(9)If H1varies from 0 to ∞, σ( = H1/H2) varies from 0 to 1. The following table gives values of the two estimates of the discharge for different values of σ:—H1/H2= σ.Q2/Q1.H1/H2= σ.Q2/Q1.0.0.9430.8.9990.2.9790.9.9990.5.9951.01.0000.7.998Hence it is obvious that, except for very small values of σ, the simpler equation (5) gives values sensibly identical with those of (8). When σ < 0.5 it is better to use equation (8) with values of c determined experimentally for the particular proportions of orifice which are in question.Fig. 44.§ 40.Large Jets having a Circular Section from Orifices in a Vertical Plane Surface.—Let fig. 44 represent the section of the jet, OO being the free surface level in the reservoir. The discharge through the horizontal strip aabb, of breadth aa = b, between the depths h1+ y and h1+ y + dy, isdQ = b √ {2g (h1+ y) } dy.The whole discharge of the jet isQ =∫d0b √ { 2g (h1+ y) } dy.But b = d sin φ; y =1⁄2d (1 − cos φ); dy =1⁄2d sin φ dφ. Let ε = d/(2h1+ d), thenQ =1⁄2d2√ { 2g (h1+ d/2) }∫π0sin2φ √1 − ε cos φdφ.From eq. (5), putting ω = πd2/4, h = h1+ d/2, c = 1 when d is the diameter of the jet and not that of the orifice,Q1=1⁄4πd2√ {2g (h1+ d/2) },Q/Q1= 2/π∫π0sin2φ √ {1 − ε cos φ} dφ.Forh1= ∞, ε = 0 and Q/Q1= 1;and forh1= 0, ε = 1 and Q/Q1= 0.96.So that in this case also the difference between the simple formula (5) and the formula above, in which the variation of head at different parts of the orifice is taken into account, is very small.Notches and Weirs§ 41.Notches, Weirs and Byewashes.—A notch is an orifice extending up to the free surface level in the reservoir from which the discharge takes place. A weir is a structure over which the water flows, the discharge being in the same conditions as for a notch. The formula of discharge for an orifice of this kind is ordinarily deduced by putting H1= 0 in the formula for the corresponding orifice, obtained as in the preceding section. Thus for a rectangular notch, put H1= 0 in (8). ThenQ =2⁄3cB √(2g) H3/2,(11)where H is put for the depth to the crest of the weir or the bottom of the notch. Fig. 45 shows the mode in which the discharge occurs in the case of a rectangular notch or weir with a level crest. As, the free surface level falls very sensibly near the notch, the head H should be measured at some distance back from the notch, at a point where the velocity of the water is very small.Since the area of the notch opening is BH, the above formula is of the formQ = c × BH × k √(2gH),where k is a factor depending on the form of the notch and expressing the ratio of the mean velocity of discharge to the velocity due to the depth H.Fig. 45.§ 42.Francis’s Formula for Rectangular Notches.—The jet discharged through a rectangular notch has a section smaller than BH, (a) because of the fall of the water surface from the point where His measured towards the weir, (b) in consequence of the crest contraction, (c) in consequence of the end contractions. It may be pointed out that while the diminution of the section of the jet due to the surface fall and to the crest contraction is proportional to the length of the weir, the end contractions have nearly the same effect whether the weir is wide or narrow.J. B. Francis’s experiments showed that a perfect end contraction, when the heads varied from 3 to 24 in., and the length of the weir was not less than three times the head, diminished the effective length of the weir by an amount approximately equal to one-tenth of the head. Hence, if l is the length of the notch or weir, and H the head measured behind the weir where the water is nearly still, then the width of the jet passing through the notch would be l − 0.2H, allowing for two end contractions. In a weir divided by posts there may be more than two end contractions. Hence, generally, the width of the jet is l − 0.1nH, where n is the number of end contractions of the stream. The contractions due to the fall of surface and to the crest contraction are proportional to the width of the jet. Hence, if cH is the thickness of the stream over the weir, measured at the contracted section, the section of the jet will be c(l − 0.1nH)H and (§ 41) the mean velocity will be2⁄3√(2gH). Consequently the discharge will be given by an equation of the formQ =2⁄3c (l − 0.1nH) H √2gH= 5.35c (l − 0.1nH) H3/2.This is Francis’s formula, in which the coefficient of discharge c is much more nearly constant for different values of l and h than in the ordinary formula. Francis found for c the mean value 0.622, the weir being sharp-edged.§ 43.Triangular Notch(fig. 46).