1911 Encyclopædia Britannica/Hydraulics

by William Cawthorne Unwin

HYDRAULICS (Gr. ὕδωρ, water, and αὐλός, a pipe), the branch of engineering science which deals with the practical applications of the laws of hydromechanics.

I. THE DATA OF HYDRAULICS^[1]

§ 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, &c., 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.

Fig. 1.

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. 1 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 abcd will take the form of the parallelogram a′b′cd. Further, the resistance to the motion of ab may be expressed in the form

$\mathrm {R} =\kappa \omega \mathrm {V}$,

(1)

where $\kappa$ is a coefficient the nature of which remains to be determined.

If we suppose the liquid between ab and cd divided into layers as shown in fig. 2, 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 nT; 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,

$\mathrm {R} =\kappa '\omega \left(n\mathrm {V} \right).$

(2)

But equations (1) and (2) may both be expressed in one equation if $\kappa$ and $\kappa '$ are replaced by a constant varying inversely as the thickness of the layer. Putting ${\begin{aligned}\kappa &=\mu /\mathrm {T} ,\kappa '=\mu /n\mathrm {T} ,\\\mathrm {R} &=\mu \omega \mathrm {V} /\mathrm {T} ;\\\end{aligned}}$

or, for an indefinitely thin layer,

$\mathrm {R} =\mu \omega d\mathrm {V} /dt,$

(3)

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

According to J. Clerk Maxwell, the value of μ for air at θ° Fahr. in pounds, when the velocities are expressed in feet per second, is

$\mu =0.0000000256\left(461^{\circ }+\theta \right);$

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 H. von Helmholtz and G. Piotrowski,

$\mu =0.0000188,$

the units being the same as before. For water μ decreases rapidly with increase of temperature.

Fig. 2.

§ 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. V. J. Boussinesq has shown that the central filament in a semicircular canal of 1 metre radius, and inclined at a slope of only 0.0001, would have a velocity of 187 metres per second,^[2] 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 merely on the general velocities of translation at different points of the fluid (or what 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. Units of Volume.—In practical calculations the cubic foot and gallon are largely used, and in metric countries the litre and cubic metre (= 1000 litres). The imperial gallon is now exclusively used in England, but the United States have retained the old English wine gallon.

1 cub. ft.	= 6·236 imp. gallons	= 7·481 U.S. gallons.
1 imp. gallon	= 0·1605 cub. ft.	= 1·200 U.S. gallons.
1 U.S. gallon	= 0·1337 cub. ft.	= 0·8333 imp. gallon.
1 litre	= 0·2201 imp. gallon	= 0·2641 U.S. gallon.

Density of Water.—Water at 53° F. and ordinary pressure contains 62·4 ℔ per cub. ft., or 10 ℔ per imperial gallon at 62° F. The litre contains one kilogram of water at 4° C. or 1000 kilograms per cubic metre. River and spring water is not sensibly denser than pure water. But average sea water weighs 64 ℔ per cub. ft. at 53° F. The weight of water per cubic unit will be denoted by G. Ice free from air weighs 57·28 ℔ per cub. ft. (Leduc).

§ 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 ℔ per sq. in., the compression is proportional to the pressure. The chief results of experiment are given in the following table. Let V₁ be the volume of a liquid in cubic feet under a pressure p₁ ℔ per sq. ft., and V₂ its volume under a pressure p₂. Then the cubical compression is (V₂ − V₁)/V₁, and the ratio of the increase of pressure p₂ − p₁ to the cubical compression is sensibly constant. That is, k = (p₂ − p₁)V₁/(V₂ − V₁) is constant. This constant is termed the elasticity of volume. With the notation of the differential calculus,

k = dp / ( −	dV	) = − V	dp	.
	V		dV

Elasticity of Volume of Liquids.

	Canton.	Oersted.	Colladon and Sturm.	Regnault.
Water	45,990,000	45,900,000	42,660,000	44,000,000
Sea water	52,900,000	··	··	··
Mercury	705,300,000	··	626,100,000	604,500,000
Oil	44,090,000	··	··	··
Alcohol	32,060,000	··	23,100,000	··

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.

Temperature.		ρ Density of Water.	G  Weight of  1 cub. ft. in ℔.
Cent.	Fahr.
0	32·0	·999884	62·417
1	33·8	·999941	62·420
2	35·6	·999982	62·423
3	37·4	1·000004	62·424
4	39·2	1·000013	62·425
5	41·0	1·000003	62·424
6	42·8	·999983	62·423
7	44·6	·999946	62·421
8	46·4	·999899	62·418
9	48·2	·999837	62·414
10	50·0	·999760	62·409
11	51·8	·999668	62·403
12	53·6	·999562	62·397
13	55·4	·999443	62·389
14	57·2	·999312	62·381
15	59·0	·999173	62·373
16	60·8	·999015	62·363
17	62·6	·998854	62·353
18	64·4	·998667	62·341
19	66·2	·998473	62·329
20	68·0	·998272	62·316
22	71·6	·997839	62·289
24	75·2	·997380	62·261
26	78·8	·996879	62·229
28	82·4	·996344	62·196
30	86	·995778	62·161
35	95	·99469	62·093
40	104	·99236	61·947
45	113	·99038	61·823
50	122	·98821	61·688
55	131	·98583	61·540
60	140	·98339	61·387
65	149	·98075	61·222
70	158	·97795	61·048
75	167	·97499	60·863
80	176	·97195	60·674
85	185	·96880	60·477
90	194	·96557	60·275
100	212	·95866	59·844

The weight per cubic foot has been calculated from the values of ρ, on the assumption that 1 cub. ft. of water at 39·2° Fahr. is 62·425 ℔. 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 ℔ per cub. ft., which is its density at 53° Fahr. It may be noted also that ice at 32° Fahr. contains 57·3 ℔ per cub. ft. 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 (fig. 3). 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

$p\omega =\mathrm {G} h\omega$℔,
$p/\mathrm {G} =h$;

that is, h is the height due to the pressure at p. The level OO will be termed the free surface level corresponding to the pressure at P.

Relation of Pressure, Temperature, and Density of Gases

     Fig. 3.

§ 9. Relation of Pressure, 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 p and volume V of a given quantity of air is a constant (Boyle’s law).

Let $p_{0}$ be mean atmospheric pressure (2116·8 ℔ per sq. ft.), $\mathrm {V} _{0}$ the volume of 1 ℔ of air at 32° Fahr. under the pressure $p_{0}$. Then

$p_{0}\mathrm {V} _{0}=26214$.

(1)

If $G_{0}$ is the weight per cubic foot of air in the same conditions,

$\mathrm {G} _{0}=1/\mathrm {V} _{0}=2116.8/26214=.0807$.

(2)

For any other pressure $p$, at which the volume of 1 ℔ is $\mathrm {V}$ and the weight per cubic foot is $\mathrm {G}$, the temperature being 32° Fahr.,

$p\mathrm {V} =p/\mathrm {G} =26214$; or $\mathrm {G} =p/26214$.

(3)

Change of Pressure or Volume by Change of Temperature.—Let p₀, V₀, G₀, 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 p, V, G be the same quantities at a temperature t (measured strictly by the air thermometer, the degrees of which differ a little from those of a mercurial thermometer). Then, by experiment,

$p\mathrm {V} =p_{0}\mathrm {V} _{0}(460.6+t)/(460.6+32)=p_{0}\mathrm {V} _{0}\tau /\tau _{0}$,

(4)

where $\tau ,\tau _{0}$ are the temperatures t and 32° reckoned from the absolute zero, which is −460·6° Fahr.;

$p/\mathrm {G} =p_{0}\tau /\mathrm {G} _{0}\tau _{0}$;

(4a)

$\mathrm {G} =p\tau _{0}\mathrm {G} _{0}/p_{0}\tau$.

(5)

If $\tau _{0}=2116.8,\mathrm {G} _{0}=.08075,\tau _{0}=460.6+32=492.6$, then

$p/\mathrm {G} =53.2\tau$.

(5a)

Or quite generally $p/\mathrm {G} =\mathrm {R} \tau$ for all gases, if $\mathrm {R}$ is a constant varying inversely as the density of the gas at 32° F. For steam $\mathrm {R}$ = 85·5.

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.

Fig. 4.

If, in following a given path ab (fig. 4), 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. 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 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.

Fig. 5.

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. 5) 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.

Fig. 6.

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. 6) be any ideal plane surface, of area ω, 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. 7), and in that case

Q = ωV cos θ.

Fig. 7.

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 dω is the area and v, or v cos θ, the normal velocity for this element of the surface, the volume of flow is

Q = ∫ v dω, or ∫ v cos θ 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 a positive and the outflow with a negative sign, and estimating the volume of flow Q for all the boundaries,

ΣQ = 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 A₁, A₂ are the areas of two normal cross sections of a stream, and V₁, V₂ are the velocities of the stream at those sections, then from the principle of continuity,

V₁A₁ = V₂A₂;

V₁/V₂ = A₂/A₁

(2)

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 A₁, A₂ and between A₁, A₂ by stream lines (fig. 8), then, as there is no flow across the stream lines,

V₁/V₂ = A₂/A₁,

as in a stream with rigid boundaries.

Fig. 8.

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 p₁, p₂ be the pressures, v₁, v₂ the velocities, G₁, G₂ the weight per cubic foot of fluid, at cross sections of a stream of areas A₁, A₂. The volumes of inflow and outflow are

A₁v₁ and A₂v₂,

and, if the weights of these are the same,

G₁A₁v₁ = G₂A₂v₂;

and hence, from (5a) § 9, if the temperature is constant,

p₁A₁v₁ = p₂A₂v₂.

(3)

Fig. 9.

Fig. 10.	Fig. 11.	Fig. 12.

Fig. 13.

§ 15. Stream Lines.—The characteristic of a perfect fluid, that is, a fluid free from viscosity, is that the pressure between any two parts into which it is divided by a plane must be normal to the plane. One consequence of this is that the particles can have no rotation impressed upon them, and the motion of such a fluid is irrotational. A stream line is the line, straight or curved, traced by a particle in a current of fluid in irrotational movement. In a steady current each stream line preserves its figure and position unchanged, and marks the track of a stream of particles forming a fluid filament or elementary stream. A current in steady irrotational movement may be conceived to be divided by insensibly thin partitions following the course of the stream lines into a number of elementary streams. If the positions of these partitions are so adjusted that the volumes of flow in all the elementary streams are equal, they represent to the mind the velocity as well as the direction of motion of the particles in different parts of the current, for the velocities are inversely proportional to the cross sections of the elementary streams. No actual fluid is devoid of viscosity, and the effect of viscosity is to render the motion of a fluid sinuous, or rotational or eddying under most ordinary conditions. At very low velocities in a tube of moderate size the motion of water may be nearly pure stream line motion. But at some velocity, smaller as the diameter of the tube is greater, the motion suddenly becomes tumultuous. The laws of simple stream line motion have hitherto been investigated theoretically, and from mathematical difficulties have only been determined for certain simple cases. Professor H. S. Hele Shaw has found means of exhibiting stream line motion in a number of very interesting cases experimentally. Generally in these experiments a thin sheet of fluid is caused to flow between two parallel plates of glass. In the earlier experiments streams of very small air bubbles introduced into the water current rendered visible the motions of the water. By the use of a lantern the image of a portion of the current can be shown on a screen or photographed. In later experiments streams of coloured liquid at regular distances were introduced into the sheet and these much more clearly marked out the forms of the stream lines. With a fluid sheet 0.02 in. thick, the stream lines were found to be stable at almost any required velocity. For certain simple cases Professor Hele Shaw has shown that the experimental stream lines of a viscous fluid are so far as can be measured identical with the calculated stream lines of a perfect fluid. Sir G. G. Stokes pointed out that in this case, either from the thinness of the stream between its glass walls, or the slowness of the motion, or the high viscosity of the liquid, or from a combination of all these, the flow is regular, and the effects of inertia disappear, the viscosity dominating everything. Glycerine gives the stream lines very satisfactorily.

Fig. 9 shows the stream lines of a sheet of fluid passing a fairly shipshape body such as a screwshaft strut. The arrow shows the direction of motion of the fluid. Fig. 10 shows the stream lines for a very thin glycerine sheet passing a non-shipshape body, the stream lines being practically perfect. Fig. 11 shows one of the earlier air-bubble experiments with a thicker sheet of water. In this case the stream lines break up behind the obstruction, forming an eddying wake. Fig. 12 shows the stream lines of a fluid passing a sudden contraction or sudden enlargement of a pipe. Lastly, fig. 13 shows the stream lines of a current passing an oblique plane. H. S. Hele Shaw, “Experiments on the Nature of the Surface Resistance in Pipes and on Ships,” Trans. Inst. Naval Arch. (1897). “Investigation of Stream Line Motion under certain Experimental Conditions,” Trans. Inst. Naval Arch. (1898); “Stream Line Motion of a Viscous Fluid,” Report of British Association (1898).

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

Fig. 14.

§ 16. 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. 14) 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

v = √2gh.

(1)

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 = ωv = ω√2gh nearly.

(2)

This is sometimes quite improperly called the theoretical discharge for any kind of orifice. Except for a well-formed conoidal orifice the result is not approximate even, so that if it is supposed to be based on a theory the theory is a false one.

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. 14 descends through a height h to the orifice, we may say there are h ft. of head above the orifice. Still more generally any mass of liquid h ft. above a horizontal plane may be said to have h ft. 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 p/G = h, the quantity p/G may be termed the pressure head at the orifice. Lastly, the velocity v is connected with h by the relation v²/2g = h, so that v²/2g may be termed the head due to the velocity v.

§ 17. Coefficients of Velocity and Resistance.—As the actual velocity of discharge differs from √2gh by a small quantity, let the actual velocity

= v_a = c_v √2gh,

(3)

where c_v 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 h causing outflow consists of two parts—one part h_e expended effectively in producing the velocity of outflow, another h_r in overcoming the resistances due to viscosity and friction. Let

h_r = c_rh_e,

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_e = √ { 2gh / (1 + c_r) }.

(4)

The relation between c_v and c_r for any orifice is easily found:—

v_a = c_v√2gh = √ { 2gh / (1 + c_r) }

   c_v = √ { 1 / (1 + c_r) }

(5)

cr = 1 / cv2 − 1.

(5a)

Thus if c_v = 0.97, then c_r = 0.0628. That is, for such an orifice about 61/4% of the head is expended in overcoming frictional resistances to flow.

Fig. 15.

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. 15), 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 plate in which the aperture is formed is less than the diameter of the jet. But if the thickness is greater the condition shown at c will occur.

When the discharge occurs as at a or b, the filaments converging towards the orifice continue to converge beyond it, so that the section of the jet where the filaments have become parallel is smaller than the section of the orifice. 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_cω 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 of the area of the orifice.

Fig. 16.

Coefficient of Discharge.—In applying the general formula Q = ωv 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_a = c_vv × c_cω = c_cc_vω √(2gh),

or simply, if c = c_vc_c,

Q_a = cω √(2gh),

where c is called the coefficient of discharge. Thus for a sharp-edged plane orifice c = 0.97 × 0.64 = 0.62.

§ 18. Experimental Determination of _v, c_c, and c.—The coefficient of contraction c_c is directly determined by measuring the dimensions of the jet. For this purpose fixed screws of fine pitch (fig. 16) 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. 17) 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 O to A, then

x = v_at; y = 1/2 gt².

Eliminating t,

v_a = √ (gx²/2y).

Then

c_v = v_a √ (2gh) = √ (x²/4yh).

In the case of large orifices such as weirs, the velocity can be directly determined by using a Pitot tube (§ 144).

Fig. 17.

The coefficient of discharge, which for practical purposes is the most important of the three coefficients, is best determined by tank measurement of the flow from the given orifice in a suitable time. If Q is the discharge measured in the tank per second, then

c = Q/ω √ (2gh).

Measurements of this kind though simple in principle are not free from some practical difficulties, and require much care. In fig. 18 is shown an arrangement of measuring tank. The orifice is fixed in the wall of the cistern A and discharges either into the waste channel BB, or into the measuring tank. There is a short trough on rollers C which when run under the jet directs the discharge into the tank, and when run back again allows the discharge to drop into the waste channel. D is a stilling screen to prevent agitation of the surface at the measuring point, E, and F is a discharge valve for emptying the measuring tank. The rise of level in the tank, the time of the flow and the head over the orifice at that time must be exactly observed.

Fig. 18.

For well made sharp-edged orifices, small relatively to the water surface in the supply reservoir, the coefficients under different conditions of head are pretty exactly known. Suppose the same quantity of water is made to flow in succession through such an orifice and through another orifice of which the coefficient is required, and when the rate of flow is constant the heads over each orifice are noted. Let h₁, h₂ be the heads, ω₁, ω₂ the areas of the orifices, c₁, c₂ the coefficients. Then since the flow through each orifice is the same

Q = c₁ω₁ √ (2gh₁) = c₂ω₂ √ (2gh₂).

c₂ = c₁ (ω₁/ω₂) √ (h₁/h₂).

Fig. 19.

§ 19. 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 = 1. 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. 19 shows the proportions of such a bellmouth or bell-mouthed orifice, which approximates to the form of the contracted jet sufficiently for any practical purpose.

For such an orifice L. J. Weisbach found the following values of the coefficients with different heads.

Head over orifice, in ft. = h	·66	1·64	11·48	55·77	337·93
Coefficient of velocity = cv	·959	·967	·975	·994	·994
Coefficient of resistance = cr	·087	·069	·052	·012	·012

As there is no contraction after the jet issues from the orifice, c_c = 1, c = c_v; and therefore

Q = c_vω √ (2gh) = ω √ { 2gh / (1 + c_r }.

§ 20. Coefficients for Sharp-edged or virtually Sharp-edged Orifices.—There are a very large number of measurements of discharge from sharp-edged orifices under different conditions of head. An account of these and a very careful tabulation of the average values of the coefficients will be found in the Hydraulics of the late Hamilton Smith (Wiley & Sons, New York, 1886). The following short table abstracted from a larger one will give a fair notion of how the coefficient varies according to the most trustworthy of the experiments.

Coefficient of Discharge for Vertical Circular Orifices, Sharp-edged,
with free Discharge into the Air. Q = cω √ (2gh).

Head  measured to  Centre of Orifice.	Diameters of Orifice.
	·02	·04	·10	·20	·40	·60	1·0
	Values of C.
0·3	..	..	·621	..	..	..	..
0·4	..	·637	·618	..	..	..	..
0·6	·655	·630	·613	·601	·596	·588	..
0·8	·648	·626	·610	·601	·597	·594	·583
1·0	·644	·623	·608	·600	·598	·595	·591
2·0	·632	·614	·604	·599	·599	·597	·595
4·0	·623	·609	·602	·599	·598	·597	·596
8·0	·614	·605	·600	·598	·597	·596	·596
20·0	·601	·599	·596	·596	·596	·596	·594

At the same time it must be observed that differences of sharpness in the edge of the orifice and some other circumstances affect the results, so that the values found by different careful experimenters are not a little discrepant. When exact measurement of flow has to be made by a sharp-edged orifice it is desirable that the coefficient for the particular orifice should be directly determined.

The following results were obtained by Dr H. T. Bovey in the laboratory of McGill University.

Coefficient of Discharge for Sharp-edged Orifices.

Head in  ft.	Form of Orifice.
	Circular.	Square.		Rectangular Ratio  of Sides 4:1		Rectangular Ratio  of Sides 16:1		Tri-  angular.
		Sides Vertical.	Diagonal vertical.	Long Sides vertical.	Long Sides hori- zontal.	Long Sides vertical.	Long Sides  hori-  zontal.
1	·620	·627	·628	·642	·643	·663	·664	·636
2	·613	·620	·628	·634	·636	·650	·651	·628
4	·608	·616	·618	·628	·629	·641	·642	·623
6	·607	·614	·616	·626	·627	·637	·637	·620
8	·606	·613	·614	·623	·625	·634	·635	·619
10	·605	·612	·613	·622	·624	·632	·633	·618
12	·604	·611	·612	·622	·623	·631	·631	·618
14	·604	·610	·612	·621	·622	·630	·630	·618
16	·603	·610	·611	·620	·622	·630	·630	·617
18	·603	·610	·611	·620	·621	·630	·629	·616
20	·603	·609	·611	·620	·621	·629	·628	·616

The orifice was 0·196 sq. in. area and the reductions were made with g = 32·176 the value for Montreal. The value of the coefficient appears to increase as (perimeter) / (area) increases. It decreases as the head increases. It decreases a little as the size of the orifice is greater.

Very careful experiments by J. G. Mair (Proc. Inst. Civ. Eng. lxxxiv.) on the discharge from circular orifices gave the results shown on top of next column.

The edges of the orifices were got up with scrapers to a sharp square edge. The coefficients generally fall as the head increases and as the diameter increases. Professor W. C. Unwin found that the results agree with the formula

c = 0·6075 + 0·0098 / √ h − 0·0037d,

where h is in feet and d in inches.

Coefficients of Discharge from Circular Orifices. Temperature 51° to 55°.

Head in feet h.	Diameters of Orifices in Inches (d).
	1	11/4	11/2	13/4	2	21/4	21/2	23/4	3
	Coefficients (c).
·75	·616	·614	·616	·610	·616	·612	·607	·607	·609
1·0	·613	·612	·612	·611	·612	·611	·604	·608	·609
1·25	·613	·614	·610	·608	·612	·608	·605	·605	·606
1·50	·610	·612	·611	·606	·610	·607	·603	·607	·605
1·75	·612	·611	·611	·605	·611	·605	·604	·607	·605
2·00	·609	·613	·609	·606	·609	·606	·604	·604	·605

The following table, compiled by J. T. 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.

