏es p e t ween 13 and 20, and the ratio of (1 - air the depth at which the maximum velocity is found to the whole depth, =α, ranges from zero to 0 - 2, except in some of the artificial channels, where it reached 0 ‘35. The Mississippi experiments give different results, and Bazin inclines to believe that the method of experimenting was untrustworthy.

The extreme difference V - r b between the maximum and bottom velocity is found by Bazin to range from ⅛ V to ⅜ V in artificial channels, being greater the greater the roughness of the .sides. In natural streams it is more generally ⅜ V, but in the Rhine at Basel it reached ⅜V, the bed being covered with boulders.

Soileau's Formulas.— Boileau also assumes the vertical velocity curve to be a parabola; below the filament of greatest velocity the curve is expressed by the relation

ι>=A-By 2 (1).

That is, the velocity curve is a parabola having its axis at the free surface of the stream. Above the filament of greatest velocity this law fails, and the velocities diminish instead of increasing. The vertical velocity curve is therefore such a curve as M'MM" (fig. 116), where the part MM" is a parabola having its vertex at S, and DM is the maximum velocity. The part M'M does not follow the parabolic law. Let Vbe the maximum velocity DM, CMits depth= θ. Draw at M the tangent and normal to the parabola. Then PC is the half parameter A . Let OS = V + c, then CT = 2CS = 2c.

‰-L>

B = — 1 , andCM 2 = PC∙ CT;

2 PC

... b = àÇt = A.

2 CM 2 e 2

Boileau finds c=Bθ 2 to be nearly constant for very different streams. Thus from two experiments of his own on streams 0 - 2 and 0'3 metres deep, c=OO107θ and 0 , 01072. In Hennocque’s experiments on the Rhine, 2‘45 metres deep, c=0O107; and in the Mississippi experiments with a depth of 32 metres, c=0 ’ 0093 to 0∙0113. Replacing A and B in (1) by the values now given v - v " c ⅞ + c (2 ).

an equation which gives the velocity v at any depth y from the surface in the region below the filament of maximum velocity. For the region above the filament of maximum velocity Boileau assumes λ∕2 g 7 ,, = V'-C-∣y + (VV' + C)y .... (3), where υ is the velocity at the depth y and V' is the surface velocity.

102. Ratio of Mean to Greatest Surface Velocity, for the whole Cross Section in Trapezoidal Channels.— It is often very important to be able to deduce the mean velocity, and thence the discharge, from observation of the greatest surface velocity. The simplest method of gauging small streams and channels is to observe the greatest surface velocity by floats, and thence to deduce the mean velocity. Now, for channels not widely differing from those experimented on by Bazin, the expression obtained by him for the ratio of surface to mean velocity may be relied on as at least a good approximation to the truth. Let v 0 be the greatest surface velocity, r m the mean velocity of the stream. Then, according to Bazin,

‰=v 0 - 25 ’ 4√m.

But v n ≈c ∖ ∕mi, where c is a coefficient, the values of which have been already given in the table in § 90. Hence c ι∙ = vn ·

• c+25∙4°

Q

Values of Coefficient c + 254 tn ^ ιe, Formula v ∞=°^^25 7 4 v ° ’ [table]

103. River Bends.— In rivers flowing in alluvial plains, the windings which already exist tend to increase in curvature by the scouring away of material from the outer bank and the deposition of detritus along the inner bank. The sinuosities sometimes increase till a loop is formed with only a narrow strip of land between the two encroaching branches of the river. Finally a “cut off” may occur, a waterway being opened through the strip of land and the loop left separated from the stream, forming a horse-shoe shaped lagoon or marsh. Professor James Thomson has pointed out {Proc. RoyaP Soc. 1877, p. 356; Proc. Inst. of Meeh. Engineers, 1879, p. 456) that the usual supposition is that the water tending to go forwards in a straight line, rushes against the outer bank and scours it, at the same time creatingdeposits at the inner bank.

That view is very far from a complete account of the matter, and Professor Thomson hasgiven a much more ingenious account of· the action at the bend, which he has completely confirmed by experiment.

When water moves round a circular curve under the action of gravity only, it takes a motion like that in a free vortex. Its velocity is greater parallel to the axis of the stream at the inner than at the outer side of the bend. Hence the scouring at the outer side and the deposit at the inner side of the bend are not due to mere difference of velocity of flow in the general direction of the stream; but, in virtue of the centrifugal force, the water passing round the bend presses outwards, and the free surface in a radial cross section has a slope from the inner side upwards to the outer side (fig. 118). For the greater part of the water flowing in curved paths, this difference of pressure produces no tendency to transverse motion. But the water immediately in contact with the rough bottom and sides of the channel is retarded, and its centrifugal force is insufficient to balance the pressure due to the greater depth at the outside of the bend. It therefore flows inwards towards the inner side of the bend, carrying with it detritus which is deposited at the inner bank. Conjointly with this flow inwards along the bottom and sides, the general mass of water must flow outwards to take its place. Fig. 117 shows the directions of flow as observed in a small artificial stream, by means of light seeds and specks of aniline dye. The lines CC show the directions of flow immediately in contact with the sides and bottom. The dotted line AB shows the direction of motion of floating particles on the surface of the stream.

Steady· Motion of Water in Open Channels of Varying Cross Section and Slope.

104. In every stream the discharge of which is constant, or may be regarded as constant for the time considered, the velocity at different places depends on the slope of the bed. Except at certain exceptional points the velocity will be greater as the slope of the bed is greater, and, as the velocity and cross section of the stream vary inversely, the section of the stream will be least where the velocity and slope are greatest. If in a stream of tolerably uniform slope an obstruction such as a weir is built, that will cause an alteration of flow similar to that of an alteration of the slope of the bed for a greater or less distance above the weir, and the originally uniform cross section of the stream will become a varied one. In such cases it is often of much practical importance to determine the longitudinal section of the stream.

The cases now considered will be those in which the changes of velocity and cross section are gradual and not abrupt, and in which the only internal work which needs to be taken into account is that due to the friction of the stream bed, as in cases of uniform motion. Further, the motion will be supposed to be steady, the mean velocity at each given cross section remaining constant, though it varies from section to section along the course of the stream.

Let fig. 119 represent a longitudinal section of the stream, A 0 A 1 being the water surface, B 0 B 1 the stream bed. Let A 0 B 0 , A 1 B 1 be cross sections normal to e direction of flow. Suppose the mass of water A 0 B 0 AjB 1 comes in a short time θ to C 0 D 0 C 1 D 1 , and let the work done on the mass be equated to its change of kinetic energy during that period. Let l be the length A 0 A 1 of the portion of the stream considered, and z the fall of surface level in that distance. Let Q be the discharge of the stream per second.

Change of Kinetic Energy. — At the end of the time θ there are as many particles possessing the same velocities in the space C 0 D 0 A 1 B 1 as at the beginning. The change of kinetic energy is therefore the difference of the kinetic energies of A 0 B 0 C 0 D β and A 1 B 1 C 1 D 1 .

Let fig. 120 represent the cross section A 0 B n , and let w be a small element of its area at a point where the velocity is v. Let ∩ 0 be the whole area of the cross s θction a ∏4 ⅝ R ιe mean velocity for the whole cross section. From the definition of mean velocity we have Let v = w 0 + w, where w is the difference between the velocity at the small element ω and the mean velocity. For the whole cross section, Sω∞=0.

The mass of fluid passing through the element of section «, in θ seconds, is — ωvθ, and its kinetic energy is -ωv 3 θ. For the whole J section, the kinetic energy of the mass A 0 B 0 C 0 D 0 passing in θ seconds is

∙‰<>≈,

⅜7

Ge = ^2fS ω ( u o a + 3w 0 2 w + 3w 0 w 2 + w 3 ), = s ~2f {⅝ 3 Ω + 2ωw 2 (3w 0 + w)j- .

The factor 3w 0 + w is equal to 2w 0 + ι ’, a quantity necessarily positive. Consequently 2ωv 3 > ∩ 0 w 0 3 , and consequently the kinetic energy of A 0 B 0 C 0 D 0 is greater than

⅛W or ⅜√, which would be its value if all the particles passing the section had the same velocity u 0 , Let the kinetic energy be taken at α ⅛ i W = *¾Q m <> 2 ’ *g ±g where a is a corrective factor, the value of which has been estimated by Belanger at 1 ’ 1. Its precise value is not of great importance.

In a similar way we should obtain for the kinetic energy of A 1 B 1 C 1 D 1 the expression

Gø _ , Gø λ 2 "⅛⅛ ’-⅛⅛ ’ >.

where ∩ 1 , u 1 are the section and mean velocity at A 1 B 1 , and where a may be taken to have the same value as before without any important error.

Hence the change of kinetic energy in the whole mass A 0 B 0 A 1 B 1 in θ seconds is Cθ β ^-Q(V-w 0 2 ) (U-

Motive Work of the Weight and Pressures.— Consider a small filament α 0 α 1 which comes in θ seconds to c 0 c 1 . The work done by gravity during that movement is the same as if the portion α 0 c 0 were carried to α 1 c 1 . Let dQ j θ be the volume of a 0 c 0 or α 1 c 1 , and y 0 , y 1 the depths of α 0 , α 1 from the surface of the stream. Then the volume dQø or G<ZQθ pounds falls through a vertical height z+y 1 -y 0 > ar) d the work done by gravity is

GdQ θ(z + y 1 - y 0 ).

Putting p a for atmospheric pressure, the whole pressure per unit of area at α 0 is Gy 0 +p α , and that at a 1 is -(Gy 1 +p a ). The work of these pressures is G (yo + CΓ ~ V1 ~ ⅜) d ⅛ θ ’

^G(y 0 -y 1 )dQθ.

Adding this to the work of gravity, the whole work is GzdQθ ; or, for the whole cross section,

GzQØ (2).

Work expended in Overcoming the Friction of the Stream Bed.— Let A'B', A"B" be two cross sections at distances s and s + ds from A 0 B 0 . Between these sections the velocity may be treated as uniform, because by hypothesis the changes of velocity from section to section are gradual. Hence, to this short length of stream the equation for uniform motion is applicable. But in that case the work in overcoming the friction of the stream bed between A'B' and A"B" is gq0 ⅞ n ds, where u, χ, ∩ are the mean velocity, wetted perimeter, and section at A'B'. Hence the whole work lost in friction from A 0 B 0 to A 1 B 1 will be gq Z , ≡⅛⅜* < s >∙

Equating the work given in (2) and (3) to the change of kinetic energy given in (1),

“^ ( <- W » 2) = GQ ' 0 - GQ0 / ?C S Q d °'' j< i ∆y ιι

-- √ l ¾ ’ -⅝ ’ > I ∕'f2i, ⅛

’ ‘ ⅛ *J. l ¾ a d *- 105. Fundamental Differential Equation of Steady Varied Motion. —Suppose the equation just found to be applied to an indefinitely short length ds of the stream, limited by the end sections ab, α,5 1 , taken for simplicity normal to the stream bed (fig. 121). For that short length of stream the fall of surface level, or difference of level of a and α 1 , may be written dz. Also, if we write u for u 0 , and u+du for u 1 , the term u *÷- u I becomes . Hence the eouation applicable to an indefinitely short length of the stream is

*-⅛⅛ + ⅞ f ⅛ b (1).

g Ω s 2g

From this equation some general conclusions may be arrived at as to the form of the longitudinal section of the stream, but, as the investigation is somewhat complicated, it is convenient to simplify it by restricting the conditions of the problem.

Modification of the Formula for the Restricted Case of a Stream flowing in a Prismatic Stream Bed of Constant Slope.— Let i be the constant slope of the bed. Draw ad parallel to the bed, and ac horizontal. Then dz is sensibly equal to a'c. The depths of the stream, h and h + dh, are sensibly equal to ab and a'b', and therefore dh = a'd. Also cd is the fall of the bed in the distance ds, and is equal to ids. Hence dz=a'd=cd-a'c≈ids-dh (2).

Since the motion is steady—

Q = ∩w = constant.

Differentiating, Ωdu + udΩ — 0;

*---β-∙

Let x be the width of the stream, then dΩ,≈xdh very nearly. Inserting this value, du≈- υ ^dh (3).

Putting the values of du and dz found in (2) and (3) in equation (1), ., „ u i x ,, ∙v , u 2 , ids - dh=--dh + * ζ-- ds. gΩ Ω 2g i - ×(- dh Ω s 2g ίΛΛ ds~, u 2 X ' λ g Ω

Further Restriction to the Case of a Stream of Rectangular Section and of Indefinite Width.— The equation might be discussed in the form just given, but it becomes a little simpler if restricted in the way just stated. For, if the stream is rectangular, xh=Ω, and if x is large compared with h, — = — ≈h nearly. Then equation (4) becomes

1 -ζ— dh . 2qih ,-. d⅛--≈~^≠ - < 5 >∙

1 ^^gh

106. General Indications as to the Form of Water Surface furnished by Equation (5).—Let A 0 A 1 (fig. 122) be the water surface, B 0 B 1 the bed in a longitudinal section of the stream, and ab any section at a distance s from B o , the depth ab being h. Suppose B 0 B 1 , B 0 A 0 taken as rectangular coordinate axes, then is the trigonometric tangent of the angle which the surface of the stream at a makes w.th the axis B 0 B 1 . This tangent will Ire positive, if the stream is increasing in depth in the direction B 0 B 1 ; negative, if the stream is diminishing in depth from B o towards B 1 . If ∙^^~ s θ> the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4)

⅛-o k it i-⅜f≠-0 i ds ’ Ω 2g ’ . u* Ω .

ς-≈- =—ι=m, 2g χ which is the well-known general equation for uniform motion, based on the same assumptions as the equation for varied steady motion now being considered. The case of uniform motion is therefore a limiting case between two different kinds of varied motion.

Consider the possible changes of value of the fraction

1 - ζ U *

1 ' a 2gih 1 ∙^^τ∙

gh

As h tends towards the limit 0, and consequently u is large, the numerator tends to the limit - ∞. On the other hand if h = ∞, in which case u is small, the numerator becomes equal to 1. For a value H of h given by the equation 1 -⅛ar 0 - κ -f⅞-

we fall upon the case of uniform motion. The results just stated may be tabulated thus :—

For h= 0 H >H ∞

the numerator has the value - oo 0 ≥0 1.

Next consider the denominator. If h becomes very small, in which case u must be very large, the denominator tends to the limit - ∞. As h becomes very large and u consequently very small, the denominator tends to the limit 1. For h= —, or u≈ ∖ fgh, the g denominator becomes zero. Hence, tabulating these results as before :—

λ n w 2 w 2 ror λ=0 — >>— oo g 9 the denominator becomes - oo 0 ≥0 1.

107. Case 1.—Suppose Λ≥- , and also A≥H, or the depth greater than that corresponding to uniform motion. In this case is ds positive, and the stream increases in depth in the direction of flow. In fig. 123 let B 0 B 1 be the bed, C 0 C 1 a line parallel to the bed and at a height above it equal to H. By hypothesis, the surface A 0 A l of the stream is above C 0 Cj, and it has just been shown that the depth of the stream increases from B o towards B 1 . But going up stream h approaches more and more nearly the value H, and therefore —

approaches the limit 0, or the surface of the stream is asymptotic to C 0 C 1 . Going down stream h increases and u diminishes, the 1 ζ⅛ih numerator and denominator of the fraction both tend

ι w dh ^9⅛ towards the limit 1, and to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line D 0 D 1 .

The form of water surface here discussed is produced when the flow of a stream originally uniform is altered by the construction of a weir. The raising of the water surface above the level C 0 C 1 is termed the backwater due to the weir.

dh

108. Case 2.—Suppose ⅛≥y, and also Λ<Η. Then is negative, and the stream is diminishing in depth in the direction of flow. In fig. 124 let B 0 B 1 be the stream bed as before; C 0 C 1 a line drawn parallel to B 0 B 1 at a height above it equal to H. By hypothesis the surface A 0 A 1 of the stream is below C 0 C 1 , and the depth has just been shown to diminish from B o towards B 1 . Going up stream h approaches the limit H, and tends to the limit zero. That is, up stream A 0 A 1 is asymptotic to C 0 C 1 . Going down stream h diminishes and u increases; the inequality ∕ι>-diminishes; the ι-fX- g denominator of the fraction tends to the limit zero, and

ι u dh consequently tends to ∞. That is, down stream A 0 A 1 tends to a direction perpendicular to the bed. Before, however, this limit was reached the assumptions on which the general equation is based would cease to be even approximately true, and the equation would cease to be applicable. The filaments would have a relative motion, which would make the influence of internal friction in the fluid too important to be neglected. A stream surface of this form may be produced if there is an abrupt fall in the bed of the stream (fig. 125).

On the Ganges canal, as originally constructed, there were abrupt falls precisely of this kind, and it appears that the lowering of the water surface and increase of velocity which such falls occasion, for a distance of some miles up stream, was not foreseen. The result was that, the velocity above the falls being greater than was intended, the bed was scoured and considerable damage was done to the works. “When the canal was first opened the water was allowed to pass freely over the crests of the overfalls, which were laid on the level of the bed of the earthen channel; erosion of bed and sides for some miles up rapidly followed, and it soon became apparent that mea,ns must be adopted for raising the surface of the stream at those points (that is, the crests of the falls). Planks were accordingly fixed in the grooves above the bridge arches, or temporary weirs were formed over which the water was allowed to fall; in some cases the surface of the water was thus raised above its n∩rmal height, causing a backwater in the channel above” (Crofton’s Report on the Ganges Canal, p. 14). Fig. 126 represents in an exaggerated form what probably occurred, the diagram being intended to represent some miles’length of the canal bed above the fall. AA parallel to the canal bed is the level corresponding to uniform motion with the intended velocity of the canal. In consequence of the presence of the ogee fall, however, the water surface would take some such form as BB, corresponding to Case 2 above, and the velocity would be greater than the intended velocity, nearly in the inverse ratio of the actual to the intended depth. By constructing a weir on the crest of the fall, as shown by dotted lines, a new water surface CC corresponding to Case 1 would be produced, and by suitably choosing the height of the weir this might be made to agree approximately with the intended level AA.

109. Case 3.—Suppose a stream flowing uniformly with a depth f lβ ’

H ∙≤—. For a stream in uniform motion ζ- ι r-=mi, or if the stream *7

u 2

is of indefinitely great width, so that τn=H, then = ιΗ, and Vj

H = f ——. Consequently the condition stated above involves that

. u i .ω 2 , .. f s? · <.—, or that ι ≥ -⅜- ∙ ⅛gι g 2

If such a stream is interfered with by the construction of a weir which raises its level, so that its depth at the weir becomes ½ 1 , then for a portion of the stream the depth h will satisfy the conditions A<— and Λ≥H, which are not the same as those assumed 9 in the two previous cases. At some point of the stream above the weir the depth h becomes equal to —, and at that point — becomes infinite, or the surface of the stream is normal to the bed. It is obvious that at that point the influence of internal friction will be too great to be neglected, and the general equation will cease to represent the true conditions of the motion of the water. It is known that, in cases such as this, there occurs an abrupt rise of the free surface of the stream, or a standing wave is formed, the conditions of motion in which will be examined presently.

It appears that the condition necessary to give rise to a standing wave is that ζ>^κ^∙ Now £ depends for different channels on the Ji roughness of the channel and its hydraulic mean depth. Μ. Bazin has calculated the values of ζ for channels of different degrees of roughness and different depths given in the following table, and the corresponding .minimum values of i for which the exceptional case of the production of a standing wave may occur. [table]

Standing Waves.

110. The formation of a standing wave was first observed by Bidone. Into a small rectangular masonry channel, having a slope of 0 023 feet per foot, he admitted water till it flowed uniformly with a depth of 0 - 2 feet. He then placed a plank across the stream which raised the level just above the obstruction to 0 95 feet. He found that the stream above the obstruction was sensibly unaffected up to a point 15 feet from it. At that point the depth suddenly increased from 0 ’ 2 feet to 0 - 56 feet. The velocity of the stream in the part unaffected by the obstruction was 5 ’ 54 feet per second. Above the point where the abrupt change of depth occurred u 2 = 5 ’ 54 2 =30‘7, and gh = 32 '2 × 0 - 2 = 6 '44; hence u 2 was ~j>-gh. Just be-

0 ’2

low the abrupt change of depth = 5 - 54 × —— = 1‘97; % 2 = 3 - 88; 0’56

and gh= 32 ’ 2 × 0’56 = 18 03; hence at this point u-<fjh. Between these two points, therefore, u=gh ; and the condition for the production of a standing wave occurred.

The change of level at a standing wave may be found thus. Let fig. 127 represent the longitudinal section of a stream and ab, cd cross sections normal to the bed, which for the short distance considered may be assumed horizontal. Suppose the mass of water abed to come to db'dd' in a short time t ; and let w- 0 , u 1 be the velocities at ab and cd, Ω 0 ,Ω 1 the areas of the cross sections. The force causing change of momentum in the mass abed estimated horizontally is simply the difference of the pressures on ab and cd. Putting Λ 0 ,Λ 1 for the depths of the centres of gravity of ab and cd measured down from the free water surface, the force is G(Λ 0 Ω 0 - Λ 1 Ω 1 ) pounds, and the impulse in t seconds is G (Λ 0 Ω 0 - Λ 1 Ω 1 ) t second pounds. The horizontal change of momentum is the difference of the momenta of cddd' and aba'b' ; that is,

2(Ω 1 u 1 2 -Ω 0 w, 0 2 )i.

Hence, equating impulse and change of momentum,

G(Λ 0 Ω 0 - A 1 Ω 1 )i = — (Ω 1 u 1 2 - Ω 0 w 0 2 )i; J

.∙. ½ 0 Ω 0 -A 1 Ω 1 = 9d≠Z⅛ s (I,.

For simplicity let the section be rectangular, of breadth B and depthsH o andH 1 , at the two cross sections considered; then h 0 = - H o , 2

and A 1 =-H 1 . Hence 1 2 1

9 H 0 2 -H 1 2 =^(H 1 u 1 2 -H 0 u 0 2 ).

But, since Ω 0 ‰=O 1 u 1 , we have

H 2 u 1 — 2 ’

9f∕ 2 ∕∏ 2 ∖

H 0 2 -H 1 ≡=ψ(⅛L-H 0 ) .... (2).

