§ 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 termedstream 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 isQ = 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 caseQ = 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 isQ = ∫ 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, A2are the areas of two normal cross sections of a stream, and V1, V2are 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, A2and between A1, A2by 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, p2be the pressures, v1, v2the velocities, G1, G2the weight per cubic foot of fluid, at cross sections of a stream of areas A1, A2. The volumes of inflow and outflow areA1v1and 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 EXPERIMENTSFig. 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 hr(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 equationv = √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√2ghnearly.(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 termheadis 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 √2ghby a small quantity, let the actual velocity= va= cv√2gh,(3)where cvis a coefficient to be determined by experiment, called thecoefficient 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 heexpended effectively in producing the velocity of outflow, another hrin overcoming the resistances due to viscosity and friction. Lethr= crhe,where cris a coefficient determined by experiment, and called thecoefficient of resistanceof the orifice. It is tolerably constant for different heads with well-formed orifices. Thenva= √2ghe= √ { 2gh / (1 + cr) }.(4)The relation between cvand crfor any orifice is easily found:—va= cv√2gh= √ { 2gh / (1 + cr) }cv= √ { 1 / (1 + cr) }(5)cr= 1 / cv2− 1.(5a)Thus if cv= 0.97, then cr= 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 ccω the area of the jet at the point where convergence ceases; then ccis a coefficient to be determined experimentally for each kind of orifice, called thecoefficient of contraction. When the orifice is a sharp-edged orifice in a plane surface, the value of ccis 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 isQa= cvv × ccω = cccvω √(2gh),or simply, if c = cvcc,Qa= cω √(2gh),where c is called thecoefficient of discharge. Thus for a sharp-edged plane orifice c = 0.97 × 0.64 = 0.62.§ 18.Experimental Determination ofcv, cc,andc.—The coefficient of contraction ccis 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 vais the velocity at the orifice, and t the time in which a particle moves from O to A, thenx = vat; y =1⁄2gt2.Eliminating t,va= √ (gx2/2y).Thencv= va√ (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, thenc = 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, h2be the heads, ω1, ω2the areas of the orifices, c1, c2the coefficients. Then since the flow through each orifice is the sameQ = 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, ccmust be put = 1. It is often desirable to bellmouth the ends of pipes, to avoid the loss of head which occurs if this isnot 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.661.6411.4855.77337.93Coefficient of velocity = cv.959.967.975.994.994Coefficient of resistance = cr.087.069.052.012.012As there is no contraction after the jet issues from the orifice, cc= 1, c = cv; and thereforeQ = cvω √ (2gh) = ω √ { 2gh / (1 + cr}.§ 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 theHydraulicsof 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).Headmeasured toCentre ofOrifice.Diameters of Orifice..02.04.10.20.40.601.0Values of C.0.3.....621........0.4...637.618........0.6.655.630.613.601.596.588..0.8.648.626.610.601.597.594.5831.0.644.623.608.600.598.595.5912.0.632.614.604.599.599.597.5954.0.623.609.602.599.598.597.5968.0.614.605.600.598.597.596.59620.0.601.599.596.596.596.596.594At 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 inft.Form of Orifice.Circular.Square.Rectangular Ratioof Sides 4:1Rectangular Ratioof Sides 16:1Tri-angular.SidesVertical.DiagonalVertical.LongSidesVertical.LongSideshori-zontal.LongSidesVertical.LongSidesHori-zontal.1.620.627.628.642.643.663.664.6362.613.620.628.634.636.650.651.6284.608.616.618.628.629.641.642.6236.607.614.616.626.627.637.637.6208.606.613.614.623.625.634.635.61910.605.612.613.622.624.632.633.61812.604.611.612.622.623.631.631.61814.604.610.612.621.622.630.630.61816.603.610.611.620.622.630.630.61718.603.610.611.620.621.630.629.61620.603.609.611.620.621.629.628.616The 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 formulac = 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 infeeth.Diameters of Orifices in Inches (d).111⁄411⁄213⁄4221⁄421⁄223⁄43Coefficients (c)..75.616.614.616.610.616.612.607.607.6091.0.613.612.612.611.612.611.604.608.6091.25.613.614.610.608.612.608.605.605.6061.50.610.612.611.606.610.607.603.607.6051.75.612.611.611.605.611.605.604.607.6052.00.609.613.609.606.609.606.604.604.605The 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 toCentre ofOrifice.Ratio of Height to Width.4211⁄213⁄41⁄21⁄41⁄8Feet.4 ft. high.1 ft. wide.2 ft. high.1 ft. wide.11⁄2ft. 