—Consider a lamina issuing between the depths h and h + dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is √(2gh). Hence the discharge for this lamina isb√2ghdh.ButB/b = H / (H − h); b = B (H − h) / H.Hence discharge of lamina= B(H − h) √(2gh) dh/H;and total discharge of notch= Q = B √(2g)∫H0(H − h) h1/2dh/H=4⁄15B √(2g) H3/2.or, introducing a coefficient to allow for contraction,Q =4⁄15cB √(2g) H1/2,Fig. 46.When a notch is used to gauge a stream of varying flow, the ratio B/H varies if the notch is rectangular, but is constant if the notch is triangular. This led Professor James Thomson to suspect that the coefficient of discharge, c, would be much more constant with different values of H in a triangular than in a rectangular notch, and this has been experimentally shown to be the case. Hence a triangular notch is more suitable for accurate gaugings than a rectangular notch. For a sharp-edged triangular notch Professor J. Thomson found c = 0.617. It will be seen, as in § 41, that since1⁄2BH is the area of section of the stream through the notch, the formula is again of the formQ = c ×1⁄2BH × k √(2gH),where k =8⁄15is the ratio of the mean velocity in the notch to the velocity at the depth H. It may easily be shown that for all notches the discharge can be expressed in this form.Coefficients for the Discharge over Weirs, derived from the Experiments of T. E. Blackwell. When more than one experiment was made with the same head, and the results were pretty uniform, the resulting coefficients are marked with an(*).The effect of the converging wing-boards is very strongly marked.Heads ininchesmeasuredfrom stillWater inReservoir.Sharp Edge.Planks 2 in. thick,square on Crest.Crests 3 ft. wide.3 ft. long.10 ft. long.3 ft. long.6 ft. long.10 ft. long.10 ft. long,wing-boardsmaking anangle of 60°.3 ft. long.level.3 ft. long,fall 1 in 18.3 ft. long,fall 1 in 12.6 ft. long.level.10 ft. long.level.10 ft. long,fall 1 in 18.1.677.809.467.459.4354.754.452.545.467...381.4672.675.803.509*.561.585*.675.482.546.533...479*.495*3.630.642*.563*.597*.569*...441.537.539.492*....4.617.656.549.575.602*.656.419.431.455.497*...5155.602.650*.588.601*.609*.671.479.516.....518..6.593...593*.608*.576*...501*...531.507.513.5437.....617*.608*.576*...488.513.527.497....8...581.606*.590*.548*...470.491.....468.5079...530.600.569*.558*...476.492*.498.480*.486..10.....614*.539.534*.........465*.455..12.......525.534*.........467*....14.......549*................Fig. 47.§ 44.Weir with a Broad Sloping Crest.—Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a movement sensibly rectilinear and uniform (fig. 47). Let the inner edge be so rounded as to prevent a crest contraction. Consider a filament aa′, the point a being so far back from the weir that the velocity of approach is negligible. Let OO be the surface level in the reservoir, and let a be at a height h″ below OO, and h′ above a′. Let h be the distance from OO to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at a is Gh″; at a′ it is Gz. If v be the velocity at a′,v2/2g = h′ + h″ − z = h − e;Q = be √2g (h − e).Theory does not furnish a value for e, but Q = 0 for e = 0 and for e = h. Q has therefore a maximum for a value of e between 0 and h, obtained by equating dQ/de to zero. This gives e =2⁄3h, and, inserting this value,Q = 0.385 bh √2gh,as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately applicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonrysluice openings than the ordinary weir formula for sharp-edged weirs. It should be remembered, however, that the friction on the sides and crest of the weir has been neglected, and that this tends to reduce a little the discharge. The formula is equivalent to the ordinary weir formula with c = 0.577.Special Cases of Discharge from Orifices§ 45.Cases in which the Velocity of Approach needs to be taken into Account.Rectangular Orifices and Notches.—In finding the velocity at the orifice in the preceding investigations, it has been assumed that the head h has been measured from the free surface of still water above the orifice. In many cases which occur in practice the channel of approach to an orifice or notch is not so large, relatively to the stream through the orifice or notch, that the velocity in it can be disregarded.Fig. 48.Let h1, h2(fig. 48) be the heads measured from the free surface to the top and bottom edges of a rectangular orifice, at a point in the channel of approach where the velocity is u. It is obvious that a fall of the free surface,ɧ = u2/2ghas been somewhere expended in producing the velocity u, and hence the true heads measured in still water would have been h1+ ɧ and h2+ ɧ. Consequently the discharge, allowing for the velocity of approach, isQ =2⁄3cb √2g{ (h2+ ɧ)3/2− (h1+ ɧ)3/2}.