(6)Hence the initial velocity in the pipe isu0= √ [{gcτ (p02− p12)} / {p02(ζ l/m + log (p0/ p1) }].(7)When l is great, log p0/p1is comparatively small, and thenu0= √ [ (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 approximatelyu0= (1.1319 − 0.7264 p1/p0) √ (gcτd / 4ζ l).(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, 21⁄4in. (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. Puttingζ = α/v + β,(8)he obtained the following values—Diameter of Pipein feet.αβζ for 100 ft.per second.1.64.00129.00483.004841.07.00972.00640.00650.83.01525.00704.00719.338.03604.00941.00977.266.03790.00959.00997.164.04518.01167.01212It 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 formulaζ = 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 valueζ = 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 thenz0− z1− p0/G0− p1/G1− ζ (v2/2g) (l/m) = 0,(10)precisely equivalent to the equation for the flow of water, z0and z1being the elevations of the two ends of the pipe above any datum, p0and p1the pressures, G0and G1the 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, isp = p0√ (1 − ζ lu02/ mgcτ);(11)which is of the formp = √ (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, u0be 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 √ (gd5(p02− p12) / ζ lcτ),= .611 √ (d5(p02− p12) / ζ lτ).(13)ApproximatelyW = (.6916p0− .4438p1) (d5/ 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 only3⁄4oz. 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 =∫l0dl/u,From (4a) neglecting dH/H, and putting m = d/4,dl = gdΩ2p dp / 2ζW2cr.From (1) and (3)u = Wcτ / pΩ;dl/u = gdΩ3p2dp / 2ζW3c2τ2;t =∫p0p1g dΩ3p2dp / 2ζW3c2τ2,= g dΩ3(p03− p13) / 6ζW3c2τ2.(14)ButW = p0u0Ω / cτ;∴ t = gdcτ (p03− p13) / 6ζp03u03,= ζ1/2l3/2(p03− p13) / 6(gcτd)1/2(p02− p12)3/2;(15)If τ = 521°, corresponding to 60° F.,t = .001412 ζ1/2l3/2(p03− p13) / d1/2(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:—AbsolutePressures in℔ per sq. in.Mean Velocities for Tubes of alength in feet.p0p110002000300040005000VacuumWorking15599.470.357.449.744.5151067.247.538.834.430.1PressureWorking201557.240.533.028.625.6251574.652.743.137.333.3301584.760.049.042.437.9Limiting Velocity in the Pipe when the Pressure at one End is diminished indefinitely.—If in the last equation there be put p1= 0, thenu′mean= 0.708 √ (d / ζ l);where the velocity is independent of the pressure p0at 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 normalto 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 formv = √ (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 f is the fall in feet per mile,f = 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 =1⁄41⁄2√ (2mf).(3)The flow down the stream per second, or discharge of the stream, isQ = 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 formζ = α (1 + β/v),(5)and from 255 experiments obtained for the constants the valuesα = 0.007409; β = 0.1920.This gives the following values at different velocities:—v =0.30.50.7111⁄223571015ζ =0.012150.010250.009440.008830.008360.008120.907880.007690.007610.007550.00750In 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 formζ = 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.αβI. Very smooth channels, sides of smooth cement or planed timber.002940.10II. Smooth channels, sides of ashlar, brickwork, planks.003730.23III. Rough channels, sides of rubble masonry or pitched with stone.004710.82IV. Very rough canals in earth.005494.10V. Torrential streams encumbered with detritus.007855.74The 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 ofcinv = c √ (mi),deduced from Darcy and Bazin’s Values.HydraulicMean.Depth = m.Very SmoothChannels.Cement.SmoothChannels.Ashlar orBrickwork.RoughChannels.RubbleMasonry.Very RoughChannels.Canals inEarth.ExcessivelyRough Channelsencumberedwith Detritus..2512595572618.5.5135110723625.6.75139116814230.81.0141119874834.91.5143122945641.22.0144124986246.02.514512610167..3.014512610470533.514612710573..4.014612810676584.514612810778..5.014612810880625.514612910982..6.014712911084656.514712911085..7.014712911086677.514712911187..8.014713011188698.514713011289..9.014713011290719.514713011290..10.014713011291721114713011392..1214713011393741314713011394..1414713011395..1514713011496771614713011497..1714713011497..1814713011498..20147131114988025148131115100..301481311151028340148131116103855014813111610486∞14813111710891§ 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 someSwiss 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 foundc = 256 for an inclination of0.0034 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 formulav = √ (2g / ζ) √ (mi) = c √ (mi),Ganguillet and Kutter proceeded in a purely empirical way. They found that an expression of the formc = α / (1 + β/√ m)could be made to fit