This equation is satisfied if H 0 = H 1 , which corresponds to the case of uniform motion. Dividing by H o - H 1 , the equation becomes gl(H 0 + H 1 ).⅛ (3); n 0 g

... H 1 -√j⅛.⅛<, + -λH.∙j -∣H,. . . (4).

In Bidone’s experiment w 0 = 5 ’ 54, and H =0 ’ 2. Hence H 1 = 0 ’ 52, which agrees very well with the observed height.

111. A standing wave is frequently produced at the foot of a weir. Thus in the ogee falls originally constructed on the Ganges canal a standing wave was observed as shown in fig. 128. The water falling over the weir crest A acquired a very high velocity on the steep slope AB, and the section of the stream at B became very small. It easily happened, therefore, that at B the depth h < — . In flowing along the rough apron of the weir the 9 velocity u diminished and the depth h increased. At a point C, where h became equal to — , the conditions for producing the standing wave occurred. Beyond C the free surface abruptly rose to the level corresponding to uniform motion with the assigned slope of the lower reach of the canal.

A standing wave is sometimes formed on the down stream side of bridges the piers of which obstruct the flow of the water. Some interesting cases of this kind are described in a paper on the “Floods in the Nerbudda Valley” in the Proc. Inst, of Civil Engineers, vol. xxvii. p. 222, by Mr A. C. Howden. Fig. 129 is compiled from the data given in that paper. It represents the section of the stream at pier 8 of the Towal> Viaduct, during the flood of 1865. The ground level is not exactly given by Mr Howden, but has been inferred from data given on another drawing. The velocity of the stream was not observed, but the author states it was probably the same as at the Gunjal river during a similar flood, that is 16’58 feet per second. Now, taking the depth on the down stream face of the pier at 26 feet, the velocity necessary for the production of a standing wave would be u = ∖ fgh = √(32 ’ 2 × 26) = 29 feet per second nearly. But the velocity at this point was probably from Mr Howden’s statements 16'58 ×⅜θ = 25’5 feet, an agreement as close as the approximate character of the data would lead us to expect.

XI. ON STREAMS AND RIVERS.

112. Catchment Basin.— A stream or riveris the channel for the discharge of the available rainfall of a district, termed its catchment basin. The catchment basin is surrounded by a ridge or watershed line, continuous except at the point where the river finds an outlet. The area of the catchment basin may be determined from a suitable contoured map on a scale of at least 1 in 100,000. Of the whole rainfall on the catchment basin, a part only finds its way to the stream. Part is directly re-evaporated, part is absorbed by vegetation, part may escape by percolation into neighbouring districts. The following table gives the relation of the average stream discharge to the average rainfall on the catchment basin (Tiefenbacher). [table]

113. Flood Discharge.—Υhe flood discharge can generally only be determined by examining the greatest height to which floods have been known to rise. To produce a flood the rainfall must be heavy and widely distributed, and to produce a flood of exceptional height the duration of the rainfall must be so great that the flood waters of the most distant affluents reach the point considered, simultaneously with those from nearer points. The larger the catchment basin the less probable is it that all the conditions tending to pro - duce a maximum discharge should simultaneously occur. Further, lakes and the river bed itself act as storage reservoirs during the rise of water level and diminish the rate of discharge, or serve as flood moderators. The influence of these is often important, because very heavy rain storms are in most countries of comparatively short-duration. Tiefenbacher gives the following estimate of the flood discharge of streams in Europe:— [table]

It has been attempted to express the decrease of the rate of flood discharge with the increase of extent of the catchment basin by empirical formulas. Thus Colonel O’Connell proposes the formula ι∕ = M∖∕a ’, where Mis a constant called the modulus of thé river, the value of which depends on the amount of rainfall, the physical characters of the basin, and the extent to which the floods are moderated by storage of the water. If M is small for any given river, it shows that the rainfall is small, or that the permeability or slope of the sides of the valley is such that the water does not drain rapidly to the river, or that lakes and river bed moderate the rise of the floods. If values of M are known for a number of rivers, they may be used in inferring the probable discharge of other similar rivers. For British rivers M varies from 0 - 43 for a small stream draining meadow land to 37 for the Tyne. Generally it is about 15 or 20. For large European rivers M varies from 16 for the Seine to 67 - 5 for the Danube. For the Nile M = 11, a low value which results from the immense length of the Nile throughout which it receives no affluent, and probably also from the influence of lakes. For different tributaries of the Mississippi M varies from 13 to 56. For various Indian rivers it varies from 40 to 303, this variation being due to the great variations of rainfall, slope, and character of Indian rivers.

In some of the tank projects in India, the flood discharge has been calculated from the formula D = C ∖ Ffi, where D is the discharge in cubic yards per hour from n square miles of basin. The constant C was taken = 61,523 in the designs for the Ekrooka tank, = 75,000 on Ganges and Godavery works, and=10,000 on Madras works.

114. Action of a Stream on its Bed.—Tf the velocity of a stream exceeds a certain limit, depending on its size, and on the size, heaviness, form, and coherenceof the materialof which its bed is composed, it scours its bed and carries forward the materials. The quantity of material which a given stream can carry in suspension depends on the size and density of the particles in suspension, and is greater as the velocity of the stream is greater. If in one part of its course the velocity of a stream is great enough to scour the bed and the water becomes loaded with silt, and in a subsequent part of the river’s course the velocity is diminished, then part of the transported material must be deposited. Probably deposit and scour go on simultaneously over the whole river bed, but in some parts the rate of scour is in excess of the rate of deposit, and in other parts the rate of deposit is in excess of the rate of scour. Deep streams appear to have the greatest scouring power at any given velocity. It is possible that the difference is strictly a difference of transporting, not of scouring action. Let fig. 130 represent a section of a stream. The material lifted at a will be diffused through the mass of the stream and deposited at different distances down stream. The average path of a particle lifted at a will be some such curve as abc, and the average distance of transport each time a particle is lifted will be represented by ac. In a deeper stream such as that in fig. 131, the average height to which particles are lifted, and, since the rate of vertical fall through the water may be assumed the same as before, the average distance a'd of transport, will be greater. Consequently, although the scouring action may be identical in the two streams, the velocity of transport of material down stream is greater as the depth of the stream is greater. The effect is that the deep stream excavates its bed more rapidly than the shallow stream.

115. Bottom Velocity at which Scour commences.— The following bottom velocities were determined by Dubuat to be the maximum velocities consistent with stability of the stream bed for different materials.

Darcy and Bazin give, for the relation of the mean velocity v m and bottom velocity ι⅜,

_ ’ ι⅛=t⅛ +10’87√τni.

But ∖ hmi≈v m . ∕ X;

V 2g _ .∙. Φm=n÷^l-10∙87^∕X)∙

Taking a mean value for £ we get t⅛ = l∙312v t , and from this the following values of the mean velocity are obtained :— [table]

The following table of velocities which should not be exceeded in channels is given in the Ingenieurs Taschenbuch of the Vereiu “Hütte”:— [table]

116. Regime of a River Channel.— À river channel is said to be in a state of regime, or stability, when it changes little in draught or form in a series of years. In some rivers the deepest part of the channel changes its position perpetually, and is seldom found in the same place in two successive years. The sinuousness of the river also changes by the erosion of the banks, so that in time the position of the river is completely altered. In other rivers the change from year to year is very small, but probably the regime is never perfectly stable except where the rivers flow over a rocky bed.

If a river had a constant discharge it would gradually modify its bed till a permanent regime was established. But as the volume discharged is constantly changing, and therefore the velocity, silt is deposited when the velocity decreases, and scour goes on when the velocity increases in the same place. When the scouring and silting are considerable, a perfect balance between the two is rarely established, and hence continual variations occur in the form of the river and the direction of its currents. In other cases, where the action is less violent, a tolerable balance may be established, and the deepening of the bed by scour at one time is compensated by the silting at another. In that case the general regime is permanent, though alteration is constantly going on. This is more likely to happen if by artificial means the erosion of the banks is prevented. If a river flows in soil incapable of resisting its tendency to scour it is necessarily sinuous (§ 103), for the slightest deflexion of the current to either side begins an erosion which increases progressively till a considerable bend is formed. If such a river is straightened it becomes sinuous again unless its banks are protected from scour.

117. Longitudinal Section of River Bed.— The declivity of rivers decreases from source to mouth. In their higher parts rapid and torrential, flowing over beds of gravel or boulders, they enlarge in volume by receiving affluent streams, their slope diminishes, their bed consists of smaller materials, and finally they reach the sea. Fig. 132 shows the length in miles, and the surface fall in feet per mile, of the Tyne and its tributaries.

The decrease of the slope is due to two causes. (1) The action of the transporting power of the water, carrying the smallest debris the greatest distance, causes the bed to be less stable near the mouth than in the higher parts of the river; and, as the river adjusts its slope to the stability of the bed by scouring or increasing its sinous-ness when the slope is too great, and by silting or straightening its course if the slope is too small, the decreasing stability of the bed would coincide with a decreasing slope. (2) The increase of volume and section of the river leads to a decrease of slope; for the larger the section the less slope is necessary to ensure a given velocity.

The following investigation, though it relates to a purely arbitrary case, is not without interest. Let it be assumed, to make the conditions definite—(1) that a river flows over a bed of uniform resistance to scour, and let it be further assumed that to maintain stability the velocity of the river in these circumstances is constant from source to mouth · (2) suppose the sections of the river at all points are similar, so that, b being the breadth of the river at any point, its hydraulic mean depth is ab and its section is cδ 2 r where a and c are constants applicable to all parts of the river; (3) let us further assume that the discharge increases uniformly in consequence of the supply from affluents, so that, if l is the length of the river from its source to any given point, the discharge there will be kl, where k is another constant applicable to all points in the course of the river.

Let AB (fig. 133) be the longitudinal section of the river, whose source is at A; and take A for the origin of vertical and horizontal coordinates. Let C be a point whose ordinates are x and y, and let the river at C have the breadth b, the slope i, and the velocity v.

Since velocity × area of section = discharge, vcb i *=kl, or δ =

Hydraulic mean depth = ab≈ α* ∕ — .

v cv

But, by the ordinary formula for the flow of rivers, mi= ζv i ;

. i c≤*/Z.

' m a y lcl'

But i is the tangent of the angle which the curve at C makes with the axis of X, and is therefore=⅛. Also, as the slope is small, dx r ’ I =AC = AD = x nearly.

. ⅛'=⅛∕Ξ. ' ' dx a V k x ’ and, remembering that v is constant, 2ζyi ∕ ex . a k , or y 2 = constant × x; so that the curve is a common parabola, of which the axis is horizontal and the vertex at the source. This may be considered an ideal longitudinal section, to which actual rivers approximate more or less, with exceptions due to the varying hardness of their beds, and the iιτegular manner in which their volume increases.

118. Surface Level of River.— The surface level of a river is a plane changing constantly in position from changes in the volume of water discharged, and more slowly from changes in the river bed, and the circumstances affecting the drainage into the river.

For the purposes of the engineer, it is important to determine (1) the extreme low water level, (2) the extreme high water or flood level, and (3) the highest navigable level.

(1) Low Water Level cannot be absolutely known, because a river reaches its lowest level only at rare intervals, and because alterations in the cultivation of the land, the drainage, the removal of forests, the removal or erection of obstructions in the river bed, &c., gradually alter the conditions of discharge. The lowest level of which records can be found is taken as the conventional or approximate low water level, and allowance is made for possible changes.

(2) High Water or Flood Level.— The engineer assumes as the highest flood level the highest level of which records can be obtained. In forming a judgment of the data available, it must be remembered that the highest level at one point of a river is not always simultaneous with the attainment of the highest level at other points, and that the rise of a river in flood is very different in different parts of its course. In temperate regions, the floods of rivers seldom rise more than 20 feet above low-water level, but in the tropics the rise of floods is greater.

(3) Highest Navigable Level.— When the river rises above a certain level, navigation becomes difficult from the increase of the velocity of the current, or from submersion of the tow paths, or from the headway under bridges becoming insufficient. Ordinarily the highest navigable level may be taken to be that at which the river begins to overflow its banks.

119. Relative Value of Different Materials for Submerged Works.— That the power of water to remove and transport different materials depends on their density has an important bearing on the selection of materials for submerged works. In many cases, as in the aprons or floorings beneath bridges, or in front of locks or falls, and in the formation of training walls and breakwaters by pierres perdus, which have to resist a violent current, the materials of which the structures are composed should be of such a size and weight as to be able individually to resist the scouring action of the water. The heaviest materials will therefore be the best; and the different value of materials in this respect will appear much more striking, if it is remembered that all materials lose part of their weight in water. A block whose volume is V cubic feet, and whose density in air is w ft per cubic foot, weighs in air røV ft, but in water only (w- 62 - 4) V ft. [table]

120. Inundation Deposits from a River.— When a river carrying silt periodically overflows its banks, it deposits silt over the area flooded, and gradually raises the surface of the country. The silt is depositedin greatest abundance where the water first leaves the river. It hence results that the section of the country assumes a peculiar form, the river flowing in a trough along the crest of a ridge, from which the land slopes downwards on both sides. The silt deposited from the water forms two wedges, having their thick ends towards the river (fig. 134).

This is strikingly the case with the Mississippi, and that river is now kept from flooding immense areas by artificial embankments or levees. In India, the term deltaic segment is sometimes applied to that portion of a river running through deposits formed by inundation, and having this characteristic section. The irrigation of the country in this case is very easy, a comparatively slight raising of the river surface by a weir or annicut gives a command of level which permits the water to be conveyed to any part of the district.

121. Deltas.— The name delta was originally given to the Δ-shaped portion of Lower Egypt, included between seven branches of the Nile. It is now given to the whole of the alluvial tracts round river mouths formed by deposition of sediment from the river, where its velocity is checked on its entrance to the sea. The characteristic feature of these alluvial deltas is that the river traverses them, not in a single channel, but in two or many bifurcating branches. Each branch has a tract of the delta under its influence, and gradually raises the surface of that tract, and extends it seaward. As the delta extends itself seaward, the conditions of discharge through the different branches change. The water finds the passage through one of the branches less obstructed than through the others; the velocity and scouring action in that branch are increased; in the others they diminish. The one channel gradually absorbs the whole of the water supply, while the other branches silt up. But as the mouth of the new main channel extends seaward the resistance increases both from the greater length of the channel and the formation of shoals at its mouth, and the river tends to form new bifurcations AC or AD (fig. 135), and one of these may in time become the main channel of the river.

122. Field Operations preliminary to a Study of River Improvement.— There are required (1) a plan of the river, on which the positions of lines of levelling and cross sections are marked; (2) a longitudinal section and numerous cross sections of the river; (3) a series of gaugings of the discharge at different points and in different conditions of the river.

Longitudinal Section. —This requires to be carried out with great accuracy. A line of stakes is planted, following the sinuosities of the river and chained and levelled. The cross sections are refeιτed to the line of stakes, both as to position and direction. To determine the slope of the water surface great care is necessary.

123. Cross Sections. — A stake is planted flush with the water, and its level relatively to some point on the line of levels is determined. Then the depth of the water is determined at a series of points (if possible at uniform distances) in a line starting from the stake and perpendicular to the thread of the stream. To obtain these, a wire may be stretched across with equal distances marked on it by hanging tags. The depth at each of these tags may be obtained by a light wooden staff, with a disk-shaped shoe 4 to 6 inches in diameter. If the depth is great, soundings may be taken by a chain and weight. To ensure the wire being perpendicular to the thread of the stream, it is desirable to stretch two other wires similarly graduated, one above and the other below, at a distance of 20 to 40 yards. A number of floats being then thrown in, it is observed whether they pass the same graduation on each wire.

For large and rapid rivers the cross section is obtained by sounding in the following way. Let AC (fig. 136) be the line on which soundings are required. A base line AB is measured out at right angles to AC, and ranging staves are set up at AB and at D in line with AC. A boat is allowed to drop down stream, and, at the moment it comes in line with A D, the lead is dropped and an observer in the boat takes, with a box sextant, the angle AEB subtended by AB. The sounding line may have a weight of 14 lb of lead, and, it the boat drops down stream slowly, it may hang near the bottom, so that the observation is made instantly. In extensive surveys of the Mississippi observers with theodolites were stationed at A and B. The theodolite at A was directed towards C, that at B was kept on the boat. When the boat came on the line AC, the observer at A signalled, the sounding line was dropped, and the observer at B read off the angle ABE. By repeating observations a number of soundings are obtained, which can be plotted in their proper position, and the form of the river bed drawn by connecting the extremities of the lines. From the section can be measured the sectional area of the stream Ω and its wetted perimeter χ ; and from these the hydraulic mean depth m can be calculated.

124. Measurement of the Discharge of Rivers.— The area of cross section multiplied by the mean velocity gives the discharge of the stream. The height of the river with reference to some fixed mark should be noted whenever the velocity is observed, as the velocity an I area of cross section are different in different states of the river. To determine the mean velocity various methods may be adopted; and, since no method is free from liability to error, either from the difficulty of the observations or from uncertainty as to the ratio of the mean velocity to the velocity observed, it is desirable that more than one method should be used.

Instruments for Measuring the Velocity of Water.

125. Surface Floats are convenient for determining the surface velocities of a stream, though their use is difficult near the banks. 1 he floats may be small balls of wood, of wax, or of hollow metal, so loaded as to float nearly flush with the water surface. To render them visible they may have a vertical painted stem. In experiments on the Seine, cork balls If inches diameter were used, loaded to float flush with the water, and provided with a stem. In Captain Cunningham’s observations at Roorkce, the floats were thin circular disks of English deal, 3 inches diameter and .∣ inch thick. For observations near the banks, floats 1 inch diameter and ⅜ inch thick were used. To render them visible a tuft of cotton wool was used loosely fixed in a hole at the centre.

The velocity is obtained by allowing the float to be carried down, and noting the time of passage over a measured length of the stream. If V is the velocity of any float, t the time of passing over a length l, then v=—, To mark out distinctly the length of stream over which the floats pass, two ropes may be stretched across the stream at a distance apart, which varies usually from 50 to 250 feet, according to the size and rapidity of the river. In the Roorkee experiments a length of run of 50 feet was found best for the central two-fifths of the width, and 25 feet for the remainder, except very close to the banks, where the run was made 12⅛ feet only. The longer the run the less is the proportionate error of the time observations, but on the other hand the greater the deviation of the floats from a straight course parallel to the axis of the stream. To mark the precise position at which the floats cross the ropes, Captain Cunningham used short white rope pendants, hanging so as nearly to touch the surface of the water. In this case the streams were 80 to 180 feet in width. In wider streams the use of ropes to mark the length of run is impossible, and recourse must be had to box sextants or theodolites to mark the path of tlιe floats.

Let AB (fig. 137) be a measured base line strictly parallel to the thread of the stream, and AA 1 , BB 1 lines at right angles to AB marked out by ranging rods at A 1 and B 1 . Suppose observers stationed at A and B with sextants or theodolites, and let CD be the path of any float down stream. As the float approaches ΛA 1 , the observer at B keeps it on the cross wire of his instrument. The observer at A observes the instant of the float reaching the line AA 1 , and signals to B who then reads off the angle ABC. Similarly, as the float approaches BB∣, the observer at A keeps it in sight, and when signalled to by B reads the angle BAD. The data so obtained are sufficient for plotting the path of the float and determining the distances AC, BD.

The time taken by the float in passing over the measured distance may be observed by a chronograph, started as the float passes the upper rope or line, and stopped when it passes the lower. In Captain Cunningham’s observations two chronometers were sometimes used, the time of passing one end of the πιn being noted on one, and that of passing the other end of the run being noted on the other. The chronometers were compared immediately before the observations. In other cases a single chronometer was used placed midway of the ran. The moment of the floats passing the ends of the run was signalled to a timekeeper at the chronometer by shouting. It was found quite possible to count the chronometer beats to the nearest half second, and in some cases to the nearest quarter second.

126. Sub-surface Floats.— The velocity at different depths below the surface of a stream may be obtained by sub-surface floats, used precisely in the same way as surface floats. The most usual arrangement is to have a large float, of slightly greater density than water, connected with a small and very light surface float. The motion of the combined arrangement is not sensibly different from that of the large float, and the small surface float enables an observer to note the path and velocity of the sub-surface float. The instrument is, however, not free from objection. If the large submerged float is made of very nearly the same density as water, then it is liable to be thrown upwards by very slight eddies in the water, and it does not maintain its position at the depth at which it is intended to float. On the other hand, if the large float is made sensibly heavier than water, the indicating or surface float must be made rather large, and then it to some extent influences the motion of the submerged float Fig. 138 shows one form of sub-surface float. It consists of a couple of tin plates bent at a right angle and soldered together at the angle. This is connected with a wooden ball at the surface by a very thin wire or cord. As the tin alone makes a heavy submerged float, it is better to attach to the tin float some pieces of wood to diminish its weight in water. Fig. 139 shows the form of submerged float used by Captain Cunningham. It consists of a hollow metal ball connected to a slice of cork, which serves as the surface float.

127. Twin Floats. —Suppose two equal and similar floats (fig. 140) connected by a wire. Let one float be a little lighter and the other a little heavier than water. Then the velocity of the combined floats will be the mean of the surface velocity and the velocity at the depth at which the heavier float swims, which is determined by the length of the connecting wire. Thus if v, is the surface velocity and rd the velocity at the depth to which the lower float is sunk, the velocity of the combined floats will be V, + rd v 2 ·

Consequently, if υ is observed, and v, determined by an exp niment with a single float, vd=2v-v,.

According to Captain Cunningham, the twin float gives better ι ’.suits than the sub-surface float.

128. Velocity Rods.— Another form of float is shown in fig. 141. This consists of a cylindrical rod loaded at the lower end so ι s to float nearly vertical in water. A wooden rod, with a metal cap at the bottom in which shot can be placed, answers better than anything else, and sometimes the wooden rod is made in lengths which can be screwed together so as to suit streams of different depths. A tuft of cotton wool at the top serves to make the float more easily visible. Such a rod, so adjusted in length that it sinks nearly to the bed of the stream, gives directly the mean velocity of the whole vertical section in which it floats.