(1)And for a rectangular notch for which h1= 0, the discharge isQ =2⁄3cb √2g{ (h2+ ɧ)3/2− ɧ3/2}.(2)In cases where u can be directly determined, these formulae give the discharge quite simply. When, however, u is only known as a function of the section of the stream in the channel of approach, they become complicated. Let Ω be the sectional area of the channel where h1and h2are measured. Then u = Q/Ω and ɧ = Q2/2g Ω2.This value introduced in the equations above would render them excessively cumbrous. In cases therefore where Ω only is known, it is best to proceed by approximation. Calculate an approximate value Q′ of Q by the equationQ′ =2⁄3cb √2g{h23/2− h13/2}.Then ɧ = Q′2/2gΩ2nearly. This value of ɧ introduced in the equations above will give a second and much more approximate value of Q.Fig. 49.§ 46.Partially Submerged Rectangular Orifices and Notches.—When the tail water is above the lower but below the upper edge of the orifice, the flow in the two parts of the orifice, into which it is divided by the surface of the tail water, takes place under different conditions. A filament M1m1(fig. 49) in the upper part of the orifice issues with a head h′ which may have any value between h1and h. But a filament M2m2issuing in the lower part of the orifice has a velocity due to h″ − h″′, or h, simply. In the upper part of the orifice the head is variable, in the lower constant. If Q1, Q2are the discharges from the upper and lower parts of the orifice, b the width of the orifice, thenQ1=2⁄3cb √2g{ h3/2− h13/2}Q2= cb (h2− h) √2gh.(3)In the case of a rectangular notch or weir, h1= 0. Inserting this value, and adding the two portions of the discharge together, we get for a drowned weirQ = cb √2gh(h2− h/3),(4)where h is the difference of level of the head and tail water, and h2is the head from the free surface above the weir to the weir crest (fig. 50).From some experiments by Messrs A. Fteley and F.P. Stearns (Trans. Am. Soc. C.E., 1883, p. 102) some values of the coefficient c can be reducedh3/h2ch3/h2c0.10.6290.70.5780.20.6140.80.5830.30.6000.90.5960.40.5900.950.6070.50.5821.000.6280.60.578If velocity of approach is taken into account, let ɧ be the head due to that velocity; then, adding ɧ to each of the heads in the equations (3), and reducing, we get for a weirQ = cb √2g[ (h2+ ɧ) (h + ɧ)1/2−1⁄3(h + ɧ)3/2−2⁄3ɧ3/2];(5)an equation which may be useful in estimating flood discharges.Fig. 50.Bridge Piers and other Obstructions in Streams.—When the piers of a bridge are erected in a stream they create an obstruction to the flow of the stream, which causes a difference of surface-level above and below the pier (fig. 51). If it is necessary to estimate this difference of level, the flow between the piers may be treated as if it occurred over a drowned weir. But the value of c in this case is imperfectly known.§ 47.Bazin’s Researches on Weirs.—H. Bazin has executed a long series of researches on the flow over weirs, so systematic and complete that they almost supersede other observations. The account of them is contained in a series of papers in theAnnales des Ponts et Chaussées(October 1888, January 1890, November 1891, February 1894, December 1896, 2nd trimestre 1898). Only a very abbreviated account can be given here. The general plan of the experiments was to establish first the coefficients of discharge for a standard weir without end contractions; next to establish weirs of other types in series with the standard weir on a channel with steady flow, to compare the observed heads on the different weirs and to determine their coefficients from the discharge computed at the standard weir. A channel was constructed parallel to the Canal de Bourgogne, taking water from it through three sluices 0.3 × 1.0 metres. The water enters a masonry chamber 15 metres long by 4 metres wide where it is stilled and passes into the canal at the end of which is the standard weir. The canal has a length of 15 metres, a width of 2 metres and a depth of 0.6 metres. From this extends a channel 200 metres in length with a slope of 1 mm. per metre. The channel is 2 metres wide with vertical sides. The channels were constructed of concrete rendered with cement. The water levels were taken in chambers constructed near the canal, by floats actuating an index on a dial. Hook gauges were used in determining the heads on the weirs.Fig. 51.Standard Weir.