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42
[STEADY MOTION OF FLUIDS
HYDRAULICS


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.

Let al, ag (fig. 24) be two points in an orifice at depths hi, h2 from the free surface. The filaments issuing at ai. az will have the different velocities x/ 2 gh; and / 2gh2.

L - Consequently they will

tend to describe parabolic

paths a, cb1 and azcbz of

  • T

ii:-'l - different horizontal range,

"' /H and intersecting in the

/'s I point 6. But since two

I L filaments cannot simul:”

F Q, taneously flow through the

g Q; 2 same point, they must

exercise mutual pressure,

6, and will be deflected out of

the paths they tend to

1-. 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 Oriyices.-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 I to 1% ft. the coefficients were:-

Fro. 24.

Temperature F ....... C.

205° ..... -594

62° ...... -598

For a conoidal or bell-mouthed orifice I cm. diameter the effect of C€ITlp€I'2'tt.ll'C WPIS gl'€3.t€l' i""

Temperature F ....... C.

190; . . 0-987

130° ....... 0-974

60

0' 2

an increase in velocity of discharge of 4% when th24temperature increased 130°.

]. G. iIair repeated these experiments on a much larger scale (Proc. Inst. Civ. Eng. lxxxiv.). For a sharp-edged orifice 2% in. diameter, with a head of I-75 ft., the coefficient was 0-604 at 57° and 0~6o7 at 179° F., a very small difference. With a conoidal orifice the coefficient was 0-961 at 55° and o~981 at 170° F. The corresponding coefficients of resistance are O'O828 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 O'983 for %-in. nozzle; o-982 for Q5 in.; o~972 for 1 in.; 0-976 for 1% in.; and o~971 for 1% 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) lf the motion is rectilinear and uniform, the variation of Dressure 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 simfply the hydrostatic pressure due to the depth below the free sur ace.

(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 or ENERGY IN INco1v11>nEss1B1.a Ftums. § 29. Application of the Principle of the Conservation of Energy to Cases of Stream Line M ation.—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 Bernoullis theorem.

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 0 0

A' B B

iff-is, ~ L /-V+

-3{, ..;-..-.- . . ....:. . .; -.2 -.. FIG. 25.

w be the area of a normal cross section, 'v the velocity, p the intensity of pressure, and 2 the elevation above XX, of the elementary stream AB at A, and wi, pi, 111, zi 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, "vit, and the volumes of fluid AA', BB' are the equal inflow and outfiow=Qt=wvt=w1z:1t, 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-21). T he work of the pressures on the ends, reckoning that.at B negative, because it is opposite to the direction of motion, is (pw><'vt) - (p1wiXvit) == Qz(p-pl). 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) (1112/2 -112/2). Equating this to the work done on the mass AB,

GQt(z-zf) -I-Qt(p-pi) = (GQt/g) (viz/2 -112/2). Dividing by GQt and rearranging the terms, . 11"/2g-l~1>/G-|12='vi2/2g+1>i/G4-21; (I) or, as A and B are any two points,

21”/2g+p/G-1-z=constant=H. ' (2)

Now v'/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 lb of fluid falling to the datum line, and similarly p/G and vi'/2g are the quantities of work which would be done by I lb of fluid due to the pressure p and velocity 21. The expression on the left of the equation is, therefore, the total energy of the stream at the

section considered, per lb of fluid, estimated with reference to the