129. Revfs Current Meter. — No instrument has been so much used in directly determining the velocity of a stream at a given point as the screw current meter. Of this there are a dozen varieties at least. As an example of the instrument in its simplest form, Mr Revy’s meter may be selected. This is an ordinary screw meter of a larger size than usual, more carefully made, and with its details carefully studied (figs. 142, 143). It was designed after experience in gauging the great South American rivers. The screw, which is actuated by the water, is 6 inches in diameter, and is of the type of the Griffiths screw used in ships. The hollow spherical boss serves to make the weight of the screw sensibly equal to its displacement, so that friction is much reduced. On the axis aa of the screw is a worm which drives the counter. This consists of two worm wheels g and h fixed on a common axis. The worm wheels are carried on a frame attached to the pin l. By means of a string attached to l they can be pulled into gear with the worm, or dropped out of gear and stopped at any instant. A nut m can be screwed up, if necessary, to keep the counter permanently in gear. The worm is two-threaded, and the worm wheel g has 200 teeth. Consequently it makes one rotation for 100 rotations of the screw, and the number of rotations up to 100 is marked by the passage of the graduations on its edge in front of a fixed index. The second worm wheel has 196 teeth, and its edge is divided into 49 divisions. Hence it falls behind the first wheel one division for a complete rotation of the latter. The number of hundreds of rotations of the screw are therefore shown by the number of divisions on h passed over by an index fixed to g. One difficulty in the use of the ordinary screw meter is that particles of grit, getting into the working parts, very sensibly alter the friction, and therefore the speed of the meter. Mr Revy obviates this by enclosing the counter in a brass box with a glass face. This box is filled with pure water, which ensures a constant coefficient of friction for the rubbing parts, and prevents any mud or grit finding its way in. In order that the meter may place itself with the axis parallel to the current, it is pivoted on a vertical axis and directed by a large vane shown in fig. 143. To give the vane more directing power the vertical axis is nearer the screw than in ordinary meters, and the vane is larger. A second horizontal vane is attached by the screws x,x, the object of which is to allow the meter to rest on the ground without the motion of the screw being interfered with. The string or wire for starting and stopping the meter is carried through the centre of the vertical axis, so that the strain on it may not tend to pull the meter oblique to the current. The pitch of the screw is about 9 inches. The screws at x serve for filling the meter with water. The whole apparatus is fixed to a rod (fig. 143), of a length proportionate to the depth, or for very great depths it is fixed to a weighted bar lowered by ropes, a plan invented by Mr Revy. The instrument is generally used thus. The reading of the counter is noted, and it is put out of gear. The meter is then lowered into the water to the required position from a platform between two boats, or better from a temporary bridge. Then the counter is put into gear for one, two, or five minutes. Lastly, the instrument is raised and the counter again read. The velocity is deduced from the number of rotations in unit time by the formulae given below. For surface velocities the counter may be kept permanently in gear, the screw being started and stopped by hand.

130. The Harlacher Current Meter.— A very interesting modification of the current meter is that made by Amsler Laffon of Schaffhausen, which is described in Heusinger von Wald egg (Handb. der Ingenieur - Wissenschaften, iii. p. 284). In this the ordinary counting apparatus is abandoned. A worm drives a worm-wheel, which makes an electrical contact once for each 100 rotations of the worm. This contact gives a signal above water. With this arrangement, a series of velocity observations can be made, without removing the instrument from the water, and a number of practical difficulties attending the accurate starting and stopping of the ordinary counter are entirely got rid of. Fig 144 shows the meter. The worm wheel z makes one rotation for 100 of the screw.

' A pin moving the lever x makes the electrical contact. The wires b, c are led through a gas pipe B; this also serves to adjust the meter to any required position on the wooden rod dd. The rudder or vane is shown at WH. The galvanic current acts on the electromagnet m, which is fixed in a small metal box containing also the battery. The magnet exposes and withdraws a coloured disk at an opening in the cover of the box.

Moore's Current Meter.— The difficulties in using the ordinary current meter have been overcome to a great extent by an arrangement of another kind, invented by Mr B. T. Moore (Proc. Inst. Civil Eng., xlv. 220). This instrument (fig. 145) can be lowered into the water to any required depth by a light cord or chain. The counting arrangement inside the meter can be started or stopped at any instant. The instrument consists of a light brass frame carry ing an ogival head, and long rudder, cross-shaped in section. The frame is suspended in a stirrup, and if necessary a lead weight can be suspended below the meter. A rotating cylinder with screw blades is placed behind the ogival head. The centre of gravity of the instrument is accurately in the intersection of the axis of the stirrup-bearings and the longitudinal axis of the instrument. The rotating cylinder is started by releasing a spring by a cord. The recording mechanism is inside the rotating cylinder. The instrument is put in motion by a very small force. Some experiments made by towing it in still water gave the following equations :—

For speeds giving more than sixty rotations per minute,

ι>=Γ2R, where v is the velocity of the water relatively to the instrument in feet per minute, and R the number of rotations per minute. For lower speeds, v=R + 12.

It would appear therefore that the instrument will record velocities down to 12 feet per minute. Mr Moore states that a velocity at any depth down to 20 feet can be taken in five minutes, the meter being raised and lowered much more easily than when it is attached to a rod.

Determination of the Coefficients of the Current Meter.— Suppose a series of observations have been made, by towing the meter in still water, at different speeds, and it is required from these to ascertain the coefficients of the meter. A formula must be assumed to connect the observed velocities v with the number of rotations per second n. Then, in determining the coefficients of the formula from the given observations, the condition to be fulfilled is that the sum of the squares of the differences between the observed results and those given by the formula should be a minimum.

Let the formula assumed be of the form v=an + β (1).

Then the difference in any case between the observed and calculated quantity is v - an - β ; and therefore 2(v - an - β) 2 is to be a minimum.

The coefficients being independent, we must equate separately to zero the differential coefficient s of the expression with respect to the two coefficients.

2 [ (υ - an - β)n J = 0;

2[(v-ατι-0)] =0; whence

2(n 8 )a + 2(n)0=2(wi);

2(π)a+2(n)j8 = 2(r); from which a and β are easily determined.

Exner has shown {Zeitschrift fur Bauwesen, 1875) that the relation between the velocity of the water and the number of rotations of the meter is better expressed by the formula r=√(α 2 r 2 +v 0 2 ) (2)

than by that generally used, r 0 is sensibly equal to the velocity at which the meter just ceases to revolve; and a is a constant determined by experiments at different sj>eeds. Other expressions have been given, but they are more complicated and not more accurate than (1) and (2).

131. Darcy Gauge or Modified Pitot Tube.— A very old instrument for measuring velocities, invented or used by Pitot, consisted simply of a vertical glass tube with a right-angled bend, placed so that its mouth was normal to the direction of flow (fig. 146).

The impact of the stream on the mouth of the tube balances a column in the tube, the height of which is approximately h = — , where v is

2ø

the velocity at the depth x. Placed with its mouth parallel to the stream the water inside the tube is nearly at the same level as the surface of the stream, and turned with the mouth down v s stream, the fluid sinks a depth nearly, though the tube in that case interferes with the free flow of the liquid and somewhat modifies the result. Pitot expanded the mouth of the tube so as toform a funnel or bell mouth. In that case he found by experiment *-≤∙

The objection to this is that the motion of the stream is interfered with, and it is no longer certain that the velocity in front of the orifice is exactly the velocity of the unobstructed stream. Darcy preferred to make the mouth of the tube very small, partly to avoid interference with the stream, partly to check oscillations of the eolumn. In that case he found the difference of level of two tubes, such as A and B, to be almost exactly

One objection to the Pitot tube in its original form was the great difficulty and inconvenience ofreading the height 7⅛in 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 147 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 horizontally. 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 supported 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 required 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 rn, 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 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.

132. Hydrodynamometer of Μ. Perrodil.— This consists of a frame abed (fig. 148) 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 abed. To raise and lower the apparatus easily, it is not fixed directly to the rod mn, but to a tube kl sliding on wn.

Suppose the apparatus arranged so that the disk a: is at that level in the stream where the velocity is to be determined. The plane abed 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 abed 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, l 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 m-e ∣ i4; i/ where E e 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 = ⅛τrr 4 . 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

Fδ-fcδ⅛rW, where G is the heaviness of water and k an experimental coefficient. Then

E,l4-^⅛rR s v a ∙

; 2ÿ

For any given instrument, φ ≡ c √- 5 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 [table]

For a thin circular plate, the coefficient ⅛=1'12. In the actual Instrument the torsion rod was a brass wire 0 ‘06 inch diameter and 6⅜ feet long. Supposing a measured in degrees, we get by calculation r=0∙885√β; 0T15√^ ι 0∙042√^ for the three disks.

Very careful experiments were made with the instrument. It was fixed to a wooden turning bridge, revolving over a circular channel of 2 feet width, and about 76 feet 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^fa,

1st disk, c=0 - 3126 for velocities of 3 to 16 feet

2d „ ΟΊ177 „ · „ 1⅛ to 3⅛ „

3d ,, 0 ’ 0349 ,, ,, less than 1⅛ ,, or values little different from the values calculated from the torsion.

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. At very low velocities the current meter fails altogether.

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.

Processes for Gauging Streams.

133. Gauging by Observation of the Maximum Surface Velocity.— The method of gauging which involves the least trouble is to determine the surface velocity at the thread of the stream, and to deduce from it the mean velocity of the whole cross section. The maximum surface velocity may be determined by floats or by a current meter. Unfortunately, however, the ratio of the maximum surface to the mean velocity is extremely variable. Thus putting v 0 for the surface velocity at the thread of the stream, and v n for the mean velocity of the whole cross section, — has been found to have the following , v o values :— [table]

Various formulae, either empirical or based on some theory of the vertical and horizontal velocity curves, have been proposed for determining the ratio ⅛ . Bazin found from his experiments the empirical expression v» * s * vQ ~ 25 '4 ’ ∖ fmi J where m is the hydraulic mean depth and ithe slope of the stream. In article 101, it has already been shown how from this formula the ratio — can be obtained for different kinds of channels.

v °

In the case of irrigation canals and rivers, it is often important to determine the discharge either daily or at other intervals of time, while the depth and consequently the mean velocity is varying. Captain Cunningham, R. E. {Roorkee Prof. Papers, vol. iv. p. 47), has shown that, for a given part of such a stream, where the bed is regular and.of . permanent section, a simple formula may be found for the variation of the central surface velocity with the depth. When once the constants of this formula have been determined by measuring the central surface velocity and depth, in different conditions of the stream, the surface velocity can be obtained by simply observing the depth of the stream, and from this the mean velocity and discharge cun be calculated. Let z be the depth of the stream, and t⅛ the surface velocity, both measured at the thread of the stream. Then ⅞ 2 =cz; where c is a constant which for the Solani aqueduct had the values 1 - 9 to 2, the depths being 6 to 10 feet, and the velocities 3⅜ to 4∣ feet. Without any assumption of a formula, however, the surface velocities, or still better the mean velocities, for different conditions of the stream may be plotted on a diagram in which the abscissae are depths and the ordinates velocities. The continuous curve through points so found would then always give the velocity for any observed depth of the stream, without the need of making any new float or current meter observations.

134. Mean Velocity determined by observing a Scries of Surface Velocities.— The ratio of the mean velocity to the surface velocity in one longitudinal section is better ascertained than the ratio of the central surface velocity to the mean velocity of the whole cross section. Suppose the river divided into a number of compartments by equidistant longitudinal planes, and the surface velocity observed in each compartment. From this the mean velocity in each compartment and the discharge can be calculated. The sum of the partial discharges will be the total discharge of the stream. When wires or ropes can be stretched across the stream, the compartments can be marked out by tags attached to them. Suppose two such ropes stretched across the stream, and floats dropped in above the upper rope. By observing within which compartment the path of the float lies, and noting the time of transit between the ropes, the surface velocity in each compartment can be ascertained. The mean velocity in each compartment is 0 85 to 0 91 of the surface velocity in that compartment. Putting k for this ratio, and v 1 , υ. 2 . . . for the observed velocities, in compartments of area β 1 , ß 2 . . . then the total discharge is

•Q = ⅛(β 1 υ 1 + Ω 2 υ 2 + . . . ) .

If several floats are allowed to pass over each compartment, the mean of all those corresponding to one compartment is to be taken as the surface velocity of that compartment.

This method is very applicable in the case of large streams or rivers too wide to stretch a rope across. The paths of the floats are then ascertained in this way. Let fig. 149 represent a portion of the river, which should be straight and free from obstructions. Suppose a base line AB measured parallel to the thread of the stream, and let the mean cross section of the stream be ascertained either by sounding the terminal cross sections AE, BF, or by sounding a series of equidistant cross sections. The cross sections are taken at right angles to the base line. Observers are placed at A and B with theodolites or box sextants. The floats are dropped in from a boat above AE, and picked up by another boat below BF. An observer with a chronograph or watch notes the time in which each float passes from AE to BF. The method of proceeding is this. The observer A sets his theodolite in the direction AE, and gives a signal to drop a float. B keeps his instrument on the float as it comes down. At the moment the float arrives at C in the line AE, the observer at A calls out. B clamps his instrument and reads off the angle ABC, and the time observer begins to note the time of transit. B now points his instrument in the direction BF, and A keeps the float on the cross wire of his instrument. At the moment the float arrives at D in the line BF, the observer B calls out, A clamps his instrument and reads off the angle BAD, and the time observer notes the time of transit from C to D. Thus all the data are determined for plotting the path CP of the float and determining its velocity. By dropping in a series ot floats, a number of surface velocities can be determined. When all these have been plotted, the river can be divided into convenient compartments. The observations belonging to each compartment are then averaged, and the mean velocity and discharge calculated. It is obvious that, as the surface velocity is greatly altered by wind, experiments of this kind should be made in very calm weather.

The ratio of the surface velocity to the mean velocity in the same vertical can be ascertained from the formulae for the vertical velocity curve already given (§ 101). Exner, in Erbkam's Zeitschrift for 1875, has given the following convenient formula. Let v be the mean and V the surface velocity in any given vertical longitudinal section, the depth of which is h υ l + θ∙1478√⅛,

V - ι + o-2216√7i

If vertical velocity rods are used instead of common floats, the mean velocity is directly determined for the vertical section in which the rod floats. No formula of reduction is then necessary, The observed velocity has simply to be multiplied by the area of the compartment to which it belongs.

135. Mean Velocity of the Stream from a Scries of Mid Depth Velocities.— In the gaugings of the Mississippi it was found that the mid depth velocity differed by only a very small quantity from the mean velocity in the vertical section, and it was uninfluenced by wind. If therefore a series of mid depth velocities are determined by double floats or by a current meter, they may be taken to be the mean velocities of the compartments in which they occur, and no formula of reduction is necessary. If floats are used, the method is precisely the same as that described in the last paragraph for surface floats. The paths of the double floats are observed and plotted, and the mean taken of those corresponding to each of the compartments into which the river is divided. The discharge is the sum of the products of the observed mean mid depth velocities and the areas of the compartments.

136. Boileau’s Process for Gauging Streams.— Let U be the mean velocity at a given section of a stream, V the maximum velocity, or that of the principal filament, which is generally a little below the surface, W and w the greatest and least velocities at the surface. The distance of the principal filament from the surface is generally less than one-fourth of the depth of the stream; W is a little less than V; and U lies between W and w. As the surface velocities change continuously from the centre towards the sides, there are at the surface two filaments having a velocity equal to U. The determination of the position of these filaments, which Boileau terms the gauging filaments, cannot be effected entirely by theory. But, for sections of a stream in which there are no abrupt changes of depth, their position can be very approximately assigned. Let Δ and l be the horizontal distances of the surface filament, having the velocity W, from the gauging filament, which has the velocity U, and from the bank on one side. Then

Δ 41 W + 2m> ^T~ c √7(W-w)'

c being a numerical constant. From gaugings by Humphreys and Abbot, Bazin, and Baumgarten, the values c=0 ’ 919, 0 - 922, and 0 - 925 are obtained. Boileau adopts as a mean value 0’922. Hence, if W and w are determined by float gauging or otherwise, Δ can be found, and then a single velocity observation at Δ feet from the filament of maximum velocity gives, without need of any reduction, the mean velocity of the stream. More conveniently W, w, and U can be measured from a horizontal surface velocity curve, obtained from a series of float observations.

137. Direct Determination of the Mean Velocity by a Current Meter or Darcy Gauge. — The only method of determining the mean velocity at a cross section of a stream which involves no assumption of the ratio of the mean velocity to other quantities is this—a plank bridge is fixed across the stream near its surface. F rom this, velocities are observed at a sufficient number of points in the cross section of the stream, evenly distributed over its area. The mean of these is the true mean velocity of the stream. In Darcy and Bazin’s experiments on small streams, the velocity was thus observed at 36 points in the cross section.

When the stream is too large to fix a bridge across it, the observations may be taken from a boat, or from a couple of boats with a gangway between them, anchored successively at a series of points across the width of the stream. The position of the boat for each series of observations is fixed by angular observations to a base line on shore.

138. Harlacher's Graphic Method of determining the Discharge from a Series of Current Meter Observations.— Let ABC (fig. 150) be the cross section of a river at which a complete series of current meter observations have been taken. Let I., II., III. . . . be the verticals at different points of which the velocities were measured. Suppose the depths at I., II., III., . . . (fig. 150), set off as vertical ordinates in fig. 151, and on these vertical ordinates suppose the velocities set off horizontally at their proper depths. Thus, if v is the measured velocity at the depth h from the surface in fig. 150, on vertical marked 11I., then at III. in fig. 151 take cd≈h and αc=v.

Thend is a point in the vertical velocity curve for the vertical III., and, all the velocities for that ordinate being similarly set off, the curve can be drawn. Suppose all the vertical velocity curves I. . . . V. (fig. 151), thus drawn. On each of these figures draw verticals corresponding to velocities of x. 2x, 3a;. . . feet per second. Then for instance cd at III. (fig. 151) is the depth at which a velocity of 2x feet per second existed on the vertical III. in fig. 150, and if cd is set off at III. in fig. 150 it gives a point in a curve passing through points of the section where the velocity was 2x feet per second. Set off on each of the verticals in fig. 150 all the depths thus found in the corresponding diagram in fig. 151. Curves drawn through the corresponding points on the verticals are curves of equal velocity.

The discharge of the stream per second may be regarded as a solid having the cross section of the river (fig. 150), as a base, and cross sections normal to the plane of fig. 150 given by the diagrams in fig. 151. The curves of equal velocity may therefore be considered as contour lines of the solid whose volume is the discharge of the stream per second. Let Ω o be the area of the cross section of the liver, Ω 1 , Ω 2 . . . . the areas contained bythe successive curves of equal velocity, or, if these cut the surface of the stream, by the curves and that surface. Let x be the difference of velocity for which the successive curves are drawn, assumed above for simplicity at 1 foot per second. Then the volume of the successive layers of the solid body whose volume represents the discharge, limited by successive planes passing through the contour curves, will be

⅛r(Ω 0 + Ω 1 ), ⅛x(Ω 1 + Ω a ), and so on.

Consequently the discharge is

Q = x j ⅛+⅛ + ω i + ω 2 + . . . + Ω,- 1 ∣ .

The areas Ω 0 ,Ω 1 . . . . are easily ascertained by means of the polar planimeter. A slight difficulty arises in the part of the solid lying above the last contour curve. This will have generally a height which is not exactly x, and a form more rounded than the other layers and less like a conical frustum. The volume of this may be estimated separately, and taken to be the area of its base (the area Ω n ) multiplied by ⅛ to ⅜ its height.

Fig. 152 shows the results of one of Professor Harlacher’s gaugings worked out in this way. The upper figure shows the section of the river and the positions of the verticals at which the soundings and gaugings were taken. The lower gives the curves of equal velocity, worked out from the current meter observations, by the aid of vertical velocity curves. The vertical scale in this figure is ten times as great as in the other. The discharge calculated from the contour curves is 14 '1087 cubic metres per second. In the lower figure some other interesting curves are drawn. Thus, the uppermost dotted curve is the curve through points at which the maximum velocity was found; it shows that the maximum velocity was always a little below the surface, and at a greater depth at the centre than at the sides. The next curve shows the depth at which the mean velocity for each vertical was found. The next is the curve of equal velocity corresponding to the mean velocity of the stream; that is, it passes through points in the cross section where the velocity was identical with the mean velocity of the stream.

XII. IMPACT AND REACTION OF WATER.

When a stream of fluid impinges on a solid surface, it presses on the surface with a force equal and opposite to that by which the velocity and direction of motion of the fluid are changed. Generally, in problems on the impact of fluids, it is necessary to neglect the effect of friction between the fluid and the surface on which it moves.

139. During Impact the Velocity of the Fluid relatively to the Surface on which it impinges remains unchanged in Magnitude.— Con-aider a mass of fluid flowing in contact with a solid surface also in motion, the motion of both fluid and solid being estimated relatively to the earth. Then the motion of the fluid may be resolved into two parts, one a motion equal to that of the solid, and in the same direction, the other a motion relatively to the solid. The motion which the fluid has in common with the solid cannot at all be influenced by the contact. The relative component of the motion of the fluid can only be altered in direction, but not in magnitude. The fluid moving in contact with the surface can only have a relative motion parallel to the surface, while the pressure between the fluid and solid, if friction is neglected, is normal to the surface. The pressure therefore can only deviate the fluid, without altering the magnitude of the relative velocity. The unchanged common component and, combined with it, the deviated relative component give the resultant final velocity, which may differ greatly in magnitude and direction from the initial velocity.

From the principle of momentum the impulse of any mass of fluid reaching the surface in any given time is equal to the change of momentum estimated in the same direction. The pressure between the fluid and surface, in any direction, is equal to the change of momentum in that direction of so much fluid as reaches the surface in one second. If P α is the pressure in any direction, m the mass of fluid impinging per second, v a the change of velocity in the direction of P β due to impact, then P a = ιnva .

If ν 1 (fig. 153) is the velocity and direction of motion before impact, V» that after impact, then v is the total change of motion due to impact. The resultant pressure of the fluid on the surface is in the direction of v, and is e<pιal to v multiplied by the mass impinging per second. That is, putting P for the resultant pressure, P=mv.