Head to  Centre of  Orifice.	Ratio of Height to Width.
	4	2	11/2	1	3/4	1/2	1/4	1/8
Feet.	4 ft. high. 1 ft. wide.	2 ft. high. 1 ft. wide.	11/2 ft. high. 1 ft. wide.	1 ft. high. 1 ft. wide.	0·75 ft. high. 1 ft. wide.	0·50 ft. high. 1 ft. wide.	0·25 ft. high. 1 ft. wide.	0·125 ft. high.  1 ft. wide.
0·2	..	..	..	..	..	..	..	·6333
·3	..	..	..	..	..	..	·6293	·6334
·4	..	..	..	..	..	·6140	·6306	·6334
·5	..	..	..	..	·6050	·6150	·6313	·6333
·6	..	..	..	·5984	·6063	·6156	·6317	·6332
·7	..	..	..	·5994	·6074	·6162	·6319	·6328
·8	..	..	·6130	·6000	·6082	·6165	·6322	·6326
·9	..	..	·6134	·6006	·6086	·6168	·6323	·6324
1·0	..	..	·6135	·6010	·6090	·6172	·6320	·6320
1·25	..	·6188	·6140	·6018	·6095	·6173	·6317	·6312
1·50	..	·6187	·6144	·6026	·6100	·6172	·6313	·6303
1·75	..	·6186	·6145	·6033	·6103	·6168	·6307	·6296
2	..	·6183	·6144	·6036	·6104	·6166	·6302	·6291
2·25	..	·6180	·6143	·6029	·6103	·6163	·6293	·6286
2·50	·6290	·6176	·6139	·6043	·6102	·6157	·6282	·6278
2·75	·6280	·6173	·6136	·6046	·6101	·6155	·6274	·6273
3	·6273	·6170	·6132	·6048	·6100	·6153	·6267	·6267
3·5	·6250	·6160	·6123	·6050	·6094	·6146	·6254	·6254
4	·6245	·6150	·6110	·6047	·6085	·6136	·6236	·6236
4·5	·6226	·6138	·6100	·6044	·6074	·6125	·6222	·6222
5	·6208	·6124	·6088	·6038	·6063	·6114	·6202	·6202
6	·6158	·6094	·6063	·6020	·6044	·6087	·6154	·6154
7	·6124	·6064	·6038	·6011	·6032	·6058	·6110	·6114
8	·6090	·6036	·6022	·6010	·6022	·6033	·6073	·6087
9	·6060	·6020	·6014	·6010	·6015	·6020	·6045	·6070
10	·6035	·6015	·6010	·6010	·6010	·6010	·6030	·6060
15	·6040	·6018	·6010	·6011	·6012	·6013	·6033	·6066
20	·6045	·6024	·6012	·6012	·6014	·6018	·6036	·6074
25	·6048	·6028	·6014	·6012	·6016	·6022	·6040	·6083
30	·6054	·6034	·6017	·6013	·6018	·6027	·6044	·6092
35	·6060	·6039	·6021	·6014	·6022	·6032	·6049	·6103
40	·6066	·6045	·6025	·6015	·6026	·6037	·6055	·6114
45	·6054	·6052	·6029	·6016	·6030	·6043	·6062	·6125
50	·6086	·6060	·6034	·6018	·6035	·6050	·6070	·6140

§ 21. 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. 20). The plan of all the orifices is shown at S. The planks forming the orifice and sluice were each 2 in. thick, and the orifices were all 24 in. wide. The heads were measured immediately over the orifice. In this case,

Q = cb (H − h) √ { 2g(H + h)/2 }.

§ 22. 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. 21), the convergence for so much of the edge is suppressed. For such cases

G. Bidone found the following empirical formulae applicable:—

Table of Coefficients of Discharge for Rectangular Vertical Orifices in Fig. 20.

Head h above upper  edge of  Orifice in feet.	Height of Orifice, H − h, in feet.
	1.31			0.66			0.16			0.10
	P	Q	R	P	Q	R	P	Q	R	P	Q	R
0.328	0.598	0.644	0.648	0.634	0.665	0.668	0.691	0.664	0.666	0.710	0.694	0.696
.656	0.609	0.653	0.657	0.640	0.672	0.675	0.685	0.687	0.688	0.696	0.704	0.706
.787	0.612	0.655	0.659	0.641	0.674	0.677	0.684	0.690	0.692	0.694	0.706	0.708
.984	0.616	0.656	0.660	0.641	0.675	0.678	0.683	0.693	0.695	0.692	0.709	0.711
1.968	0.618	0.649	0.653	0.640	0.676	0.679	0.678	0.695	0.697	0.688	0.710	0.712
3.28	0.608	0.632	0.634	0.638	0.674	0.676	0.673	0.694	0.695	0.680	0.704	0.705
4.27	0.602	0.624	0.626	0.637	0.673	0.675	0.672	0.693	0.694	0.678	0.701	0.702
4.92	0.598	0.620	0.622	0.637	0.673	0.674	0.672	0.692	0.693	0.676	0.699	0.699
5.58	0.596	0.618	0.620	0.637	0.672	0.673	0.672	0.692	0.693	0.676	0.698	0.698
6.56	0.595	0.615	0.617	0.636	0.671	0.672	0.671	0.691	0.692	0.675	0.696	0.696
9.84	0.592	0.611	0.612	0.634	0.669	0.670	0.668	0.689	0.690	0.672	0.693	0.693

For rectangular orifices,

Cc = 0.62 (1 + 0.152 n/p);

and for circular orifices,

Cc = 0.62 (1 + 0.128 n/p);

when 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 cc, when the border extends over 1/4, 1/2, or 3/4 of the whole perimeter:—

n/p	Cc  Rectangular Orifices	Cc  Circular Orifices
0.25	0.643	.640
0.50	0.667	.660
0.75	0.691	.680

Fig. 20.	Fig. 21.

For larger values of n/p the formulae are not applicable. C. R. Bornemann has shown, however, that these formulae for suppressed contraction are not reliable.

§ 23. 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.

Fig. 22.

§ 24. 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 in. wide, and small in proportion to the channel of approach (fig. 22, A, B, C).

h2 − h1 in feet		h1 in feet.
		.0656	.164	.328	.656	1.640	3.28	4.92	6.56	9.84
A	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$0.656	.480	.511	.542	.574	.599	.601	.601	.601	.601
B		.480	.510	.538	.506	.592	.600	.602	.602	.601
C		.527	.553	.574	.592	.607	.610	.610	.609	.608
A	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$0.164	.488	.577	.624	.631	.625	.624	.619	.613	.606
B		.487	.571	.606	.617	.626	.628	.627	.623	.618
C		.585	.614	.633	.645	.652	.651	.650	.650	.649

Fig. 23.

§ 25. 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 G. Bidone (1781–1839); subsequently H. G. Magnus (1802–1870) measured jets from different orifices; and later Lord Rayleigh (Proc. Roy. Soc. xxix. 71) investigated them anew.

Fig. 23 shows some forms, the upper figure giving the shape of the orifices, and the others sections of the jet. 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 normal 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.

Fig. 24.

Let a₁, a₂ (fig. 24) be two points in an orifice at depths h₁, h₂ from the free surface. The filaments issuing at a₁, a₂ will have the different velocities √ 2gh₁ and √ 2gh₂. Consequently they will tend to describe parabolic paths a₁cb₁ and a₂cb₂ 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 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. 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 H. Buff (1805–1878), namely, that it is due to surface tension.

§ 26. Influence of Temperature on Discharge of Orifices.—Professor W. C. Unwin found (Phil. Mag., October 1878, p. 281) that for sharp-edged orifices temperature has a very small influence on the discharge. For an orifice 1 cm. in diameter with heads of about 1 to 11/2 ft. the coefficients were:—

Temperature F.	C.
205°	.594
62°	.598

For a conoidal or bell-mouthed orifice 1 cm. diameter the effect of temperature was greater:—

Temperature F.	C.
190°	0.987
130°	0.974
60°	0.942

an increase in velocity of discharge of 4% when the temperature increased 130°.

J. G. Mair repeated these experiments on a much larger scale (Proc. Inst. Civ. Eng. lxxxiv.). For a sharp-edged orifice 21/2 in. diameter, with a head of 1.75 ft., the coefficient was 0.604 at 57° and 0.607 at 179° F., a very small difference. With a conoidal orifice the coefficient was 0.961 at 55° and 0.981 at 170° F. The corresponding coefficients of resistance are 0.0828 and 0.0391, showing that the resistance decreases to about half at the higher temperature.

§ 27. Fire Hose Nozzles.—Experiments have been made by J. R. Freeman on the coefficient of discharge from smooth cone nozzles used for fire purposes. The coefficient was found to be 0.983 for 3/4-in. nozzle; 0.982 for 7/8 in.; 0.972 for 1 in.; 0.976 for 11/8 in.; and 0.971 for 11/4 in. The nozzles were fixed on a taper play-pipe, and the coefficient includes the resistance of this pipe (Amer. Soc. Civ. Eng. xxi., 1889). Other forms of nozzle were tried such as ring nozzles for which the coefficient was smaller.

IV. THEORY OF THE STEADY MOTION OF FLUIDS.

§ 28. 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.

(b) 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.

(d) 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.

§ 29. 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 Bernoulli’s theorem.

Fig. 25.

Let AB (fig. 25) 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 OO be the free surface and XX any horizontal datum line. Let ω be the area of a normal cross section, v the velocity, p the intensity of pressure, and z the elevation above XX, of the elementary stream AB at A, and ω₁, p₁, v₁, z₁ 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, v₁t, and the volumes of fluid AA′, BB′ are the equal inflow and outflow = Qt = ωvt = ω₁v₁t, 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 AA′ to BB′; that is, GQt (z − z₁). 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) − (p₁ω₁ × v₁t) = Qt(p − p₁). 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 AA′ and BB′ is GQt/g, and the change of kinetic energy is therefore (GQt/g) (v₁²/2 − v²/2). Equating this to the work done on the mass AB,

GQt (z − z₁) + Qt (p − p₁) = (GQt/g) (v₁²/2 − v²/2).

Dividing by GQt and rearranging the terms,

v²/2g + p/G + z = v₁²/2g + p₁/G + z₁;

(1)

or, as A and B are any two points,

v²/2g + p/G + z = constant = H.

(2)

Now v²/2g is the head due to the velocity v, p/G is the head equivalent to the pressure, and z is the elevation above the datum (see § 16). 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 ℔ of fluid falling to the datum line, and similarly p/G and v²/2g are the quantities of work which would be done by 1 ℔ 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 ℔ 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 ℔ of fluid is uniformly distributed along the stream line. If the free surface of the fluid OO is taken as the datum, and −h, −h₁ are the depths of A and B measured down from the free surface, the equation takes the form

v²/2g + p/G − h = v₁²/2g + p₁/G − h₁;

(3)

or generally

v²/2g + p/G − h = constant.

(3a)

Fig. 26.

§ 30. Second Form of the Theorem of Bernoulli.—Suppose at the two sections A, B (fig. 26) 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 b = p/G, and b′ = p₁/G. Consequently the tops of the pressure columns A′ and B′ will be at total heights b + c = p/G + z and b′ + c′ = p₁/G + z₁ above the datum line XX. The difference of level of the pressure column tops, or the fall of free surface level between A and B, is therefore

ξ = (p − p₁) / G + (z − z₁);

and this by equation (1), § 29 is (v₁² − v²)/2g. 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 ft. above XX by the quantities a = v²/2g and a′ = v₁²/2g, when friction is absent.

§ 31. Illustrations of the Theorem of Bernoulli. In a lecture to the mechanical section of the British Association in 1875, W. Froude gave some experimental illustrations of the principle of Bernoulli. He remarked that it was a common but erroneous impression that a fluid exercises in a contracting pipe A (fig. 27) an excess of pressure against the entire converging surface which it meets, 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.

Fig. 27.

Fig. 28 shows a pipe so formed that a contraction is followed by an enlargement, and fig. 29 one in which an enlargement is followed by a contraction. The vertical pressure columns show the decrease of pressure at the contraction and increase of pressure at the enlargement. The line abc in both figures shows the variation of free surface level, supposing the pipe frictionless. In actual pipes, however, work is expended in friction against the pipe; the total head diminishes in proceeding along the pipe, and the free surface level is a line such as ab₁c₁, falling below abc.

Froude further pointed out that, if a pipe contracts and enlarges again to the same size, the resultant pressure on the converging part exactly balances the resultant pressure on the diverging part so that there is no tendency to move the pipe bodily when water flows through it. Thus the conical part AB (fig. 30) presents the same projected surface as HI, and the pressures parallel to the axis of the pipe, normal to these projected surfaces, balance each other. Similarly the pressures on BC, CD balance those on GH, EG. In the same way, in any combination of enlargements and contractions, a balance of pressures, due to the flow of liquid parallel to the axis of the pipe, will be found, provided the sectional area and direction of the ends are the same.

Fig. 28.

Fig. 29.

The following experiment is interesting. Two cisterns provided with converging pipes were placed so that the jet from one was exactly opposite the entrance to the other. The cisterns being filled very nearly to the same level, the jet from the left-hand cistern A entered the right-hand cistern B (fig. 31), shooting across the free space between them without any waste, except that due to indirectness of aim and want of exact correspondence in the form of the orifices. In the actual experiment there was 18 in. of head in the right and 201/2 in. of head in the left-hand cistern, so that about 21/2 in. were wasted in friction. It will be seen that in the open space between the orifices there was no pressure, except the atmospheric pressure acting uniformly throughout the system.

Fig. 30.

Fig. 31.

§ 32. Venturi Meter.—An ingenious application of the variation of pressure and velocity in a converging and diverging pipe has been made by Clemens Herschel in the construction of what he terms a Venturi Meter for measuring the flow in water mains. Suppose that, as in fig. 32, a contraction is made in a water main, the change of section being gradual to avoid the production of eddies. The ratio ρ of the cross sections at A and B, that is at inlet and throat, is in actual meters 5 to 1 to 20 to 1, and is very carefully determined by the maker of the meter. Then, if v and u are the velocities at A and B, u = ρv. Let pressure pipes be introduced at A, B and C, and let H₁, H, H₂ be the pressure heads at those points. Since the velocity at B is greater than at A the pressure will be less. Neglecting friction

H₁ + v²/2g = H + u²/2g,

H₁ − H = (u² − v²) / 2g = (ρ² − 1) v² 2g.

Let h = H₁ − H be termed the Venturi head, then

u = √ { ρ².2gh / (ρ² − 1) },

from which the velocity through the throat and the discharge of the main can be calculated if the areas at A and B are known and h observed. Thus if the diameters at A and B are 4 and 12 in., the areas are 12.57 and 113.1 sq. in., and ρ = 9,

u = √ 81/80 √ (2gh) = 1.007 √ (2gh).

If the observed Venturi head is 12 ft.,

u = 28 ft. per sec.,

and the discharge of the main is

28 × 12.57 = 351 cub. ft. per sec.

Fig. 32.

Fig. 33.

Hence by a simple observation of pressure difference, the flow in the main at any moment can be determined. Notice that the pressure height at C will be the same as at A except for a small loss h_f due to friction and eddying between A and B. To get the pressure at the throat very exactly Herschel surrounds it by an annular passage communicating with the throat by several small holes, sometimes formed in vulcanite to prevent corrosion. Though constructed to prevent eddying as much as possible there is some eddy loss. The main effect of this is to cause a loss of head between A and C which may vary from a fraction of a foot to perhaps 5 ft. at the highest velocities at which a meter can be used. The eddying also affects a little the Venturi head h. Consequently an experimental coefficient must be determined for each meter by tank measurement. The range of this coefficient is, however, surprisingly small. If to allow for friction, u = k √ {ρ²/(ρ² − 1)} √(2gh), then Herschel found values of k from 0.97 to 1.0 for throat velocities varying from 8 to 28 ft. per sec. The meter is extremely convenient. At Staines reservoirs there are two meters of this type on mains 94 in. in diameter. Herschel contrived a recording arrangement which records the variation of flow from hour to hour and also the total flow in any given time. In Great Britain the meter is constructed by G. Kent, who has made improvements in the recording arrangement.

In the Deacon Waste Water Meter (fig. 33) a different principle is used. A disk D, partly counter-balanced by a weight, is suspended in the water flowing through the main in a conical chamber. The unbalanced weight of the disk is supported by the impact of the water. If the discharge of the main increases the disk rises, but as it rises its position in the chamber is such that in consequence of the larger area the velocity is less. It finds, therefore, a new position of equilibrium. A pencil P records on a drum moved by clockwork the position of the disk, and from this the variation of flow is inferred.

§ 33. Pressure, Velocity and Energy in Different Stream Lines.—The equation of Bernoulli gives the variation of pressure and velocity from point to point along a stream line, and shows that the total energy of the flow across any two sections is the same. Two other directions may be defined, one normal to the stream line and in the plane containing its radius of curvature at any point, the other normal to the stream line and the radius of curvature. For the problems most practically useful it will be sufficient to consider the stream lines as parallel to a vertical or horizontal plane. If the motion is in a vertical plane, the action of gravity must be taken into the reckoning; if the motion is in a horizontal plane, the terms expressing variation of elevation of the filament will disappear.^[3]

Fig. 34.

Let AB, CD (fig. 34) be two consecutive 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,

H = z + p/G + v²/2g.

Differentiating, we get

dH = dz + dp/G + v dv/g,

(1)

for the increment of energy between Q and P. But

dz = PQ cos φ = ds cos φ;

∴ dH = dp/G + v dv/g + ds cos φ,

(1a)

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 Gω ds. Hence, taking the components of the forces parallel to PQ—

ωdp = Gv²ω ds/gρ − Gω cos φds,

where ρ is the radius of curvature of the stream line at Q. Consequently, introducing these values in (1),

dH = v²ds/gρ + vdv/g = (v/g) (v/ρ + dv/ds) ds.

(2)

Currents

§ 34. Rectilinear Current.—Suppose the motion is in parallel straight stream lines (fig. 35) in a vertical plane. Then ρ is infinite, and from eq. (2), § 33,

dH = vdv/g.

Comparing this with (1) we see that

dz + dp/G = 0;

∴ z + p/G = constant;

(3)

Fig. 35.

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 r₁ and r₂, where the velocities are v₁ and v₂, and the pressures p₁ and p₂. Since the flow across each cylindrical section of the current is the same,

Q = 2πr₁ δv₁ = 2πr₂ δv₂

r₁v₁ = r₂v₂

 r₁/r₂ = v₂/v₁.

(4)

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 = p₁/G + v₁²/2g = p₂/G + v₂²/2g;

= p₂/G + r₁² + v₁²/r₂²2g;

(p₂ − p₁)/G = v₁² (1 − r₁²/r₂²)/2g;

(5)

p₂/G = H − r₁²v₁²/r₂²2g.

(6)

Hence the pressure increases from the interior outwards, in a way indicated by the pressure columns in fig. 36, the curve through the free surfaces of the pressure columns being, in a radial section, the quasi-hyperbola of the form xy² = c³. This curve is asymptotic to a horizontal line, H ft. above the line from which the pressures are measured, and to the axis of the current.

Fig. 36.

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 = p/G + v²/2g = constant;

∴ dH = dp/G + vdv/g = 0.

(7)

Consider two stream lines at radii r and r + dr (fig. 36). Then in (2), § 33, ρ = r and ds = dr,

v² dr/gr + vdv/g = 0,

dv/v = −dr/r,

v ∞ 1/r,

(8)

precisely as in a radiating current; and hence the distribution of pressure is the same, and formulae 5 and 6 are applicable to this case.

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.

§ 35. Forced Vortex.—If the law of motion in a rotating current is different 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), § 33, considering two circular stream lines of radii r and r + dr (fig. 37), we have ρ = r, ds = dr. If the angular velocity is α, then v = αr and dv = αdr. Hence

dH = α²rdr/g + α²rdr/g = 2α²rdr/g.

Comparing this with (1), § 33, and putting dz = 0, because the motion is horizontal,

dp/G + α²rdr/g = 2α²rdr/g,

dp/G = α²rdr/g,

p/G = α²/2g + constant.

(9)

Let p₁, r₁, v₁ be the pressure, radius and velocity of one cylindrical section, p₂, r₂, v₂ those of another; then

p₁/G − α²r₁²/2g = p₂/G − α²r₂²/2g;

(p₂ − p₁)/G = α² (r₂² − r₁²)/2g = (v₂² − v₁²)/2g.

(10)

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. 37).

Fig. 37.

Dissipation of Head in Shock

§ 36. Relation of Pressure and Velocity in a Stream in Steady Motion when the Changes of Section of the Stream are Abrupt.—When a stream changes section abruptly, rotating eddies are formed 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. 38 to represent a stream having such an abrupt change of section. Let AB, CD be normal sections at points where ordinary stream line motion has not been disturbed and where it has been re-established. Let ω, p, v be the area of section, pressure and velocity at AB, and ω₁, p₁, v₁ corresponding quantities at CD. Then if no work were expended internally, and assuming the stream horizontal, we should have

p/G + v²/2g = p₁/G + v₁²/2g.

(1)

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

pω + p₀ (ω₁ − ω) − p₁ω₁,

where p₀ is put for the unknown pressure on the annular space between AB and EF. The impulse of that force is

{ pω + p₀ (ω₁ − ω) − p₁ω₁ } t.

Fig. 38.

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 Qt = ωvt = ω₁v₁t, and the momentum of these masses is (G/g) Qvt and (G/g) Qv₁t. The change of momentum is therefore (G/g) Qt (v₁ − v). Equating this to the impulse,

{ pω + p₀ (ω₁ − ω) − p₁ω₁ } t = (G/g) Qt (v₁ − v).

Assume that p₀ = 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 = ω₁v₁)

(p − p₁) = (G/g) v₁ (v₁ − v);

p/G − p₁/G = v₁ (v₁ − v)/g;

(2)

p/G + v²/2g = p₁/G + v₁²/2g + (v − v₁)²/2g.

(3)

This differs from the expression (1), § 29, obtained for cases where no sensible internal work is done, by the last term on the right. That is, (v − v₁)²/2g has to be added to the total head at CD, which is p₁/G + v₁²/2g, to make it equal to the total head at AB, or (v − v₁)²/2g is the head lost in shock at the abrupt change of section. But (v − v₁) 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 (v − v₁)²/2g foot-pounds of energy is wasted for each pound of fluid. Experiment verifies this result, so that the assumption that p₀ = p appears to be admissible.

If there is no shock,

p₁/G = p/G + (v² − v₁²)/2g.

If there is shock,

p₁/G = p/G − v₁ (v₁ − v)/g.

Hence the pressure head at CD in the second case is less than in the former by the quantity (v − v₁)²/2g, or, putting ω₁v₁ = ωv, by the quantity

(v²/2g) (1 − ω/ω₁)².

(4)

V. THEORY OF THE DISCHARGE FROM ORIFICES AND MOUTHPIECES

Fig. 39.

§ 37. 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. 39 represent a vessel with vertical sides, OO being the free water surface, at which 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 Ω be the area of the mouthpiece AB, ω that of the contracted jet aa Suppose that in a short time t, the mass OOaa comes to the position O′O′ 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 OE, and so for all other portions of the vertical surfaces of the mass, excepting the portion EF opposite the mouthpiece and the surface AaaB of the jet. On EF the pressure is simply the hydrostatic pressure due to the depth, that is, (p_a + Gh). On the surface and section AaaB 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 −p_aΩ. Hence the resultant horizontal force for the whole mass OOaa is (p_a + Gh) Ω − p_aΩ = GhΩ. Its impulse in the time t is GhΩt. Since the motion is steady there is no change of momentum between O′O′ and aa. The change of horizontal momentum is, therefore, the difference of the horizontal momentum lost in the space OOO′O′ 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 space is (G/g)ωvt; its momentum is (G/g)ωv²t. Equating impulse to momentum gained,

GhΩt = (G/g) ωv²t;

∴ ω/Ω = gh/v²

But

v² = 2gh, and ω/Ω = c_c;

∴ ω/Ω = 1/2 = c_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.