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.....6134.6006.6086.6168.6323.63241.0.....6135.6010.6090.6172.6320.63201.25...6188.6140.6018.6095.6173.6317.63121.50...6187.6144.6026.6100.6172.6313.63031.75...6186.6145.6033.6103.6168.6307.62962...6183.6144.6036.6104.6166.6302.62912.25...6180.6143.6029.6103.6163.6293.62862.50.6290.6176.6139.6043.6102.6157.6282.62782.75.6280.6173.6136.6046.6101.6155.6274.62733.6273.6170.6132.6048.6100.6153.6267.62673.5.6250.6160.6123.6050.6094.6146.6254.62544.6245.6150.6110.6047.6085.6136.6236.62364.5.6226.6138.6100.6044.6074.6125.6222.62225.6208.6124.6088.6038.6063.6114.6202.62026.6158.6094.6063.6020.6044.6087.6154.61547.6124.6064.6038.6011.6032.6058.6110.61148.6090.6036.6022.6010.6022.6033.6073.60879.6060.6020.6014.6010.6015.6020.6045.607010.6035.6015.6010.6010.6010.6010.6030.606015.6040.6018.6010.6011.6012.6013.6033.606620.6045.6024.6012.6012.6014.6018.6036.607425.6048.6028.6014.6012.6016.6022.6040.608330.6054.6034.6017.6013.6018.6027.6044.609235.6060.6039.6021.6014.6022.6032.6049.610340.6066.6045.6025.6015.6026.6037.6055.611445.6054.6052.6029.6016.6030.6043.6062.612550.6086.6060.6034.6018.6035.6050.6070.6140§ 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 haboveupperedge ofOrificein feet.Height of Orifice, H − h, in feet.1.310.660.160.10PQRPQRPQRPQR0.3280.5980.6440.6480.6340.6650.6680.6910.6640.6660.7100.6940.696.6560.6090.6530.6570.6400.6720.6750.6850.6870.6880.6960.7040.706.7870.6120.6550.6590.6410.6740.6770.6840.6900.6920.6940.7060.708.9840.6160.6560.6600.6410.6750.6780.6830.6930.6950.6920.7090.7111.9680.6180.6490.6530.6400.6760.6790.6780.6950.6970.6880.7100.7123.280.6080.6320.6340.6380.6740.6760.6730.6940.6950.6800.7040.7054.270.6020.6240.6260.6370.6730.6750.6720.6930.6940.6780.7010.7024.920.5980.6200.6220.6370.6730.6740.6720.6920.6930.6760.6990.6995.580.5960.6180.6200.6370.6720.6730.6720.6920.6930.6760.6980.6986.560.5950.6150.6170.6360.6710.6720.6710.6910.6920.6750.6960.6969.840.5920.6110.6120.6340.6690.6700.6680.6890.6900.6720.6930.693For 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 over1⁄4,1⁄2, or3⁄4of the whole perimeter:—n/pCcRectangular OrificesCcCircular Orifices0.250.643.6400.500.667.6600.750.691.680Fig. 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− h1in feeth1in feet..0656.164.328.6561.6403.284.926.569.84A0.656.480.511.542.574.599.601.601.601.601B.480.510.538.506.592.600.602.602.601C.527.553.574.592.607.610.610.609.608A0.164.488.577.624.631.625.624.619.613.606B.487.571.606.617.626.628.627.623.618C.585.614.633.645.652.651.650.650.649Fig. 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, h2from the free surface. The filaments issuing at a1, a2will have the different velocities √ 2gh1and √ 2gh2. Consequently they will tend to describe parabolic paths a1cb1and a2cb2of 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 VV. 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 11⁄2ft. the coefficients were:—Temperature F.C.205°.59462°.598For a conoidal or bell-mouthed orifice 1 cm. diameter the effect of temperature was greater:—Temperature F.C.190°0.987130°0.97460°0.942an 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 21⁄2in. 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 0.98l 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 for3⁄4-in. nozzle; 0.982 for7⁄8in.; 0.972 for 1 in.; 0.976 for 11⁄8in.; and 0.971 for 11⁄4in. 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, z1the 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.
§ 11.Steady and Unsteady, Uniform and Varying, Motion.—There are two quite distinct ways of treating hydrodynamical questions. We may either fix attention on a given mass of fluid and consider its changes of position and energy under the action of the stresses to which it is subjected, or we may have regard to a given fixed portion of space, and consider the volume and energy of the fluid entering and leaving that space.
If, in following a given path ab (fig. 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.