Let P be resolved into two components,

N and T, normal and tangential to the direction of motion of the solid on which the fluid impinges. Then N is a lateral force producing a pressure on the supports of the solid, T is an effort which does work on the solid. If u is the velocity of the solid, Tu is the work done per second by the fluid in moving the solid surface.

Let Q be the volume, and GQ the weight of the fluid impinging per second, and let v 1 be the initial velocity of the fluid before striking the surface. Then ⅜-Qv 1 2 is the original kinetic energy of Q cubic feet of fluid, and the efficiency of the stream considered as an arrangement for moving the solid surface is

Tu n=—p ·

4 >' s

140. Jet deviated entirely in one Direction—Geometrical Solution

(fig. 154).—Suppose a jet of water impinges on a surface ac with a velocity ab, and let it be wholly deviated in planes parallel to the figure. Also let ae be the velocity and direction of motion of the surface. Join eb ; then the water moves with respect to the surface in the direction and with the velocity eb. As this relative velocity is unaltered by contact with the surface, take cd=eb, then cd is the relative motion of the water with respect to the surface ate. Take df equal and parallel to ae. Then fc (obtained by compounding the relative motion of water to surface and common velocity of water and surface) is the absolute velocity and direction of the water leaving the surface. Take ag equal and parallel to fc. Then, since ab is the initial and ag the final velocity and direction of motion, gb is the total change of motion of the water. The resultant pressure on the plane is in the direction gb. Join eg. In the triangle gae, ae is equal and parallel to df, and ag to fc. Hence eg is equal and parallel to cd. But cd=eb = relative motion of water and surface. Hence the change of motion of the water is represented in magnitude and direction by the third side of an isosceles triangle, of which the other sides are equal to the relative velocity of the water and surface, and parallel to the initial and final directions of relative motion.

Special Cases.

141. (1) A Jet impinges on a plane surface at rest, in a direction normal to theplane (fig. 155).—Let ajet whose section is ω impinge with a velocity v on a plane surface at rest, in a direction normal to the plane. The particles approach the plane, are gradually deviated, and finally flow away parallel to the plane, having then no velocity in the original direction of the jet. The quantity of water impinging per second is a>ν. The pressure on the plane, which is equal to the change of j . G ο momentum per second, is — ωv.

(2) If theplane is moving in the direction of the jet with the velocity ±u, the quantity impinging per second is ⅛ z Fii). The momentum of this quantity before impact is — ω(rψu) u After impact, the water still possesses the velocity ±w in the direction of the jet; and the momentum, in that direction, of so much water as impinges in one second, after impact, is - t — ω{v^u)u. The pressure on the plane, which is the change of momentum per second, is the difference of these quantities or — α>(i)ψii) 3 . This differs from the expression obtained in the previous case, in that the relative velocity of the water and plane v÷Vιι is substituted for v. The expression may be written P=2 ×Gxω ’ w ^ lere ^ ιe ^ ast term is the volume of a prism of water whose section is the area of the jet and whose length is the head due to the relative velocity. The pressure on the plane is twice the weight of that prism of water. The work done on the plane in this case is Pu = — ω (vψu) 2 foot-pounds per second. There issue from the jet a>v cubic feet per second, and the energy of this quantity before impact is ωv 3 . The efficiency of the jet is therefore η = 2 u. The value of u which makes this a maximum is found by differentiating and equating the differential coefficient to zero :—

<⅞ 0 ∕% 2 ψ4vu + 3u 2 ∖ _

du ∖ v 3 J ’

.∙. u=uor⅛v.

The former gives a minimum, the latter a maximum efficiency. Putting u = ⅛v in the expression above, η max. =∙⅛ ·

(3) If, instead of one plane moving before the jet, a series of planes are introduced at short intervals at the same point, the quantity of water impinging on the series will be ωv instead of ω{v - u), and the whole pressure = —α>v(v - u). The work done is ωvu{υ - u). The efficiency ωvu(v-u) ÷~ωv 3 = 2u^ v . This becomes a maximum for — 2(v - 2u) = 0, or u = ⅜v. This result is often used as an approximate expression for the velocity of greatest efficiency when a jet of water strikes the floats of a water wheel. The work wasted in this case is half the whole energy of the jet when the floats run at the best speed.

142. (4) Case of a Concave Cup Vane, velocity of water v, velocity of vane in the same direction u (fig. 156).

If the cup is hemispherical, the water leaves the cup in a direction parallel to the jet. Its relative velocity is v- u when approaching the cup, and - (v - u) when leaving it. Hence its absolute velocity when leaving the cup is u - (v - u) = 2u - v. The change of momen-G G G

tum per second = -ω(pu)v ω (p- u)(2u - v) ≈ 2~- a>(v - u) a .

Comparing this with case 2, it is seen that the pressure on a hemispherical cup is double that on a flat plane. The work done on Q the cup≈2- ω (v-u) 2 u foot-pounds per second. The efficiency of the jet is greatest when v≈2u∙, in that case the efficiency ⅜* If a series of cup vanes are introduced in front of the jet, so that the quantity of water acted upon is ωv instead of ω(v - w), then the whole pressure on the chain G G G

of cups is —ωv 2 uv(2u-v) = 2-av(v-u). In this case the efficiency is greatest when v=2u, and the maximum efficiency is unity, or all the energy of the water is expended on the cups.

143. (5) Case of a Flat Vane oblique to the Jet (fig. 157).—This case presents some difficulty. The water spreading on the plane in all directions from the point of impact, different particles leave the plane with different absolute velocities. Let AB = ι = velocity of water, AC = u= velocity of plane. Then, completing the parallelogram, AD represents in magnitude and direction the relative velocity of water and plane. Draw AE normal to the plane and DE parallel to the plane. Then the relative velocity AD may be regarded as consisting of two components, one AE normal, the other DE parallel to the plane. On the assumption that friction is insensible, DE is unaffected by impact, but AE is destroyed. Hence AE represents the entire change of velocity due to impact and the direction of that change. The pressure on the plane is in the direction AE, and its amount is = mass of water impinging per second × AE.

Let DAE = 0, and let AD = v r . Then AE = v r cos 0; DE = v r sin 0. If Q is the volume of water impinging on the plane per second, the change of momentum is — Qv r cos 0. Let AC = u = velocity of the plane, and let AC make the angle CAE = δ with the normal to the plane. The velocity of the plane in the direction AE = u cos δ. The work of the jet on the plane = θ Qι⅛ cos 0 u cos δ. The same problem may be thus treated algebraically (fig. 158). Let BAF = α, and CAF = δ. The velocity v of the water may be decomposed into AF = v cos a normal to the plane, and F B = v sin a parallel to the plane. Similarly the velocity of the plane = u = AC = BD can be decomposed into BG = FE = u cos δ normal to the plane, and DG = u sin δ parallel to the plane. As friction is neglected, the velocity of the water parallel to the plane is unaffected by the impact, but its component v cos α normal to the plane becomes after impact the same as that of the plane, that Is, u cos δ. Hence the change of velocity during impact = AE = v cos a-u cos δ. The change of momentum per second, and consequently the normal pressure on the plane is N = — Q (v cos a - u cos δ). The pressure in the direction in which the plane is moving is P = N cos δ = — Q(υ cos α - u cos δ) cos δ, and the work done on the plane is g G Pιι≈^-Q(v cos a - u cos δ) u cos δ, which is the same expression as before, since AE = υ r cos θ== V cos a-u cos δ.

In one second the plane moves so that the point A (fig. 159) comes to C, or from the position shown in full lines to .the position shown in dotted lines. If the plane remained stationary, a length AB = v of the jet would impinge on the plane, but, since the plane moves in the same direction as the jet, only the length HB=AB-AH impinges on the plane.

n 1 ,τr i r, cos δ cos δ ,, ττr , cos δ τ .

But AH = AC = u , and therefore HB = ν - u . Let cos α cos a cos a

ω= sectional area of jet; volume impinging on plane per second r , ( cos δ∖ HCOSα -iZCOS5 τ ,. ,,. . ,.

= Q = ω v-u =ω . Inserting this in the

∖ cos a J cos a t, formulae above, we get

N = — (ν cos au cos δ) 2 .... (1); g cosα κ '

τ, G W COS δ . . ∖ λ /ox P= V n (υ cn3a ~ u cos δ) (2); τ , G cos δ . ...

Ptt=-BB (v cos au cos δ) 2 .... (3).

g cos a

Three cases may be distinguished :—

(a) The plane is at rest. Then «=0, N = — ωv n ∙ cos a ; and the work done on the plane and the efficiency of the jet are zero.

(⅛) The plane moves parallel to the jet. Then δ = α, and G

P«”— ωu cos 2 α (v-w) 2 , which is a maximum when u=⅛v.

G

When u≈⅛υ then Ph max. =√ r — w υ 3 cos 2 α, and the efficiency = ιj = I cos 2 α.

(c) The plane moves perpendicularly to the jet. Then δ = 90°- a; cos δ = sin a; and Pm = — ωtι (v cos a - u sin a) 2 . This is a g cos a ' maximum when u- ∖ v cos a.

When ω = Jυ cos a, the maximum work and the efficiency are the same as in the last case.

144. Best Form of Vane to receive Water.— When water impinges normally or obliquely on a plane, it is scattered in all directions after impact, and the work carried away by the water is then generally lost, from the impossibility of dealing afterwards with streams of water deviated in so many directions. By suitably forming the vane, however, the water may be entirely deviated in one direction, and the loss of energy from agitation of the water is entirely avoided.

Let AB (fig. 160) be a vane, on which a jet of water impinges at the point A and in the direction AC. Take AC = υ= velocity of water, and let AD represent in magnitude and direction the velocity of the vane. Completing the parallelogram, DC or AE represents the direction in which the water is moving relatively to the vane. If the lip of the vane at A is tangential to AE, the water will not have its direction suddenly changed when it impinges on the vane, and will therefore have no tendency to spread laterally. On the contrary it will be so gradually deviated that it will glide up the vane in the direction AB. This is sometimes expressed by saying that the vane receives the water without shock.

145. Floats of Poncelet Water Wheels.— Let AC (fig. 161) represent the direction of a thin horizontal stream of water having the velocity'^. Let AB be a curved float moving horizontally with velocity u. The relative motion of water and float is then initially horizontal, and equal to vu.

In order that the float may receive the water without shock, it is necessary and sufficient that the lip of the float at A should be tangential to the direction AC of relative motion. At the end of seconds the float moving with the velocity u comes to the position A∣B 1 , and during this time a particle of water received at A and gliding up the float with the relative velocity v - u, attains a iy <jA2

height DE= — o -. At E the water comes to relative rest. It then descends along the float, and when after ~ seconds the g float has come to A 2 B 2 the water will again have reached the lip at A 2 and will quit it tangentially, that is, in the direction CA a , with a relative velocity - (v-u)= - √2yDE acquired under the influence of gravity. The absolute velocity of the water leaving the float is therefore u -(υ-u y ) = 2u - v. If u = ∣ v, the water will drop off the bucket deprived of all energy of motion. The whole of the work of the jet must therefore have been expended in driving the float. The water will have been received without shock and discharged without velocity. This is the principle of the Poncelet wheel, but in that case the floats move over an arc of a large circle; the stream of water has considerable thickness (about 8 inches); in order to get the water into and out of the wheel, it is then necessary that the lip of the float should make a small angle (about 15°) with the direction of its motion. The water quits the wheel with a little of its energy of motion remaining.

146. Pressure on a Curved Surface when the Water is deviated wholly in one Direction. — When a jet of water impinges on a curved surface in such a direction that it is received without shock, the pressure on the surface is due to its gradual deviation from its first direction. On any portion of the area the pressure is equal ami opposite to the force required to cause the deviation of so much water as rests on that surface. In common language, it is equal to the centrifugal force of that quantity of water.

Case 1. Surface Cylindrical and Stationary. — Let AB (fig. 162) be the surface, having its axis at 0 and its radius = r. Let the water impinge at A tangentially, and quit the surface tangentially at B. Since the surface is at rest, v is both the absolute velocity of the water and the velocity relatively to the surface, and this remains unchanged during contact with the surface, because the deviating force is at each point perpendicular to the direction of motion. The water is deviated through an angle BCD = AOB = φ. Each particle of water of weight p exerts radially a centrifugal force L— . Let the thickness of rg the stream =t feet. Then the weight of water resting on unit of surface = Gt lb; and the normal pressure per unit of surface = sil =- . The resultant of the radial pressures uniformly distributed from A to B will be a force acting in the direction OC bisecting AOB, and its magnitude will equal that of a force of intensity = n, acting on the projection of AB on a plane perpendicular to the direction OC. The length of the chord AB = 2rsin-~-j let b = breadth of the surface perpendicular to the plane of the figure. The resultant pressure on surface

= R~2rδ sin ≠ × — . — - 2 — bt v i sir. f- ,

2 g v g 2

whieh is independent of the radius of curvature. It may be inferred that the resultant pressure is the same for any curved surface of the same projected area, which deviates the water through the same angle.

Case 2. Cylindrical Surface moving in the Direction AC with Velo city u.-The relative velocity ≈v-u. The final velocity BF (fig. 163) is found by combining the relative velocity BD = ∙v-w tangential to the surface with the velocity BE=u of the surface. The intensity of normal pressure, as in the last case, is — -~~' The resultant normal pressure R = 2^W (t>-w) 1 sin-^-∙ This resultant pressure may be resolved into two components P and L, one parallel and the other perpendicular to the direction of the vane’s motion. The former is an effort doing work on the vane. The latter is a lateral force which does no work.

P = Rsin⅜=- bt(p-u)* (l-cos ≠); λ fl

L=R cos -π^=- bt(p-u)* sin φ. 9 G

The work done by the jet on the vane is Pw = — btu(p - w) 2 (l - cos φ), which is a maximum when u=⅛v. This result can also be obtained by considering that the work done on the plane must be equal to the energy lost by the water, when friction is neglected.

If ≠ = 180°, cos ≠=-l, l-cosø = 2; then P = 2 — bt(v-u) 9 , the same result as for a concave cup.

147. Position which a Movable Plane takes in Flowing Water.— When a rectangular plane, movable about an axis parallel to one of its sides, is placed in an indefinite current of fluid, it takes a position such that the resultant of the normal pressures on the two sides of the axis passes through the axis. If, therefore, planes pivoted so that the ratio (fig. 164) b is varied are placed in water, and the angle they make with the direction of the stream is observed, the position of the resultant of the pressures on the plane is determined for different angular positions. Experiments of this kind have been made by Herr Hagen. Some of his results are given in the following table :— [table]

148. Effect of Friction during Impulse.— Thus far the effect of the friction between the water and the surface which deviates it has been neglected. Nothing precise is known of its mode of action, and the following investigation is in part conjectural (Rankine, Steam Engine, p. 171, § 146).

Let it be assumed that the friction causes a loss of energy per second proportional to the height due to the velocity of the water relatively to the surface; that is, the head due to the relative velocity being the loss of head due to friction will be/U; the ⅛ ig V whole energy due to the relative head being GQ^, the loss of 2^2

energy due to friction will be GQ/—-.

Cylindrical Surface with Water deviated wholly in one Direction, Friction taken into account.— In Case 2, discussed in § 146, the velocity of the water relatively to the surface is v - u. The quantity of water impinging per second is blip - u). The loss of head due to friction is f ~ -- . The loss of energy due to friction is 2g Gbtf . The energy exerted on the surface, after deducting the loss due to friction, is Pu = — bt(v - u) 2 u(l - cos φ^) - Gbtf g ⅛l = — bt(p - u) 2 j ∙u(l - cos φ y ) -f v - i f±- I · g ∖ z '

The efficiency when friction is taken into account becomes Pu (v ~ i 2 “(1 - c0s ~ J Ί = >

GW(V-u)^

and this becomes a maximum if

1 - cos φ +f

O∕ = L - W .

2-2cos≠+∕ ’

being greater than the speed when friction is neglected in the ratio 2(1 - cos φ) +f : 3(1 - cos φ +∕).

Suppose that the speed of greatest efficiency u has been found by experiment;

, (2u - v)(l - cos φ) ∙*~ v-u

149. Direct Action distinguished from Reaction (Rankine, Steam Engine, § 147).

The pressure which a jet exerts on a vane can be distinguished into two parts, viz :—

(1) The pressure arising from changing the direct component of the velocity of the water into the velocity of the vane. In fig. 154, § 140, ab cos bae is the direct component of the water’s velocity, or component in the direction of motion of vane. This is changed into the velocity αe of the vane. The pressure due to direct impulse is then p _ g q ab cos bae - ae 1 9 ‘ '

For a flat vane moving normally, this direct action is the only action producing pressure on the vane.

(2) The term reaction is applied to the additional action due to the direction and velocity with which the Vater glances off the vane. It is this which is diminished by the friction between the water and the vane. In Case 2, § 146, the direct pressure is P 1 - gw ⅛→. 1 s

That due to reaction is

P s ≈-G¾ —cos φ. g ψ

If ≠<≤90°, the direct component of the water’s motion is not wholly converted into the velocity of the vane, and the whole pressure due to direct impulse is not obtained. If ψ≥90°, cos φ is negative and an additional pressure due to reaction is obtained.

150. Reaction of a Jet issuing from a Vessel. —Suppose a vessel filled with water (fig. 165), having an orifice of area ω, from which water issues horizontally with a velocity v≈ ∖ ∕2gh. The volume discharged per second, neglecting contraction, = a>v. The momentum generated per second in a horizontal direction = — ωv i ∙, and this is equal to the g force producing the change of momentum.

Hence the horizontal force or reaction R, acting on the side of the vessel, opposite to the orifice, and equal and opposite to the force producing the momentum, is

R = — ωι 2 = 2Gα>7⅛; g this is the weight of a column of water the section of which is the area of the orifice, and the height is twice the head.

If the vessel moves in a direction opposite to that of the jet, with the velocity u, the absolute velocity of the water leaving the vessel is v - u. The momentum generated per second is — ωvlp - u) = R∙

Jet Propeller. — In the case of vessels propelled by a jet of water (fig. 166), driven sternwards from orifices at the side of the vessel, the water, originally at rest outside the vessel, is drawn into the ship and caused to move with the forward velocity V of the ship. Afterwards it is projected sternwards from the jets with a velocity v relatively to the ship, or V - V relatively to the earth. If Ω is the total sectional area of the jets, Ωυ is the quantity of water discharged per second. The momentum generated per second in a stern ward direction is — Ωv(v - V), and this is equal to the forward acting reaction P which propels the ship.

The energy carried away by the water

= ⅜y∙ Ωι>(t>-V)≡ (1).

The useful work done on the ship

PV=-2-Ωφ-V)V (2).

Adding (1) and (2), we get the whole work expended on the water, neglecting friction :—

τrr G _ v 2 -V 2

W =—Ωv— ö — · g 2

Hence the efficiency of the jet propeller is

ZX=JΣ- . (3).

W « + V v

This increases towards unity as υ approaches V. In other words, the less the velocity of the jets exceeds that of the ship, and therefore the greater the area of the orifice of discharge, the greater is the efficiency of the propeller.

In the “Waterwitch” υ was about twice V. Hence in this case the theoretical efficiency of the propeller, friction neglected, was. about ⅜.

151. Pressure of a Steady Stream in a Uniform Pipe on a Plane normal to the Direction of Motion.— Let CD (fig. 167) be a plane placed normally to the stream which, for simplicity, may be supposed to flow horizontally. The fluid filaments are deviated in front of the plane, form a contraction at A 1 A 1 , and converge again, leaving-a mass of eddying water behind the plane. Suppose the section A 0 A 0 taken at a point where the parallel motion has not begun to be disturbed, and A 2 A 2 where the parallel motion is reestablished. Then, since the same quantity of water with the same velocity passes A 0 A 0 , A 2 A 2 in any given time, the external forces produce no change of momentum on the mass A 0 A 0 A 3 A 2 , and must therefore be in equilibrium. If Ω is the section of the stream at A 0 A 0 or A 2 A 2 , and ω the area of the plate CD, the area of the contracted section of the stream at A 1 A 1 will be c c (Ω-ω), where c e is the coefficient of contraction. Hence, if v is the velocity at A 0 A 0 or A t A 2 , and ν γ the velocity at A 1 A 1 ,

τ∙Ω=c e v 1 (Ω-ω);

Ω zι λ ’ vχ ^ υ ^fμ-ω) (1) ’

Let p 0 , p 1 , ρ s be the pressures at the three sections. Applying Bernoulli’s theorem to the sections A 0 A 0 and A 1 A 1 ,

G + 2g G + 2< z ,

Also, for the sections A 1 A 1 and A ij A 2 , allowing that the head due to the relative velocity v 1 - υ is lost in shock :—

ft. ⅛L Pi . v * , (⅜⅛-ι>) a . G^ 1 ^2g G^ t 2y 2g ’ .'. Po-^ 3 =G¾^ (2); or, introducing the value in (1), y ∙-*-⅛(ι√0 t ⅛- 1 ) ’ * ,t · is) ·

Now the external forces in the direction of motion acting on the mass A 0 A 0 A 2 A 2 are the pressures n 0 Ω, - n 2 Ω at the ends, and the reaction - R of the plane on the water, which is equal and opposite to the pressure of the water on the plane. As these are in equilibrium, (Po-2⅛)Ω-R = O; ·· r = go (⅛⅛→) , 2 7 · · < 4 > i an expression like that for the pressure of an isolated jet on an indefinitely extended plane, with the addition of the term in brackets winch depends only on the areas of the stream and the plane. For a given plane, the expression in brackets diminishes as Ω increases.

If —=p, the equation (4) becomes κ ^ α "S i ’ ⅛⅛⅛- ι )Ι · · · · < te >> which is of the form

2^2

R = Gω δ -×K, where K depends only on the ratio of the sections of the stream and plane.

For example, let e c =0*85, a value which is probable, if we allow that the sides of the pipe act as internal borders to an orifice. Then M 117 ⅛- 1 ) ’ ∙ [table]

The assumption that the coefficient of contraction c c is constant for different values of p is probably only true when p is not very large. Further, the increase of K for large values of p is contrary to experience, and hence it may be inferred that the assumption that all the filamentshave a common velocity υ 1 at the section A 1 A 1 and a common velocity v at the section A 2 A 2 is not true when the stream is very much larger than the plane. Hence, in the expression

R = KGα>∣J,

K must be determined by experiment in each special case.