the experiments somewhat better than Darcy’s expression. Inverting this, we get1/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.LetA = a + p/i.Thenc = (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 ofRoughness n.Well-planed timber0.009Cement plaster0.010Plaster of cement with one-third sand0.011Unplaned planks0.012Ashlar and brickwork0.013Canvas on frames0.015Rubble masonry0.017Canals in very firm gravel0.020Rivers and canals in perfect order, free from stones or weeds0.025Rivers and canals in moderately good order, not quite freefrom stones and weeds0.030Rivers and canals in bad order, with weeds and detritus0.035Torrential streams encumbered with detritus0.050Ganguillet 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:—iValues of M forn=0.0100.0120.0150.0170.0200.0250.030.000013.22603.87124.83905.48426.45208.06509.6780.000021.82102.18522.73153.09573.64204.55255.4630.000041.11851.34221.67771.90142.23702.79623.3555.000060.88431.06121.32641.50331.76862.21072.6529.000080.76720.92061.15081.30421.53441.91802.3016.000100.69700.83641.04551.18491.39401.74252.0910.000250.52840.63410.79260.89831.05681.32101.5852.000500.47220.56660.70830.80270.94441.18051.4166.000750.45350.54420.68020.77090.90701.13371.3605.001000.44410.53290.66610.75500.88821.11021.3323.002000.43000.51600.64500.73100.86001.07501.2900.003000.42540.51050.63810.72320.85081.06351.2762A 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 ordern = 0.02II.Canals in earth, above the average in ordern = 0.0225III.Canals in earth, in fair ordern = 0.025IV.Canals in earth, below the average in ordern = 0.0275V.Canals in earth, in rather bad order, partiallyovergrown with weeds and obstructed by detritus.n = 0.03Ganguillet 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/v2tends 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/v2tends 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 =0.109II.Smooth sides, planks, brickwork0.290III.Rubble masonry sides0.833IV.Sides of very smooth earth, or pitching1.539V.Canals in earth in ordinary condition2.353VI.Canals in earth exceptionally rough3.168§ 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; v0is the surface; vzthe maximum and vdthe 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 isvm=1⁄3[2vz+ vd+ (dz/d) (v0− vd)].Fig. 104.The Horizontal Velocity Curve.—Similarly if at each point along a horizontal representing the width of the stream the velocities areplotted, 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 dzat 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 at3⁄10ths 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 at3⁄10ths 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 expressionh′ / 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 v0is the surface and vmthe mean velocity at a vertical vm=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 vmdis the mid-depth velocity, then on the average vm= 0.98vmd.§ 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 v0be the surface, vmthe mean, and vxdthe velocity at the depth xd; thenvm=1⁄4(v0+ 3v2/3d)=1⁄2(v.211d+ v.789d).§ 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 v0be the greatest surface velocity, vmthe mean velocity of the stream. Then, according to Bazin,vm= v0− 25.4 √ (mi).Butvm= c √ (mi),where c is a coefficient, the values of which have been already given in the table in § 98. Hencevm= cv0/ (c + 25.4).Values of Coefficientc/(c + 25.4)in the Formulavm= cv0/(c + 25.4).HydraulicMean Depth= m.VerySmoothChannels.Cement.SmoothChannels.Ashlar orBrickwork.RoughChannels.RubbleMasonry.Very RoughChannels.Canals inEarth.ChannelsencumberedwithDetritus.0.25.83.79.69.51.420.5.84.81.74.58.500.75.84.82.76.63.551.0.85...77.65.582.0...83.79.71.643.0.....80.73.674.0.....81.75.705.0.......76.716.0...84...77.727.0.......78.738.0..........9.0.....82...7410.0..........15.0.......79.7520.0.......80.7630.0.....82...7740.0..........50.0..........∞.........79Fig. 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 + 8.2H2in metric measures, or Q = 696H + 26.8H2in 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, b0the 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 νr2, where κ, μ, and ν have the following values:—Depth of water interms of radiusκ =.01.05.10.15.20.25.30.35.40.45.50.55.60.65.70.75.80.85.90.951.0Hydraulic mean depthin terms of radiusμ =.00668.0321.0523.0963.1278.1574.1852.2142.242.269.293.320.343.365.387.408.429.449.466.484.500Waterway in terms ofsquare of radiusν =.00189.0211.0598.1067.1651.228.294.370.450.532.614.709.795.885.9791.0751.1751.2761.3711.4701.571Fig. 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 areζ = α(1 + β/m);
(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/p1is 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 / 4ζ l).