Fig. 40.	Fig. 41.

§ 38. Velocity of Filaments issuing in a Jet.—A jet is composed of fluid 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, fig. 40 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 p₁, v₁, h₁, be the pressure, velocity and depth below datum at M; p, v, h, the corresponding quantities at m. Then § 29, eq. (3a),

v₁²/2g + p₁/G − h₁ = v²/2g + p/G − h

(1)

But at M, since the velocity is insensible, the pressure is the hydrostatic pressure due to the depth; that is v₁ = 0, p₁ = p_a + Gh₁. At m, p = p_a, the atmospheric pressure round the jet. Hence, inserting these values,

0 + p_a/G + h₁ − h₁ = v²/2g + p_a/G − h;

v2/2g = h;

(2)

or

v = √ (2gh) = 8.025V √ h

(2a)

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.—Let the orifice discharge below the level of the tail water. Then using the notation shown in fig. 41, we have at M, v₁ = 0, p₁ = Gh; + p_a at m, p = Gh₃ + p_a. Inserting these values in (3), § 29,

0 + h₁ + p_a/G − h₁ = v²/2g + h₃ − h₂2 + p_a/G;

v²/2g = h₂ − h₃ = h,

(3)

where h is the difference of level of the head and tail water, and may be termed the effective head producing flow.

Fig. 42.

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 p₀ into a vessel in which the pressure is p, fig. 42. The pressure p₀ will produce the same effect as a layer of fluid of thickness p₀/G added to the head water; and the pressure p, will produce the same effect as a layer of thickness p/G added to the tail water. Hence the effective difference of level, or effective head producing flow, will be

h = h₀ + p₀/G − p/G;

and the velocity of discharge will be

v = √ [ 2g { h₀ + (p₀ − p) / G } ].

(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 = cωv = cω√ (2gh),

(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

Fig. 43.

§ 39. 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. 43) be the breadth of the jet, h₁ and h₂ 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 bdh, and the velocity of discharge √2gh. The discharge per second in this lamina is therefore b√2gh dh, and that of the whole jet is therefore

${\mbox{Q}}=\int _{h_{1}}^{h_{2}}b{\sqrt {(}}2gh)dh$

= 2/3 b √2g { h₂^3/2 − h₁^3/2 },

(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, H₁, H₂, the depths to the upper and lower edges of the orifice. Put

b (h ₂^3/2 − h₁^3/2) / B (H₂^3/2 − H₁^3/2) = c.

(7)

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

Q = 2/3 cB √2g (H₂^3/2 − H₁^3/2),

(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

b (h ₂ − h₁) / B (H₂ − H₁),

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 values of B/H₂ and B/H₁, and must be ascertained experimentally.

Relation between the Expressions (5) and (8).—For a rectangular orifice the area of the orifice is ω = B(H₂ − H₁), and the depth measured to its centre is 1/2 (H₂ + H₁). Putting these values in (5),

Q₁ = cB (H₂ − H₁) √ {g (H₂ + H₁) }.

From (8) the discharge is

Q₂ = 2/3 cB √2g (H₂^3/2 − H₁^3/2).

Hence, for the same value of c in the two cases,

Q₂/Q₁ = 2/3 (H₂^3/2 − H₁^3/2) / [ (H₂ − H₁) √ { (H₂ + H₁)/2} ].

Let H₁/H₂ = σ, then

Q₂/Q₁ = 0.9427 (1 − σ^3/2) / {1 − σ √ (1 + σ) }.

(9)

If H₁ varies from 0 to ∞, σ( = H₁/H₂) varies from 0 to 1. The following table gives values of the two estimates of the discharge for different values of σ:—

H1/H2 = σ.	Q2/Q1.	H1/H2 = σ.	Q2/Q1.
0.0	.943	0.8	.999
0.2	.979	0.9	.999
0.5	.995	1.0	1.000
0.7	.998

Hence it is obvious that, except for very small values of σ, the simpler equation (5) gives values sensibly identical with those of (8). When σ < 0.5 it is better to use equation (8) with values of c determined experimentally for the particular proportions of orifice which are in question.

Fig. 44.

§ 40. Large Jets having a Circular Section from Orifices in a Vertical Plane Surface.—Let fig. 44 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 depths h₁ + y and h₁ + y + dy, is

dQ = b √ {2g (h₁ + y) } dy.

The whole discharge of the jet is

Q = $\int _{0}^{d}$ b √ { 2g (h ₁ + y) } dy.

But b = d sin φ; y = 1/2d (1 − cos φ); dy = 1/2d sin φ dφ. Let ε = d/(2h₁ + d), then

Q = 1/2d ² √ { 2g (h ₁ + d/2) } $\int _{0}^{\pi }\sin ^{2}\phi {\sqrt {1-\epsilon \cos \phi }}\ d\phi .$

From eq. (5), putting ω = πd ²/4, h = h₁ + d/2, c = 1 when d is the diameter of the jet and not that of the orifice,

Q₁ = 1/4πd ² √ {2g(h₁ + d/2) },

Q/Q₁ = 2/π $\int _{0}^{\pi }$ sin² φ √ {1 − ε cos φ} dφ.

For

h₁ = ∞, ε = 0 and Q/Q₁ = 1;

and for

h₁ = 0, ε = 1 and Q/Q₁ = 0.96.

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

§ 41. 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 H₁ = 0 in the formula for the corresponding orifice, obtained as in the preceding section. Thus for a rectangular notch, put H₁ = 0 in (8). Then

Q = 2/3 cB √(2g) H^3/2,

(11)

where H is put for the depth to the crest of the weir or the bottom of the notch. Fig. 45 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 × k √(2gH),

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.

§ 42. 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, (b) 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.

Fig. 45.

J. B. Francis’s experiments showed that a perfect end contraction, when the heads varied from 3 to 24 in., 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 l − 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 − 0.1nH, 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(l − 0.1nH)H and (§ 41) the mean velocity will be 2/3 √(2gH). Consequently the discharge will be given by an equation of the form

Q = 2/3 c (l − 0.1nH) H √2gH

= 5.35c (l − 0.1nH) H^3/2.

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 weir being sharp-edged.

§ 43. Triangular Notch (fig. 46).—Consider a lamina issuing between the depths h and h + dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is √(2gh). Hence the discharge for this lamina is

b√2gh dh.

But

B/b = H / (H − h); b = B (H − h) / H.

Hence discharge of lamina

= B(H − h) √(2gh) dh/H;

and total discharge of notch

= Q = B √(2g) ∫H0 (H − h) h^1/2 dh/H

= 4/15 B √(2g) H^3/2.

or, introducing a coefficient to allow for contraction,

Q = 4/15 cB √(2g) H^1/2,

Fig. 46.

When a notch is used to gauge a stream of varying flow, the ratio B/H 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 § 41, that since 1/2BH is the area of section of the stream through the notch, the formula is again of the form

Q = c × 1/2BH × k √(2gH),

where k = 8/15 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.

Coefficients for the Discharge over Weirs, derived from the Experiments of T. E. 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.

Heads in inches measured from still Water in Reservoir.	Sharp Edge.		Planks 2 in. thick, square on Crest.				Crests 3 ft. wide.
	3 ft. long.	10 ft. long.	3 ft. long.	6 ft. long.	10 ft. long.	10 ft. long, wing-boards making an angle of 60°.	3 ft. long. level.	3 ft. long, fall 1 in 18.	3 ft. long, fall 1 in 12.	6 ft. long. level.	10 ft. long. level.	10 ft. long, fall 1 in 18.
1	.677	.809	.467	.459	.435[4]	.754	.452	.545	.467	..	.381	.467
2	.675	.803	.509*	.561	.585*	.675	.482	.546	.533	..	.479*	.495*
3	.630	.642*	.563*	.597*	.569*	..	.441	.537	.539	.492*	..	..
4	.617	.656	.549	.575	.602*	.656	.419	.431	.455	.497*	..	.515
5	.602	.650*	.588	.601*	.609*	.671	.479	.516	..	..	.518	..
6	.593	..	.593*	.608*	.576*	..	.501*	..	.531	.507	.513	.543
7	..	..	.617*	.608*	.576*	..	.488	.513	.527	.497	..	..
8	..	.581	.606*	.590*	.548*	..	.470	.491	..	..	.468	.507
9	..	.530	.600	.569*	.558*	..	.476	.492*	.498	.480*	.486	..
10	..	..	.614*	.539	.534*	..	..	..	..	.465*	.455	..
12	..	..	..	.525	.534*	..	..	..	..	.467*	..	..
14	..	..	..	.549*	..	..	..	..	..	..	..	..

Fig. 47.

§ 44. Weir with a Broad Sloping Crest.—Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a movement sensibly rectilinear and uniform (fig. 47). 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 OO be the surface level in the reservoir, and let a be at a height h″ below OO, and h′ above a′. Let h be the distance from OO 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 Gh″; at a′ it is Gz. If v be the velocity at a′,

v²/2g = h′ + h″ − z = h − e;

Q = be √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 h, obtained by equating dQ/de to zero. This gives e = 2/3h, and, inserting this value,

Q = 0.385 bh √2gh,

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 weir formula with c = 0.577.

Special Cases of Discharge from Orifices

§ 45. Cases in which the Velocity of Approach needs to be taken into Account. Rectangular Orifices and Notches.—In finding the velocity at the orifice in the preceding investigations, it has been assumed that the head h has been measured from the free surface of still water above the orifice. In many cases which occur in practice the channel of approach to an orifice or notch is not so large, relatively to the stream through the orifice or notch, that the velocity in it can be disregarded.

Fig. 48.

Let h₁, h₂ (fig. 48) be the heads measured from the free surface to the top and bottom edges of a rectangular orifice, at a point in the channel of approach where the velocity is u. It is obvious that a fall of the free surface,

ɧ = u²/2g

has been somewhere expended in producing the velocity u, and hence the true heads measured in still water would have been h₁ + ɧ and h₂ + ɧ. Consequently the discharge, allowing for the velocity of approach, is

Q = 2/3 cb √2g { (h ₂ + ɧ)^3/2 − (h ₁ + ɧ)^3/2 }.

(1)

And for a rectangular notch for which h₁ = 0, the discharge is

Q = 2/3 cb √2g { (h ₂ + ɧ)^3/2 − ɧ^3/2 }.

(2)

In cases where u can be directly determined, these formulae give the discharge quite simply. When, however, u is only known as a function of the section of the stream in the channel of approach, they become complicated. Let Ω be the sectional area of the channel where h₁ and h₂ are measured. Then u = Q/Ω and ɧ = Q²/2g Ω².

This value introduced in the equations above would render them excessively cumbrous. In cases therefore where Ω only is known, it is best to proceed by approximation. Calculate an approximate value Q′ of Q by the equation

Q′ = 2/3 cb √2g {h₂^3/2 − h₁^3/2 }.

Then ɧ = Q′²/2gΩ² nearly. This value of ɧ introduced in the equations above will give a second and much more approximate value of Q.

Fig. 49.

§ 46. Partially Submerged Rectangular Orifices and Notches.—When the tail water is above the lower but below the upper edge of the orifice, the flow in the two parts of the orifice, into which it is divided by the surface of the tail water, takes place under different conditions. A filament M₁m₁ (fig. 49) in the upper part of the orifice issues with a head h′ which may have any value between h₁ and h. But a filament M₂m₂ issuing in the lower part of the orifice has a velocity due to h″ − h″′, or h, simply. In the upper part of the orifice the head is variable, in the lower constant. If Q₁, Q₂ are the discharges from the upper and lower parts of the orifice, b the width of the orifice, then

Q₁ = 2/3 cb √2g { h^3/2 − h₁^3/2 }

Q₂ = cb (h ₂ − h) √2gh.

(3)

In the case of a rectangular notch or weir, h₁ = 0. Inserting this value, and adding the two portions of the discharge together, we get for a drowned weir

Q = cb √2gh (h ₂ − h/3),

(4)

where h is the difference of level of the head and tail water, and h₂ is the head from the free surface above the weir to the weir crest (fig. 50).

From some experiments by Messrs A. Fteley and F.P. Stearns (Trans. Am. Soc. C.E., 1883, p. 102) some values of the coefficient c can be reduced

h3/h2	c	h3/h2	c
0.1	0.629	0.7	0.578
0.2	0.614	0.8	0.583
0.3	0.600	0.9	0.596
0.4	0.590	0.95	0.607
0.5	0.582	1.00	0.628
0.6	0.578

If velocity of approach is taken into account, let ɧ be the head due to that velocity; then, adding ɧ to each of the heads in the equations (3), and reducing, we get for a weir

Q = cb √2g [ (h ₂ + ɧ) (h + ɧ)^1/2 − 1/3 (h + ɧ)^3/2 − 2/3 ɧ^3/2 ];

(5)

an equation which may be useful in estimating flood discharges.

Fig. 50.

Bridge Piers and other Obstructions in Streams.—When the piers of a bridge are erected in a stream they create an obstruction to the flow of the stream, which causes a difference of surface-level above and below the pier (fig. 51). If it is necessary to estimate this difference of level, the flow between the piers may be treated as if it occurred over a drowned weir. But the value of c in this case is imperfectly known.

§ 47. Bazin’s Researches on Weirs.—H. Bazin has executed a long series of researches on the flow over weirs, so systematic and complete that they almost supersede other observations. The account of them is contained in a series of papers in the Annales des Ponts et Chaussées (October 1888, January 1890, November 1891, February 1894, December 1896, 2nd trimestre 1898). Only a very abbreviated account can be given here. The general plan of the experiments was to establish first the coefficients of discharge for a standard weir without end contractions; next to establish weirs of other types in series with the standard weir on a channel with steady flow, to compare the observed heads on the different weirs and to determine their coefficients from the discharge computed at the standard weir. A channel was constructed parallel to the Canal de Bourgogne, taking water from it through three sluices 0.3 × 1.0 metres. The water enters a masonry chamber 15 metres long by 4 metres wide where it is stilled and passes into the canal at the end of which is the standard weir. The canal has a length of 15 metres, a width of 2 metres and a depth of 0.6 metres. From this extends a channel 200 metres in length with a slope of 1 mm. per metre. The channel is 2 metres wide with vertical sides. The channels were constructed of concrete rendered with cement. The water levels were taken in chambers constructed near the canal, by floats actuating an index on a dial. Hook gauges were used in determining the heads on the weirs.

Fig. 51.

Standard Weir.—The weir crest was 3.72 ft. above the bottom of the canal and formed by a plate 1/4 in. thick. It was sharp-edged with free overfall. It was as wide as the canal so that end contractions were suppressed, and enlargements were formed below the crest to admit air under the water sheet. The channel below the weir was used as a gauging tank. Gaugings were made with the weir 2 metres in length and afterwards with the weir reduced to 1 metre and 0.5 metre in length, the end contractions being suppressed in all cases. Assuming the general formula

Q = mlh √(2gh),

(1)

Bazin arrives at the following values of m:—

Coefficients of Discharge of Standard Weir.

Head h metres.	Head h feet.	m
0.05	.164	0.4485
0.10	.328	0.4336
0.15	.492	0.4284
0.20	.656	0.4262
0.25	.820	0.4259
0.30	.984	0.4266
0.35	1.148	0.4275
0.40	1.312	0.4286
0.45	1.476	0.4299
0.50	1.640	0.4313
0.55	1.804	0.4327
0.60	1.968	0.4341

Bazin compares his results with those of Fteley and Stearns in 1877 and 1879, correcting for a different velocity of approach, and finds a close agreement.

Influence of Velocity of Approach.—To take account of the velocity of approach u it is usual to replace h in the formula by h + au²/2g where α is a coefficient not very well ascertained. Then

Q = μl (h + αu²/2g) √ { 2g (h + αu²/2g) }

= μlh √(2gh) (1 + αu²/2gh)^3/2.

(2)

The original simple equation can be used if

m = μ (1 + αu²/2gh)^3/2

or very approximately, since u²/2gh is small,

m = μ (1 + 3/2αu²/2gh).

(3)

Fig. 52.

Now if p is the height of the weir crest above the bottom of the canal (fig. 52), u = Q/l(p + h). Replacing Q by its value in (1)

u²/2gh = Q² / {2ghl²(p + h)²} = m² {h/(p + h) }²,

(4)

so that (3) may be written

m = μ [1 + k {h/(p + h)}² ].

(5)

Gaugings were made with weirs of 0.75, 0.50, 0.35, and 0.24 metres height above the canal bottom and the results compared with those of the standard weir taken at the same time. The discussion of the results leads to the following values of m in the general equation (1):—

m = μ (1 + 2.5u²/2gh)

= μ [1 + 0.55 {h/(p + h)}² ].

Values of μ—

Head h metres.	Head h feet.	μ
0.05	.164	0.4481
0.10	.328	0.4322
0.20	.656	0.4215
0.30	.984	0.4174
0.40	1.312	0.4144
0.50	1.640	0.4118
0.60	1.968	0.4092

An approximate formula for μ is:

μ = 0.405 + 0.003/h (h in metres)
μ = 0.405 + 0.01/h (h in feet).  

Inclined Weirs.—-Experiments were made in which the plank weir was inclined up or down stream, the crest being sharp and the end contraction suppressed. The following are coefficients by which the discharge of a vertical weir should be multiplied to obtain the discharge of the inclined weir.

			Coefficient.
Inclination up stream		1 to 1	0.93
”	”	3 to 2	0.94
”	”	3 to 1	0.96
Vertical weir		..	1.00
Inclination	down stream	3 to 1	1.04
”	”	3 to 2	1.07
”	”	1 to 1	1.10
”	”	1 to 2	1.12
”	”	1 to 4	1.09

The coefficient varies appreciably, if h/p approaches unity, which case should be avoided.

Fig. 53.

Fig. 54.

In all the preceding cases the sheet passing over the weir is detached completely from the weir and its under-surface is subject to atmospheric pressure. These conditions permit the most exact determination of the coefficient of discharge. If the sides of the canal below the weir are not so arranged as to permit the access of air under the sheet, the phenomena are more complicated. So long as the head does not exceed a certain limit the sheet is detached from the weir, but encloses a volume of air which is at less than atmospheric pressure, and the tail water rises under the sheet. The discharge is a little greater than for free overfall. At greater head the air disappears from below the sheet and the sheet is said to be “drowned.” The drowned sheet may be independent of the tail water level or influenced by it. In the former case the fall is followed by a rapid, terminating in a standing wave. In the latter case when the foot of the sheet is drowned the level of the tail water influences the discharge even if it is below the weir crest.

Weirs with Flat Crests.—The water sheet may spring clear from the upstream edge or may adhere to the flat crest falling free beyond the down-stream edge. In the former case the condition is that of a sharp-edged weir and it is realized when the head is at least double the width of crest. It may arise if the head is at least 11/2 the width of crest. Between these limits the condition of the sheet is unstable. When the sheet is adherent the coefficient m depends on the ratio of the head h to the width of crest c (fig. 53), and is given by the equation m = m₁ [0.70 + 0.185h/c], where m₁ is the coefficient for a sharp-edged weir in similar conditions. Rounding the upstream edge even to a small extent modifies the discharge. If R is the radius of the rounding the coefficient m is increased in the ratio 1 to 1 + R/h nearly. The results are limited to R less than 1/2 in.

Drowned Weirs.—Let h (fig. 54) be the height of head water and h₁ that of tail water above the weir crest. Then Bazin obtains as the approximate formula for the coefficient of discharge

m = 1.05m₁ [1 + 1/5 h₁/p] ³√ { (h − h₁) / h },

Fig. 55.

where as before m₁ is the coefficient for a sharp-edged weir in similar conditions, that is, when the sheet is free and the weir of the same height.

§ 48. 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. 55 shows one of these separating weirs in the form in which they were first introduced on the Manchester Waterworks; fig. 56 a more modern weir of the same kind designed by Sir A. Binnie for the Bradford Waterworks. When the quantity of water coming down the stream is not excessive, it drops over the 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.

Fig. 56.

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

2/3 √(2gh),

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. 57). Then, if a particle passes from a to b in t seconds,

y = 1/2 gt², x = 2/3 √(2gh)t;

∴ y = 9/16 x²/h,

which gives the width x for any given difference of level y and head h, which the jet will just pass over the orifice. Set off ad vertically and equal to 1/2g on any scale; af horizontally and equal to 2/3 √(gh). 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.

Fig. 57.

Mouthpieces—Head Constant

§ 49. Cylindrical Mouthpieces.—When water issues from a short cylindrical pipe or mouthpiece of a length at least equal to 11/2 times its smallest transverse dimension, the stream, 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. 58 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 Ω be the area of the mouthpiece, ω the area of the stream at the contracted section EF. Let v, p be the velocity and pressure at EF, and v₁, p₁ the same quantities at GH. If the discharge is into the air, p₁ 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 + p_a/G; at EF the total head is v²/2g + p/G; at GH it is v₁²/2g + p₁/G.

Between EF and GH there is a loss of head due to abrupt change of velocity, which from eq. (3), § 36, may have the value

(v − v₁)²/2g.

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 + p_a/G = v²/2g + p/G = v₁²/2g + p₁/G + (v − v₁)²/2g.

But ωv = Ωv₁, and ω = c_cΩ, if c_c is the coefficient of contraction within the mouthpiece. Hence

v = Ωv₁/ω = v₁/c_c.

Supposing the discharge into the air, so that p₁ = p_a,

h + p_a/G = v₁²/2g + p_a/G + (v₁²/2g) (1/c_c − 1)²;

(v₁/2g) {1 + (1/c_c − 1)²} = h;

∴ v₁ = √(2gh) / √ {1 + (1/c_c − 1)² };

(1)

Fig. 58.

where the coefficient 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

c_v = 1/√ {1 + (1/c_c − 1)² } = 0.87

(2)

The actual value of c_v, 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_v = 0.82   c_c = 1.00   c = 0.82,

Q = 0.82Ω √(2gh).

It is easy to see from the equations that the pressure p at EF is less than atmospheric pressure. Eliminating v₁, we get

(p_a − p)/G = 3/4 h nearly;

(3)

or

p = p_a − 3/4 Gh ℔ 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. 59), the water will rise in this pipe to a height

KL = (p_a − p) / G = 3/4 h nearly.