152. Pressure on a Cylindrical Body of a Length about three times its Diameter.— A contraction of the stream is formed at A 1 A 1

(fig. 168). Let the same notation be used, the subscript figures indicating the section to which the quantities belong.

For sections A 0 A 0 , A 1 A 1 ,

£14.11! ·

G ^ t ^2y G^2g , for sections A 1 A 1 and A 2 A 2 , allowing for the abrupt enlargement of the stream,

114.1L ft i v ' i < t, ι~⅞) 8 ·

G 2g G^ v 2g 2g ’ and for sections A 2 A 2 , A 3 A 3 , allowing for another abrupt enlargement, Pi l . i ι,a - (⅞→') a G 2 i7 G^2y 2g '

Adding the three equations,

„ n r H t, ι-*⅛) 2 ∣ (¾→)Ι

j>.-λ-o ∣ -⅛-+- v - ∣ ∙

From the principle of momentum, (-Po-P3) β -R=θ; κ . 00 j(⅛(-⅛⅛T Putting « for the section of the body, c c for the coefficient of contraction, c c (Ω - «) for the area of the stream at A 1 A 1 ,

Ω Ω .

vι = v —7n ∖ > ®a = î, 7S ’

1 c c (Ω - «) 2 Ω - α>

Ω or, putting p = — > P P ν 1 = ν ~7—TΓ > r.>=υ- j 1 - · 1 c e (p-l) ’ 2 p-1

Then r 2 R=K 1 Gα>^, where

χ.-d(A) ’ GrO÷(⅛- 1 )'i∙

Taking c c = 0 - 85 and p = 4, K 1 =0 ’ 467, a value less than before. Hence there is less pressure on the cylinder than on the thin plane.

153. Distribution of Pressure on a Surface on which a Jet impinges normally.— The principle of momentum gives readily enough the total or resultant pressure of a jet impinging on a plane surface, but in some cases it is useful to know the distribution of the pressure. The problem in the case in which the plane is struck normally, and the jet spreads in all directions, is one of great complexity, but even in that case the maximum intensity of the pressure is easily assigned. Each layer of water flowing from an orifice is gradually deviated (fig. 169) by contact with the surface, and during deviation exercises a centrifugal pressure towards the axis of the jet. The force exerted by each small mass of water is normal to its path, and inversely as the radius of curvature of the path. Hence the greatest pressure on the plane must be at the axis of the jet, and the pressure must decrease from the axis outwards, in some such way as is shown by the curve of pressure in fig. 170, the branches of the curve being probably asymptotic to the plane.

For simplicity suppose the jet is a vertical one. Let h 1 be the depth of the orifice from the free surface, and v 1 the velocity of dis-

charge. Then, if ω is the area of the orifice, the quantity of water .impinging on the plane is obviously

Q = α>r 1 = ω∖∕2<∕Λ 1 ; that is, supposing the orifice rounded, and neglecting the coefficient of discharge. ·

The velocity with which the fluid reaches the plane is, however, greater than this, and may reach the value ν = ∖ l2gh; where h is the depth of the plane below the free surface. The external layers of fluid subjected throughout, after leaving the orifice, to the atmospheric pressure will attain the velocity ν, and will flow away with this velocity unchanged except by friction. The layers towards the interior of the jet, being subjected to a pressure greater than atmospheric pressure, will attain a less velocity, and so much less as they are nearer the centre of the jet.

But the pressure can in no case exceed the pressure or h measured in feet of water, or the direction of motion of the water λvould be reversed, and there would be reflux. Hence the maximum intensity of the pressure of the jet on the plane is h feet 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=— ωv-,υ: t) Ύ = 2ω ∖ ∕hh 1 .

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

Some very interesting experiments on the distribution of pressure on a surface struck by a jet have been made by Mr J. S. Beresford (Prof. Papers on Indian Engineering, No. cccxxii. ), with a view to afford imformation as to the forces acting on the aprons of weirs. Cylindrical jets ⅛ inch to 2 inches diameter, issuing from a vessel in which the water level was constant, were allowed to fall vertically on a brass plate 9 inches 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 - 5 ⅛ inch 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— . 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 171 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. [table]

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 (1)∙

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

∕" 0 °

V=∕ 2ιrxyda!

√o r∞ -z 2 = ΊτΙιΙ b xdx "' 0

τr⅞ r - ,- χ2 η°° "lθg e ⅛L Jθ

Th

Using the condition already stated, 2 " v "⅛-⅛% i k ≡- i -⅛∖∕⅜ (2) ·

Putting the value of b in (2) in eq. (1), and also r for the radius of the jet at the orifice, so that w≈τrr 2 , the equation to the pressure curve is

ι χt y≈hi i V z , ι ,, i .

154. 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, Ω 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=∕GΩ — pounds =∕GΩA; where/is a coefficient having about the value 13 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 ,l unfair" form, that is, in which the surfaces over which the water slides are Dot 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/. The method is not free from objection, as the centrifugal force causes a flow outwards across the plate. [table]

There is a steady increase of resistance with the size of the plate, in part or wholly due to centrifugal action.

Dubuat made experiments on a plane one foot square, moved in a straight line in water at 3 to 6⅜ feet per second. Calling m the coefficient of excess oi pressure in front, and n the coefficient of deficiency of pressure behind, so that ∕= 7>ι+ w, he found' the following values :—

ffl=l; n=0∙4S3; ∕=1∙433.

The pressures were measured by pressure columns. Experiments by Morin, Piobert, and Didion on-plates of 0 - 3 to 2'7 square feet area, drawn vertically through water, gave /=2’18; but the experiments were made in a reservoir of comparatively small depth. For similar plates moved through air they found∕= 1 - 36, a result more in accordance with those which precede.

For a fixed plane in a moving current of water Mariotte found ∕= 1 - 25.. Dubuat, in experiments in a current of water like those mentioned above,, obtained the value» τη = 1Τ86; 7! = 0'670;

∕=1 ’ 856. Thibault exposed to wind pressure planes of 1T7 and 2 - fr square feet area, and found /to vary from 1’568 to 2'125, the mean value being ∕= 1’834, a result agreeing well with Dubuat.

155. 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.

Let AB (fig. 172) 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 ψ; L = N cos 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

R = P(sin ≠) 1 842 cosφ ∙

A simpler and more convenient expression given by Colonel Duchemin is

R=P 2 sin *≠- .

1+ sin 2 ≠

Consequently, the total pressure between the fluid and plane is

2 sin ⅜>, 2P ∣

1 + sin 2 ≠ cosec φ + sin φ ’ and the lateral force is j j -P 2 sin φ cos φ

1 + sin 2 ≠

In 1872 some experiments were made for the Aeronautical Society on the pressure of air on oblique planes. These plates, of 1 to 2 feet square, were balanced by ingenious mechanism designed by Mr Wenham and Mr 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 inches square. The pressure of the blast varied from ⅛ to 1 inch 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 :— [table]

Resistance of Ships.

156. Down to a recent period the resistance of ships was supposed to be due to a difference between the pressure on the bow and stem, caused by the pushing aside of the water, precisely as in the case of the “unfair” bodies whose resistance has just been discussed. Hence the resistance was supposed to be proportional to the immersed midship sectional area of the vessel. It will be shown immediately, however, that in a “fair” body, completely immersed, there is no resistance of this kind, the pressure of the water closing in behind exactly balancing the excess of pressure on the bow. In such a body, therefore, the resistance is almost entirely due to the frictional drag of the water on the surface of the body, and is proportional to its skin area. In a ship, which is only partially immersed, a further resistance, which in some cases becomes very large, is due to the alteration of the surface level of the water causing a dissipation of energy in producing waves.

157. Stream Line Motion of a Fluid past a submerged Body.— Consider a shipshape body, or body of fair form, that is, bounded by surfaces of continuous curvature, moving below the surface of a fluid, and for the moment let the friction of the fluid against the surface of the body be supposed absent. In such conditions, the particles of fluid are gradually deviated sideways as the body passes, and gradually close together again behind it. They are left after the operation in their original position with momentum unchanged; there is, therefore, in this case no resistance due to the direct action of the inertia of the water.

The nature of the action is more conveniently studied by supposing the body at rest and the fluid flowing past it. Let S, fig 173, represent the immersed body surrounded by fluid which is flowing past it. The fluid particles, arriving at A in the direction shown by the arrow, are gradually deviated as they approach S, gradually unite again after passing it; and, if the body is of fair form, that is, if it presents no abrupt changes of section or discontinuity of curvature, the stream lines or paths of the particles will be continuous lines, which take, at a sufficient distance B sternwards of S, their original direction of motion. The fluid surrounding S may then be conceived to be divided into an infinite number of elementary streams of continuous curvature. Suppose, for simplicity, S is a solid of revolution. Then, from the similarity of conditions in all directions, the elementary streams will be in planes drawn through the axis of S.

Each elementary stream may be conceived as a mass of fluid flowing steadily in an infinitely thin frictionless pipe. But it has already been shown that in a tortuous pipe, the ends of which are in the same direction, there is no resultant force due to the motion of the fluid which tends to displace the pipe, either due to its curvature or its changes of section. Consequently the whole mass of fluid exerts no resultant pressure on the body S past which it is flowing. Nor, if the fluid is at rest, will there be any resistance to the uniform motion of the body S through it. The resistance of the ship therefore cannot be due, like that of an unshipshape body, to the forward momentum impressed directly on the fluid. In a frictionless fluid, and for a uniformly moving and wholly immersed body of fair form, the resistance would be nil.

With a fluid which is not perfectly frictionless, however, a resistance may be generated in this way. The particles of water exert a drag on the surface of the body over which they slide. They receive, either in consequence qf their adhesion to the surface, or in consequence of impact on the roughnesses which project from it, a forward momentum, and the velocity at B is no longer, as in a frictionless fluid, the same as the velocity at A.

In the case of a ship which is only in part immersed there is another source of resistance. Considering the elementary streams already defined as flowing along indefinitely thin frictionless pipes, it is obvious that there would be greater pressure in those parts where the cross section was large and the velocity small, and less pressure where the section was small and the velocity high. It will be seen from the diagram that the streams are large in cross section in the neighbourhood of the bow and stern, and small along the sides. There will therefore be an excess of pressure at bow and stern, and a diminution at the sides. But the free surface of the water in which the ship floats is a surface of uniform pressure. Hence the water will be forced up at the bow and stern, and sink down in the space between, the variation of the hydrostatic pressure due to depth balancing the variation of pressure in the stream lines. There are thus formed waves accompanying the ship. So far as the ship in its passage through the water has to supply the waste of energy due to the diffusion of this wave motion in the surrounding liquid, it suffers a resistance which may be termed the wavemaking resistance. This resistance would arise even in a frictionless' fluid.

It will be seen from the foregoing that the two principal causes of the resistance to the motion of a ship are the skin friction and the production of waves. The frictional resistance depends on the immersed surface of the ship, its roughness, and the velocity of the water relatively to the surface. Mr Froude concludes that no sensible error is committed if the frictional resistance is taken to be equivalent to that of a rectangular surface of equal area and of length (in the line of motion) equal to that of the ship and moving at the same speed. For such a rectangular surface Mr Froude’s experiments already described furnish the means of calculating the resistance.

Experiments made on H.M. ship “Greyhound” appear to show that in well-formed, clean-bottomed ships, at speeds not exceeding 8 knots per hour, the frictional resistance is from 80 to 90 per cent, of the whole resistance, and that at the greatest speeds of the quickest ships the frictional resistance is from 60 to 70 per cent, of the whole resistance. For ships with foul bottoms the frictional resistance is a still larger fraction of the whole resistance.

The wave-making resistance is not yet fully understood, and involves considerations beyond the scope of the present article. For any given length of ship, with given proportions of entrance, middle body, and ran, there is a limit of speed beyond which the resistance due to dissipation of energy in waves rapidly increases. Below that limit the resistance, being chiefly due to friction, increases nearly as the square of the speed. Above that limit the resistance increases as a higher power of the speed. In the trials of the “Greyhound” the resistance varied nearly as the square of the speed up to 8 knots per hour, as the cube of the speed at 10 knots, and as the fourth power of the speed at 12 knots.

158. Ratio of the Resistance of Models and of Actual Ships. — It will be understood from the foregoing explanations that the laws of the resistance of ships are complicated and at present imperfectly knoλvn. Mr Froude has, however, indicated a way in which experiments, on comparatively small models, may be made so as to furnish very useful data as to the resistance of ships. In order that experiments on models may be serviceable, it is necessary that their resistance should be measured at speeds for which the different resistances bear the same proportion to each other as in the actual ship. Let d be the ratio of tne dimensions of the model to that of the ship. Let R 1 , R 2 , R 3 . . . be the resistances of the model at speeds v 1 , ν. 2 , v 3 . . . Then it may be expected that the actual ship at speeds vpfd, v i S∕d, υ 3 ∖ ∕d . . . will have resistances d 3 R 1 , (Z 3 R 2 , d 3 R 3 .

This law, however, is not strictly applicable to that part of the resistance which is due to friction, because of the diminution of the coefficient of friction for a given surface as the length increases. Hence, when the resistance of the model has been ascertained, a correction must be made to allow for the different coefficient of friction of the ship. The frictional resistances of the model and of the ship are calculated from their immersed surfaces, using the coefficients of friction suitable for their respective lengths. Deducting the former and adding the latter to the observed resistance at the corresponding speeds, the total resistance of the ship is ascertained.

XIII. HYDRAULIC MACHINERY.

159. Hydraulic machinery may be broadly divided into hydraulic motor machines and pumps. In the former class, a quantity of water descending from a higher to a lower level, or from a higher to a lower pressure, drives a machine which receives energy from the water, and applies it to overcoming the resistances of other machines doing useful work. In the latter class, work done on the machine by a steam engine or other source of energy is employed in lifting water from a lower to a higher level. A few machines such as the ram and jet pump combine the functions of motors and pumps.

Water Motors.

In every system of machinery deriving energy from a natural water fall 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 feet 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 on or 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.

Water Motors with Artificial Sources of Energy.— The great convenience and simplicity of water motors has led to their adoption in certain cases, where no natural source of water power is available. In these cases, an artificial source of w iter power is created by using a steam engine to pump water to a reservoir at a great elevation, or to pump water into a closed reservoir in which there is great pressure. The water flowing from the reservoir through hydraulic engines gives back the energy expended, less so much as has been wasted in friction. Such arrangements are most useful where a continuously acting steam engine stores up energy by pumping the water, while the work done by the hydraulic engines is done intermittently.

160. Energy ofα Water Fall. —Let H i be the total fall of level from the point where the water is taken from a natural stream to the point where it is discharged into it again. Of this total fall a portion, which can be estimated independently, is expended in overcoming the resistances of the head and tail races or the supply and discharge pipes. Let this portion of head wasted be f‰. Then the available head to work the motor is H = Ht -1>. 11 is this available head which should be used in all calculations of the proportions of the motor. Let Q be the supply of water per second. Then

GQH foot-pounds per second is the gross available work of the fall. The power of the fall may be rendered available in three ways. The GQ pounds of water may be placed on a machine at the highest level, and descending in contact with it a distance of H feet, the work done will be (neglecting losses from friction or leakage)

GQH foot-pounds per second.

Or the water may descend in a closed pipe from the higher to the lower level, in which case, with the same reservation as before, the pressure at the foot of the pipe will be j p = GH pounds per square foot. If the water with this pressure acts on a movable piston like that of a steam engine, it will drive the piston so that the volume described is Q cubic feet per second. Then the work done will be y>Q = GHQ foot-pounds per second as before. O r la stly, the water may be allowed to acquire the velocity v = √2gH by its descent. The kinetic energy of Q cubic feet will then be — Q θ =GQH, and if the water is allowed to Ims' i p⅛ge on surfaces suitably curved which bring it finally to rest, it will impart to these the same energy as in the previous cases. Generally, if Q feet per second of water act by weight through a distance ½ 1 , at a pressure p due to ⅛ 2 feet of fall, and with a velocity v due to Λ 8 feet of fall, so that h^+h 3 + h 3 = H, then, apart from energy wasted by friction or leakage or imperfction of the machine, the work done will be

G ι> 2

GQΛ 1 +pQ + y ∙Q^ = GQH foot pounds, the same as if the water acted simply by its weight while descending H feet.

161. Site for Water Motor.— Wherever a stream flows from a higher to a lower level it is possible to erect a water motor. The amount of power obtainable depends on the available head and the supply of water. In choosing a site the engineer will select a portion of the stream where there is an abrupt natural fall, or at least a considerable slope of the bed. He will have regard to the facility of constructing the channels which are to convey the water, and will take advantage of any bend in the river which enables him to shorten them. He will have accurate measurements made of the quantity of water flowing in the stream, and he will endeavour to ascertain the average quantity available throughout the year, the minimum quantity in dry seasons, and the maximum for which bye-wash channels must be provided. In many cases the natural fall can be increased by a dam or weir thrown across the stream. The engineer will also examine to what extent the head will vary in different seasons, and whether it is necessary to sacrifice part of the fall and give a steep slope to the tail race to prevent the motor being drowned by backwater in floods.

In designing or selecting a water motor it is not sufficient to consider only its efficiency in normal conditions of working. It is generally quite as important to know how it will act with a scanty water supply or a diminished head. The greatest difference in water motors is in their adaptability to varying conditions of working.

162. Aetion of Water in a Water Motor.— Water motors may be divided into water-pressure enginès, water wheels, and turbines.

Water-pressure engines are machines with a cylinder and piston or ram, in principle identic d with the corresponding part of a steam engine. The water is alternately admitted to and discharged from the cylinder, causing a reciprocating action of the piston or ram. It is admitted at a high pressure and discharged at a low one, and consequently work is done on the piston. The water in these machines never acquires a high velocity, and for the most part the kinetic energy of the water is wasted. The useful work is due to the difference of the pressure of admission and discharge, whether that pressure is due to the weight of a columh of water of more or less considerable height, or is artificially produced in ways to be described presently.

Water wheels are large vertical wheels driven by water falling from a higher to a lower level. In most water wheels, the water acts directly by its weight loading one side of the wheel and so causing rotation. But in all water wheels a portion and in some a considerable portion of the work due to gravity is first employed to generate kinetic energy in the water; during its action on the water wheel the velocity of the water diminishes, and the wheel is therefore in part driven by the impulse due to the change of the water’s momentum. Water wheels are therefore motors on which the water acts, partly by weight, partly by impulse.

Turbines are wheels, generally of small size compared with water wheels, driven chiefly by the impulse of the water. Before entering the moving part of the turbine, the water is allowed to acquire a considerable velocity; during its action on the turbine this velocity is diminished, and the impulse due to the change of momentum drives the turbine.

Roughly speaking, the fluid acts in a water-pressure engine directly by its pressure, in a water wheel chiefly by its weight causing a pressure, but in part by its kinetic energy, and in a turbine chiefly by its kinetic energy, which again causes a pressure.

Water-Pressure Engines.

163. In these water acts by simple pressure due to the height of the column in the supply pipe or the pressure in the supply reservoir. The water acts on a piston or ram which it displaces. When the height of the column exceeds 100 or 200 feet, or there is a pressure equivalent to this, water wheels are inapplicable, and turbines have the disadvantage that in such circumstances their speed is very great. Then water-pressure engines maybe very conveniently adopted. In other cases they are generally too cumbrous.

When an incompressible fluid such as water is used to actuate piston engines, two special difficulties arise. One is that the waste of work in friction is very great, if the water attains considerable velocity; another is that there is great straining action on the machinery. The violent straining action due to the more or less sudden arrest of the motion of water in machinery is termed hydraulic shock. For these reasons the maximum velocity of flow of water in hydraulic machines should generally not exceed 5 to 10 feet per second. Under very high pressure, where there is less object in economizing energy, and it is very important to keep the dimensions of the machinery small, Mr Anderson gives 24 feet per second as the limiting velocity. In large water-pressure engines used for pumping mines the average piston speed does not exceed ∣ to 2 feet per second.

Direct-Acting Hydraulic Lift (fig. 174).—This is the simplest of all kinds of hydraulic motor. A cage W is lifted directly by water pressure acting in a cylinder C, the length of which is a little greater than the lift. A ram or plunger R of the same length is attached to the cage. The water pressure admitted by a cock to the cylinder forces up the ram, and when the supply valve is closed and the discharge valve opened, the ram descends. In this case the ram is 9 inches diameter, with a stroke of 49 feet. It consists of lengths of wroughtiron pipe screwed together perfectly water-tight, the lower end being closed by a cast-iron plug. The ram works in a cylinder 11 inches diameter, of 9 feet lengths of flanged cast-iron pipe. The ram passes water-tight through the cylinder cover, which is provided with double hat leathers to prevent leakage outwards or inwards. As the weight of the ram and cage is much more than sufficient to cause a descent of the cage, part of the weight is balanced. A chain attached to the cage passes over a pulley at the top of the lift, and carries at its free end a balance weight B, working in T iron guides. Water is admitted to the cylinder from a 4-inch supply pipe through a two-way slide, worked by a rack, spindle, and endless rope. The lift works under 73 feet of head, and lifts 1350 ft at 2 feet per second. The efficiency is given by Mr Anderson at 75 to 80 per cent.

The principal prejudicial resistance to the motion of a ram of this kind is the friction of the cup leathers, which make the joint between the cylinder and ram. Some experiments by Mr John Hick give for the friction of these leathers the following formula. Let F= the total friction in pounds; <Z=∙ diameter of ram in feet; p≈ water pressure in pounds per square foot; k a coefficient. T=k p d fc = 0 00393 if the leathers are new or badly lubricated;

= 0Ό0262 if the leathers are in good condition and well lubricated.