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 termedstream lines) will have fixed positions in space.
Periodic Unsteady Motion.—In ordinary streams with rough boundaries, it is observed that at any given point the velocity varies from moment to moment in magnitude and direction, but that the average velocity for a sensible period (say for 5 or 10 minutes) varies very little either in magnitude or velocity. It has hence been conceived that the variations of direction and magnitude of the velocity are periodic, and that, if for each point of the stream the mean velocity and direction of motion were substituted for the actual more or less varying motions, the motion of the stream might be treated as steady stream line or steady laminar motion.
§ 13.Volume of Flow.—Let A (fig. 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.
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, A2are the areas of two normal cross sections of a stream, and V1, V2are 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, A2and between A1, A2by 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.
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, p2be the pressures, v1, v2the velocities, G1, G2the weight per cubic foot of fluid, at cross sections of a stream of areas A1, A2. The volumes of inflow and outflow are
A1v1and 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)
§ 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
§ 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 hr(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√2ghnearly.
(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 termheadis 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 √2ghby a small quantity, let the actual velocity
= va= cv√2gh,
(3)
where cvis a coefficient to be determined by experiment, called thecoefficient 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 heexpended effectively in producing the velocity of outflow, another hrin overcoming the resistances due to viscosity and friction. Let
hr= crhe,
where cris a coefficient determined by experiment, and called thecoefficient of resistanceof the orifice. It is tolerably constant for different heads with well-formed orifices. Then
va= √2ghe= √ { 2gh / (1 + cr) }.
(4)
The relation between cvand crfor any orifice is easily found:—
va= cv√2gh= √ { 2gh / (1 + cr) }
cv= √ { 1 / (1 + cr) }
(5)
cr= 1 / cv2− 1.
(5a)
Thus if cv= 0.97, then cr= 0.0628. That is, for such an orifice about 61⁄4% of the head is expended in overcoming frictional resistances to flow.
Coefficient of Contraction—Sharp-edged Orifices in Plane Surfaces.—When a jet issues from an aperture in a vessel, it may either spring clear from the inner edge of the orifice as at a or b (fig. 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 ccω the area of the jet at the point where convergence ceases; then ccis a coefficient to be determined experimentally for each kind of orifice, called thecoefficient of contraction. When the orifice is a sharp-edged orifice in a plane surface, the value of ccis on the average 0.64, or the section of the jet is very nearly five-eighths of the area of the orifice.
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
Qa= cvv × ccω = cccvω √(2gh),
or simply, if c = cvcc,
Qa= cω √(2gh),
where c is called thecoefficient of discharge. Thus for a sharp-edged plane orifice c = 0.97 × 0.64 = 0.62.
§ 18.Experimental Determination ofcv, cc,andc.—The coefficient of contraction ccis 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 vais the velocity at the orifice, and t the time in which a particle moves from O to A, then
x = vat; y =1⁄2gt2.
Eliminating t,
va= √ (gx2/2y).
Then
cv= va√ (2gh) = √ (x2/4yh).
In the case of large orifices such as weirs, the velocity can be directly determined by using a Pitot tube (§ 144).
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.
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, h2be the heads, ω1, ω2the areas of the orifices, c1, c2the coefficients. Then since the flow through each orifice is the same
Q = c1ω1√ (2gh1) = c2ω2√ (2gh2).
c2= c1(ω1/ω2) √ (h1/h2).
§ 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, ccmust be put = 1. It is often desirable to bellmouth the ends of pipes, to avoid the loss of head which occurs if this isnot 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.
As there is no contraction after the jet issues from the orifice, cc= 1, c = cv; and therefore
Q = cvω √ (2gh) = ω √ { 2gh / (1 + cr}.
§ 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 theHydraulicsof 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).
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.
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°.
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.
§ 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.
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 over1⁄4,1⁄2, or3⁄4of the whole perimeter:—
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.
§ 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).
§ 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.
Let a1, a2(fig. 24) be two points in an orifice at depths h1, h2from the free surface. The filaments issuing at a1, a2will have the different velocities √ 2gh1and √ 2gh2. Consequently they will tend to describe parabolic paths a1cb1and a2cb2of 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 VV. 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 11⁄2ft. the coefficients were:—
For a conoidal or bell-mouthed orifice 1 cm. diameter the effect of temperature was greater:—
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 21⁄2in. 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 0.98l 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 for3⁄4-in. nozzle; 0.982 for7⁄8in.; 0.972 for 1 in.; 0.976 for 11⁄8in.; and 0.971 for 11⁄4in. 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.
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, z1the 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.