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.

§ 50. 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 = c_vc_cΩ √(2gh),

(4)

where Ω is the area of the external end of the mouthpiece, and c_cΩ the section of the contracted jet beyond the mouthpiece.

Convergent Mouthpieces (Castel’s Experiments).—Smallest diameter of
orifice = 0.05085 ft. Length of mouthpiece = 2.6 Diameters.

Angle of Convergence.	Coefficient of Contraction, cc	Coefficient of Velocity, cv	Coefficient of Discharge, c
0°  0′	.999	.830	.829
1° 36′	1.000	.866	.866
3° 10′	1.001	.894	.895
4° 10′	1.002	.910	.912
5° 26′	1.004	.920	.924
7° 52′	.998	.931	.929
8° 58′	.992	.942	.934
10° 20′	.987	.950	.938
12°  4′	.986	.955	.942
13° 24′	.983	.962	.946
14° 28′	.979	.966	.941
16° 36′	.969	.971	.938
19° 28′	.953	.970	.924
21°  0′	.945	.971	.918
23°  0′	.937	.974	.913
29° 58′	.919	.975	.896
40° 20′	.887	.980	.869
48° 50′	.861	.984	.847

The maximum coefficient of discharge is that for a mouthpiece with a convergence of 13°24′.

Fig. 59.	Fig. 60.

The values of c_v and c_c 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_c diminishes, and on the other hand the loss of energy diminishes, so that c_v increases with the convergence, there is an angle for which the product c_c c_v, and consequently the discharge, is a maximum.

§ 51. 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. 60. 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 Ω, v₁, p₁ 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,

h + p_a/G = v²/2g + p/G
= v₁²/2g + p₁/G.

If the jet discharges into the air, p₁ = p_a; and

v₁²/2g = h;

v₁ = √(2gh);

or, if a coefficient is introduced to allow for friction,

v₁ = c_v √(2gh);

where c_v is about 0.97 if the mouthpiece is smooth and well formed.

Q = Ωv₁ = c_v Ω √(2gh).

Fig. 61.

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. 61).

From the equations,

p/G = p_a/G − (v² − v₁²) / 2g.

Let Ω/ω = m. Then

v = v₁m;

p/G = p_a/G − v₁² (m² − 1) / 2g

= p_a/G − (m² − 1)h;

whence we find that p/G will become zero or negative if

Ω/ω ≧ √ {(h + p_a/G) / h }
= √ {1 + p_a/Gh};

or, putting p_a/G = 34 ft., if

Ω/ω ≧ √ { (h + 34)/h}.

In practice there will be an interruption of the full bore flow with a less ratio of Ω/ω, because of the disengagement of air from the water. But, supposing this does not occur, the maximum discharge of a mouthpiece of this kind is

Q = ω √ {2g (h + p_a/G) };

that is, the discharge is the same as for a well-bell-mouthed mouthpiece of area ω, and without the expanding part, discharging into a vacuum.

§ 52. Jet Pump.—A divergent mouthpiece may be arranged to act as a pump, as shown in fig. 62. 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.

Fig. 63 shows the whole arrangement in a diagrammatic way. A is the reservoir which supplies the water that effects the pumping; B is the reservoir of water to be pumped; C is the reservoir into which the water is pumped.

Fig. 63.

Discharge with Varying Head

§ 53. Flow 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.

Fig. 64.

Time of Emptying or Filling a Vertical-sided Lock Chamber.—Suppose the lock chamber, which has a water surface of Ω square ft., is emptied through a sluice in the tail gates, of area ω, 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. 64) 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 cω √(2gh)dt. Hence the differential equation connecting h and t is

cω √(2gh)dt + Ωh = 0.

For the time t, during which the initial head H diminishes to any other value h,

−{Ω/(cω √2g) } $\int _{H}^{h}$ dh/√h = $\int _{0}^{t}$ dt.

∴ t = 2Ω (√H − √h) / {cω √(2g)}

= (Ω/cω) {√(2H/g) − √(2h/g) }.

For the whole time of emptying, during which h diminishes from H to 0,

T = (Ω/cω) √(2H/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

§ 54. 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 water-inch. A common estimate of its value was 14 pints per minute, or 677 English cub. ft. in 24 hours. An experiment by C. Bossut gave 634 cub. ft. in 24 hours (see Navier’s edition of Belidor’s Arch. Hydr., p. 212).

L. J. 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.

Fig. 65.

Pin 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 service-pipe. 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. 65 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.

§ 55. 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. 66). Then if at any moment the head is h, the discharge is, for a rectangular notch of breadth b,

Q = 2/3 cbh√2gh

where c = 0.62; or, better, the formula in § 42 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. 67), and then the edge may be protected by a metal plate about 1/10th in. 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.

Fig. 66.

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, E. Sang 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 = 1, 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.

Fig. 67.

Fig. 68.

The accurate measurement of the discharge of a stream by means of a weir is, however, in practice, rather more difficult than might be inferred from the simplicity of the principle of the operation. Apart 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 ft. long, with 1 ft. depth of water flowing over, an error of 1-1000th 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 cub. ft. per minute.

Hook Gauge.—For the determination of the surface level of water, the most accurate instrument is the hook gauge used first by U. Boyden of Boston, in 1840. It consists of a fixed frame with scale and vernier. In the instrument in fig. 68 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 reflection 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.

§ 56. 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.

Fig. 69.

§ 57. 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.

Fig. 70.—Scale 1/100.

In fig. 69, 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 ω₁ be the area of the orifice through the sluice at A, and ω₂ that of the fixed orifice at B; let h₁ be the difference of level between the surface of the water in the canal and regulating chamber; h₂ 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 = c₁ω₁ √(2gh₁) = c₂ω₂ √(2gh₂),

where c₁ and c₂ are the coefficients of discharge suitable for the two orifices. Hence

c₁ω₁ / c₂ω₂ = √(h ₂/h₁).

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 1.12 ft. high, 0.12416 ft. wide, with a head of 0.32 ft. above the top edge of the orifice, or .88 ft. above the centre. This corresponds to a discharge of about 0.6165 cub. ft. per second.

Fig. 71.

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. 70, 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 ft. long, with a breadth 1.64 ft. 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 0.655 ft. wider than the orifice, splaying out till it is 1.637 ft. wider than the orifice, and about 18 ft. in length.

§ 58. 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. 72) 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 no 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. 71) be the radius of the fixed opening, r the radius 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 (R² − r ²) √(2gh).

Taking c = 0.63,

Q = 15.88 (R² − r ²) √h;

r = √ {R² − Q/15.88 √h}.

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. 72 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 Sir C. Scott-Moncrieff’s Irrigation in Southern Europe.

§ 59. 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 throughout the year. This compensation water requires to be measured in such a way that the millowners 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.

Fig. 72.

Fig. 74 shows an arrangement designed for the Manchester water works. The water enters from the reservoir at chamber A, the object of which is to still the irregular motion of the water. The 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 maintained by adjusting the sluices at b, b, b. At any time the millowners 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 (fig. 73) 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.

Fig. 73.—Scale 1/120.

Fig. 74.—Scale 1/500.

§ 60. Professor Fleeming Jenkin’s 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. 75 shows a valve of this kind suitable for a 6-in. water main. The water arriving by the main C passes through an equilibrium valve D into the chamber A, and thence through a sluice O, 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 B₂ connected by a pipe B₁ 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 pω 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 w/ω. By making w 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 1/2 in. of water have been constructed.

Fig. 75.—Scale 1/24.

VI. STEADY FLOW OF COMPRESSIBLE FLUIDS.

 

Fig. 76.

§ 61. 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 J. P. 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 1 ℔ of air confined behind a piston of 1 sq. ft. area (fig. 76). Let the initial pressure be p₁ and the volume of the air v₁, and suppose this to expand to the pressure p₂ and volume v₂. 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 v₁ to v₂, represented by the area abcd, is

$\int _{v_{1}}^{v_{2}}pdv.$

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

$pv=p_{1}v_{1}$.

Work done during expansion per pound of air

= $\int _{v_{1}}^{v_{2}}pdv=p_{1}v_{1}\int _{v_{1}}^{v_{2}}dv/v$

= p1v1 logε v2/v1 = p1v1 logε p1/p2.

(1)

Since the weight per cubic foot is the reciprocal of the volume per pound, this may be written

(p1/G1) logε G1/G2.

(1a)

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

pv^γ = p₁v₁^γ,

where γ is the ratio of the specific heats of air at constant pressure and volume. Its value for air is 1.408, and for dry steam 1.135.

Work done during expansion per pound of air.

= $\int _{v_{1}}^{v_{2}}$pdv = p₁v₁^γ $\int _{v_{1}}^{v_{2}}$ dv/v^γ

= −{p₁v₁^γ / (γ − 1)} {1/v₂^γ−1 − 1/v₁^γ−1}

= {p₁v₁^γ / (γ − 1)} {1/v₁^γ−1 − 1/v₂^γ−1}

={p1v1 / (γ − 1)} {1 − (v1/v2) γ−1}.

(2)

The value of p₁v₁ for any given temperature can be found from the data already given.

As before, substituting the weights G₁, G₂ per cubic foot for the volumes per pound, we get for the work of expansion

(p1/G1) {1/(γ − 1)} {1 − (G2/G1) γ−1},

(2a)

= p1v1 {1/(γ − 1)} {1 − (p2/p1) γ−1/γ}.

(2b)

 

Fig. 77.

§ 62. 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 p₁, ω₁, v₁, G₁ be the pressure, sectional area of stream, velocity and weight of a cubic foot at A, and p₂, ω₂, v₂, G₂ 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:

G₁ω₁v₁t = G₂ω₂v₂t.

Let z₁, z₂ 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

G₁ω₁v₁t (z₁ − z₂) = W (z₁ − z₂) 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

p₁ω₁v₁t − p₂ω₂v₂t,

= (p₁/G₁ − p₂/G₂) Wt.

The work done by expansion of Wt ℔ of fluid between A and B is Wt$\int _{v_{1}}^{v_{2}}$pdv The change of kinetic energy as before is (W/2g) (v₂² − v₁²) t. Hence, equating work to change of kinetic energy,

W(z₁ − z₂) t + (p₁/G₁ − p₂/G₂)Wt + Wt $\int _{v_{1}}^{v_{2}}$ pdv = (W/2g) (v₂² − v₁²) t;

∴ z1 + p1/G1 + v12/2g = z2 + p2/G2 + v22/2g − $\int _{v_{1}}^{v_{2}}$pdv.

(1)

Now the work of expansion per pound of fluid has already been given. If the temperature is constant, we get (eq. 1a, § 61)

Z₁ + P₁/G₁ + v₁²/2g = z₂ + p²/G₂ + v₂²/2g − (p₁/G₁) log_ε (G₁/G₂).

But at constant temperature p₁/G₁ = p₂/G₂;

∴ z1 + v12/2g = z2 + v22/2g − (p1/G1) logε (p1/p2),

(2)

or, neglecting the difference of level,

(v22 − v12) / 2g = (p1/G1) logε (p1/p2).

(2a)

Similarly, if the expansion is adiabatic (eq. 2a, § 61),

z1 + p1/G1 + v12/2g = z2 + p2/G2 + v22/2g − (p1/G1) {1/(γ − 1) } {1 − (p2/p1)(γ−1)/γ};

(3)

or, neglecting the difference of level,

(v22 − v12)/2g = (p1/G1) [1 + 1/(γ − 1) {1 − (p2/p1)(γ−1)/γ)} ] − p2/G2.

(3a)

It will be seen hereafter that there is a limit in the ratio p₁/p₂ beyond which these expressions cease to be true.

§ 63. Discharge of Air from an Orifice.—The form of the equation of work for a steady stream of compressible fluid is

z₁ + p₁/G₁ + v₁²/2g = z₂ + p₂/G₂ + v₂²/2g−(p₁/G₁) {1/(γ − 1)} {1 − (p₂/p₁^(γ−1)/γ},

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 p₁ and the velocity sensibly zero, through an orifice, into a space where the pressure is p₂. Let v₂ 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 p₂. As air is light, the work of gravity will be small compared with that of the pressures and expansion, so that z₁z₂ may be neglected. Putting these values in the equation above—

p₁/G₁ = p₂/G₂ + v₂²/2g − (p₁/G₁) {1/(γ − 1)} {1 − (p₂/p₁)^(γ−1)/γ;

v₂²/2g = p₁/G₁ − p₂/G₂ + (p₁/G₁) {1/(γ − 1)} {1 − (p₂/p₁)^(γ−1)/γ}

= (p₁/G₁) {γ/(γ − 1) − (p₂/p₁)^{γ−1 /γ} / (γ − 1)} − p₂/G₂.

v22/2g = (p1/G1) {γ/(γ − 1)} {1 − (p2/p1)(γ−1)/γ};

(1)

an equation commonly ascribed to L. J. Weisbach (Civilingenieur, 1856), though it appears to have been given earlier by A. J. C. Barre de Saint Venant and L. Wantzel.

It has already (§ 9, eq. 4a) been seen that

p₁/G₁ = (p₀/G₀) (τ₁/τ₀)

where for air p₀ = 2116.8, G₀ = .08075 and τ₀ = 492.6.

v22/2g = {p0τ1γ / G0τ0 (γ − 1)} {1 − (p2/p1)(γ−1)/γ};

(2)

or, inserting numerical values,

v22/2g = 183.6τ1 {1 − (p2/p1)0.29};

(2a)

which gives the velocity of discharge v₂ in terms of the pressure and absolute temperature, p₁, τ₁, in the vessel from which the air flows, and the pressure p₂ 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 p₂ and temperature τ₂ is

Q₂ = cωv₂ = cω √ [(2gγp₁ / (γ − 1) G₁) (1 − (p₂/p₁)^(γ−1)/γ)]

= 108.7cω √ [τ1 {1 − (p2/p1)0.29}].

(3)

If the volume discharged is measured at the pressure p₁ and absolute temperature τ₁ in the vessel from which the air flows, let Q₁ be that volume; then

p₁Q₁^γ = p₂Q₂^γ;

Q₁ = (p₂/p₁)^1/γ Q₂;

Q₁ = cω √ [ {2gγp₁ / (γ − 1) G₁} {(p₂/p₁)^2/γ − (p₂/p₁)^(γ+1)/γ}].

Q₁ = cω √ [2gγp₁ψ / (γ − 1) G₁]

= 108.7cω √ (τ1ψ).

(4)

The weight of air at pressure p₁ and temperature τ₁ is

G₁ = p₁/53.2τ₁ ℔ per cubic foot.

Hence the weight of air discharged is

W = G₁Q₁ = cω √ [2gγp₁G₁ψ / (γ − 1)]

= 2.043cωp1 √ (ψ/τ1).

(5)

Weisbach found the following values of the coefficient of discharge c:—

Conoidal mouthpieces of the form of the	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$
contracted vein with effective pressures		c =
of .23 to 1.1 atmosphere		0.97	to	0.99
Circular sharp-edged orifices		0.563	”	0.788
Short cylindrical mouthpieces		0.81	”	0.84
The same rounded at the inner end		0.92	”	0.93
Conical converging mouthpieces		0.90	”	0.99

§ 64. 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 p₁ to the pressure p₂, while passing from the vessel to the section of the jet considered in estimating the area ω. Hence p₂ 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 p₂ was identical with the general pressure external to the orifice. R. D. Napier first discovered that, when the ratio p₂/p₁ exceeded a value which does not 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_cω of the contracted section of the jet, and may be greater than the area ω of the orifice. Napier made experiments with steam which showed that, so long as p₂/p₁ > 0.5, the formulae above were trustworthy, when p₂ was taken to be the general external pressure, but that, if p₂/p₁ < 0.5, then the pressure at the contracted section was independent of the external pressure and equal to 0.5p₁. Hence in such cases the constant value 0.5 should be substituted in the formulae for the ratio of the internal and external pressures p₂/p₁.

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

p₂/p₁ = {2/(γ + 1)}^γ−1/γ
= 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.

A. F. Fliegner showed (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 p₂/p₁, for which Weisbach’s formula, as originally understood, ceases to apply, is for air 0.5767; and this is the number to be substituted for p₂/p₁ in the formulae when p₂/p₁ falls below that value. For later researches on the flow of air, reference may be made to G. A. Zeuner’s paper (Civilingenieur, 1871), and Fliegner’s papers (ibid., 1877, 1878).

VII. FRICTION OF LIQUIDS.

§ 65. 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. C. H. 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 in. 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 1/2 ft. 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 on 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. H. P. G. 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.

§ 66. Ordinary Expressions for Fluid Friction at Velocities not Extremely Small.—Let f be the frictional resistance estimated in pounds per square foot of surface at a velocity of 1 ft. 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 = fωv2

(1)

where f is a quantity approximately constant for any given surface. If

ξ = 2gf/G,

R = ξGωv2/2g,

(2)

where ξ is, like f, 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.

	Coefficient  of Friction, ξ	Frictional  Resistance in  ℔ per sq. ft. f
New well-painted iron plate	.00489	.00473
Painted and planed plank (Beaufoy)	.00350	.00339
Surface of iron ships (Rankine)	.00362	.00351
Varnished surface (Froude)	.00258	.00250
Fine sand surface (Froude)	.00418	.00405
Coarser sand surface (Froude)	.00503	.00488

The distance through which the frictional resistance is overcome is v ft. per second. The work expended in fluid friction is therefore given by the equation—

Work expended = fωv3 foot-pounds per second = ξGωv3/2g    〃    〃$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$

(3).

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

ξ = α + β/v

in which expression the second term is of greatest importance at very low velocities, and of comparatively little importance at velocities over about 1/2 ft. 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.

§ 67. 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.

Fig. 78 shows Coulomb’s apparatus. LK supports the wire and disk: ag is the brass wire, the torsion of which causes the oscillations; 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 4.7 to 7.7 in. 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 in. per second, the resistance was sensibly proportional to the velocity.

Beaufoy’s Experiments.—Towards the end of the 18th century Colonel Mark Beaufoy (1764–1827) made an immense mass of experiments on the resistance of bodies moved through water (Nautical 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 sq. ft., at a velocity of 10 ft. per second, the difference of resistance, measured on the difference of area, was 0.339 ℔ per square foot. Also the resistance varied as the 1.949th power of the velocity.

§ 68. Froude’s Experiments.—The most important direct experiments on fluid friction at ordinary velocities are those made by William Froude (1810–1879) 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 extension 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 O, 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 3/16 in. thick, 19 in. deep, and from 1 to 50 ft. 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 Froude may be summarized as follows:—

Fig. 79.

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 ft. long, at 10 ft. per second, was—

Tinfoil or varnish	0.25	℔ per sq. ft.
Calico	0.47	〃   〃
Fine sand	0.405	〃   〃
Coarser sand	0.488	〃   〃

2. The power of the velocity to which the friction is proportional varies for different surfaces. Thus, with short boards 2 ft. long,

For tinfoil the resistance varied as v^2.16.
For other surfaces the resistance varied as v^2.00.

With boards 50 ft. long,

For varnish or tinfoil the resistance varied as v^1.83.
For sand the resistance varied as v^2.00.

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,

			Mean Resistance in ℔ per sq. ft.
Varnished surface	2	ft. long	0.41
	50	〃	0.25
Fine sand surface	2	〃	0.81
	50	〃	0.405

This remarkable result is explained thus by 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.”

§ 69. The following table gives a general statement of 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.

	Length of Surface, or Distance from Cutwater, in feet.
	2 ft.			8 ft.			20 ft.			50 ft.
	A	B	C	A	B	C	A	B	C	A	B	C
Varnish	2.00	.41	.390	1.85	.325	.264	1.85	.278	.240	1.83	.250	.226
Paraffin	..	.38	.370	1.94	.314	.260	1.93	.271	.237	..	..	..
Tinfoil	2.16	.30	.295	1.99	.278	.263	1.90	.262	.244	1.83	.246	.232
Calico	1.93	.87	.725	1.92	.626	.504	1.89	.531	.447	1.87	.474	.423
Fine sand	2.00	.81	.690	2.00	.583	.450	2.00	.480	.384	2.06	.405	.337
Medium sand	2.00	.90	.730	2.00	.625	.488	2.00	.534	.465	2.00	.488	.456
Coarse sand	2.00	1.10	.880	2.00	.714	.520	2.00	.588	.490	..	..	..

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 ft., they indicate what would be the resistances of longer surfaces. For at 50 ft. 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 ft. per second diminishes from 0.41 to 0.32 ℔ per square foot when the length is increased from 2 to 8 ft., but it only diminishes from 0.278 to 0.250 ℔ per square foot for an increase from 20 ft. to 50 ft.

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 sternwards, 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.

§ 70. Friction of Rotating Disks.—A rotating disk is virtually a surface of unlimited extent and it is convenient for experiments on friction with different surfaces at different speeds. Experiments carried out by Professor W. C. Unwin (Proc. Inst. Civ. Eng. lxxx.) are useful both as illustrating the laws of fluid friction and as giving data for calculating the resistance of the disks of turbines and centrifugal pumps. Disks of 10, 15 and 20 in. diameter fixed on a vertical shaft were rotated by a belt driven by an engine. They were enclosed in a cistern of water between parallel top and bottom fixed surfaces. The cistern was suspended by three fine wires. The friction of the disk is equal to the tendency of the cistern to rotate, and this was measured by balancing the cistern by a fine silk cord passing over a pulley and carrying a scale pan in which weights could be placed.

If ω is an element of area on the disk moving with the velocity v, the friction on this element is fωvⁿ, where f and n are constant for any given kind of surface. Let α be the angular velocity of rotation, R the radius of the disk. Consider a ring of the surface between r and r + dr. Its area is 2πrdr, its velocity αr and the friction of this ring is f 2πrdrαⁿr ⁿ. The moment of the friction about the axis of rotation is 2παⁿfr ⁿ⁺² dr, and the total moment of friction for the two sides of the disk is

M = 4πα_nf $\int _{0}^{\text{R}}$ r ⁿ⁺² dr = {4παⁿ/(n + 3) } f Rⁿ⁺³.

If N is the number of revolutions per sec.,

M = {2ⁿ⁺² πⁿ⁺¹ Nⁿ/(n + 3) } f Rⁿ⁺³,

and the work expended in rotating the disk is

Mα = {2ⁿ⁺³ πⁿ⁺² Nⁿ⁺¹/(n + 3) } f Rⁿ⁺³ foot ℔ per sec.