Since the total pressure on the ram is y <B t p, the fraction of the total pressure expended in overcoming the friction of the leathers is i⅛ to i≡, d being in feet.

d d

Let II be the height of the pressure column measured from the free surface of the supply reservoir to the bottom of the ram in its lowest position, H& the height from the discharge reservoir to the same point, h the height of the ram above its lowest point at any moment, S the length of stroke, Ω the area of the ram, W the weight of cage, R the weight of ram, B the weight of balance weight, τo the weight of balance chain per foot run, F the friction of the cup leather and slides. Then, neglecting fluid friction, if the ram is rising the accelerating force is

P 1 =G(H-Λ)Ω-R-W + B-w(S-Λ) + wΛ-F, and if the ram is descending

P 2 = -G(H i -A)Ω + W + R-B÷w(S-A)-wΛ-F.

If w =∣ GΩ, P 1 and P 2 are constant throughout the stroke; and the moving force in ascending and descending is the same, if b=w+r+wS-gω i ⅛ it6 .

Using the values just found for w and B, p 1 -p 8 = ⅜GΩ(H-H 4 )-F.

•Let W + R + u·S + B = U, and let P be the constant accelerating force acting on the system, then the acceleration is

The velocity at the end of the stroke is (assuming the friction to be constant)

χ∕( 2 τ ss ) i and the mean velocity of ascent is ⅛ v.

164. Self-Acting Hydraulic Engines.— The admission and discharge valve in the lift just described is worked by hand at the required times. It is easy to see that mechanism like that used in steam engines can be applied to actuate the admission and discharge valves periodically, and the lift is then converted into a continuously acting engine.

Let II be the available fall to work the engine after deducting the loss of head in the supply and discharge pipes, Q the supply of water in cubic feet per second, and p the efficiency of the engine. Then the horse-power of the engine is p ηGQH H ‘ 550 ·

The efficiency of large slow-moving pressure engines is η = - 66 to ’ 8. In small motors of this kind probably ∙η is not greater than - 5. Let v be the mean velocity of the piston, then its diameter d is given by the relation

Q = ~ d 2 ν in double-acting engines,

= - θ- d 2 v in single-acting engines, o

If there are n cylinders put for Q in these equations.

The mean velocity v is from j to 2 feet per second in large engines. Smaller engines working on high lifts may be run at a greater speed, but with a sacrifice of efficiency. The usual piston speed of Messrs Hastie’s engines described below is 100 feet per minute. For pressures of less than 200 feet of head, the speed is less. The velocity of the water in the supply pipes may be 3 to 6 feet per second.

In large engiues the admission and discharge valves are of very large size, and require very considerable force to move them. It is also desirable that they should open and close more rapidly than the eccentric-moved valves used in steam engines. In these engines the valves are made cylindrical, so that the water pressure causes no friction of the valve on its seating. They are moved by a weight which is released at the proper moment, or by a subsidiary water-pressure engine, the valves of which being small can be actuated automatically. Tolerably full details of engines with mechanism of this kind are to be found in Weisbach’s Mechanics of Engineering.

Small pressure engines form extremely convenient motors for hoists, capstans, or winches, and for driving small machinery. They are usually rotative engines, and may be single or double acting. The single-acting engine has the advantage that the pressure of the piston on the crank pin is always in one direction; there is then no knocking as the dead centres are passed. Generally three singleacting cylinders are used, so that the engine will readily start in all positions, and the driving effort on the crank pin is very uniform.

Mr Brotherhood’s well-known three-cylinder steam engine has been modified so as to be used as a water-pressure engine. The three cylinders are formed in one casting. The valve is a circular revolving disc with segmental ports, which pass over corresponding apertures in the valve seating during rotation. The valve seating is of lign um vitae. i

Fig. 175 shows a similar engine made by Messrs Hastie of Greenock. G, G, G are the three plungers which pass out of the cylinders through cup leathers, and act on the same crank pin. A is the inlet pipe which communicates with the cock B. This cock controls the action of the engine, being so constructed that it acts as a reversing valve when the handle C is in its extremepositions and as a brake when in its middle position. With the handle in its middle position, the ports of the cylinders are in communication with the exhaust. Two passages are formed in the framing leading from the cock B to the ends of the cylinders, one being in communication with the supply pipe A, the other with the discharge pipe Q. These passages end as shown at E. The oscillation of the cylinders puts them alternately in communication with each of these passages, and thus the water is alternately admitted and exhausted.

In any ordinary rotative engine the length of stroke is invariable. Consequently the consumption of water depends simply on the speed of the engine, irrespective of the effort overcome. If the power of the engine must be varied without altering the number of rotations, then the stroke must be made variable. Messrs Hastie have contrived an exceedingly ingenious method of varying the stroke automatically, in proportion to the amount of work to be done (fig. 176). The crank pin I is carried in a slide H moving in a disk Μ. In this is a double cam K acting on two small steel rollers J, L attached to the slide H. If the cam rotates it moves the slide and increases or decreases the radius of the circle in which the crankpin I rotates. The disk M is keyed on a hollow shaft surrounding the driving shaft P, to which the cams are attached. The hollow shaft N has two snugs to which the chains RR are attached (fig. 177). The shaft P carries the spring case S3 to which also are attached the other ends of the chains. When the engine is at rest the springs extend themselves, rotating the hollow shaft N and the frame M, so as to place the crank pin I at its nearest position to the axis of rotation. When a resistance has to be overcome, the shaft N rotates relatively to P, compressing the springs, till their resistance balances the pressure due to the resistance to the rotation of P. The engine then commences to work, the crank pin being in the position in which the turning effort just overcomes the resistance. If the resistance diminishes, the springs force out the chains and shorten the stroke of the plungers, and vice versa. The following experiments, on an engine of this kind working a hoist, show how the automatic arrangement adjusted the water used to the work done. The lift was 22 feet and the water pressure in the cylinders 80 It> per square inch.

Weightlifted, ∣ Chain ∣ 427 633 745 g57 969 1 θ 81 1193 in pounds 1 only j

Wateruse ∣,in ) 7 , 10 14 16 17 20 21 22 gallons ⅛ 2

165. Accumulator Machinery.— It has already been pointed out that it is in some cases convenient to use a steam engine to create an artificial head of water, which is afterwards employed in driving water-pressure machinery. Where power is required intermittently, for short periods, at a number of different points, as, for instance, in moving the cranes, lock gates, <fcc., of a dockyard, a separate steam engine and boiler at each point is very inconvenient; nor can engines worked from a common boiler be used, because of the great loss of heat and the difficulties which arise out of condensation in the pipes. If a tank, into which water is continuously pumped, can be placed at a great elevation, the water can then be used in hydraulic machinery in a very convenient way. Each hydraulic machine is put in communication with the tank by a pipe, and on opening a valve it commences work, using a quantity of water directly proportional to the work done. No attendance is required when the machine is not working.

A site for such an elevated tank is, however, sei dr τ available, and in place of it a beautiful arrangement termed r accumulator, invented by Sir W. Armstrong, is used. This consists of a tall vertical cylinder; into this v orks a solid ram through cup leathers or hemp packing, and the ram is loaded by fixed weights, so that the pressure in the cylinder is 700 lb or 800 ft» per square inch. The pumping engines which supply the energy that is stored in the accumulator should be a pair coupled at right angles, so as to start in any position. The engines pump into the accumulator cylinder till the ram is at the top of its stroke, when by a catch arrangement the engines are stopped. If the accumulator ram descends, in consequence of water being taken to work machinery, the engines immediately recommence working. Pipes lead from the accumulator to each of the machines requiring to be driven. These pipes do not require to be of large size, as the pressure is so great. They are generally flanged pipes about 1∣ inches bore, the joints being made by a gutta-percha ring.

Fig. 178 shows in a diagrammatic way the scheme of a system of accumulator machinery. A is the accumulator, with its ram carrying a cylindrical wrought-iron tank ΛV, in which weights are placed to load the accumulator. At R is one of the pressure engines worked from the accumulator, discharging the water after use into the tank T. In this case the pressure engine is shown working a set of blocks, the fixed block being on the ram cylinder, the running block on the ram. The chain running over these blocks works a lift cage C, the speed of which is as many times greater than that of the ram as there are plies of chain on the block tackle. B is the balance weight of the cage.

In the use of accumulators on shipboard for working gun gear or steering gear, the accumulator ram is loaded by springs, or by steam pressure acting on a piston much larger than the ram.

Mr Tweddell has used accumulators with a pressure of 2000 lb per square inch to work hydraulic riveting machinery.

The amount of energy stored in the accumulator, having a ram cl inches in diameter, a stroke of 8 feet, and working at p pounds pressure per square inch, is

2fp<PS foot-pounds.

Thus, if the ram is 9 inches, the stroke 20 feet, and the pressure 800 Bö per square inch, the work stored in the accumulator when the ram is at the top of the stroke is 1,017,600 footpounds, that is, enough to drive a machine requiring one horse power for about half an hour. As, however, the pumping engine replaces water as soon as it is drawn off, the working capacity of the accumulator is very much greater than this.

Water Wheels.

166. Overshot and High Breast Wheels.— When a water fall ranges between 10 and 70 feet, and the water supply is from 3 to 25 cubic feet per second, it is possible to construct a bucket wheel on which the water acts chiefly by its weight. If the variation of the head-water level does not exceed 2 feet, an overshot wheel may be used (fig. 179). The water is then projected over the summit of the wheel, and falls in a parabolic path into the buckets. With greater variation of head-water level, a pitch-back or high breast wheel is better. The water falls over the top of a sliding sluice into the wheel, on the same side as the head race channel. By adjusting the height of the sluice, the requisite supply is given to the wheel in all positions of the head-water level.

The wheel consists of a cast-iron or wrought-iron axle C supporting the weight of the wheel. To this are attached two sets of arms A of wood or iron, which support circular segmental plates termed shrouds B. A cylindrical sole plate dd extends between the shrouds on the inner side. The buckets are formed by wood planks or curved wrought-iron plates extending from shroud to shroud, the back of the buckets being formed by the sole plate.

The efficiency may be taken at 0 ’ 75. Hence, if h.p. is the effective horse power, H the available fall, and Q the available water supply per second,

Ap.=0∙75⅞⅞≤ = 0 ∙085 QH. 550

If the peripheral velocity of the water wheel is too great, water is thrown out of the buckets before reaching the bottom of the fall. In practice, the circumferential velocity of water wheels of the kind now described is from 4⅜ to 10 feet per second, about 6 feet being the usual velocity of good iron wheels not of very small size. In order that the water may enter the buckets easily, it must have a greater velocity than the wheel. Usually the velocity of the water at the point where it enters the wheel is from 9 to 12 feet per second, and to produce this it must enter the wheel at a point 16 to 27 inches below the head-water level. Hence the diameter of an overshot wheel may be

D = H-1⅜ to H-2⅜ feet.

Overshot and high breast wheelswork badly in back-water, and hence if the tail-water level varies, it is better to reduce the diameter oi the wheel so that its greatest immersion in flood is not more than 1 foot. The depth d of the shrouds is about 10 to 16 inches. The number of buckets may be about

N-≤.

d

Let υ be the peripheral velocity of the wheel. Then the capacity of that portion of the wheel which passes the sluice in one second is Q 1 =g(Dd-d 2 )

= υ b d nearly, b being the breadth of the wheel between the shrouds. If, however, this quantity of water were allowed to pass on to the wheel the buckets would begin to spill their contents almost at the top of the fall. To diminish the loss from spilling, it is not only necessary to give the buckets a suitable form, but to restrict the water supply to One-fourth or one-third of the gross bucket capacity. Let m be the value of this ratio; then, Q being the supply of water per second, Q, = mQ 1 = mbdv.

This gives the breadth of the wheel if the water supply is known. The form of the buckets should be determined thus. The outer element of the bucket should be in the direction of motion of the water entering relatively to the wheel, so that the water may enter without splashing or shock. The buckets should retain the water as long as possible, and the width of opening of the buckets should be 2 or 3 inches greater than the thickness of the sheet of water entering.

For a wooden bucket (fig. 180, A), take ab — distance between two buckets on periphery of wheel. Make cd≈⅛ eb, and 5c = ⅜ to ∣ ab. Join cd. For an iron bucket (fig. 180, B), take ed=⅜ eb ; δc≡∙⅞ ab. Draw cθ making an angle of 10° to 15° with the radius at c. On Oc take a centre giving a circular arc passing near d, and round the curve into the radial part of the bucket de.

There are two ways in which the power of a water wheel is given off to the machinery driven. In wooden wheels and wheels with rigid arms, a spur or bevil wheel keyed on the axle of the turbine will transmit the power to the shafting. It is obvious that the whole turning moment due to the weight of the water is then transmitted through the arms and axle of the water wheel. When the water wheel is an iron one, it usually has light iron suspension arms incapable of resisting the bending action due to the transmission of the turning effort to the axle. In that case spur segments are bolted to one of the shrouds, and the pinion to which the power is transmitted is placed so that the teeth in gear are, as nearly as may be, on the line of action of the resultant of the weight of the water in the loaded arc of the wheel.

167. The Poncelet Water Wheel.— When the fall does not exceed 6 feet, the best water motor to adopt in many cases is the Poncelet undershot water wheel. In this the water acts very ’ nearly in the same way as in a turbine, and the Poncelet wheel, although slightly less efficient than the best turbines, in normal conditions of working, is superior to most of them when working with a reduced supply of water. A general notion of the action of the water on a Poncelet wheel has already been given in § 145. Fig. 181 shows its construction. The water penned back between the side walls of the wheel pit is allowed to flow to the wheel under a movable sluice, at a velocity nearly equal to the velocity due to the whole fall. The water is guided down a slope of 1 in 10, or a curved race, and enters the wheel without shock. Gliding up the curved floats it comes to rest, falls back, and acquires at the point of discharge a backward velocity relative to the wheel nearly equal to the forward velocity of the wheel. Consequently it leaves the wheel deprived of nearly the whole of its original kinetic energy.

Taking the efficiency at 0 - 60, and putting H for the available fall, h.p. for the horse-power, and Q for the water supply per second, h.p. =0'068 QH.

The diameter D of the wheel does not depend on the fall. With, a straight channel of approach the smallest convenient diameter is about 14 feet, with a curved channel 10 feet. The diameter is often taken at four times the fall.

Let H' be the fall measured from the free surface of the head-water to the point F where the mean layer enters the w heel; then the velocity at which the water enters is v= ∖ j2gH', and the best circumferential velocity of the wheel is V = 0 - 55v to 0’6®. The number of rotations of the wheel per second is N = -Σ- .

ιrD

The thickness of the sheet of water entering the wheel is very important. The best thickness according to experiment is 8 to 10 inches. The maximum thickness should not exceed 12 to 15 inches, when there is a surplus water supply. Let e be the thickness of the sheet of water entering the wheel, and b its width; then

Z>it>≈Q; or ό = — .

ev Graslιof takes e = ffi, and then i -⅛⅜a∙

Allowing for the contraction of the stream, the area of opening through the sluice may be 1 '25 be to 1 '3 be. The inside width of the wheel is made about 4 inches greater than b.

Several constructions have been given for the floats of Poncelet wheels. One of the simplest is that shown in figs. 181, 182.

Let OA (fig. 181) be the vertical radius of the wheel. Set off OB, OD making angles of 15° with OA. Then BD may be the length of the close breasting fitted to the wheel. Draw the bottom of the head race BC at a slope of 1 in 10. Parallel to this, at distances ⅜e and e, draw EF and GH. Then EF is the mean layer and GH the surface layer entering the wheel. Join OF, and make OFK = 23°. Take FK, = 0∙5 to 07 H. Then K is the centre from which the bucket curve is struck and KF is the radius. The depth of the shrouds must be sufficient to prevent the water from rising over the top of the float. It is ⅛H to ⅜H. The number of buckets is not very important. They are usually 1 foot apart on the circumference of the wheel.

The efficiency of a Poncelet wheel has been found in experiments to reach 0 - 68. It is better to take it at 0’6 in estimating the power of the wheel, so as to allow some margin.

In fig. 182 vi is the initial and r 0 the final velocity of the water, v r parallel to the vane the relative velocity of the water and wheel, and. V the velocity of the wheel.

Turbines.

168. The name turbine was originally given in France to any water motor which revolved in a horizontal plane, the axis being vertical. The rapid development of this class of motors dates from 1827, when a prize was offered by the Société d’Encouragement for a motor of this kind, which should be an improvement on certain wheels then in use. The prize was ultimately awarded to M. Fourneyron, whose turbine, but little modified, is still constructed.

Classification of Turbines.— In some turbines the whole available energy of the water is converted into kinetic energy before the water acts on the moving part of the turbine. Such turbines are termed Impulse Turbines, and they are distinguished by this that the wheel passages are never entirely filled by the water. To ensure this condition they must be placed a little above the tail water and discharge into free air.

Turbines in which part only of the available energy is converted into kinetic energy, before the water enters the turbine wheel, may be termed Reaction Turbines. In these the pressure is greater at the inlet than at the outlet ends of the wheel passages. The wheel passages must therefore be entirely filled, and the wheel may be and generally is placed below the tail-water level.

Next there is a difference of constructive arrangement of turbines, which does not very essentially alter the mode of action of the water. In axial flow or so called parallel flow turbines, the water enters and leaves the turbine in a direction parallel to the axis of rotation, and the paths of the molecules lie on cylindrical surfaces concentric with that axis. In radial outward and inward flow turbines, the water enters and leaves the turbine in directions normal to the axis of rotation, and the paths of the molecules lie exactly or nearly in planes normal to the axis of rotation. In outward flow turbines the general direction of flow is away from the axis, and in inward flow turbines towards the axis. There are also mixed flow turbines in which the water enters normally and is discharged parallel to the axis of rotation.

Another difference of construction is this, that the water may be admitted equally to every part of the circumference of the turbine wheel or to a portion of the circumference only. In the former case, the condition of the wheel passages is always the same; they receive water equally in all positions during rotation. In the latter case, they receive water during a part of the rotation only. The former may be termed turbines with complete admission, the latter turbines with partial admission. A reaction turbine should always have complete admission. An impulse turbine may have complete or partial admission.

When two turbine wheels similarly constructed are placed on the same axis, in order to balance the pressures and diminish journal friction, the arrangement may be termed a twin turbine.

If the water, having acted on one turbine wheel, is then passed through a second on the same axis, the arrangement may be termed a compound turbine. The object of such an arrangement would be to diminish the speed of rotation.

Many forms of reaction turbine may be placed at any height not exceeding 30 feet above the tail water. They then discharge into an air-tight suction pipe. The weight of the column of water in this pipe balances part of the atmospheric pressure, and the difference of pressure, producing the flow through the turbine, is the same as if the turbine were placed at the bottom of the fall. [table]

169. The Simple Reaction Wheel.— It has been shown, in § 151, that, when water issues from a vessel, there is a reaction on the vessel tending to cause motion in a direction opposite to that of the jet. This principle was applied in a rotating water motor at a very early period, and the Scotch turbine, at one time much used, differs in no essential respect from the older form of reaction wheel.

The old reaction wheel consisted of a vertical pipe balanced on a vertical axis, and supplied with water (fig. 183). From the bottom, of the vertical pipe two or more hollow horizontal arms extended, at the ends of which were orifices from which the water was discharged. The reaction of the jets caused the rotation of the machine.

Let H be the available fall measured from the level of the water in the vertical pipe to the centres of the orifices, r the radius from the axis of rotation to the centres of the orifices, v the . velocity of discharge through the jets, α the angular velocity of the machine. When the machine is at rest the water issues from the orifices with the velocity √2<∕H (friction being neglected). But when the machine rotates the water in the arms rotates a.so, and is in the condition of a forced vortex, all the particles having the same angular velocity. Consequently the pressure in the arms at the orifices is ττ α 2 r s h+ v feet of water, and the velocity of discharge through the orifices is v=√2pH + α 2 r a ∙

If the total area of the orifices is ω, the quantity discharged from the wheel per second is

Q = ωv — ω√2^H + α 2 r 2 .

While the water passes through the orifices with the velocity v, the orifices are moving in the opposite direction with the velocity ar. The absolute velocity of the water is therefore υ-ar = √2yH+ αM-αr.

GQ

The momentum generated per second is — (v - αr) , which is numerically equal to the force driving the motor at the radius r. The work done by the water in rotating the wheel is therefore, per second,

CO

-y (v - ar)ar foot-pounds.

The work expended by the water fall is GQH foot-pounds per second. Consequently the efficiency of the motor is {v - ar)ar { V2g ’ H + α 2 r 2 - ar } ar 1 ’ ~ i/H

Let √2yH + a 2 r 2 = ar + * ar 2a 3 r 3 then γj = i-∣iL + 2ar which increases towards the limit 1 as ar increases towards infinity. Neglecting friction, therefore, the maximum efficiency is reached when the wheel has an infinitely great velocity of rotation. But this condition is impracticable to realize, and even, at practicable but high velocities of rotation, the friction would considerably reduce the efficiency. Experiment seems to show that the best efficiency is reached when ar= √2⅛jH . Then the efficiency apart from friction is ( ∖ Z2a 3 r i - ar) ar V== gH

.04U^. 0 , 828 about 17 per cent, of the energy of the fall being carried away by the water discharged. The actual efficiency realized appears to be about 60 per cent., so that about 21 per cent, of the energy of the fall is lost in friction, in addition to the energy carried away by the water.

170. General Statement of Hydrodynamical Principles necessary for the Theory of Turbines.

1. When water flows through any pipe-shaped passage, such as the passage between the vanes of a turbine wheel, the relation between the changes of pressure and velocity is given by Bernoulli’s theorem (§ 26). Suppose that, at a section A of such a passage, Λ 1 is the pressure measured in feet of water, v l the velocity, and z 1 the elevation above any horizontal datum plane, and that at a section B the same quantities are denoted by Λ 2 , v 2 , z 2 . Then , , v 2 3 - v 1 ’.,.

Ä i-Ä a —+ (1).

If the flow is horizontal, z 2 ≈z 1 ; and ν , s _ 2

= . . . . . . (la).

2 . When there is an abrupt change of section of the passage, or an abrupt change of section of the stream due to a contraction, then, in applying Bernoulli’s equation allowance must be made for the loss of head in shock (§ 32). Let u 1 , v 2 be the velocities before and after the abrupt change, then a stream of velocity v 1 impinges on a stream at a velocity r a and the relative velocity is r 1 - v 2 . The head lost is · Then equation (la) becomes } l -h - v ι~ v ' frι~ t ⅛) a ⅛~*⅛) (2 )

2 1 2 fl r 2g g · · · ( Ι-

Το diminish as much as possible the loss of energy from irregular eddying motions, the change of section in the turbine passages must be very gradual, and the curvature without discontinuity.