—The weir crest was 3.72 ft. above the bottom of the canal and formed by a plate1⁄4in. thick. It was sharp-edged with free overfall. It was as wide as the canal so that end contractions were suppressed, and enlargements were formed below the crest to admit air under the water sheet. The channel below the weir was used as a gauging tank. Gaugings were made with the weir 2 metres in length and afterwards with the weir reduced to 1 metre and 0.5 metre in length, the end contractions being suppressed in all cases. Assuming the general formulaQ = mlh √(2gh),(1)Bazin arrives at the following values of m:—Coefficients of Discharge of Standard Weir.Head h metres.Head h feet.m0.05.1640.44850.10.3280.43360.15.4920.42840.20.6560.42620.25.8200.42590.30.9840.42660.351.1480.42750.401.3120.42860.451.4760.42990.501.6400.43130.551.8040.43270.601.9680.4341Bazin compares his results with those of Fteley and Stearns in 1877 and 1879, correcting for a different velocity of approach, and finds a close agreement.Influence of Velocity of Approach.—To take account of the velocity of approach u it is usual to replace h in the formula by h + au2/2g where α is a coefficient not very well ascertained. ThenQ = μl (h + αu2/2g) √ { 2g (h + αu2/2g) }= μlh √(2gh) (1 + αu2/2gh)3/2.(2)The original simple equation can be used ifm = μ (1 + αu2/2gh)3/2or very approximately, since u2/2gh is small,m = μ (1 +3⁄2αu2/2gh).(3)Fig. 52.Now if p is the height of the weir crest above the bottom of the canal (fig. 52), u = Q/l(p + h). Replacing Q by its value in (1)u2/2gh = Q2/ {2ghl2(p + h)2} = m2{h/(p + h) }2,(4)so that (3) may be writtenm = μ [1 + k {h/(p + h)}2].(5)Gaugings were made with weirs of 0.75, 0.50, 0.35, and 0.24 metres height above the canal bottom and the results compared with those of the standard weir taken at the same time. The discussion of the results leads to the following values of m in the general equation (1):—m = μ (1 + 2.5u2/2gh)= μ [1 + 0.55 {h/(p + h)}2].Values of μ—Head h metres.Head h feet.μ0.05.1640.44810.10.3280.43220.20.6560.42150.30.9840.41740.401.3120.41440.501.6400.41180.601.9680.4092An approximate formula for μ is:μ = 0.405 + 0.003/h (h in metres)μ = 0.405 + 0.01/h (h in feet).Inclined Weirs.—-Experiments were made in which the plank weir was inclined up or down stream, the crest being sharp and the end contraction suppressed. The following are coefficients by which the discharge of a vertical weir should be multiplied to obtain the discharge of the inclined weir.Coefficient.Inclinationup stream1 to 10.93””3 to 20.94””3 to 10.96Vertical weir..1.00Inclinationdown stream3 to 11.04””3 to 21.07””1 to 11.10””1 to 21.12””1 to 41.09The coefficient varies appreciably, if h/p approaches unity, which case should be avoided.Fig. 53.Fig. 54.In all the preceding cases the sheet passing over the weir is detached completely from the weir and its under-surface is subject to atmospheric pressure. These conditions permit the most exact determination of the coefficient of discharge. If the sides of the canal below the weir are not so arranged as to permit the access of air under the sheet, the phenomena are more complicated. So long as the head does not exceed a certain limit the sheet is detached from the weir, but encloses a volume of air which is at less than atmospheric pressure, and the tail water rises under the sheet. The discharge is a little greater than for free overfall. At greater head the air disappears from below the sheet and the sheet is said to be “drowned.” The drowned sheet may be independent of the tail water level or influenced by it. In the former case the fall is followed by a rapid, terminating in a standing wave. In the latter case when the foot of the sheet is drowned the level of the tail water influences the discharge even if it is below the weir crest.Weirs with Flat Crests.—The water sheet may spring clear from the upstream edge or may adhere to the flat crest falling free beyond the down-stream edge. In the former case the condition is that of a sharp-edged weir and it is realized when the head is at least double the width of crest. It may arise if the head is at least 11⁄2the width of crest. Between these limits the condition of the sheet is unstable. When the sheet is adherent the coefficient m depends on the ratio of the head h to the width of crest c (fig. 53), and is given by the equation m = m1[0.70 + 0.185h/c], where m1is the coefficient for a sharp-edged weir in similar conditions. Rounding the upstream edge even to a small extent modifies the discharge. If R is the radius of the rounding the coefficient m is increased in the ratio 1 to 1 + R/h nearly. The results are limited to R less than1⁄2in.Drowned Weirs.