The experiments give directly the values of M for the disks corresponding to any speed N. From these the values of f and n can be deduced, f being the friction per square foot at unit velocity. For comparison with Froude’s results it is convenient to calculate the resistance at 10 ft. per second, which is F = f10ⁿ.

The disks were rotated in chambers 22 in. diameter and 3, 6 and 12 in. deep. In all cases the friction of the disks increased a little as the chamber was made larger. This is probably due to the stilling of the eddies against the surface of the chamber and the feeding back of the stilled water to the disk. Hence the friction depends not only on the surface of the disk but to some extent on the surface of the chamber in which it rotates. If the surface of the chamber is made rougher by covering with coarse sand there is also an increase of resistance.

For the smoother surfaces the friction varied as the 1.85th power of the velocity. For the rougher surfaces the power of the velocity to which the resistance was proportional varied from 1.9 to 2.1. This is in agreement with Froude’s results.

Experiments with a bright brass disk showed that the friction decreased with increase of temperature. The diminution between 41° and 130° F. amounted to 18%. In the general equation M = cNⁿ for any given disk,

c_t = 0.1328 (1 − 0.0021t),

where c_t is the value of c for a bright brass disk 0.85 ft. in diameter at a temperature t° F.

The disks used were either polished or made rougher by varnish or by varnish and sand. The following table gives a comparison of the results obtained with the disks and Froude’s results on planks 50 ft. long. The values given are the resistances per square foot at 10 ft. per sec.

Froude’s Experiments.		Disk Experiments.
Tinfoil surface	0.232	Bright brass	0.202 to 0.229
Varnish	0.226	Varnish	0.220 to 0.233
Fine sand	0.337	Fine sand	0.339
Medium sand	0.456	Very coarse sand	0.587 to 0.715

VIII. STEADY FLOW OF WATER IN PIPES OF UNIFORM SECTION.

§ 71. 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.

 

Fig. 80.

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 ft. above any horizontal datum line xx, the pressures at the cross sections being p and p + dp ℔ per square foot. Further, let Q be the volume of flow or discharge of the pipe per second, Ω 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 ℔, and fall through a height −dz ft. The work done by gravity is then

−GQ dz;

a positive quantity if dz is negative, and vice versa. The resultant pressure parallel to the axis of the pipe is p − (p + dp) = −dp ℔ per square foot of the cross section. The work of this pressure on the volume Q is

−Q dp.

The only remaining force doing work on the system is the friction against the surface of the pipe. The area of that surface is χdl.

The work expended in overcoming the frictional resistance per second is (see § 66, eq. 3)

−ζGχ dl v³/2g,

or, since Q = Ωv,

−ζG (χ/Ω) Q (v²/2g)dl;

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 dz − Q dp − ζG (χ/Ω) Q (v ²/2g)dl = 0.

Dividing by GQ,

dz + dp/G + ζ (χ/Ω) (v ²/2g)dl = 0.

Integrating,

z + p/G + ζ (χ/Ω) (v 2/2g)l = constant.

(1)

§ 72. Let A and B (fig. 81) be any two sections of the pipe for which p, z, l have the values p₁, z₁, l₁, and p₂, z₂, l₂, respectively. Then

z₁ + p₁/G + ζ (χ/Ω) (v ²/2g)l₁ = z₂ + p₂/G + ζ (χ/Ω) (v ²/2g)l₂;

or, if l₂ − l₁ = L, rearranging the terms,

ζv 2/2g = (1/L) {(z1 + p1/G) − (z2 + p2/G)} Ω/χ.

(2)

Fig. 81.

Suppose pressure columns introduced at A and B. The water will rise in those columns to the heights p₁/G and p₂/G due to the pressures p₁ and p₂ at A and B. Hence (z₁ + p₁/G) − (z₂ + p₂/G) 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 DE/AB = h/L is termed the virtual slope 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 Ω/χ which appears in many hydraulic equations is called the hydraulic mean radius of the pipe. It will be denoted by m.

Introducing these values,

ζv 2/2g = mh/L = mi.

(3)

For pipes of circular section, and diameter d,

m = Ω/χ = 1/4πd ²/πd = 1/4d.

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 4ζL/d. It is assumed above that the atmospheric pressure at C and D is the same, and this is usually nearly the case. But if C and D are at greatly different levels the excess of barometric pressure at C, in feet of water, must be added to p₂/G.

§ 73. Hydraulic Gradient or 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 joint 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 ft. 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.

Fig. 82.

§ 74. 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 v the velocity, in a pipe of length L and diameter d. The whole work done per second is virtually the removal of Q cub. ft. of water from the surface of the upper reservoir to the surface of the lower reservoir, that is GQh foot-pounds. This is expended in three ways. (1) The head v ²/2g, corresponding to an expenditure of GQv ²/2g foot-pounds of work, is employed in giving energy of motion to the water. This is ultimately wasted in eddying motions in the lower reservoir. (2) A portion of head, which experience shows may be expressed in the form ζ₀v ²/2g, corresponding to an expenditure of GQζ₀v ²/2g 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 ζ(4L/d) (v ²/2g) corresponding to GQζ (4L/d) (v ²/2g) foot-pounds of work. Hence

GQh = GQv ²/2g + GQζ₀v ²/2g + GQζ·4L·v ²/d·2g;

h = (1 + ζ0 + ζ·4L/d)v2/2g.       v = 8.025 √ [hd / {(1 + ζ0)d + 4ζL}].$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$

(5)

If the pipe is bell-mouthed, ζ₀ is about = .08. If the entrance to the pipe is cylindrical, ζ₀ = 0.505. Hence 1 + ζ₀ = 1.08 to 1.505. In general this is so small compared with ζ4L/d that, for practical 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 ft. per second, and the pipe is bellmouthed. Then 1.08v ²/2g = 0.067 ft. In pipes for towns’ supply v may range from 2 to 41/2 ft. per second, and then 1.5v ²/2g = 0.1 to 0.5 ft. 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 v 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

h = (4ζ/d) [{ (1 + ζ0)d/4ζ} + L] v 2/2g;

(6)

from which it is clear that the head expended at the mouthpiece is equivalent to that of a length

(1 + ζ₀)d/4ζ

of the pipe. Putting 1 + ζ₀ = 1.505 and ζ = 0.01, the length of pipe equivalent to the mouthpiece is 37.6d nearly. This may be added to the actual length of the pipe to allow for mouthpiece resistance in approximate calculations.

§ 75. Coefficient of Friction for Pipes discharging Water.—From the average of a large number of experiments, the value of ζ for ordinary iron pipes is

ζ = 0.007567.

(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 ζ on the velocity. Having regard to the difference of the law of resistance at very low and at ordinary velocities, they assumed that ζ might be expressed in the form

ζ = a + β/v.

The following are the best numerical values obtained for ζ so expressed:—

	α	β
R. de Prony (from 51 experiments)	0.006836	0.001116
J. F. d’Aubuisson de Voisins	0.00673	0.001211
J. A. Eytelwein	0.005493	0.00143

Weisbach proposed the formula

4ζ = α + β/√v = 0.003598 + 0.004289/√v.

(8)

§ 76. Darcy’s Experiments on Friction in Pipes.—All previous experiments on the resistance of pipes were superseded by the remarkable researches carried out by H. P. G. Darcy (1803–1858), the Inspector-General of the Paris water works. 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 cast iron it is about 11/2 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 incrusted 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. 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 ζ = α + β/v; but in pipes which have been some time in use it is sufficiently accurate to take ζ = α₁ simply, where α₁ 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;

ζ = α (1 + β/d).

(9)

	α	β
For drawn wrought-iron or smooth cast-iron pipes	.004973	.084
For pipes altered by light incrustations	.00996	.084

These coefficients may be put in the following very simple form, without sensibly altering their value:—

For clean pipes	ζ = .005 (1 + 1/12d)	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$
For slightly incrusted pipes	ζ = .01 (1 + 1/12d)

(9a)

Darcy’s Value of the Coefficient of Friction ζ for Velocities not less
than 4 in. per second.

Diameter of Pipe  in Inches.	ζ		Diameter of Pipe  in Inches.	ζ
	New Pipes.	Incrusted  Pipes.		New Pipes.	Incrusted  Pipes.
2	0.00750	0.01500	18	.00528	.01056
3	.00667	.01333	21	.00524	.01048
4	.00625	.01250	24	.00521	.01042
5	.00600	.01200	27	.00519	.01037
6	.00583	.01167	30	.00517	.01033
7	.00571	.01143	36	.00514	.01028
8	.00563	.01125	42	.00512	.01024
9	.00556	.01111	48	.00510	.01021
12	.00542	.01083	54	.00509	.01019
15	.00533	.01067

These values of ζ are, however, not exact for widely differing velocities. To embrace all cases Darcy proposed the expression

ζ = (α + α₁/d) + (β + β₁/d ²) /v ;

(10)

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

α	= .004346
α1	= .0003992
β	= .0010182
β1	= .000005205.

It has become not uncommon to calculate the discharge of pipes by the formula of E. Ganguillet and W. R. Kutter, which will be discussed under the head of channels. For the value of c in the relation v = c √(mi), Ganguillet and Kutter take

c =	41.6 + 1.811/n + .00281/i
	1 + [ (41.6 + .00281/i) (n/ √m) ]

where n is a coefficient depending only on the roughness of the pipe. For pipes uncoated as ordinarily laid n = 0.013. The formula is very cumbrous, its form is not rationally justifiable and it is not at all clear that it gives more accurate values of the discharge than simpler formulae.

§ 77. Later Investigations on Flow in Pipes.—The foregoing statement gives the theory of flow in pipes so far as it can be put in a simple rational form. But the conditions of flow are really more complicated than can be expressed in any rational form. Taking even selected experiments the values of the empirical coefficient ζ range from 0.16 to 0.0028 in different cases. Hence means of discriminating the probable value of ζ are necessary in using the equations for practical purposes. To a certain extent the knowledge that ζ decreases with the size of the pipe and increases very much with the roughness of its surface is a guide, and Darcy’s method of dealing with these causes of variation is very helpful. But a further difficulty arises from the discordance of the results of different experiments. For instance F. P. Stearns and J. M. Gale both experimented on clean asphalted cast-iron pipes, 4 ft. in diameter. According to one set of gaugings ζ = .0051, and according to the other ζ = .0031. It is impossible in such cases not to suspect some error in the observations or some difference in the condition of the pipes not noticed by the observers.

It is not likely that any formula can be found which will give exactly the discharge of any given pipe. For one of the chief factors in any such formula must express the exact roughness of the pipe surface, and there is no scientific measure of roughness. The most that can be done is to limit the choice of the coefficient for a pipe within certain comparatively narrow limits. The experiments on fluid friction show that the power of the velocity to which the resistance is proportional is not exactly the square. Also in determining the form of his equation for ζ Darcy used only eight out of his seventeen series of experiments, and there is reason to think that some of these were exceptional. Barré de Saint-Venant was the first to propose a formula with two constants,

dh/4l = mVⁿ,

where m and n are experimental constants. If this is written in the form

log m + n log v = log (dh/4l),

we have, as Saint-Venant pointed out, the equation to a straight line, of which m is the ordinate at the origin and n the ratio of the slope. If a series of experimental values are plotted logarithmically the determination of the constants is reduced to finding the straight line which most nearly passes through the plotted points. Saint-Venant found for n the value of 1.71. In a memoir on the influence of temperature on the movement of water in pipes (Berlin, 1854) by G. H. L. Hagen (1797–1884) another modification of the Saint-Venant formula was given. This is h/l = mv ⁿ/d ^x, which involves three experimental coefficients. Hagen found n = 1.75; x = 1.25; and m was then nearly independent of variations of v and d. But the range of cases examined was small. In a remarkable paper in the Trans. Roy. Soc., 1883, Professor Osborne Reynolds made much clearer the change from regular stream line motion at low velocities to the eddying motion, which occurs in almost all the cases with which the engineer has to deal. Partly by reasoning, partly by induction from the form of logarithmically plotted curves of experimental results, he arrived at the general equation h/l = c (vⁿ/d ³⁻ⁿ) P²⁻ⁿ, where n = l for low velocities and n = 1.7 to 2 for ordinary velocities. P is a function of the temperature. Neglecting variations of temperature Reynold’s formula is identical with Hagen’s if x = 3−n. For practical purposes Hagen’s form is the more convenient.

Values of Index of Velocity.

Surface of Pipe.	Authority.	Diameter of Pipe  in Metres.	Values of n.
Tin plate	Bossut	.036	1.697	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$1.72
		.054	1.730
Wrought iron (gas pipe)	Hamilton Smith	.0159	1.756	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$1.75
		.0267	1.770
Lead	Darcy	.014	1.866	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$1.77
		.027	1.755
		.041	1.760
Clean brass	Mair	.036	1.795	1.795
Asphalted	Hamilton Smith	.0266	1.760	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$1.85
	Lampe.	.4185	1.850
	W. W. Bonn	.306	1.582
	Stearns	1.219	1.880
Riveted wrought iron	Hamilton Smith	.2776	1.804	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$1.87
		.3219	1.892
		.3749	1.852
Wrought iron (gas pipe)	Darcy	.0122	1.900	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$1.87
		.0266	1.899
		.0395	1.838
New cast iron	Darcy	.0819	1.950	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$1.95
		.137	1.923
		.188	1.957
		.50	1.950
Cleaned cast iron	Darcy	.0364	1.835	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,}$2.00
		.0801	2.000
		.2447	2.000
		.397	2.07
Incrusted cast iron	Darcy	.0359	1.980	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$2.00
		.0795	1.990
		.2432	1.990

Fig. 83.

In 1886, Professor W. C. Unwin plotted logarithmically all the most trustworthy experiments on flow in pipes then available.^[5] Fig. 83 gives one such plotting. The results of measuring the slopes of the lines drawn through the plotted points are given in the table.

It will be seen that the values of the index n range from 1.72 for the smoothest and cleanest surface, to 2.00 for the roughest. The numbers after the brackets are rounded off numbers.

The value of n having been thus determined, values of m/d^x were next found and averaged for each pipe. These were again plotted logarithmically in order to find a value for x. The lines were not very regular, but in all cases the slope was greater than 1 to 1, so that the value of x must be greater than unity. The following table gives the results and a comparison of the value of x and Reynolds’s value 3−n.

Kind of Pipe.	n	3−n	x
Tin plate	1.72	1.28	1.100
Wrought iron (Smith)	1.75	1.25	1.210
Asphalted pipes	1.85	1.15	1.127
Wrought iron (Darcy)	1.87	1.13	1.680
Riveted wrought iron	1.87	1.13	1.390
New cast iron	1.95	1.05	1.168
Cleaned cast iron	2.00	1.00	1.168
Incrusted cast iron	2.00	1.00	1.160

With the exception of the anomalous values for Darcy’s wrought-iron pipes, there is no great discrepancy between the values of x and 3−n, but there is no appearance of relation in the two quantities. For the present it appears preferable to assume that x is independent of n.

It is now possible to obtain values of the third constant m, using the values found for n and x. The following table gives the results, the values of m being for metric measures.

Here, considering the great range of diameters and velocities in the experiments, the constancy of m is very satisfactorily close. The asphalted pipes give rather variable values. But, as some of these were new and some old, the variation is, perhaps, not surprising. The incrusted pipes give a value of m quite double that for new pipes but that is perfectly consistent with what is known of fluid friction in other cases.

Kind of Pipe.	Diameter  in Metres.	Value of m.	Mean Value of m.	Authority.
Tin plate	0.036	.01697	.01686	Bossut
	0.054	.01676
Wrought iron	0.016	.01302	.01310	Hamilton Smith
	0.027	.01319
Asphalted pipes	0.027	.01749	.01831	Hamilton Smith
	0.306	.02058		W. W. Bonn
	0.306	.02107		W. W. Bonn
	0.419	.01650		Lampe
	1.219	.01317		Stearns
	1.219	.02107		Gale
Riveted wrought iron	0.278	.01370	.01403	Hamilton Smith
	0.322	.01440
	0.375	.01390
	0.432	.01368
	0.657	.01448
New cast iron	0.082	.01725	.01658	Darcy
	0.137	.01427
	0.188	.01734
	0.500	.01745
Cleaned cast iron	0.080	.01979	.01994	Darcy
	0.245	.02091
	0.297	.01913
Incrusted cast iron	0.036	.03693	.03643	Darcy
	0.080	.03530
	0.243	.03706

General Mean Values of Constants.

The general formula (Hagen’s)—h/l = mvⁿ/d^x·2g—can therefore be taken to fit the results with convenient closeness, if the following mean values of the coefficients are taken, the unit being a metre:—

Kind of Pipe.	m	x	n
Tin plate	.0169	1.10	1.72
Wrought iron	.0131	1.21	1.75
Asphalted iron	.0183	1.127	1.85
Riveted wrought iron	.0140	1.390	1.87
New cast iron	.0166	1.168	1.95
Cleaned cast iron	.0199	1.168	2.0
Incrusted cast iron	.0364	1.160	2.0

The variation of each of these coefficients is within a comparatively narrow range, and the selection of the proper coefficient for any given case presents no difficulty, if the character of the surface of the pipe is known.

It only remains to give the values of these coefficients when the quantities are expressed in English feet. For English measures the following are the values of the coefficients:—

Kind of Pipe.	m	x	n
Tin plate	.0265	1.10	1.72
Wrought iron	.0226	1.21	1.75
Asphalted iron	.0254	1.127	1.85
Riveted wrought iron	.0260	1.390	1.87
New cast iron	.0215	1.168	1.95
Cleaned cast iron	.0243	1.168	2.0
Incrusted cast iron	.0440	1.160	2.0

§ 78. Distribution of Velocity in the Cross Section of a Pipe.—Darcy made experiments with a Pitot tube in 1850 on the velocity at different points in the cross section of a pipe. He deduced the relation

V − v = 11.3 (r^3/2/R) √i,

where V is the velocity at the centre and v the velocity at radius r in a pipe of radius R with a hydraulic gradient i. Later Bazin repeated the experiments and extended them (Mém. de l’Académie des Sciences, xxxii. No. 6). The most important result was the ratio of mean to central velocity. Let b = Ri/U², where U is the mean velocity in the pipe; then V/U = 1 + 9.03 √b. A very useful result for practical purposes is that at 0.74 of the radius of the pipe the velocity is equal to the mean velocity. Fig. 84 gives the velocities at different radii as determined by Bazin.

Fig. 84.

§ 79. Influence of Temperature on the Flow through Pipes.—Very careful experiments on the flow through a pipe 0.1236 ft. in diameter and 25 ft. long, with water at different temperatures, have been made by J. G. Mair (Proc. Inst. Civ. Eng. lxxxiv.). The loss of head was measured from a point 1 ft. from the inlet, so that the loss at entry was eliminated. The 11/2 in. pipe was made smooth inside and to gauge, by drawing a mandril through it. Plotting the results logarithmically, it was found that the resistance for all temperatures varied very exactly as v1.795, the index being less than 2 as in other experiments with very smooth surfaces. Taking the ordinary equation of flow h = ζ (4L/D) (v²/2g), then for heads varying from 1 ft. to nearly 4 ft., and velocities in the pipe varying from 4 ft. to 9 ft. per second, the values of ζ were as follows:—

Temp. F.	ζ	Temp. F.	ζ
57	.0044 to .0052	100	.0039 to .0042
70	.0042 to .0045	110	.0037 to .0041
80	.0041 to .0045	120	.0037 to .0041
90	.0040 to .0045	130	.0035 to .0039
		160	.0035 to .0038

This shows a marked decrease of resistance as the temperature rises. If Professor Osborne Reynolds’s equation is assumed h = mLVⁿ/d³⁻ⁿ, and n is taken 1.795, then values of m at each temperature are practically constant—

Temp. F.	m.	Temp. F.	m.
57	0.000276	100	0.000244
70	0.000263	110	0.000235
80	0.000257	120	0.000229
90	0.000250	130	0.000225
		160	0.000206

where again a regular decrease of the coefficient occurs as the temperature rises. In experiments on the friction of disks at different temperatures Professor W. C. Unwin found that the resistance was proportional to constant × (1 − 0.0021t) and the values of m given above are expressed almost exactly by the relation

m = 0.000311 (1 − 0.00215 t).

In tank experiments on ship models for small ordinary variations of temperature, it is usual to allow a decrease of 3% of resistance for 10° F. increase of temperature.

§ 80. Influence of Deposits in Pipes on the Discharge. 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. In pipes which convey certain kinds of water, oxidation proceeds rapidly and the discharge is considerably diminished. A main laid at Torquay in 1858, 14 m. in length, consists of 10-in., 9-in. and 8-in. pipes. It was not protected from corrosion by any coating. But it was found to the surprise of the engineer that in eight years the discharge had diminished to 51% of the original discharge. J. G. Appold suggested an apparatus for scraping the interior of the pipe, and this was constructed and used under the direction of William Froude (see “Incrustation of Iron Pipes,” by W. Ingham, Proc. Inst. Mech. Eng., 1899). It was found that by scraping the interior of the pipe the discharge was increased 56%. The scraping requires to be repeated at intervals. After each scraping the discharge diminishes rather rapidly to 10% and afterwards more slowly, the diminution in a year being about 25%.

Fig. 85 shows a scraper for water mains, similar to 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 561/2%, at Oswestry 541/2%. The increased discharge is due to the diminution of the friction of the pipe by removing the roughnesses due to oxidation. The scraper can be easily followed when the mains are about 3 ft. deep by the noise it makes. The average speed of the scraper at Torquay is 21/3 m. per hour. At Torquay 49% of the deposit is iron rust, the rest being silica, lime and organic matter.

Fig. 85. Scale 1/25.

In the opinion of some engineers it is inadvisable to use the scraper. The incrustation is only temporarily removed, and if the use of the scraper is continued the life of the pipe is reduced. The only treatment effective in preventing or retarding the incrustation due to corrosion is to coat the pipes when hot with a smooth and perfect layer of pitch. With certain waters such as those derived from the chalk the incrustation is of a different character, consisting of nearly pure calcium carbonate. A deposit of another character which has led to trouble in some mains is a black slime containing a good deal of iron not derived from the pipes. It appears to be an organic growth. Filtration of the water appears to prevent the growth of the slime, and its temporary removal may be effected by a kind of brush scraper devised by G. F. Deacon (see “Deposits in Pipes,” by Professor J. C. Campbell Brown, Proc. Inst. Civ. Eng., 1903–1904).

§ 81. Flow of Water through Fire Hose.—The hose pipes used for fire purposes are of very varied character, and the roughness of the surface varies. Very careful experiments have been made by J. R. Freeman (Am. Soc. Civ. Eng. xxi., 1889). It was noted that under pressure the diameter of the hose increased sufficiently to have a marked influence on the discharge. In reducing the results the true diameter has been taken. Let v = mean velocity in ft. per sec.; r = hydraulic mean radius or one-fourth the diameter in feet; i = hydraulic gradient. Then v = n √(ri).