3. Equality of Angular Impulse and Change of Angular Momentum. — Suppose that a couple, the moment of which is M, acts on a body of weight W for t seconds, during which it moves from A 1 to A s (fig. 184). Let t∙ 1 be the velocity of the body at A 1 , ι> a its velocity at A s ,, and let p 1 , p 2 be the perpendiculars from C on v 1 and v 2 . Then Mi is termed the angular impulse of the couple, and the quantity

W

— (υ 2 p 2 - r 1 p 1 ) is the change of angular momentum relatively to C. Then, from the equality of angular impulse and change of angular momentum

W

= ( v 2P2- v 1P1) , or, if the change of momentum is estimated for one second,

W

M = V ( v 2P2~ v 1P1)∙ if

Let r 1 , r 2 be the radii drawn from C to A 1 , A a , and let w 1 , w 2 be the components of t⅛ t⅛, perpendicular to these radii, making angles β and a with v 1 , v 2 . Then v 1 =w- i sec β; v 2 =w 2 sec a; Pι=vι cos β; p 2 = r 2 cos a.

W

·· M = —(^ 3 i∙ s -w 1 r 1 )∙ (3), if where the moment of the couple is expressed in terms of the radii drawn to the positions of the body at the beginning and end of a second, and the tangential components of its velocity at those points.

Now the water flowing through a turbine enters at the admission surface and leaves at the discharge surface of the wheel, with its angular momentum relatively to the axis of the wheel changed. It therefore exerts a couple - M tending to rotate the wheel, equal and opposite to the couple M which the wheel exerts on the water. Let Q cubic feet enter and leave the wheel per second, and let w v w 2 be the tangential components of the velocity of the water at the receiving and discharging surfaces of the wheel, r 1 , r 2 the radii of those surfaces. By the principle above,

~ m =~ (W 2 -«Vi) (<)∙

If a is the angular velocity of the wheel, the work done by the water on the wheel is

GQ

T—Mα=-y (2z> 1 r 1 - w 2 r 2 )a foot-pounds per second . (5).

171. Total and Available Fall. — Let H i be the total difference of level from the head-water to the tail-water surface. Of this total head a portion is expended in overcoming the resistances of the head race, tail race, supply pipe, or other channel conveying the water. Let I),, be that loss of head, which varies with the local conditions in which the turbine is placed. Then

H = H t -⅛

is the available head for working the turbine, and on this the calculations for the turbine should be based. In some cases it is necessary toplace the turbine above the tail-water level, and there is then a fall I) from the centre of the outlet surface of the turbine to the tailwater level which is wasted, but which is properly one of the losses belonging to the turbine itself. In that case the velocities of the water in the turbine should be calculated for a head H - I), but the efficiency of thë turbine for the head H.

172. Gross Efficiency and Hydraulic Efficiency of a Turbine.— Let T i j be the useful work done by the turbine, in foot-pounds per second, T t the work expended in friction of the turbine shaft, gearing, &c., a quantity which varies with the local conditions in which the turbine is placed. Then the effective work done by the water in the turbine is

T = Td + T< .

The gross efficiency of the whole arrangement of turbine, races, and transmissive machinery is

”‘ = GQÏL (6) *

And the hydx ’ aulic efficiency of the turbine alone is ,, 'GQH (7) ‘

It is this last efficiency only with which the theory of turbines is concerned.

From equations (5) and (7) we get η GQH = y (w 1 r 1 - w 2 rja;

(w 1 r. - w 2 r 2 )a , fi x ' gΑ (8) ·

This is the fundamental equation in the theory of turbines. In general, u ∖ and w 2 , the tangential components of the water’s motion on entering and leaving the wheel, are completely independent. That the efficiency may be as great as possible, it is obviously necessary that w 2 =0. In that case

η=^⅛- α (9L

9 ’ H ar 1 is the circumferential velocity of the wheel at the inlet sur face. Calling this V 1 , the equation becomes

-⅛ «·

This remarkably simple equation is the fundamental equation in the theory of turbines. It was first given by Herr v. Reiche ( Turbinen-baues, 1877).

173. General Description of a Reaction Turbine.— Professor James Thomson’s inward flow or vortex turbine has been selected as the type of reaction turbines. It is one of the best even in normal conditions of working, and the mode of regulation introduced is decidedly superior to that in most reaction turbines; it might almost be said to be the only mode of regulation which satisfies the conditions of efficient working, and it has been adopted in a modified form in the Leffel turbine, which is now largely used in America.

Figs. 185 and 186 are external views of the turbine case; figs. 187 and 188 are the corresponding sections; fig. 189 is the turbine wheel. The example chosen for illustration has suction pipes, which permit the turbine to be placed at any height less than 30 feet above the tail-water level. The water enters the turbine by cast-iron supply pipes at A, and is discharged through two suction pipes S, S. The water on entering the case distributes itself through a rectangular supply chamber SC, from which it finds its way equally to the four guide-blade passages G, G, G, G. In these passages it acquires a velocity about equal to that due to half the fall, and is directed into the wheel at an angle of about 10° or 12° with the tangent to its circumference. The wheel W receives the water in equal proportions from each guide-blade passage. It consists of a centre plate p (fig. 189) keyed on the shaft αα, which passes through stuffing boxes on the suction pipes. On each side of the centre plate are the curved wheel vanes, on which the pressure of the water acts, and the vanes are bounded on each side by dished or conical coverplates c, c. Jointrings j, j on the cover plates make a sufficiently watertight joint with the casing, to prevent leakage from the guide-blade chamber into the suction pipes. The pressure near the joint rings is not very great, probably not onefourth the total head. The wheel vanes receive the water without shock, and deliver it into central spaces, from which it flows on either side to the suction pipes. The mode of regulating the power of the turbine is very simple. The guide-blades are pivoted to the case at their inner ends, and they are connected by a linkwork, so that they all open and close simultaneously and equally. In this way the area of opening through the guide-blades is altered without materially altering the angle or the other conditions of the delivery into the wheel. The guide-blade gear may be variously arranged. In this example four spindles, passing through the case, are linked to the guide-blades inside the case, and connected together by the links Z, l, l on the outside of the case. A worm wheel on one of the spindles is rotated by a worm d, the motion being thus slow enough to adjust the guide-blades very exactly. These turbines are made by Messrs Williamson Brothers of Kendal, who supplied the drawing of the turbine.

Fig. 190 shows another arrangement of the same turbine, with some adjuncts not shown in the other drawings. In this case the turbine rotates horizontally, and the turbine ease is placed entirely below the tail water. The water is supplied to the turbine by a vertical pipe, over which is a wooden pentrough, containing a strainer, which prevents sticks and other solid bodies getting into the turbine.

The turbine rests on three foundation stones, and, the pivot for the vertical shaft being under water, there is a screw and lever arrangement for adjusting it as it wears. The vertical shaft gives motion to the machinery driven by a pair of bevel wheels. On the right are the worm and wheel for working the guide-blade gear.

174. Different Forms of Turbine IVheel.— The wheel of a turbine or part of the machine on which the water acts is an annular space, furnished with curved vanes dividing it into passages exactly or roughly rectangular in cross section. For radial flow turbines the wheel may have the form A or B, fig. 191, A being most usual with inward, and B with outward flow turbines. In A the wheel vanes are fixed on each side of a centre plate keyed on the turbine shaft. The vanes are limited by slightly-coned annular cover plates. In B the vanes are fixed on one side of a disk, keyed on the shaft, and limited by a cover plate parallel to the disk. Parallel flow or axial flow turbines have the wheel as in C. The vanes are limited by two concentric cylinders.

Reaction Turbines.

175. Velocity oj Whirl and Velocity of Flow.— Let acb (fig. 192) be the path of the particles of water in a turbine wheel. That path will be in a plane normal to the axis of rotation in radial flow turbines, and on a cylindrical surface in axial flow turbines. At any point c of the path the water will have some velocity v, in the direction of a tangent to the path. That velo city may be resolved into two components, a whirling velocity w in the direction of the wheel’s rotation at the point c, and a component u at right angles to this, radial in radial flow, and parallel to the axis in axial flow turbines. This second component is termed the velocity of flow. Let v 0 , w 0 , u 0 be the velocity of the water, the whirling velocity and velocity of flow at the outlet surfaceof the wheel, and v i ,wi,uι the same quantities at the inlet surface of the wheel. Let a and β be the angles which the water’s direction of motion makes with the direction of motion of the wheel at those surfaces. Then w 0 =v 0 cos 3; sin 3 ) .... (10). wi = v 0 cos a; ui=Vi sin a )

The velocities of flow are easily ascertained independently from the dimensions of the wheel. The velocities of flow at the inlet and outlet surfaces of the wheel are normal to those surfaces. Let Ω o , Ω i be the areas of the outlet and inlet surfaces of the wheel, and Q the volume of water passing through the wheel per second; then <⅛√ i <v-⅞ < n >∙

u4θ

Using the notation in fig. 191, we have, for an inward flow turbine (neglecting the space occupied by the vanes), Ω 0 = 2πr 0 d 0 ; iiι≈2ιrrιdi .... (12α). Similarly, for an outward flow turbine,

Ω 0 = 2ιrr 0 d; Ω, = 2ιrΓid (12&); and, for an axial flow turbine,

Ω 0 = Ω t ∙= 7γ(λ 2 -γ 1 2 ) (12c).

Relative and Common Velocity of the Water and Wheel.— There is another way of resolving the velocity of the water. Let V be the velocity of the wheel at the point c, fig. 193. Then the velocity of the water may be resolved into a component V, which the water has in common with the wheel, and a component v r , which is the velocity of the water relatively to the wheel.

Velocity of Flow. —It is obvious that the frictional losses of head in the wheel passages will increase as the velocity of flow is greater, that is, the smaller the wheel is made. But if the wheel works under water, the skin friction of the wheel cover increases as the diameter of the wheel is made greater, and in any case the weight of the wheel and consequently the journal friction increase as the wheel is made larger. It is therefore desirable to choose, for the velocity of flow, as large a value as is consistent with the condition that the frictional losses in the wheel passages are a small fraction of the total head.

The values most commonly assumed in practice are these :— In axial flow turbines, tt o = w, = O∙15 to 0 - 2V z 2^H;

In outward flow turbines, rt,∙=0∙25V2<7(H-I)) , w o = 0'21 to OT7√20(H-⅛);

In inward flow turbines, w o = w,∙=O∙125√¾rH .

176. Speed of the Wheel.— The best speed of the wheel depends partly on the frictional losses, which the ordinary theory of turbines disregards. It is best, therefore, to assume for V o and V,∙ values which experiment has shown to be most advantageous.

In axial flow turbines, the circumferential velocities at the mean radius of the wheel may be taken

V o = V,∙=O∙6√¾7H to 0∙66√2gH .

In a radial outward flow turbine,

V i ≡0∙56√¾r(H-S)

V 0 = V.∙^, where r 0 , r. are the radii of the outlet and inlet surfaces.

Ina radial inward flow turbine,

V t ∙=0 - 66√2gH ,

Vo=⅞V.∙∙

If the wheel were stationary and the water flowed through it, the water would follow paths parallel to the wheel vane curves, at least when the vanes were so close that irregular motion was prevented. Similarly, when the wheel is in motion, the water follows paths relatively to the wheel, which are curves parallel to the wheel vanes. Hence the relative component, ν r , of the water’s motion at c is tangential to a wheel vane curve drawn through the point c. Let v 0 , V o , υ r0 be the velocity of the water and its common and relative components at the outlet surface of the wheel, and vι, V,∙, v r i be the same quantities at the inlet surface; and let θ and φ be the angles the wheel vanes make with the inlet and outlet surfaces; then

⅞ 2 = V0 ’ rθ 2 ÷ V 0 2 - 2V 0 'Γr 0 COS <∕>) ∣ ∩θ∖

Vi = √(v rt ≡ + V i 2 -2V i V ri cos 0) ) ’ equations which may be used to determine φ and θ.

177. Condition determining the Angle of the Vanes at the Oïitlet Surface of the Wheel. — It has been shown that, when the water leaves the wheel, it should have no tangential velocity, if the efficiency is to be as great as possible; that is, w o = O. Hence, from (10), cos 3 = 0, 3 = 90°, w 0 =v 0 , and the direction of the water’s motion is normal to the outlet surface of the wheel, radial in radial flow, and axial in axial flow turbines.

Drawing v 0 or u 0 radial or axial as the case may be, and V o tangential to the direction of motion, -ι⅛ 0 can be found by the parallelogram of velocities. From fig. 194, tan≠-⅛-(⅛ (14); but φ is the angle which the wheel vane makes with the outlet surface of the wheel, which is thus determined when the velocity of flow w 0 and velocity of the wheel V o are known. When φ is thu⅛ determined, v ro =⅜ cosec φ = V 0 ./ l+-^⅞∙ ’ ’ · (14α >∙ v v 0 Correction of the Angle φ to allow for Thickness of Vanes.— In determining φ, it is most convenient to calculate its value approximately at first, from a value of u 0 obtained by neglecting the thickness of the vanes. As, however, this angle is the most important angle in the turbine, the value should be afterwards corrected to allow for the vane thickness.

Let ψ '.tan-⅛.ta- s Q-

be the first or approximate value of φ, and let t be the thickness, and n the number of wheel vanes which reach the outlet surface of the wheel. As the vanes cut the outlet surface approximately at the angle φ', their width measured on that surface is t cosec φ'. Hence the space occupied by the vanes on the outlet surface is

For A, fig. 191, ntd 0 cosec φ )

B, fig. 191, ntd cosec φ >. . . . (15).

C, fig. 191, nt (r 2 -r 1 ) cosec φ ) Call this area occupied by the vanes ω. Then the true value of the clear discharging outlet of the wheel is Ω o - ω, and the true value of u 0 ig _9— . The corrected value of the angle of the vanes will

Ω o - ω

be

⅜>=tan'∙ W-9 (16).

r (Ω 0 - w )V 0

178. Head producing Velocity with which the Water enteis the Wheel.— Consider the variation of pressure in a wheel passage, which satisfies the condition that the sections change so gradually that there is no loss of head in shock. When the flow is in a horizontal plane, there is no work done by gravity on the water passing through the wheel. In the case of an axial flow turbine, in which the flow is vertical, the fall d between the inlet and outlet surfaces should be taken into account. [table]

At any point in the path of a portion of water, at radius r, the velocity v of the water may be resolved into a component V = ar equal to the velocity at tlιat point of the wheel, and a relative component v r . Hence the motion of the water may be considered to consist of two parts :—(a) a motion identical with that in a forced vortex of constant angular velocity a ; (ά) a flow along curves parallel to the wheel vane curves. Taking the latter first, and using Bernoulli’s theorem, the change of pressure due to flow through the wheel passages is given by the equation v ,Vr? ,, ,¾'rθ 2 . Äi+ ⅞^= ° 2< z ’ ∖ , l , vrf-vr? k ∙~ k °—⅛—

The variation of pressure due to rotation in a forced vortex is v- r - v < 2 - v ° 2 λ ∙ a °~ 2g

Consequently the whole difference of pressure at the inlet and outlet surfaces of the wheel is

¼-⅞~A' i +A" i -Λ' 0 -Λ" 0

V.∙ 2 -V 0 3 , t∙κ> 2 -^ 2 2g 2g μzλ

Case 1. Axial Flow Turbines.— ∙V,∙=V ( ,; and the first term on the right, in equation 17, disappears. Adding, however, the work of gravity due to a fall of d feet in passing through the wheel,

(17«).

Case 2. Outward Flow Turbines.— The inlet radius is less than y.2 _ y 2

the outlet radius, and —I——2- is negative. The centrifugal head diminishes the pressure at the inlet surface, and increases the velocity with which the water enters the wheel. This somewhat increases the frictional loss of head. Further, if the wheel varies in velocity from variations in the useful work done, the quantity ~ ^° 2 in-

2g creases when the turbine speed increases, and vice versa. Consequently the flow into the turbine increases when the speed increases, and diminishes when the speed diminishes, and this again augments the variation of speed. The action of the centrifugal head in an outward flow turbine is therefore prejudicial to steadiness of motion. For this reason r 0 : r i is made small, generally about 5: 4. Even then a governor is sometimes required to regulate the speed of the turbine.

Case 3. Inward Flow Turbines.— The inlet radius is greater than the outlet radius, and the centrifugal head diminishes the velocity of flow into the turbine. This tends to diminish the frictional losses, but it has a more important influence in securing steadiness of motion. Any increase of speed diminishes the flow into the turbine, and vice versa. Hence the variation of speed is less than the variation of resistance overcome. In the so-called centre vent wheels in America, the ratio r,∙: r 0 is about 5: 4, and then the influence of the centrifugal head is not very important. Professor James Thomson first pointed out the advantage of a much greater difference of radii. By making r i : r 0 = 2: 1, the centrifugal head balances about half the head in the supply chamber. Then the velo∙ c ^-J r though the guide-blades does not exceed the velocity due to half the fall, and the action of the centrifugal head in securing steadiness of speed is considerable.

Since the total head producing flow through the turbine is H - f), and of this h i - h 0 is expended in overcoming the pressure in the wheel, the velocity of flow into the wheel is j⅛(H- ξ ,-I⅛Σ. ’ + 5^⅛*)∣. . . (18), where c β may be taken 0 ’ 96.

From (14α),

*--v.√(ι + ^)∙

It will be shown immediately that

‰=ι¼∙ cosec e; or, as this is only a small term, and θ is on the average 90°, we may take, for the present purpose, v ri =u i nearly.

Inserting these values, and remembering that for an axial flow turbine V.=V 0 , f) = 0, and the fall d in the wheel is to be added»

n ~ e ∙ J i 2,, (h ^ v(1+ $) + v ^ d )! ·

For an outward flow turbine, *-<- c ∙√[¾'! h →-⅛( 1+ w)÷<Π∙

For an inward flow turbine, -*√[*l≡->÷^⅞∣]∙

179. Angle which the Guide-Blades make with the Circumference of the Wheel.— At the moment the water enters the wheel, the radial component of the velocity is u,, and the velocity is vi. Hence, if γ is the angle between the guide-blades and a tangent to the wheel y= sin“1 — .

vi

This· angle can, if necessary, be corrected to allow for the thickness of the guide-blades.

180. Condition Determining the Angle ofthe Vanes at the Inlet Surface of the Wheel.— The single condition necessary to be satisfied at the inlet surface of the wheel is that the water should enter the wheel without shock. This condition is satisfied if the direction of relative motion of the water and wheel is parallel to the first element of the wheel vanes.

Let A (fig. 195)

be a point on the inlet surface of the wheel, and let vi represent in magnitude and direction the velocity of the water entering the wheel, and V t ∙ the velocity of the wheel. Completing the parallelogram, vri is the direction of relative motion. Hence the angle between v r i and V,- is the angle θ which the vanes should make with the inlet surface of the wheel.

181. Example of the Method of Designing a Turbine. Professor James Thomson's Inward Flow Turbine.—

Let H = the available fall after deducting loss of head in pipes and channels from the gross fall;

Q=the supply of water in cubic feet per second; and ∙η = the efficiency of the turbine.

The work done per second is

ιjGQH, and the horse-power of the turbine is hn” GQH Ä ·^·—550^∙

If i) is taken at 0 ’ 75, an allowance will be made for the frictional losses in the turbine, the leakage, and the friction of the turbine shaft. Then h.p. = 0 O85QH.

The velocity of flow through the turbine (uncorrected for the space occupied by the vanes and guide-blades) may be taken u i =u n = 0 ’ 125√20H, in which case about ⅛th of the energy of the fall is carried away by the water discharged·

The areas of the outlet and inlet surface of the wheel are then

Q

2wA-2«-â- 0 . 125V2ÎH ·

If we take r 0 , so that the axial velocity of discharge from the central orifices of the wheel is equal to u 0 , we get

>∙.-0'≡√9ff-

do~ r <r

If, to obtain considerable steadying action of the centrifugal head, r i = 2r 0 , then d i =~d 0 . Speed of the Wheel.— Let V,=0∙6βV2gH, or the speed due to half the fall nearly. Then the number of rotations of the turbine per second is also vθ=-⅛V.∙=0 ∙33√¾τH .

Angle of Vanes with Outlet Surface.

Tan ≠=⅛=θ^5 -3788;

V ø V ÖÖ

≠ = 21° nearly.

If this value is revised for the vane thickness it will ordinarily become about 25°.

Velocity with which the Water enters the Wheel.— The head producing the velocity is

2<A V.∙ 2 7 2< z

= H {1 - *4356(1 + 0'0358) + O15β}

= 0'5646H.

Then the velocity is

= ∙96√2^(∙5646H) = 0∙721√¾H.

Angle of Guide-Blades. si ° 1 '-v-⅛r 0 ' 173 '

γ=10° nearly.

Tangential Velocity of Water entering Wheel. wi=v i cos γ=θ∙7101√¾H.

Angle of Vanes at Inlet Surface.

Cot ∙4008; Ui ∙125

0=68° nearly.

Hydraulic Efficiency of Wheel. w i V i .. V = — tt = ∙7101 X ∙66×2 = 0-9373.

This, however, neglects the friction of wheel covers and leakage. The efficiency from experiment has been found to be 0 ’ 75 to 0∙80.

Im r fmlse and Partial Admission Turbines.

182. The principal defect of most turbines with complete admission is the imperfection of the arrangements for working with less than the normal supply. With most forms of turbine the efficiency is considerably reduced when the regulating sluices are partially closed, but it is exactly when the supply of water is deficient that it is most important to get out of it the greatest possible amount of work. The imperfection of the regulating arrangements is, therefore, from the practical point of view, a serious defect. All turbine makers have sought by various methods to improve the regulating mechanism. Fourneyron, by dividing his wheel by horizontal diaphragms, virtually obtained three or more separate radial flow turbines, which could be successively set in action at their full power, but the arrangement is not altogether successful, because of the spreading of the water in the space between the wheel and guide-blades. M. Fontaine similarly employed two concentric axial flow turbines formed in the same casing. One was worked at full power, the other regulated. By this arrangement the loss of efficiency due to the action of the regulating sluice affected only half the water power. Many makers have adopted the expedient of erecting two or three separate turbines on the same waterfall. Then one or more could be put out of action and the others worked at full power. This is an excellent plan, but the separate turbines cost more than a single one. All these methods are rather palliatives than remedies. The movable guide-blades of Professor James Thomson meet the difficulty directly, but of course they are not applicable to every form of turbine.