(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, 21⁄4in. (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—
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, z0and z1being the elevations of the two ends of the pipe above any datum, p0and p1the pressures, G0and G1the 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 − ζ lu02/ 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.
Variation of Velocity in the Pipe.—Let p0, u0be 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;
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.
§ 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 √ (gd5(p02− p12) / ζ lcτ),= .611 √ (d5(p02− p12) / ζ lτ).
W =1⁄4π √ (gd5(p02− p12) / ζ lcτ),
= .611 √ (d5(p02− p12) / ζ lτ).
(13)
Approximately
W = (.6916p0− .4438p1) (d5/ 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 only3⁄4oz. 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 =∫l0dl/u,
From (4a) neglecting dH/H, and putting m = d/4,
dl = gdΩ2p dp / 2ζW2cr.
From (1) and (3)
u = Wcτ / pΩ;dl/u = gdΩ3p2dp / 2ζW3c2τ2;
t =∫p0p1g dΩ3p2dp / 2ζW3c2τ2,= g dΩ3(p03− p13) / 6ζW3c2τ2.
(14)
But
W = p0u0Ω / cτ;
∴ t = gdcτ (p03− p13) / 6ζp03u03,= ζ1/2l3/2(p03− p13) / 6(gcτd)1/2(p02− p12)3/2;
(15)
If τ = 521°, corresponding to 60° F.,
t = .001412 ζ1/2l3/2(p03− p13) / d1/2(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:—
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 / ζ l);
where the velocity is independent of the pressure p0at 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.
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 normalto 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.
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 f is the fall in feet per mile,
f = 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 =1⁄41⁄2√ (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:—
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:—
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 ofcinv = c √ (mi),deduced from Darcy and Bazin’s Values.
§ 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 someSwiss 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
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:—
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:—
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:—
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/v2tends 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/v2tends 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:—
§ 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; v0is the surface; vzthe maximum and vdthe 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
vm=1⁄3[2vz+ vd+ (dz/d) (v0− vd)].
The Horizontal Velocity Curve.—Similarly if at each point along a horizontal representing the width of the stream the velocities areplotted, 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.
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 dzat 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 at3⁄10ths 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 at3⁄10ths 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.
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 v0is the surface and vmthe mean velocity at a vertical vm=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 vmdis the mid-depth velocity, then on the average vm= 0.98vmd.
§ 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 v0be the surface, vmthe mean, and vxdthe velocity at the depth xd; then
vm=1⁄4(v0+ 3v2/3d)=1⁄2(v.211d+ v.789d).
§ 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 v0be the greatest surface velocity, vmthe mean velocity of the stream. Then, according to Bazin,
vm= v0− 25.4 √ (mi).
But
vm= c √ (mi),
where c is a coefficient, the values of which have been already given in the table in § 98. Hence
vm= cv0/ (c + 25.4).
Values of Coefficientc/(c + 25.4)in the Formulavm= cv0/(c + 25.4).
§ 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.
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 + 8.2H2in metric measures, or Q = 696H + 26.8H2in 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.
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, b0the 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).
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 νr2, where κ, μ, and ν have the following values:—
§ 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);