	Diameter  in Inches.	Gallons (United States)  per min.	i	v	n
Solid rubber hose $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	2.65	215	.1863	12.50	123.3
	”	344	.4714	20.00	124.0
Woven cotton, rubber lined $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	2.47	200	.2464	13.40	119.1
	”	299	.5269	20.00	121.5
Woven cotton, rubber lined $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	2.49	200	.2427	13.20	117.7
	”	319	.5708	21.00	122.1
Knit cotton, rubber lined$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	2.68	132	.0809	7.50	111.6
	”	299	.3931	17.00	114.8
Knit cotton, rubber lined$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	2.69	204	.2357	11.50	100.1
	”	319	.5165	18.00	105.8
Woven cotton, rubber lined $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	2.12	154	.3448	14.00	113.4
	”	240	.7673	21.81	118.4
Woven cotton, rubber lined $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	2.53	54.8	.0261	3.50	94.3
	”	298	.8264	19.00	91.0
Unlined linen hose$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	2.60	57.9	.0414	3.50	73.9
	”	331	1.1624	20.00	79.6

§ 82. 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.

Fig. 86.

In fig. 86 let A be the main of variable diameter, and B the equivalent uniform main. In the given main of variable diameter A, let

l1, l2	be the lengths,
d1, d2	the diameters,
v1, v2	the velocities,
i1, i2	the slopes,

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

h = i₁l₁ + i₂l₂ + . . .
= ζ (v₁² · 4l₁/2gd₁) + ζ (v₂² · 4l₂/2gd₂) + . . .

and in the uniform main

il = ζ (v² · 4l/2gd).

If the mains are equivalent, as defined above,

ζ (v² · 4l/2gd) = ζ (v₁² · 4l₁/2gd₁) + ζ (v₂² · 4l₂/2gd₂) + . . .

But, since the discharge is the same for all portions,

1/4πd ²v = 1/4πd₁²v₁ = 1/4πd₂²v₂ = . . .
v₁ = vd ²/d₁²; v₂ = vd ²/d₂² . . .

Also suppose that ζ may be treated as constant for all the pipes. Then

l/d = (d ⁴/d₁⁴) (l₁/d₁) + (d ⁴/d₂⁴) (l₂/d₂) + . . .
l = (d ⁵/d₁⁵) l₁ + (d ⁵/d₂⁵) l₂ + . . .

which gives the length of the equivalent uniform main which would have the same total loss of head for any given discharge.

 

Fig. 87.

§ 83. 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 ω₀ to an area ω₁ (fig. 87); then

v₁/v₀ = ω₀/ω₁;

or, if the section is circular,

v₁/v₀ = (d₀/d₁)².

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

ɧ_e = (v₀ − v₁)²/2g = (ω₁/ω₀ − 1)² v₁²/2g = {(d₁/d₀)² − 1}² v₁²/2g

if ζ_e is put for the expression in brackets.

ω1/ω0 =	1.1	1.2	1.5	1.7	1.8	1.9	2.0	2.5	3.0	3.5	4.0	5.0	6.0	7.0	8.0
d1/d0 =	1.05	1.10	1.22	1.30	1.34	1.38	1.41	1.58	1.73	1.87	2.00	2.24	2.45	2.65	2.83
ζe =	.01	.04	.25	.49	.64	.81	1.00	2.25	4.00	6.25	9.00	16.00	25.00	36.0	49.0

Fig. 88.	Fig. 89.

Abrupt Contraction of Section.—When water passes from a larger to a smaller section, as in figs. 88, 89, 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_cω, and the velocity (ω/c_cω) v = v/c₁, where c_c is the coefficient of contraction. Then the head lost is

ɧ_m = (v/c_c − v)² / 2g = (1/c_c − 1)² v²/2g;

and, if c_c is taken 0.64,

ɧ_m = 0.316 v²/2g.

(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 bell-mouthed, pipe from a reservoir of indefinitely large size, experiment gives

ɧ_a = 0.505 v²/2g.

(3)

If there is a diaphragm at the mouth of the pipe as in fig. 89, let ω₁ be the area of this orifice. Then the area of the contracted stream is c_cω₁, and the head lost is

ɧ_c = {(ω/c_cω₁) − 1}² v²/2g
  = ζ_cv² / 2g

(4)

if ζ, is put for {(ω/c_cω₁) − 1}². Weisbach has found experimentally the following values of the coefficient, when the stream approaching the orifice was considerably larger than the orifice:—

ω1/ω =	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
cc =	.616	.614	.612	.610	.617	.605	.603	.601	.598	.596
ζc =	231.7	50.99	19.78	9.612	5.256	3.077	1.876	1.169	0.734	0.480

Fig. 90.

When a diaphragm was placed in a tube of uniform section (fig. 90) the following values were obtained, ω₁ being the area of the orifice and ω that of the pipe:—

ω1/ω =	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
ce =	.624	.632	.643	.659	.681	.712	.755	.813	.892	1.00
ξc =	225.9	47.77	30.83	7.801	1.753	1.796	.797	.290	.060	.000

Elbows.—Weisbach considers the loss of head at elbows (fig. 91) to be due to a contraction formed by the stream. From experiments with a pipe 11/4 in. diameter, he found the loss of head

ɧe = ζεv2 / 2g;

(5)

ζ_e = 0.9457 sin² 1/2φ + 2.047 sin⁴ 1/2φ.

φ =	20°	40°	60°	80°	90°	100°	110°	120°	130°	140°
ζε =	0.046	0.139	0.364	0.740	0.984	1.260	1.556	1.861	2.158	2.431

Hence at a right-angled elbow the whole head due to the velocity very nearly is lost.

 

Fig. 91.

 

Fig. 92.

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 very satisfactorily ascertained. Weisbach obtained for the loss of head at a bend in a pipe of circular section

ɧ_b = ζ_bv² / 2g;

(6)

ζ_b = 0.131 + 1.847 (d/2ρ)^7/2,

where d is the diameter of the pipe and ρ the radius of curvature of the bend. The resistance at bends is small and at present very ill determined.

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

ɧ_v = ζ_vv² / 2g;

(7)

where v is the velocity in the pipe beyond the valve and ζ_v a coefficient determined by experiment. The following are Weisbach’s results.

Sluice in Pipe of Rectangular Section (fig. 92). Section at sluice = ω₁ in pipe = ω.

ω1/ω =	1.0	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1
ζv =	0.00	.09	.39	.95	2.08	4.02	8.12	17.8	44.5	193

Sluice in Cylindrical Pipe (fig. 93).

Ratio of height of opening to diameter of pipe${\Big \}}$	1.0	7/8	3/4	5/8	1/2	3/8	1/4	1/5
ω1/ω =	1.00	0.948	.856	.740	.609	.466	.315	.159
ζv =	0.00	0.07	0.26	0.81	2.06	5.52	17.0	97.8

Fig. 93.	Fig. 94.

Cock in a Cylindrical Pipe (fig. 94). Angle through which cock is turned = θ.

θ =	5°	10°	15°	20°	25°	30°	35°
Ratio of cross sections${\Big \}}$	.926	.850	.772	.692	.613	.535	.458
ζv =	.05	.29	.75	1.56	3.10	5.47	9.68

θ =	40°	45°	50°	55°	60°	65°	82°
Ratio of cross sections${\Big \}}$	.385	.315	.250	.190	.137	.091	0
ζv =	17.3	31.2	52.6	106	206	486	∞

Throttle Valve in a Cylindrical Pipe (fig. 95)

θ =	5°	10°	15°	20°	25°	30°	35°	40°
ζv =	.24	.52	.90	1.54	2.51	3.91	6.22	10.8

θ =	45°	50°	55°	60°	65°	70°	90°
ζv =	18.7	32.6	58.8	118	256	751	∞

Fig. 95.

§ 84. 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, § 72, and 9a, § 76) for the solution of such problems as arise:—

ζ = α (1 + 1/12d);

(1)

where α = 0.005 for new and = 0.01 for incrusted pipes.

ζv² / 2g = 1/4di.

(2)

Q = 1/4πd ²v.

(3)

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 v are required. Find ζ from (1); then v from (2); lastly Q from (3). This case presents no difficulty.

By combining equations (1) and (2), v is obtained directly:—

v = √ (gdi /2ζ) = √ (g/2α) √ [di / {1 + 1/12d}].

(4)

For new pipes	√ (g/2α) = 56.72
For incrusted pipes	= 40.13

For pipes not less than 1, or more than 4 ft. in diameter, the mean values of ζ are

For new pipes	0.00526
For incrusted pipes	0.01052.

Using these values we get the very simple expressions—

v = 55.31 √ (di) for new pipes
= 39.11 √ (di) for incrusted pipes.

(4a)

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 ζ 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); ζ from (1); lastly i from (2). If we combine (1) and (2) we get

i = ζ (v²/2g) (4/d) = 2a {1 + 1/12d} v²/gd;

(5)

and, taking the mean values of ζ for pipes from 1 to 4 ft. diameter, given above, the approximate formulae are

i = 0.0003268 v²/d for new pipes
= 0.0006536 v²/d for incrusted pipes.

(5a)

Problem 4. Given the virtual slope and the velocity, to find the diameter of the pipe and the discharge. The diameter is obtained from equations (2) and (1), which give the quadratic expression

d ² − d (2αv²/gi) − αv²/6gi = 0.

∴ d = αv²/gi + √ {(αv²/gi) (αv²/gi + 1/6)}.

(6)

For practical purposes, the approximate equations

d = 2αv²/gi + 1/12
  = 0.00031 v²/i + .083 for new pipes
  = 0.00062 v²/i + .083 for incrusted pipes

(6a)

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—

d 5 = 32ζQ2 / gπ2i.

(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 ζ given above for pipes of 1 to 4 ft. diameter we get

d = 5√ (32ζ/gπ2) 5√ (Q2/i)

(8)

= 0.2216 ⁵√ (Q²/i) for new pipes
= 0.2541 ⁵√ (Q²/i) for incrusted pipes;

equations which are interesting as showing that when the value of ζ is doubled the diameter of pipe for a given discharge is only increased by 13%.

(b) A second method is to obtain a rough value of d by assuming ζ = α. This value is

d ′ = ⁵√ (32Q²/gπ²i) ⁵√ α = 0.6319 ⁵√ (Q²/i) ⁵√ α.

Then a very approximate value of ζ is

ζ′ = α (1 + 1/12d ′);

and a revised value of d, not sensibly differing from the exact value, is

d ″ = ⁵√ (32Q²/gπ²i) ⁵√ ζ′ = 0.6319 ⁵√ (Q²/i) ⁵√ ζ′.

(c) Equation 7 may be put in the form

d = ⁵√ (32αQ²/gπ²i) ⁵√ (1 + 1/12d).

(9)

Expanding the term in brackets,

⁵√ (1 + 1/12d) = 1 + 1/60d − 1/1800d ² . . .

Neglecting the terms after the second,

d = 5√ (32α/gπ2) 5√ (Q2/i) · {1 + 1/60d } = 5√ (32α/gπ2) 5√ (Q2/i) + 0.01667;

(9a)

and

⁵√ (32α/gπ²) = 0.219 for new pipes
= 0.252 for incrusted pipes.

Fig. 96.

Fig. 97.

§ 85. Arrangement of Water Mains for Towns’ Supply.—Town mains are usually supplied oy gravitation from a service reservoir, which in turn is supplied by gravitation from a storage reservoir or by pumping from a lower level. The service reservoir should contain three days’ supply or in important cases much more. Its elevation should be such that water is delivered at a pressure of at least about 100 ft. to the highest parts of the district. The greatest pressure in the mains is usually about 200 ft., the pressure for which ordinary pipes and fittings are designed. Hence if the district supplied has great variations of level it must be divided into zones of higher and lower pressure. Fig. 96 shows a district of two zones each with its service reservoir and a range of pressure in the lower district from 100 to 200 ft. The total supply required is in England about 25 gallons per head per day. But in many towns, and especially in America, the supply is considerably greater, but also in many cases a good deal of the supply is lost by leakage of the mains. The supply through the branch mains of a distributing system is calculated from the population supplied. But in determining the capacity of the mains the fluctuation of the demand must be allowed for. It is usual to take the maximum demand at twice the average demand. Hence if the average demand is 25 gallons per head per day, the mains should be calculated for 50 gallons per head per day.

Fig. 98.

§ 86. 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_a, H_b . . . 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 AA₁, BB₁, . . . representing the minimum pressure height necessary for the adequate supply of each locality. Then A₁B₁C₁D₁ . . . is a line which should form a lower limit to the line of virtual slope. Then if heights ɧ_a, ɧ_b, ɧ_c . . . are taken representing the actual losses of head in each length l_a, l_b, l_c. . . of the main, A₀B₀C₀ will be the line of virtual slope, and it will be obvious at what points such as D₀ and E₀, 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_z − (ɧ_a + ɧ_b + . . . ɧ_z).

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 11/2 and 41/2 ft. per second. Occasionally the velocity in pipes reaches 10 ft. per second, and in hydraulic machinery working under enormous pressures even 20 ft. 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.

J. T. Fanning gives the following velocities as suitable in pipes for towns’ supply:—

Diameter in inches	4	8	12	18	24	30	36
Velocity in feet per sec.	2.5	3.0	3.5	4.5	5.3	6.2	7.0

§ 87. Branched Pipe connecting Reservoirs at Different Levels.—Let A, B, C (fig. 98) be three reservoirs connected by the arrangement of pipes shown,—l₁, d₁, Q₁, v₁; l₂, d₂, Q₂, v₂; l₃, d₃, Q₃, v₃ 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 aR, Rb, Rc 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; Q1 = Q2 + Q3.
II.	R level with b; Q1 = Q3; Q2 = 0
III.	R below b; Q1 + Q2 = Q3.

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

h_a − h′ = ζ (v₁²/2g) (4l₁/d₁)
= 32ζQ′²l₁/gπ²d₁⁵;

h′ − h_c = ζ (v₃²/2g) (4l₃/d₃)
= 32ζQ′²l₃/gπ²d₃⁵;

where Q′ is the common discharge of the two portions of the pipe. Hence

(h_a − h′)/(h′ − h_c) = l₁d₃⁵/l₃d₁⁵,

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 h_b, opening the sluice will allow flow from B, and the case is case III. If h′ = h_b, the case is case II., and is already completely solved.

The true value of h must lie between h′ and h_b. Choose a new value of h, and recalculate Q₁, Q₂, Q₃. Then if

Q₁ > Q₂ + Q₃ in case I.,

or

Q₁ + Q₂ > Q₃ in case III.,

the value chosen for h is too small, and a new value must be chosen.

If

Q₁ < Q₂ + Q₃ in case I.,

or

Q₁ + Q₂ < Q₃ in case III.,

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.

§ 88. Water Hammer.—If in a pipe through which water is flowing a sluice is suddenly closed so as to arrest the forward movement of the water, there is a rise of pressure which in some cases is serious enough to burst the pipe. This action is termed water hammer or water ram. The fluctuation of pressure is an oscillating one and gradually dies out. Care is usually taken that sluices should only be closed gradually and then the effect is inappreciable. Very careful experiments on water hammer were made by N. J. Joukowsky at Moscow in 1898 (Stoss in Wasserleitungen, St Petersburg, 1900), and the results are generally confirmed by experiments made by E. B. Weston and R. C. Carpenter in America. Joukowsky used pipes, 2, 4 and 6 in. diameter, from 1000 to 2500 ft. in length. The sluice closed in 0.03 second, and the fluctuations of pressure were automatically registered. The maximum excess pressure due to water-hammer action was as follows:—

Pipe 4-in. diameter.		Pipe 6-in. diameter.
Velocity  ft. per sec.	Excess Pressure.  ℔ per sq. in.	Velocity  ft. per sec.	Excess Pressure.  ℔ per sq. in.
0.5	31	0.6	43
2.9	168	3.0	173
4.1	232	5.6	369
9.2	519	7.5	426

In some cases, in fixing the thickness of water mains, 100 ℔ per sq. in. excess pressure is allowed to cover the effect of water hammer. With the velocities usual in water mains, especially as no valves can be quite suddenly closed, this appears to be a reasonable allowance (see also Carpenter, Am. Soc. Mech. Eng., 1893).

IX. FLOW OF COMPRESSIBLE FLUIDS IN PIPES

§ 89. Flow of Air in Long Pipes.—When air flows through a long pipe, by far the greater part of the work expended is used 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 R. S. Culley and R. Sabine (Proc. Inst. Civ. Eng. xliii.), show that the change of temperature of the air flowing along the tube is much less than it would be in adiabatic expansion.

§ 90. 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

p/G = cτ,

(1)

where c = 53.15. Taking τ = 521, corresponding to a temperature of 60° Fahr.,

cτ = 27690 foot-pounds.

(2)

 

Fig. 99.

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

GΩu = W = constant,

(3)

where Ω is the section of the pipe and u the velocity of the air. Combining (1) and (3),

Ωup/W = cτ = constant.

(3a)

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 A₀, A₁ at a distance dl (fig. 99). Let p, u be the pressure and velocity at A₀, p + dp and u + du those at A₁. Further, suppose that in a very short time dt the mass of air between A₀A₁ comes to A′₀A′₁ so that A₀A′₀ = udt and A₁A′₁ = (u + du) dt₁. Let Ω 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 A₀A′₀, and A₁A′₁ is the same. That is,

Wdt = GΩudt = GΩ (u+du)dt.

By analogy with liquids the head lost in friction is, for the length dl (see § 72, eq. 3), ζ (u²/2g) (dl/m). Let H = u²/2g. Then the head lost is ζ(H/m)dl; and, since Wdt ℔ of air flow through the pipe in the time considered, the work expended in friction is −ζ (H/m)W dl dt. The change of kinetic energy in dt seconds is the difference of the kinetic energy of A₀A′₀ and A₁A′₁, that is,

(W/g)dt{(u+du)²−u²}/2 = (W/g) u du dt = WdHdt.

The work of expansion when Ωudt cub. ft. of air at a pressure p expand to Ω(u + du) dt cub. ft. is Ωp du dt. But from (3a) u = cτW/Ωp, and therefore

du / dp = −cτW / Ωp².

And the work done by expansion is −(cτW/p) dp dt.

The work done by gravity on the mass between A₀ and A₁ 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 A₀A₁ is

pΩudt−(p+dp) Ω (u + du) dt
= −(pdu + udp) Ωdt

But from (3a)

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,

WdHdt = −(cτW/p) dp dt − ζ (H/m) W dl dt
dH + (cτ/p) dp + ζ (H/m) dl = 0,
dH/H + (cτ/Hp) dp + ζdl / m = 0

(4)

But

u = cτW / Ωp,

and

H = u²/2g = c²τ²W² / 2gΩ²p²,

∴ dH/H + (2gΩ²p / cτW²) dp + ζdl / m = 0.

(4a)

For tubes of uniform section m is constant; for steady motion W is constant; and for isothermal expansion τ is constant. Integrating,

log H + gΩ²p² / W²cτ + ζ l / m = constant;

(5)

for

l = 0, let H = H₀, and p = p₀;

and for

l = l, let H = H₁, and p = p₁.

log (H₁/H₀) + (gΩ² / W²cτ) (p₁² − p₀²) + ζ l / m = 0.

(5a)

where p₀ is the greater pressure and p₁ the less, and the flow is from A₀ towards A₁.

By replacing W and H,

log (p₀/p₁) + (gcτ / u₀²p₀²) (p₁² − p₀² + ζ l/m = 0

(6)

Hence the initial velocity in the pipe is

u₀ = √ [{gcτ (p₀² − p₁²)} / {p₀² (ζ l/m + log (p₀ / p₁) }].

(7)

When l is great, log p₀/p₁ is comparatively small, and then

u₀ = √ [ (gcτm/ζ l) {(p₀² − p₁²) / p₀²} ],

(7a)

a very simple and easily used expression. For pipes of circular section m = d/4, where d is the diameter:—

u₀ = √ [ (gcτd / 4ζ l) {(p₀² − p₁²) / p₀²} ];

(7b)

or approximately

u₀ = (1.1319 − 0.7264 p₁/p₀) √ (gcτd / 4ζl).

(7c)

§ 91. Coefficient of Friction for Air.—A discussion by Professor Unwin of the experiments by Culley and 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, 21/4 in. (0.187 ft.) in diameter, and in lengths of 2000 to nearly 6000 ft.

In some experiments on the flow of air through cast-iron pipes A. Arson found the coefficient of friction to vary with the velocity and diameter of the pipe. Putting

ζ = α/v + β,

(8)

he obtained the following values—

Diameter of Pipe  in feet.	α	β	ζ for 100 ft.  per second.
1.64	.00129	.00483	.00484
1.07	.00972	.00640	.00650
.83	.01525	.00704	.00719
.338	.03604	.00941	.00977
.266	.03790	.00959	.00997
.164	.04518	.01167	.01212

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 ft. per second, and without much error for higher velocities, these numbers agree fairly with the formula

ζ = 0.005 (1 + 3/10d),

(9)

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 later experiments on a very large scale, by E. Stockalper at the St Gotthard Tunnel, agree better with the value

ζ＝0.0028 (1 + 3/10d).

These pipes were probably less rough than 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

z0 − z1 − p0/G0 − p1/G1 − ζ (v2/2g) (l/m)＝0,

(10)

precisely equivalent to the equation for the flow of water, z₀ and z₁ being the elevations of the two ends of the pipe above any datum, p₀ and p₁ the pressures, G₀ and G₁ the densities, and v the mean velocity in the pipe. This equation may be used for the flow of coal gas.

§ 92. Distribution of Pressure in a Pipe in which Air is Flowing.—From equation (7a) it results that the pressure p, at l ft. from that end of the pipe where the pressure is p₀, is

p＝p0 √{1 − ζlu02 / mgcτ};

(11)

which is of the form

p＝√ (al+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 Sabine’s experiments, in one of which the pressure was greater than atmospheric 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 yds. in length. The pressures are given in inches of mercury.

Fig. 100.

Variation of Velocity in the Pipe.—Let p₀, u₀ be the pressure and velocity at a given section of the pipe; p, u, the pressure and velocity at any other section. From equation (3a)

up＝cτW / Ω＝constant;

so that, for any given uniform pipe,

up＝u0p0, u＝u0p0 / p;

(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 Culley and 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.

Fig. 101.