A subsidiary defect of turbines with complete admission is their very great speed of rotation on high falls. The turbine wheel cannot be increased in diameter without great increase of the fluid friction in the passages and on the surface of the wheel, and it also becomes impossible in radial flow turbines to adjust properly the vane angles, if the diameter is made very large.

M. Callon, in 1840, patented an arrangement of sluices for axial or outward flow turbines, which were to be closed successively as the water supply diminished. By preference the sluices were closed by pairs, two diametrically opposite sluices forming a pair. The wateι was thus admitted to opposite but equal arcs of the wheel, and the forces driving the turbine were symmetrically placed. As soon as this arrangement was adopted, a modification of the mode of action of the water in the turbine became necessary. If the turbine wheel passages remain full of water during the whole rotation, the water contained in each passage must be put into motion each time it passes an open portion of the sluice, and stopped each time it passes a closed portion of the sluice. It is thus put into motion and stopped twice in each rotation. This gives rise to violent eddying motions and great loss of energy in shock. To prevent this, the turbine wheel with partial admission must be placed above the tail water, and the wheel passages be allowed to clear themselves of water, while passing from one open portion of the sluices to the next.

But if the wheel passages are free of water when they arrive at the open sluices, then there can be no pressure other than atmospheric pressure in the space between the sluices and wheel. The water must issue from the sluices with the whole velocity due to the head; received on the curved vanes of the wheel, the jets must be gradually deviated and discharged with a radial velocity only, precisely in the same way as when a single jet strikes a curved vane in the free air. Turbines of this kind are therefore termed turbines of free deviation. There is no variation of pressure in the jet during the whole time of its action on the wheel, and the whole energy of the jet is imparted to the wheel,'simply by the impulse due to its gradual change of momentum. It is clear that the water may be admitted in exactly the same way to any fraction of the circumference at pleasure, without altering the efficiency of the wheel. The diameter of the wheel may be made as large as convenient, and thus the speed of rotation on high falls may be kept down to a manageable amount. The Poncelet water wheel is a turbine of free deviation, in which, however, the action of gravity causes the water to flow back along .the vanes, so that it is discharged at the same point of the wheel at which it enters.

So long as the tail-water level is invariable, no difficulty arises in adopting the system of partial admission. But if, as is more commonly the case, the tail-water level varies, then there is danger that the turbine will be drowned in flood time, and the essential condition of the system that the wheel passages should be empty when they come in front of the open sluices will not be satisfied. If the fall is considerable, a portion of it may be sacrificed without much harm, and the wheel placed sufficiently high above the tail water to secure it from being drowned; but with low falls this is impossible. The difficulty has been overcome by a method invented by M. L. D. Girard in 1849, and termed the hydropneumatic system. The turbine is placed below the tail-water level in a casing supplied with air by a small air-pump. It therefore always discharges freely into an atmosphere of air, the pressure of which, however, varies with the height of the tail-water level outside the casing. Inside the casing the free water surface is maintained at an invariable level just below the discharge orifices of the wheel.

183. General Description of an Impulse Turbine or Turbine wiih Free Deviation.— Fig. 196 shows a general sectional elevation of a Girard turbine, in which the flow is axial. The water, admitted above a horizontal floor, passes down through the annular wheel containing the guide-blades G, G, and thence into the revolving wheel WW. The revolving wheel is fixed to a hollow shaft suspended from the pivot p. The solid inter-, nal shaft ss is merely a fixed column supporting the pivot. The advantage of this is that the pivot is accessible for lubrication and adjustment. B is the mortise bevel wheel by which the power of the turbine is given off. The sluices are worked by the hand wheel h, which raises them successively, in a way to be described presently, a, α are the sluice rods. Figs. 197, 198 show the sectional form of the guideblade chamber and wheel and the curves of the wheel vanes and guideblades, when drawn on a plane development of the cylindrical section of the wheel; a, a, a are the sluices for cutting off the water; b, b, b are apertures by which the entrance or exit of air is facilitated as the buckets empty and fill. Figs. 199, 200 show the guideblade gear, a, a, a are the sluice rods as before. At the top o. each sluice rod is a small block c, having a projecting tongue, which slides in the groove of the circular cam plate d, d. This circular plate is supported on the frame e, and

When the tongues of the blocks c, c arrive at g, they slide up to a second groove, or the reverse, according as the cam plate is revolved in one direction or in the other. As this operation takes place with each sluice successively, any number of sluices can be opened or closed as desired. The turbine is of 48 horse power on 5 - 12 feet fall, and the supply of water varies from 35 to 112 cubic feet per second. The efficiency in normal working is given as 73 per cent. The mean diameter of the wheel is 6 feet, and the speed 27’4 revolutions per minute.

184. Theory of the Impulse Turbine.— The theory of the impulse turbine does not essentially differ from that of the reaction turbine, except that there is no pressure in the wheel opposing the discharge from the guide-blades. Hence the velocity with which the water enters the wheel is simply v i =O∙96√20(H-l)), where I) is the height of the top of the wheel above the tail water. If the hydropneumatic system is used, then fi = 0. Let Q m be the maximum supply of water, r 1 , r i the internal and external radii of the wheel at the inlet surface; then

Q m w, ^π(r a 2 -r 1 2 ) ’

The value of u, may be about 0 ’ 45 V2y(H -Ij), whence r j , r 2 can be determined.

The guide-blade angle is then given by the equation . ui 0 ’ 45 . i , o

81117 = — = =48; ’ v i 0∙94

7 =29°.

The value of u t should, however, be corrected for the space occupied by the guide-blades.

The tangential velocity of the entering water is wi≈vi cos 7=0 ∙82√20(H - f)).

The circumferential velocity of the wheel may be (at mean radius)

V i ≈0∙5√2<z(H-¾).

Hence the vane angle at inlet surface is given by the equation cotθ=^∑f=θ^-0 1 5 =∙71 5 ui 0 ’ 45

0=55°.

The relative velocity of the water striking the vane at the inlet edge is v r i=ui cosec 0=1'22«,. This relative velocity remains unchanged during the passage of the water over the vane; consequently the relative velocity at the point of discharge is ν r0 = 1 '22wi∙ Also in an axial flow turbine V 0 =V{.

If the final velocity of the water is axial, then

V V λ∙k c ° s ≠ = WΓ^ = i⅛⅛45≡ c0s24 ° 23 '

This should be corrected for the vane thickness. Neglecting this, u 0 =υ rβ sin ≠ = Vri sin φ = ui cosec θ sin φ = 0 ‘5«;. The discharging area of the wheel must therefore be greater than the inlet area in the ratio of at least 2 to 1. In some actual turbines the ratio is 7 to 3. This greater outlet area is obtained by splaying the wheel, as shown in the section (fig. 198).

185. The Hydraulic Pam.— The hydraulic ram is an arrangement by which a quantity of water falling a distance h forces a portion of the water to rise to a height A 1 , greater than h. It consists of a supply reservoir (A, fig. 201), into which the water enters from some natural stream. A pipe s of considerable length conducts the water to a lower level, where it is discharged intermittently through a self-acting pulsating valve at rf. The supply pipe s may be fitted with a flap-valve for stopping the ram, and this is attached in some cases to a float, so that the ram starts and stops itself automatically, according as the supply cistern fills or empties. The pipe s should be as long and as straight as possible, and as it is subjected to considerable pressure from the sudden arrest of the motion of the water, it must be strong and strongly jointed, d is an air vessel, and e the delivery pipe leading to the reservoir at a higher level than A, into which water is to be pumped. Fig. 202 shows in section the construction of the ram itself, d is the pulsating discharge valve already mentioned, which opens inwards and downwards. The stroke of the valve is regulated by the cotter through the spindle, under which are washers by which the amount of fall can be regulated. At o is a delivery valve, opening outwards, which is often a ball-valve but sometimes a flap-valve. The water which is pumped passes through this valve into the air vessel a, from which it flows by the delivery pipe in a regular stream into the cistern to which the water is to be raised. In the vertical chamber behind the outer valve a small air vessel is formed, and into this opens an aperture ∣ inch in diameter, made in a brass screw plug b. The hole is reduced to τ ⅛ inch in diameter at the outer end of the plug and is closed by a small valve opening inwards. Through this, during the rebound after each stroke of the ram, a small quantity of air is sucked in which keeps the air vessel supplied with its elastic cushion of air,

The discharge valve d is of greater weight than the statical pressure of the water on its under side. When, therefore, the water is at rest in the supply pipe this valve opens. In consequence of the flow through this valve, the water in the supply pipe acquires a gradually increasing velocity. The upward flow o the water, towards the valve d, increases the pressure tending to lift the valve, and at last, if the valve is not too heavy, lifts and closes it. The forward momentum of the column in the supply pipe being destroyed by the stoppage of the flow, the water exerts a pressure at the end of the pipe sufficient to open the delivery valve o, and to cause a portion of the water to flow into the air vessel. As the water in the supply pipe comes to rest, the valve d opens again and the operation is repeated. Bart of the energy of the descending column is employed in compressing the air at the end of the supply pipe and expanding the pipe itself. This causes a recoil of the water which momentarily diminishes the pressure in the pipe below the pressure due to the statical head. This assists in opening the valve d. Mr W. Anderson states that the recoil of the water is sufficiently great to enable a pump to be attached to the ram body instead of the direct rising pipe. With this arrangement a ram working with muddy water may be employed to raise clear spring water. Instead of lifting the delivery valve as in the ordinary ram, the momentum of the column drives a sliding or elastic piston, and the recoil brings it back. This piston lifts and forces alternately the clear water through ordinary pump valves.

Pumps.

186. The different classes of pumps correspond almost exactly to the different classes of water motors, although the mechanical details of the construction are somewhat different. They are properly reversed water motors. Ordinary reciprocating pumps correspond to water-pressure engines. Chain and bucket pumps are in principle similar to water wheels in which the water acts by weight. Scoop wheels are similar to undershot water wheels, and centrifugal pumps to turbines.

Reciprocating Pumps are single or double acting, and differ from water-pressure engines in that the valves are moved by the water instead of by automatic machinery. They may be classed thus :—

(1.) Lift Pumps.— The water drawn through a foot valve on the ascent of the pump bucket is forced through the bucket valve when it descends, and lifted by the bucket when it reascends. Such pumps give an intermittent discharge.

(2.) Plunger or Force Pumps, in which the water drawn through the foot valve is displaced by the descent of a solid plunger, and forced through a delivery valve. They have the advantage that the friction is less than that of lift pumps, and the packing round the plunger is easily accessible, whilst that round a lift pump bucket is not. The flow is intermittent.

(3.) The Double-acting Force Pump is in principle a double plunger pump. The discharge fluctuates from zero to a maximum and back to zero each stroke, but is not arrested for any appreciable time.

(4. ) Bucket and Plunger Pumps consist of. a lift pump bucket combined with a plunger of half its area. The flow varies as in a double-acting pump.

(5.) Diaphragm Pumps have been used, in which the solid plunger is replaced by an elastic diaphragm, alternately depressed into and raised out of a cylinder.

The variation of velocity of discharge would cause great waste of work in the delivery pipes when they are long, and even danger from the hydraulic ramming action of the long column of water. An air vessel is interposed between the pump and the delivery pipes, of a volume from 5 to 100 times the space described by the plunger per stroke. The air in this must be replenished from time to time, or continuously, by a special air-pump. At low speeds not exceeding 30 feet per minute the delivery of a pump is about 90 to 95 per cent, of the volume described by the plunger or bucket, from 5 to 10 per cent, of the discharge being lost by leakage. At high speeds the quantity pumped occasionally exceeds the volume described by the plunger, the momentum of the water keeping the valves open after the turn of the stroke.

The velocity of large mining pumps is about 140 feet per minute, the indoor or suction stroke being sometimes made at 250 feet per minute. Rotative pumping engines of large size have a plunger speed of 90 feet per minute. Small rotative pumps are run faster, but at some loss of efficiency. Fire-engine pumps have a speed of 180 to 220 feet per minute.

The efficiency of reciprocating pumps varies very greatly. Small reciprocating pumps, with metal valves on lifts of 15 feet, were found by Morin to have an efficiency of 16 to 40 per cent., or on the average 25 per cent. When used to pump water at considerable pressure, through hose pipes, the efficiency rose to from 28 to 57 per cent., or on the average, with 50 to 100 feet of lift, about 50 per cent. A large pump with barrels 18 inches diameter, at speeds under 60 feet per minute, gave the following results :— [table]

The very large steam-pumps employed for waterworks, with 150 feet or more of lift, appear to reach an efficiency of 90 per cent, not including the friction of the discharge pipes.

The Centrifugal Pump.

187. The efficiency of reciprocating pumps diminishes with the lift. When large quantities of water are to be raised on a low lift, no pump is so suitable as a centrifugal pump. The first pump of this kind which attracted notice was one exhibited by Mr Appold in 1851, and the special features of his pump have been retained in the best pumps since constructed. Mr Appold’s pump raised continuously a volume of water equal to 1400 times its own capacity per minute. It had no valves, and it permitted the passage of solid bodies, such as walnuts and oranges, without obstruction to its working. Its efficiency was also found to be good.

Fig. 203 shows a centrifugal pump differing from ordinary centrifugal pumps in one feature only. The water rises through a suction pipe S, which divides so as to enter the pump wheel at the centre on each side. The pump disk or wheel is very similar to a turbine wheel. It is keyed on a shaft driven by a belt on a fast and loose pulley arrangement at P. The water rotating in the pump disk presses outwards, and if the speed is sufficient a continuous flow is maintained through the pump and into the discharge pipe D. The special feature in this pump is that the water, discharged by the pump disk with a whirling velocity of not inconsiderable magnitude, is allowed to continue rotation in a chamber somewhat larger than the pump. The use of this whirlpool chamber was first suggested by Professor James Thomson. It utilizes the energy due to the whirling velocity of the water which in most pumps is wasted in eddies in the discharge pipe. In the pump shown guide-blades are also added which have the direction of the stream lines in a free vortex. They do not therefore interfere with the action of the water when pumping the normal quantity, but only prevent irregular motion. At A is a plug by which the pump case is filled before starting. If the pump is above the water to be pumped, a foot valve is required to permit the pump to be filled. Sometimes instead of the foot valve a delivery valve is used, an air-pump or steam jet pump being employed to exhaust the air from the pump case. • 188. Design and Proportions of a Centrifugal Pump.— The design of the pump disk is very simple. Let ∏, r 0 be the radii of the inlet and outlet surfaces of the pump disk, di, d 0 the clear axial width at those radii. The velocity of now through the pump may be taken the same as for a turbine. If Q is the quantity pumped, and H the lift, w i =≈0∙25√⅜rH (1).

2ιrr,∙⅛≈- . ui

Also in practice iZ i = l∙2r i . . . )

H “Ce ’ r—'2571 ∕Ξ5" «·

r.- z571,√ 7g

Usually r 0 =2r,∙, and d 0 ≈d i or ⅜d,∙

according as the disk is parallel-sided or coned. The water enters the wheel radially with the velocity «,·, and

*τ⅛ < 3 >∙

Fig. 204 shows the notation adopted for the velocities. Suppose the water enters the wheel with the velocity v,∙, while the velocity of the wheel is V,∙. Completing the parallelogram, ‰ is the relative velocity of the water and wheel, and is the properdirection of the wheel vanes. Also, by resolving, ui and wi are the component velocities of flow and velocities of whir of the velocity νι of the water. At the outlet surface, v 0 is the final velocity of discharge, and the rest of the notation is similar to that for the inlet surface.

Usually the water flows equally in all directions in the eye of the wheel, in that case r; is radial. Then, in normal conditions of working, at the inlet surface,

[table]

If the pump is raising less or more than its proper quantity, θ will not satisfy the last condition, and there is then some loss of head in shock.

At the outer circumference of the wheel or outlet surface, [table]

Variation of Pressure in the Pump Disk.— Precisely as in the case of turbines, it can be shown that the variation of pressure between the inlet and outlet surfaces of the pump is

2g 2g

Inserting the values of v r0 , in (4) and (5), we get for normal conditions of working h -h v o 2 ~ v ∙ 2 m o 2 cosec 2 φ ι⅜ 2 + Vj 2 0 ‘ = 2g 2g 2g

V 0 2 w 0 2 cosec 2 <∕> ι up

“27" 2y + 2g k , ' Hydraulic Efficiency of the Pump.— Neglecting disk friction, journal friction, and leakage, the efficiency of the pump can be found in the same way as that of turbines (§ 172). Let M be the moment of the couple rotating the pump, and a its angular velocity; w 0 , r 0 the tangential velocity of the water and radius at the outlet surface; wι, ri the same quantities at the inlet surface. Q being the discharge per second, the change of angular momentum per second is — (w 0 r 0 ~wiri). 9

Hence M = — (w o r o - u '∙ r ∙) ·

9.

I n normal working, wi = 0. Also, multiρlyingby the angular velocity, the work done per second is

Mα = - Wo α -

9

But the useful work done in pumping is GQH. Therefore the efficiency is

GQH g H g H (7).

Ma w 0 r 0 α w 0 V 0

189. Case 1. Centrifugal Pump with no Whirlpool Chamber.— When no special provision is made to utilize the energy of motion of the water leaving the wheel, and the pump discharges directly into a chamber in which the water is flowing to the discharge pipe, nearly the whole of the energy of the water leaving the disk is wasted. The water leaves the disk with the more or less considerable velocity v 0 , and impinges on a mass flowing to the discharge pipe at the much slower velocity ν,. The radial component of v 0 is almost necessarily wasted. F rom the tangential component there is a gain of pressure w 0 2 - r, 2 (w 0 - r,) 2 2g 2g m v,(w o -v,) 9 which will be small, if v, is small compared with w ö . Its greatest ’ LO value, if v 1 = ⅛w 0 , is ⅜ -^-, which will always be a small part of the whole head. Suppose this neglected. The whole variation of pressure in the pump disk then balances the lift and the head necessary to give the initial velocity of flow in the eye of the wheel.

nf , π V 0 a _ m 0 2 cosec 2 φ uf 2g + 2g 2g + 2 0 r ’

jy "V 0 2 wθ 2 cosec 2 0 I

~⅛Γ~ 2g I . . . . (8); or V o «=√(2yH + w 0 2 cosec 2 ≠) J

and the efficiency of the pump is, from (7),

„ g∏ gH

V 0 w 0 V 0 (V 0 -m 0 cot φ) - Vp 2 - u 0 i cosec φ ^°2V 0 (V 0 - w 0 cot ≠)

Forφ = 90°, π , v o~⅝ a

” 2V 0 2 ’

which is necessarily less than ∣. That is, half the work expended in driving the pump is wasted. By recurving the vanes, a plan introduced by Mr Appold, the efficiency is increased, because the velocity v 0 of discharge from the pump is diminished. If φ is very small, cosec φ = cot φ; and then n _ Υο 8 +™ο cosec ≠

n 2V 0 which may approach the value 1, as φ tends towards 0. Equation (8) shows that u 0 cosec φ cannot be greater then V o , Put tin g

⅝=O∙25√¾H

we get the following numerical values of the efficiency and the circumferential velocity of the pump :— [table]

φ cannot practically be made less than 20°; and, allowing for the frictional losses neglected, the efficiency of a pump in which Φ = 20° is found to be about ‘60.

190. Case 2. Pump with a Whirlpool Chamber, as in fig. 203.—Professor James Thomson first suggested that the energy of the water after leaving the pump disk might be utilized, if a space were left in which a free vortex could be formed. In such a free Vortex the velocity varies inversely as the radius. The gain of pressure in the vortex chamber is, putting r 0 , r„ for the radii to the outlet surface of wheel and to outside of free vortex,

⅛(1 -3)-% ’ M∙

if k=H- . ru>

The lift is then, adding this to the lift in the last case,

H = Jj V 0 2 - w 0 2 cosec 2 0 + v 0 2 (l - fc 2 ) I .

2g ( )

But v 0 2 ≈ V 0 2 - 2V 0 u 0 cot φ + u 0 2 cosec 2 ≠;

. ·. H = — j (2 - k?) V 0 2 —2XW 0 u 0 cot φ - k i u<f cosec 2 √> I (10).

2g < )

Putting this in the expression for the efficiency, we find a considerable increase of efficiency. Thus with

φ = 90 β and ⅛=⅜, n = ⅝ nearly; φ a small angle and k≈ ⅛ , η = 1 nearly.

With this arrangement of pump, therefore, the angle at the outer ends of the vanes is of comparatively little importance. A moderate angle of 30° or 40° may very well be adopted. The following numerical values of the velocity of the circumferenc e of the pump have been obtained by taking λ=⅛ , and w 0 =0'25∖ z 2^H . [table]

The quantity of water to be pumped by a centrifugal pump necessarily varies, and an adjustment for different quantities of water cannot easily be introduced. Hence it is that the average efficiency of pumps of this kind is in practice less than the efficiencies given above. The advantage of a vortex chamber is also generally neglected. The velocity in the supply and discharge pipes is also often made greater than is consistent with a high degree of efficiency. Velocities of 6 or 7 feet per second in the discharge and suction pipes, when the lift is small, cause a very sensible waste of energy; 3 to 6 feet would be much better. Centrifugal pumps of very large size have been constructed. Messrs Easton and Anderson have made pumps for the North Sea Canal in Holland which deliver each 670 tons of water per minute on a lift of 5 feet. The pump disks are 8 feet diameter. Messrs J. and H. Gwynne constructed some pumps for draining the Ferrarese Marshes, which together deliver 2000 tons per minute. A pump made under Professor J. Thomson's direction for drainage works in Barbados had a pump disk 16 feet in diameter and a whirlpool chamber 32 feet in diameter. The efficiency of centrifugal pumps when delivering less or more than the normal quantity of water is discussed in a paper in the Proc. Inst, of Civil Engineers, vol. liii.