§ 93. Weight of Air Flowing per Second.—The weight of air discharged per second is (equation 3a)—

W＝Ωu₀p₀ / cτ.

From equation (7b), for a pipe of circular section and diameter d,

W＝1/4π √ (gd 5 (p02 − p12) / ζ lcτ), ＝.611 √ (d 5 (p02 − p12) / ζ lτ).

(13)

Approximately

W＝(.6916p0 − .4438p1) (d 5 / ζ lτ)1/2.

(13a)

§ 94. 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 3/4 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,

$t=\int _{0}^{l}dl/u,$

From (4a) neglecting dH/H, and putting m = d/4,

dl＝gdΩ²p dp / 2ζW²cr.

From (1) and (3)

u＝Wcτ / pΩ;
dl/u＝gdΩ³p² dp / 2ζW³c²τ²;

t＝$\int _{p_{1}}^{p_{0}};$gdΩ³p² dp / 2ζW³c²τ²,

＝gdΩ3 (p03 − p13) / 6ζW3c2τ2.

(14)

But

W = p0u0Ω / cτ;

∴ t＝gdcτ (p03 − p13) / 6ζp03u03, ＝ζ1/2 l3/2 (p03 − p13) / 6(gcτd)1/2 (p02 − p12)3/2;

(15)

If τ = 521°, corresponding to 60° F.,

t＝.001412 ζ1/2 l3/2 (p03 − p13) / d 1/2 (p02 − p12)3/2;

(15a)

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 l/t; or, for τ = 521°,

umean＝0.708 √ {d (p02 − p12)3/2 / ζ l (p03 − p13)}.

(16)

The following table gives some results:—

		Absolute  Pressures in  ℔ per sq. in.		Mean Velocities for Tubes of a length in feet.
		p0	p1	1000	2000	3000	4000	5000
Vacuum  Working	$\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$	15	5	99.4	70.3	57.4	49.7	44.5
		15	10	67.2	47.5	38.8	34.4	30.1
Pressure  Working	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$	20	15	57.2	40.5	33.0	28.6	25.6
		25	15	74.6	52.7	43.1	37.3	33.3
		30	15	84.7	60.0	49.0	42.4	37.9

Limiting Velocity in the Pipe when the Pressure at one End is diminished indefinitely.—If in the last equation there be put p₁ = 0, then

u′_mean＝0.708 √ (d / ζl);

where the velocity is independent of the pressure p₀ 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

§ 95. 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 the 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 Ω 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.

 

Fig. 102.

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 a 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.

§ 96. Steady Flow of Water with Uniform Velocity in Channels of Constant Section.—Let aa′, bb′ (fig. 103) be two 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 Ω be the area of the cross sections, χ the wetted perimeter, pq + qr + rs, of a section. Then the quantity m = Ω/χ 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 bc/ab.

Fig. 103.

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 GΩ dl, and the component of the weight in the direction of motion is GΩdl × the cosine of the angle between Wg and ab, that is, GΩdl cos abc = GΩ dl bc/ab = GΩidl. (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 ƒ(v); ƒ(v) being the friction per sq. ft. at a velocity v. Hence the friction is −χ dl ƒ(v). Equating the sum of the forces to zero,

GΩi dl − χ dl ƒ(v) = 0, ƒ(v) / G = Ωi / χ = mi.

(1)

But it has been already shown (§ 66) that ƒ(v) = ζGv²/2g,

∴ ζv2 / 2g = mi.

(2)

This may be put in the form

v = √ (2g/ζ) √ (mi) = c √ (mi);

(2a)

where c is a coefficient depending on the roughness and form of the channel.

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 ζ 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 ƒ is the fall in feet per mile,

ƒ = 5280 i.

Putting this and the above value of ζ in (2a), we get the very simple and long-known approximate formula for the mean velocity of a stream—

v = 1/4 1/2 √ (2mf).

(3)

The flow down the stream per second, or discharge of the stream, is

Q = Ωv = Ωc √ (mi).

(4)

§ 97. 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

ζ = α (1 + β/v),

(5)

and from 255 experiments obtained for the constants the values

α = 0.007409; β = 0.1920.

This gives the following values at different velocities:—

v =  ζ =

0.3  0.01215

0.5  0.01025

0.7  0.00944

1  0.00883

11/2  0.00836

2  0.00812

3  0.90788

5  0.00769

7  0.00761

10  0.00755

15  0.00750

In using this value of ζ when v is not known, it is best to proceed by approximation.

§ 98. Darcy and Bazin’s Expression for the Coefficient of Friction.—Darcy and 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 ζ an empirical expression (similar to that for pipes) of the form

ζ = α (1 + β/m);

(6)

where m is the hydraulic mean depth. For different kinds of channels they give the following values of the coefficient of friction:—

Kind of Channel.		α	β
I.	Very smooth channels, sides of smooth cement or planed timber	.00294	0.10
II.	Smooth channels, sides of ashlar, brickwork, planks	.00373	0.23
III.	Rough channels, sides of rubble masonry or pitched with stone	.00471	0.82
IV.	Very rough canals in earth	.00549	4.10
V.	Torrential streams encumbered with detritus	.00785	5.74

The last values (Class V.) are not Darcy and Bazin’s, but are taken from experiments by Ganguillet and Kutter on Swiss streams.

The following table very much facilitates the calculation of the mean velocity and discharge of channels, when Darcy and Bazin’s value of the coefficient of friction is used. Taking the general formula for the mean velocity already given in equation (2a) above,

v = c √ (mi),

where c = √ (2g/ζ), the following table gives values of c for channels of different degrees of roughness, and for such values of the hydraulic mean depths as are likely to occur in practical calculations:—

Values of c in v = c √ (mi), deduced from Darcy and Bazin’s Values.

Hydraulic Mean.  Depth = m.	Very Smooth  Channels. Cement.	Smooth Channels. Ashlar or  Brickwork.	Rough  Channels.  Rubble Masonry.	Very Rough  Channels. Canals in Earth.	Excessively  Rough Channels  encumbered with Detritus.
.25	125	95	57	26	18.5
.5	135	110	72	36	25.6
.75	139	116	81	42	30.8
1.0	141	119	87	48	34.9
1.5	143	122	94	56	41.2
2.0	144	124	98	62	46.0
2.5	145	126	101	67	..
3.0	145	126	104	70	53
3.5	146	127	105	73	..
4.0	146	128	106	76	58
4.5	146	128	107	78	..
5.0	146	128	108	80	62
5.5	146	129	109	82	..
6.0	147	129	110	84	65
6.5	147	129	110	85	..
7.0	147	129	110	86	67
7.5	147	129	111	87	..
8.0	147	130	111	88	69
8.5	147	130	112	89	..
9.0	147	130	112	90	71
9.5	147	130	112	90	..
10.0	147	130	112	91	72
11	147	130	113	92	..
12	147	130	113	93	74
13	147	130	113	94	..
14	147	130	113	95	..
15	147	130	114	96	77
16	147	130	114	97	..
17	147	130	114	97	..
18	147	130	114	98	..
20	147	131	114	98	80
25	148	131	115	100	..
30	148	131	115	102	83
40	148	131	116	103	85
50	148	131	116	104	86
∞	148	131	117	108	91

§ 99. Ganguillet and Kutter’s Modified Darcy Formula.—Starting from the general expression v = c√mi, Ganguillet and Kutter examined the variations of c for a wider variety of cases than those discussed by Darcy and Bazin. Darcy and Bazin’s experiments were confined to channels of moderate section, and to a limited variation of slope. Ganguillet and Kutter brought into the discussion two very distinct and important additional series of results. The gaugings of the Mississippi by A. A. Humphreys and H. L. Abbot afford data of discharge for the case of a stream of exceptionally large section and or 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 and 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 appeared to indicate 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    ”     ”	0.02     ”

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 and 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 and Bazin, Ganguillet and Kutter found that the four classes of coefficients proposed by Darcy and 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 = √ (2g / ζ) √ (mi) = c √ (mi),

Ganguillet and Kutter proceeded in a purely empirical way. They found that an expression of the form

c = α / (1 + β/√ m)

could be made to fit the experiments somewhat better than Darcy’s expression. Inverting this, we get

1/c = 1/α + β/α √ m,

an equation to a straight line having 1/√m for abscissa, 1/c for ordinate, and inclined to the axis of abscissae at an angle the tangent of which is β/α.

Plotting the experimental values of 1/c and 1/√ m, the points so found indicated a curved rather than a straight line, so that β must depend on α. After much comparison the following form was arrived at—

c = (A + l/n) / (1 + An / √ m),

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 = a + p/i.

Then

c = (a + l/n + p/i) / {1 + (a + p/i) n/√ m},

the form of the expression for c ultimately adopted by Ganguillet and Kutter.

For the constants a, l, p Ganguillet and Kutter obtain the values 23, 1 and 0.00155 for metrical measures, or 41.6, 1.811 and 0.00281 for English feet. The coefficient of roughness n is found to vary from 0.008 to 0.050 for either metrical or English measures.

The most practically useful values of the coefficient of roughness n are given in the following table:—

Nature of Sides of Channel.	Coefficient of Roughness n.
Well-planed timber	0.009
Cement plaster	0.010
Plaster of cement with one-third sand	0.011
Unplaned planks	0.012
Ashlar and brickwork	0.013
Canvas on frames	0.015
Rubble masonry	0.017
Canals in very firm gravel	0.020
Rivers and canals in perfect order, free from stones or weeds	0.025
Rivers and canals in moderately good order, not quite free from stones and weeds${\big \}}$	0.030
Rivers and canals in bad order, with weeds and detritus	0.035
Torrential streams encumbered with detritus	0.050

Ganguillet and Kutter’s formula is so cumbrous that it is difficult to use without the aid of tables.

Lowis D’A. Jackson published complete and extensive tables for facilitating the use of the Ganguillet and Kutter formula (Canal and Culvert Tables, London, 1878). To lessen calculation he puts the formula in this form:—

M = n (41.6 + 0.00281/i);

v = (√ m/n) {(M + 1.811) / (M + √m)} √ (mi).

The following table gives a selection of values of M, taken from Jackson’s tables:—

i =	Values of M for n =
	0.010	0.012	0.015	0.017	0.020	0.025	0.030
.00001	3.2260	3.8712	4.8390	5.4842	6.4520	8.0650	9.6780
.00002	1.8210	2.1852	2.7315	3.0957	3.6420	4.5525	5.4630
.00004	1.1185	1.3422	1.6777	1.9014	2.2370	2.7962	3.3555
.00006	0.8843	1.0612	1.3264	1.5033	1.7686	2.2107	2.6529
.00008	0.7672	0.9206	1.1508	1.3042	1.5344	1.9180	2.3016
.00010	0.6970	0.8364	1.0455	1.1849	1.3940	1.7425	2.0910
.00025	0.5284	0.6341	0.7926	0.8983	1.0568	1.3210	1.5852
.00050	0.4722	0.5666	0.7083	0.8027	0.9444	1.1805	1.4166
.00075	0.4535	0.5442	0.6802	0.7709	0.9070	1.1337	1.3605
.00100	0.4441	0.5329	0.6661	0.7550	0.8882	1.1102	1.3323
.00200	0.4300	0.5160	0.6450	0.7310	0.8600	1.0750	1.2900
.00300	0.4254	0.5105	0.6381	0.7232	0.8508	1.0635	1.2762

A 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 0.020 to 0.035.

In Jackson’s opinion even Kutter’s numerous classes of channels seem inadequately graduated, and he proposes for artificial canals the following classification:—

I.	Canals in very firm gravel, in perfect order	n = 0.02
II.	Canals in earth, above the average in order	n = 0.0225
III.	Canals in earth, in fair order	n = 0.025
IV.	Canals in earth, below the average in order	n = 0.0275
V.	Canals in earth, in rather bad order, partially over- grown with weeds and obstructed by detritus.${\big \}}$	n = 0.03

Ganguillet and Kutter’s formula has been considerably used partly from its adoption in calculating tables for irrigation work in India. But it is an empirical formula of an unsatisfactory form. Some engineers apparently have assumed that because it is complicated it must be more accurate than simpler formulae. Comparison with the results of gaugings shows that this is not the case. The term involving the slope was introduced to secure agreement with some early experiments on the Mississippi, and there is strong reason for doubting the accuracy of these results.

§ 100. Bazin’s New Formula.—Bazin subsequently re-examined all the trustworthy gaugings of flow in channels and proposed a modification of the original Darcy formula which appears to be more satisfactory than any hitherto suggested (Étude d’une nouvelle formule, Paris, 1898). He points out that Darcy’s original formula, which is of the form mi/v² = α + β/m, does not agree with experiments on channels as well as with experiments on pipes. It is an objection to it that if m increases indefinitely the limit towards which mi/v² tends is different for different values of the roughness. It would seem that if the dimensions of a canal are indefinitely increased the variation of resistance due to differing roughness should vanish. This objection is met if it is assumed that √ (mi/v²) = α + β/√ m, so that if a is a constant mi/v² tends to the limit a when m increases. A very careful discussion of the results of gaugings shows that they can be expressed more satisfactorily by this new formula than by Ganguillet and Kutter’s. Putting the equation in the form ζv²/2g = mi, ζ = 0.002594 (1 + γ/√ m), where γ has the following values:—

I.	Very smooth sides, cement, planed plank, γ =	0.109
II.	Smooth sides, planks, brickwork	0.290
III.	Rubble masonry sides	0.833
IV.	Sides of very smooth earth, or pitching	1.539
V.	Canals in earth in ordinary condition	2.353
VI.	Canals in earth exceptionally rough	3.168

§ 101. The Vertical Velocity Curve.—If at each point along a vertical representing the depth of a stream, the velocity at that point is plotted horizontally, the curve obtained is the vertical velocity curve and it has been shown by many observations that it approximates to a parabola with horizontal axis. The vertex of the parabola is at the level of the greatest velocity. Thus in fig. 104 OA is the vertical at which velocities are observed; v₀ is the surface; v_z the maximum and v_d the bottom velocity. B C D is the vertical velocity curve which corresponds with a parabola having its vertex at C. The mean velocity at the vertical is

v_m = 1/3 [2v_z + v_d + (d_z/d) (v₀ − v_d)].

Fig. 104.

The Horizontal Velocity Curve.—Similarly if at each point along a horizontal representing the width of the stream the velocities are plotted, a curve is obtained called the horizontal velocity curve. In streams of symmetrical section this is a curve symmetrical about the centre line of the stream. The velocity varies little near the centre of the stream, but very rapidly near the banks. In unsymmetrical sections the greatest velocity is at the point where the stream is deepest, and the general form of the horizontal velocity curve is roughly similar to the section of the stream.

§ 102. Curves or Contours of Equal Velocity.—If velocities are observed at a number of points at different widths and depths in a stream, it is possible to draw curves on the cross section through points at which the velocity is the same. These represent contours of a solid, the volume of which is the discharge of the stream per second. Fig. 105 shows the vertical and horizontal velocity curves and the contours of equal velocity in a rectangular channel, from one of Bazin’s gaugings.

§ 103. Experimental Observations on the Vertical Velocity Curve.—A preliminary difficulty arises in observing the velocity at a given point in a stream because the velocity rapidly varies, the motion not being strictly steady. If an average of several velocities at the same point is taken, or the average velocity for a sensible period of time, this average is found to be constant. It may be inferred that though the velocity at a point fluctuates about a mean value, the fluctuations being due to eddying motions superposed on the general motion of the stream, yet these fluctuations produce effects which disappear in the mean of a series of observations and, in calculating the volume of flow, may be disregarded.

Fig. 105.

In the next place it is found that in most of the best observations on the velocity in streams, the greatest velocity at any vertical is found not at the surface but at some distance below it. In various river gaugings the depth d_z at the centre of the stream has been found to vary from 0 to 0.3d.

§ 104. 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 3/10ths 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 or waves 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, P. P. 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. A much more probable explanation of the diminution of the velocity at and near the free surface is 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.

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 upstream wind lowered, and a down-stream wind raised, the axis of the parabolic velocity curve. In calm weather the axis was at 3/10ths 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, f 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 Humphreys and Abbot find their results agree with the expression

h′ / m = 0.317 ± 0.06f.

Fig. 106 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.

Fig. 106.

It is impossible at present to give a theoretical rule for the vertical velocity curve, but in very many gaugings it has been found that a parabola with horizontal axis fits the observed results fairly well. The mean velocity on any vertical in a stream varies from 0.85 to 0.92 of the surface velocity at that vertical, and on the average if v₀ is the surface and v_m the mean velocity at a vertical v_m = 6/7v₀, a result useful in float gauging. On any vertical there is a point at which the velocity is equal to the mean velocity, and if this point were known it would be useful in gauging. Humphreys and Abbot in the Mississippi found the mean velocity at 0.66 of the depth; G. H. L. Hagen and H. Heinemann at 0.56 to 0.58 of the depth. The mean of observations by various observers gave the mean velocity at from 0.587 to 0.62 of the depth, the average of all being almost exactly 0.6 of the depth. The mid-depth velocity is therefore nearly equal to, but a little greater than, the mean velocity on a vertical. If v_md is the mid-depth velocity, then on the average v_m = 0.98v_md.

§ 105. Mean Velocity on a Vertical from Two Velocity Observations.—A. J. C. Cunningham, in gaugings on the Ganges canal, found the following useful results. Let v₀ be the surface, v_m the mean, and v_xd the velocity at the depth xd; then

v_m = 1/4 (v₀+3v_2/3d )
  = 1/2 (v_.211^d+v_.789^d ).

§ 106. 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. In general in streams of fairly regular section the mean velocity for the whole section varies from 0.7 to 0.85 of the greatest surface velocity. 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 v₀ be the greatest surface velocity, v_m the mean velocity of the stream. Then, according to Bazin,

v_m = v₀ − 25.4 √ (mi).

But

v_m = c √ (mi),

where c is a coefficient, the values of which have been already given in the table in § 98. Hence

v_m = cv₀ / (c + 25.4).

Values of Coefficient c/(c + 25.4) in the Formula v_m = cv₀/(c + 25.4).

Hydraulic  Mean Depth  = m.	Very Smooth  Channels.  Cement.	Smooth Channels. Ashlar or  Brickwork.	Rough  Channels.  Rubble Masonry.	Very Rough  Channels. Canals in Earth.	Channels  encumbered  with Detritus.
0.25	.83	.79	.69	.51	.42
0.5	.84	.81	.74	.58	.50
0.75	.84	.82	.76	.63	.55
1.0	.85	..	.77	.65	.58
2.0	..	.83	.79	.71	.64
3.0	..	..	.80	.73	.67
4.0	..	..	.81	.75	.70
5.0	..	..	..	.76	.71
6.0	..	.84	..	.77	.72
7.0	..	..	..	.78	.73
8.0	..	..	..	..	..
9.0	..	..	.82	..	.74
10.0	..	..	..	..	..
15.0	..	..	..	.79	.75
20.0	..	..	..	.80	.76
30.0	..	..	.82	..	.77
40.0	..	..	..	..	..
50.0	..	..	..	..	..
∞	..	..	..	..	.79

Fig. 107.

§ 107. 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 horseshoe shaped lagoon or marsh. Professor James Thomson pointed out (Proc. Roy. Soc., 1877, p. 356; Proc. Inst. of Mech. Eng., 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 creating deposits at the inner bank. That view is very far from a complete account of the matter, and Professor Thomson gave a much more ingenious account of the action at the bend, which he completely confirmed by experiment.

 

Fig. 108.

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. 108). 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. 107 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.

§ 108. Discharge of a River when flowing at different Depths.—When frequent observations must be made on the flow of a river or canal, the depth of which varies at different times, it is very convenient to have to observe the depth only. A formula can be established giving the flow in terms of the depth. Let Q be the discharge in cubic feet per second; H the depth of the river in some straight and uniform part. Then Q = aH + bH², where the constants a and b must be found by preliminary gaugings in different conditions of the river. M. C. Moquerey found for part of the upper Saône, Q = 64.7H + 8.2H² in metric measures, or Q = 696H + 26.8H² in English measures.

§ 109. Forms of Section of Channels.—The simplest form of section for channels is the semicircular or nearly semicircular channel (fig. 109), a form now often adopted from the facility with which it can be executed in concrete. It has the advantage that the rubbing surface is less in proportion to the area than in any other form.

Fig. 109.

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. 110) be the bottom breadth, b₀ 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 = (b₀ − nd) d, and the wetted perimeter χ = b + 2d √ (n² + 1).

Fig. 110.

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. 111, 112 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.

§ 110. 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 νr ², where κ, μ, and ν have the following values:—

Depth of water in terms of radius	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$κ =	.01	.05	.10	.15	.20	.25	.30	.35	.40	.45	.50	.55	.60	.65	.70	.75	.80	.85	.90	.95	1.0
Hydraulic mean depth in terms of radius	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$μ =	.00668	.0321	.0523	.0963	.1278	.1574	.1852	.2142	.242	.269	.293	.320	.343	.365	.387	.408	.429	.449	.466	.484	.500
Waterway in terms of square of radius	$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$ν =	.00189	.0211	.0598	.1067	.1651	.228	.294	.370	.450	.532	.614	.709	.795	.885	.979	1.075	1.175	1.276	1.371	1.470	1.571

Fig. 111.—Scale 20 ft. = 1 in.

Fig. 112.—Scale 80 ft. = 1 in.

Fig. 113.

§ 111. 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. 113, the numbers marked on the figure being proportional numbers.

§ 112. Problems on Channels in which the Flow is Steady and at Uniform Velocity.—The general equations given in §§ 96, 98 are

     ζ = α(1 + β/m);

(1)

ζv²/2g = mi;   

(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 Ω and m; from (1) find ζ, from (3) find v, and lastly from (2) find i.

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.

Fig. 114.

Since m lies generally between the limits m = d and m = 1/2d, where d is the depth of the stream, and since, moreover, the velocity varies as √ (m) so that an error in the value of m leads only to a much less error in the value of the velocity calculated from it, we may proceed thus. Assume a value for m, and calculate v from it. Let v₁ be this first approximation to v. Then Q/v₁ is a first approximation to Ω, say Ω₁. With this value of Ω 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 Ω. Repeat the process till the successive values of m approximately coincide.

§ 113. Problem IV. Most Economical Form of Channel for given Side Slopes.—Suppose the channel is to be trapezoidal in section (fig. 114), 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 Ω∞ ∛χ, 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. 114), then

Ω = (b + nd) d;

χ = b + 2d √ (n² + 1).

Both Ω and χ are to be minima. Differentiating, and equating to zero.

(db/dd + n) d + b + nd = 0,
db/dd + 2 √ (n² + 1) = 0;

eliminating db/dd,

{n − 2√ (n² + 1)} d + b + nd = 0;
b = 2 {√ (n² + 1) − n} d.

But

Ω / χ = (b + nd) d / {b + 2d √ (n² + 1)}.

Inserting the value of b,

m = Ω/χ = {2d √ (n² + 1) − nd} / {4d √ (n² + 1) − 2nd} = 1/2 d.

That is, with given side slopes, the section is least for 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 = 1/2d the sides of the channel are tangential to a semicircle drawn on the water line.

Since

Ω / χ = 1/2 d,

therefore

Ω = 1/2 χd.

(1)

Let ABCD be the channel (fig. 115); from E the centre of AD drop perpendiculars EF, EG, EH on the sides.

Let

AB = CD = a; BC = b; EF = EH = c; and EG = d.

Ω = area AEB + BEC + CED,
 = ac + 1/2 bd.
χ  = 2a + b.

Putting these values in (1),

ac + 1/2 bd = (a + 1/2 b) d; and hence c = d.

Fig. 115.

Fig. 116.

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 of channel. The bottom will be a horizontal tangent of that semicircle, and the sides tangents drawn at the required side slopes.

The above result may be obtained thus (fig. 116):—

χ = b + 2d / sin β.

(1)

Ω = d (b + d cot β);

Ω/d = b + d cot β;

(2)

Ω/d ² = b/d + cot β.

(3)

From (1) and (2),

χ = Ω / d − d cot β + 2d / sin β.

This will be a minimum for

dχ / dd = Ω / d ² + cot β − 2 / sin β = 0,

or

Ω/d ² = 2 cosec. β − cot β.

(4)

or

d = √ {Ω sin β / (2 − cos β)}.

From (3) and (4),

b/d = 2 (1 − cos β) / sin β = 2 tan 1/2 β.

Proportions of Channels of Maximum Discharge for given Area and Side Slopes.
Depth of channel = d; Hydraulic mean depth = 1/2d;

Area of section = Ω.

	Inclination  of Sides to  Horizon.	Ratio of  Side Slopes.	Area of  Section Ω.	Bottom  Width.	Top width =  twice length of each Side Slope.
Semicircle	..	..	1.571d 2	0	2d
Semi-hexagon	60°   0′	3   : 5	1.732d 2	1.155d	2.310d
Semi-square	90°   0′	0   : 1	2d 2	2d	2d
	75°  58′	1   : 4	1.812d 2	1.562d	2.062d
	63°  26′	1   : 2	1.736d 2	1.236d	2.236d
	53°   8′	3   : 4	1.750d 2	d	2.500d
	45°   0′	1   : 1	1.828d 2	0.828d	2.828d
	38°  40′	11/4 : 1	1.952d 2	0.702d	3.202d
	33°  42′	11/2 : 1	2.106d 2	0.606d	3.606d
	29°  44′	13/4 : 1	2.282d 2	0.532d	4.032d
	26°  34′	2  : 1	2.472d 2	0.472d	4.472d
	23°  58′	21/4 : 1	2.674d 2	0.424d	4.924d
	21°  48′	21/2 : 1	2.885d 2	0.385d	5.385d
	19°  58′	23/4 : 1	3.104d 2	0.354d	5.854d
	18°  26′	3   : 1	3.325d 2	0.325d	6.325d

Half the top width is the length of each side slope. The wetted
perimeter is the sum of the top and bottom widths.

§ 114. Form of Cross Section of Channel in which the 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.

Fig. 117.

Let fig. 117 represent the cross section of the channel. From the symmetry of the section, only half the channel need be considered. Let obac 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.

Let ob = b/2; de = y, fg = y + dy, od = x, of = x + dx; eg = ds.

The condition to be satisfied is that

v = c √ (mi)

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 obac. Hence also

Increment of section/Increment of perimeter = ydx/ds = k.

y²dx² = k²ds² = k²(dx²+dy²) and dx = kdy / √ (y² − k²).

Integrating,

x = k log_ε {y + √ (y² − k²)} + constant;

and, since y = b/2 when x = 0,

x = k log_ε [{y + √ (y² − k²)} / {1/2 b + √ (1/4 b² − k²) }].

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 ft. depth. Hence k = 2; b = 9.2; the rapid flattening of the side slopes is remarkable.

Steady Motion of Water in Open Channels of Varying Cross Section and Slope

§ 115. 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.

Fig. 118.

Let fig. 118 represent a longitudinal section of the stream, A₀A₁ being the water surface, B₀B₁ the stream bed. Let A₀B₀, A₁B₁ be cross sections normal to the direction of flow. Suppose the mass of water A₀B₀A₁B₁ comes in a short time θ to C₀D₀C₁D₁, and let the work done on the mass be equated to its change of kinetic energy during that period. Let l be the length A₀A₁ 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.

Fig. 119.

Change of Kinetic Energy.—At the end of the time θ there are as many particles possessing the same velocities in the space C₀D₀A₁B₁ as at the beginning. The change of kinetic energy is therefore the difference of the kinetic energies of A₀B₀C₀D₀ and A₁B₁C₁D₁.

Let fig. 119 represent the cross section A₀B₀, and let ω be a small element of its area at a point where the velocity is v. Let Ω₀ be the whole area of the cross section and u₀ the mean velocity for the whole cross section. From the definition of mean velocity we have

u₀ = Σ ωv / Ω₀.

Let v = u₀ + w, where w is the difference between the velocity at the small element ω and the mean velocity. For the whole cross section, Σωw = 0.

The mass of fluid passing through the element of section ω, in θ seconds, is (G/g) ωvθ, and its kinetic energy is (G/2g) ωv³θ. For the whole section, the kinetic energy of the mass A₀B₀C₀D₀ passing in θ seconds is

(Gθ / 2g) Σωv³ = (Gθ/2g) Σω (u₀³ + 3u₀²w + 3u₀² + w³),
= (Gθ / 2g) {u₀³Ω + Σωw² (3u₀ + w)}.

The factor 3u₀ + w is equal to 2u₀ + v, a quantity necessarily positive. Consequently Σωv³ > Ω₀u₀³, and consequently the kinetic energy of A₀B₀C₀D₀ is greater than

(Gθ / 2g) Ω₀u₀³ or (Gθ) / 2g) Qu₀²,

which would be its value if all the particles passing the section had the same velocity u₀. Let the kinetic energy be taken at

α (Gθ / 2g) Ω₀u₀³ = α (Gθ / 2g) Qu₀²,

where α is a corrective factor, the value of which was estimated by J. B. C. J. Bélanger at 1.1.^[6] Its precise value is not of great importance.

In a similar way we should obtain for the kinetic energy of A₁B₁C₁D₁ the expression

α (Gθ / 2g) Ω₁u₁³ = α (Gθ / 2g) Qu₁²,

where Ω₁, u₁ are the section and mean velocity at A₁B₁, 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 A₀B₀A₁B₁ in θ seconds is

α (Gθ / 2g)Q(u12−u02).

(1)

Motive Work of the Weight and Pressures.—Consider a small filament a₀a₁ which comes in θ seconds to c₀c₁. The work done by gravity during that movement is the same as if the portion a₀c₀ were carried to a₁c₁. Let dQθ be the volume of a₀c₀ or a₁c₁, and y₀, y₁ the depths of a₀, a₁ from the surface of the stream. Then the volume dQθ or GdQθ pounds falls through a vertical height z + y₁ − y₀, and the work done by gravity is

GdQθ(z + y₁ − y₀).

Putting p_a for atmospheric pressure, the whole pressure per unit of area at a₀ is Gy₀ + p_a, and that at a₁ is −(Gy₁ + p_a). The work of these pressures is

G(y₀ + p_a/G − y₁ − p_a/G) dQθ = G (y₀ − y₁) dQθ.

Adding this to the work of gravity, the whole work is GzdQθ; or, for the whole cross section,

GzQθ.

(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 A₀B₀. 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

GQθζ(u² / 2g) (χ / Ω) ds,

where u, χ, Ω are the mean velocity, wetted perimeter, and section at A′B′. Hence the whole work lost in friction from A₀B₀ to A₁B₁ will be

GQθ $\int _{0}^{l}$ζ(u2 / 2g) (χ/Ω)ds.

(3)

Equating the work given in (2) and (3) to the change of kinetic energy given in (1),

α(GQθ / 2g) (u₁² − u₀²) = GQzθ − GQθ $\int _{0}^{l}$ ζ(u² / 2g) (χ / Ω) ds;

∴ z = α(u₁² − u₀²) / 2g + $\int _{0}^{l}$ ζ(u² / 2g) (χ / Ω) ds.

Fig. 120.

§ 116. 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, a₁b₁, taken for simplicity normal to the stream bed (fig. 120). For that short length of stream the fall of surface level, or difference of level of a and a₁, may be written dz. Also, if we write u for u₀, and u + du for u₁, the term (u₀² − u₁²)/2g becomes udu/g. Hence the equation applicable to an indefinitely short length of the stream is

dz = udu/g + (χ/Ω) ζ(u²/2g) ds.

(1)

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′c = cd − a′d = i ds − dh.

(2)

Since the motion is steady—

Q = Ωu = constant.

Differentiating,

Ω du + u dΩ = 0;

∴ du = −u dΩ/Ω.

Let x be the width of the stream, then dΩ = xdh very nearly. Inserting this value,

du = −(ux / Ω) dh.

(3)

Putting the values of du and dz found in (2) and (3) in equation (1),

ids − dh = −(u²x / gΩ) dh + (χ / Ω) ζ(u² / 2g) ds.

dh/ds = {i − (χ/Ω) ζ(u2/2g)} / {1 − (u2/g) (x/Ω)}.

(4)

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, χh = Ω, and if χ is large compared with h, Ω/χ = xh/x = h nearly. Then equation (4) becomes

dh/ds = i (1 − ζu² / 2gih) / (1 − u²/gh).

(5)

§ 117. General Indications as to the Form of Water Surface furnished by Equation (5).—Let A₀A₁ (fig. 121) be the water surface, B₀B₁ the bed in a longitudinal section of the stream, and ab any section at a distance s from B₀, the depth ab being h. Suppose B₀B₁, B₀A₀ taken as rectangular coordinate axes, then dh/ds is the trigonometric tangent of the angle which the surface of the stream at a makes with the axis B₀B₁. This tangent dh/ds will be positive, if the stream is increasing in depth in the direction B₀B₁; negative, if the stream is diminishing in depth from B₀ towards B₁. If dh/ds = 0, the surface of the stream is parallel to the bed, as in cases of uniform motion. But from equation (4)

dh/ds = 0, if i − (χ/Ω) ζ(u²/2g) = 0;

∴ ζ(u²/2g) = (Ω/χ) i = mi,

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.

Fig. 121.

Consider the possible changes of value of the fraction

(1 − ζu² / 2gih) / (1 − u² / 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 − ζu² / 2giH = 0,
H = ζu²/2gi,

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 −∞, 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 = u²/g, or u = √ (gh), the denominator becomes zero. Hence, tabulating these results as before:—

For  h = 0, u²/g, > u²/g, ∞,

the denominator becomes −∞, 0, > 0, 1.

Fig. 122.

§ 118. Case 1.—Suppose h>u²/g, and also h>H, or the depth greater than that corresponding to uniform motion. In this case dh/ds is positive, and the stream increases in depth in the direction of flow. In fig. 122 let B₀B₁ be the bed, C₀C₁ a line parallel to the bed and at a height above it equal to H. By hypothesis, the surface A₀A₁ of the stream is above C₀C₁, and it has just been shown that the depth of the stream increases from B₀ towards B₁. But going up stream h approaches more and more nearly the value H, and therefore dh/ds approaches the limit 0, or the surface of the stream is asymptotic to C₀C₁. Going down stream h increases and u diminishes, the numerator and denominator of the fraction (1 − ζu²/2gih) / (1 − u²/gh) both tend towards the limit 1, and dh/ds to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line D₀D₁.

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 C₀C₁ is termed the backwater due to the weir.

Fig. 123.

§ 119. Case 2.—Suppose h > u²/g, and also h < H. Then dh/ds is negative, and the stream is diminishing in depth in the direction of flow. In fig. 123 let B₀B₁ be the stream bed as before; C₀C₁ a line drawn parallel to B₀B₁ at a height above it equal to H. By hypothesis the surface A₀A₁ of the stream is below C₀C₁, and the depth has just been shown to diminish from B₀ towards B₁. Going up stream h approaches the limit H, and dh/ds tends to the limit zero. That is, up stream A₀A₁ is asymptotic to C₀C₁. Going down stream h diminishes and u increases; the inequality h > u²/g diminishes; the denominator of the fraction (1 − ζu²/2gih) / (1 − u²/gh) tends to the limit zero, and consequently dh/ds tends to ∞. That is, down stream A₀A₁ 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. 124).

Fig. 124.

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 means 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 normal height, causing a backwater in the channel above” (Crofton’s Report on the Ganges Canal, p. 14). Fig. 125 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.

Fig. 125.

§ 120. Case 3.—Suppose a stream flowing uniformly with a depth h < u²/g. For a stream in uniform motion ζu²/2g = mi, or if the stream is of indefinitely great width, so that m = H, then ζu²/2g = iH, and H = ζu²/2gi. Consequently the condition stated above involves that

ζu² / 2gi < u² / g, or that i > ζ/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 h₁ > u²/g, then for a portion of the stream the depth h will satisfy the conditions h < u²/g and h > H, which are not the same as those assumed in the two previous cases. At some point of the stream above the weir the depth h becomes equal to u²/g, and at that point dh/ds 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 i > ζ/2. Now ζ depends for different channels on the roughness of the channel and its hydraulic mean depth. Bazin 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.

Nature of Bed of Stream.	Slope below  which a Standing  Wave is impossible in feet peer foot.	Standing Wave Formed.
		Slope in feet  per foot.	Least Depth  in feet.
Very smooth cemented surface	0.00147	0.002	0.262
		0.003	.098
		0.004	.065
Ashlar or brickwork	0.00186	0.003	.394
		0.004	.197
		0.006	.098
Rubble masonry	0.00235	0.004	1.181
		0.006	.525
		0.010	.262
Earth	0.00275	0.006	3.478
		0.010	1.542
		0.015	.919

Standing Waves

§ 121. The formation of a standing wave was first observed by Bidone. Into a small rectangular masonry channel, having a slope of 0.023 ft. per foot, he admitted water till it flowed uniformly with a depth of 0.2 ft. He then placed a plank across the stream which raised the level just above the obstruction to 0.95 ft. He found that the stream above the obstruction was sensibly unaffected up to a point 15 ft. from it. At that point the depth suddenly increased from 0.2 ft. to 0.56 ft. The velocity of the stream in the part unaffected by the obstruction was 5.54 ft. per second. Above the point where the abrupt change of depth occurred u² = 5.54² = 30.7, and gh = 32.2 × 0.2 = 6.44; hence u² was > gh. Just below the abrupt change of depth u = 5.54 × 0.2/0.56 = 1.97; u² = 3.88; and gh = 32.2 × 0.56 = 18.03; hence at this point u² < gh. Between these two points, therefore, u² = gh; and the condition for the production of a standing wave occurred.

Fig. 126.

The change of level at a standing wave may be found thus. Let fig. 126 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 abcd to come to a′b′c′d′ in a short time t; and let u₀, u₁ be the velocities at ab and cd, Ω₀, Ω₁ the areas of the cross sections. The force causing change of momentum in the mass abcd estimated horizontally is simply the difference of the pressures on ab and cd. Putting h₀, h₁ for the depths of the centres of gravity of ab and cd measured down from the free water surface, the force is G (h ₀Ω₀ − h₁Ω₁) pounds, and the impulse in t seconds is G (h ₀Ω₀ − h₁Ω₁) t second pounds. The horizontal change of momentum is the difference of the momenta of cdc′d′ and aba′b′; that is,

(G/g) (Ω₁u₁² − Ω₀u₀²) t.

Hence, equating impulse and change of momentum,

G (h₀Ω₀ − h₁Ω₁) t = (G/g) (Ω₁u₁² − Ω₀u₀²) t;

∴ h₀Ω₀ − h₁Ω₁ = (Ω₁u₁² − Ω₀u₀²) / g.

(1)

For simplicity let the section be rectangular, of breadth B and depths H₀ and H₁, at the two cross sections considered; then h₀ = 1/2H₀, and h₁ = 1/2H₁. Hence

H₀² − H₁² = (2/g) (H₁u₁² − H₀u₀²).

But, since Ω₀u₀ = Ω₁u₁, we have

u₁² = u₀²H₀² / H₁²,

H₀² − H₁² = (2u₀²/g) (H₀²/H₁ − H₀).

(2)

This equation is satisfied if H₀ = H₁, which corresponds to the case of uniform motion. Dividing by H₀ − H₁, the equation becomes

(H₁/H₀) (H₀ + H₁) = 2u₀² / g;

(3)

∴ H₁ = √ (2u₀²H₀ / g + 1/4 H₀²) − 1/2 H₀.

(4)

In Bidone’s experiment u₀ = 5.54, and H₀ = 0.2. Hence H₁ = 0.52, which agrees very well with the observed height.

Fig. 127.

§ 122. 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. 127. 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 < u²/g. In flowing along the rough apron of the weir the velocity u diminished and the depth h increased. At a point C, where h became equal to u²/g, 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.

Fig. 128.

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. Civ. Eng. vol. xxvii. p. 222, by A. C. Howden. Fig. 128 is compiled from the data given in that paper. It represents the section of the stream at pier 8 of the Towah Viaduct, during the flood of 1865. The ground level is not exactly given by 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 ft. per second. Now, taking the depth on the down stream face of the pier at 26 ft., the velocity necessary for the production of a standing wave would be u = √ (gh) = √ (32.2 × 26) = 29 ft. per second nearly. But the velocity at this point was probably from Howden’s statements 16.58 × 40/26 = 25.5 ft. per second, an agreement as close as the approximate character of the data would lead us to expect.

XI. ON STREAMS AND RIVERS

§ 123. Catchment Basin.—A stream or river is 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).

	Ratio of average Discharge to average Rainfall.	Loss by Evaporation, &c., in per cent of total Rainfall.
Cultivated land and spring-forming declivities.	3 to .33	67 to 70
Wooded hilly slopes.	.35 to .45	55 to 65
Naked unfissured mountains	.55 to .60	40 to 45

§ 124. Flood Discharge.—The 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 produce 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:—

	Flood discharge of Streams per Second per Square Mile of Catchment Basin.
In flat country	8.7 to 12.5	cub. ft.
In hilly districts	17.5 to 22.5	”
In moderately mountainous districts	36.2 to 45.0	”
In very mountainous districts	50.0 to 75.0	”

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 formulae. Thus Colonel P. P. L. O’Connell proposed the formula y = M √ x, where M is a constant called the modulus of the 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 = C³√ n², 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.

Fig. 129.

Fig. 130.

§ 125. Action of a Stream on its Bed.—If the velocity of a stream exceeds a certain limit, depending on its size, and on the size, heaviness, form and coherence of the material of 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. 129 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. 130, 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′c′ 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.

§ 126. Bottom Velocity at which Scour commences.—The following bottom velocities were determined by P. L. G. 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 v_b.

v_m = v_b + 10.87 √ (mi).

But

√ mi = v_m √ (ζ / 2g);

∴ v_m = v_b / (1 − 10.87 √ (ζ / 2g)).

Taking a mean value for ζ, we get

v_m = 1.312 v_b,

and from this the following values of the mean velocity are obtained:—

	Bottom Velocity  = vb.	Mean Velocity  = vm.
1. Soft earth	0.25	.33
2. Loam	0.50	.65
3. Sand	1.00	1.30
4. Gravel	2.00	2.62
5. Pebbles	3.40	4.46
6. Broken stone, flint	4.00	5.25
7. Chalk, soft shale	5.00	6.56
8. Rock in beds	6.00	7.87
9. Hard rock.	10.00	13.12

The following table of velocities which should not be exceeded in channels is given in the Ingenieurs Taschenbuch of the Verein “Hütte”:—

	Surface  Velocity.	Mean  Velocity.	Bottom  Velocity.
Slimy earth or brown clay	.49	.36	.26
Clay	.98	.75	.52
Firm sand	1.97	1.51	1.02
Pebbly bed	4.00	3.15	2.30
Boulder bed	5.00	4.03	3.08
Conglomerate of slaty fragments	7.28	6.10	4.90
Stratified rocks	8.00	7.45	6.00
Hard rocks	14.00	12.15	10.36

§ 127. Regime of a River Channel.—A 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.

Fig. 131.

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 (§ 107), for the slightest deflection 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.

§ 128. 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. 131 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 sinuousness 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.

 

Fig. 132.

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 cb², 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. 132) 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, vcb² = kl, or b = √ (kl/cv).

Hydraulic mean depth = ab = a √ (kl/cv).

But, by the ordinary formula for the flow of rivers, mi = ζv²;

∴ i = ζv² / m = (ζv^5/2 / a) √ (c / kl).

But i is the tangent of the angle which the curve at C makes with the axis of X, and is therefore = dy/dx. Also, as the slope is small, l = AC = AD = x nearly.

∴ dy/dx = (ζv^5/2 / a) √ (c / kx);

and, remembering that v is constant,

y = (2ζv^5/2 / a) √ (cx / k);

or

y² = 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 irregular manner in which their volume increases.

§ 129. 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 ft. 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.

§ 130. 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 ℔ per cubic foot, weighs in air wV ℔, but in water only (w−62.4)V ℔.

	Weight of a Cub. Ft. in ℔.
	In Air.	In Water.
Basalt	187.3	124.9
Brick	130.0	67.6
Brickwork	112.0	49.6
Granite and limestone	170.0	107.6
Sandstone	144.0	81.6
Masonry	116-144	53.6-81.6

§ 131. 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 deposited in 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. 133).

Fig. 133.

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.

§ 132. 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. 134), and one of these may in time become the main channel of the river.

§ 133. 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 referred to the line of stakes, both as to position and direction. The determination of the surface slope is very difficult, partly from its extreme smallness, partly from oscillation of the water. Cunningham recommends that the slope be taken in a length of 2000 ft. by four simultaneous observations, two on each side of the river.

Fig. 134.

§ 134. 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 in. 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 yds. A number of floats being then thrown in, it is observed whether they pass the same graduation on each wire.

Fig. 135.

For large and rapid rivers the cross section is obtained by sounding in the following way. Let AC (fig. 135) 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 AD, 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 ℔ of lead, and, if 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.

§ 135. 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 and 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

§ 136. Surface Floats are convenient for determining the surface velocities of a stream, though their use is difficult near the banks. The 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