—Let h (fig. 54) be the height of head water and h1that of tail water above the weir crest. Then Bazin obtains as the approximate formula for the coefficient of dischargem = 1.05m1[1 +1⁄5h1/p]3√ { (h − h1) / h },Fig. 55.where as before m1is the coefficient for a sharp-edged weir in similar conditions, that is, when the sheet is free and the weir of the same height.§ 48.Separating Weirs.—Many towns derive their water-supply from streams in high moorland districts, in which the flow is extremely variable. The water is collected in large storage reservoirs, from which an uniform supply can be sent to the town. In such cases it is desirable to separate the coloured water which comes down the streams in high floods from the purer water of ordinary flow. The latter is sent into the reservoirs; the former is allowedto flow away down the original stream channel, or is stored in separate reservoirs and used as compensation water. To accomplish the separation of the flood and ordinary water, advantage is taken of the different horizontal range of the parabolic path of the water falling over a weir, as the depth on the weir and, consequently, the velocity change. Fig. 55 shows one of these separating weirs in the form in which they were first introduced on the Manchester Waterworks; fig. 56 a more modern weir of the same kind designed by Sir A. Binnie for the Bradford Waterworks. When the quantity of water coming down the stream is not excessive, it drops over the weir into a transverse channel leading to the reservoirs. In flood, the water springs over the mouth of this channel and is led into a waste channel.Fig. 56.It may be assumed, probably with accuracy enough for practical purposes, that the particles describe the parabolas due to the mean velocity of the water passing over the weir, that is, to a velocity2⁄3√(2gh),where h is the head above the crest of the weir.Let cb = x be the width of the orifice and ac = y the difference of level of its edges (fig. 57). Then, if a particle passes from a to b in t seconds,y =1⁄2gt2, x =2⁄3√(2gh)t;∴ y =9⁄16x2/h,which gives the width x for any given difference of level y and head h, which the jet will just pass over the orifice. Set off ad vertically and equal to1⁄2g on any scale; af horizontally and equal to2⁄3√(gh). Divide af, fe into an equal number of equal parts. Join a with the divisions on ef. The intersections of these lines with verticals from the divisions on af give the parabolic path of the jet.Fig. 57.Mouthpieces—Head Constant§ 49.Cylindrical Mouthpieces.—When water issues from a short cylindrical pipe or mouthpiece of a length at least equal to l1⁄2times its smallest transverse dimension, the stream, after contraction within the mouthpiece, expands to fill it and issues full bore, or without contraction, at the point of discharge. The discharge is found to be about one-third greater than that from a simple orifice of the same size. On the other hand, the energy of the fluid per unit of weight is less than that of the stream from a simple orifice with the same head, because part of the energy is wasted in eddies produced at the point where the stream expands to fill the mouthpiece, the action being something like that which occurs at an abrupt change of section.Let fig. 58 represent a vessel discharging through a cylindrical mouthpiece at the depth h from the free surface, and let the axis of the jet XX be taken as the datum with reference to which the head is estimated. Let Ω be the area of the mouthpiece, ω the area of the stream at the contracted section EF. Let v, p be the velocity and pressure at EF, and v1, p1the same quantities at GH. If the discharge is into the air, p1is equal to the atmospheric pressure pa.The total head of any filament which goes to form the jet, taken at a point where its velocity is sensibly zero, is h + pa/G; at EF the total head is v2/2g + p/G; at GH it is v12/2g + p1/G.Between EF and GH there is a loss of head due to abrupt change of velocity, which from eq. (3), § 36, may have the value(v − v1)2/2g.Adding this head lost to the head at GH, before equating it to the heads at EF and at the point where the filaments start into motion,—h + pa/G = v2/2g + p/G = v12/2g + p1/G + (v − v1)2/2g.But ωv = Ωv1, and ω = ccΩ, if ccis the coefficient of contraction within the mouthpiece. Hencev = Ωv1/ω = v1/cc.Supposing the discharge into the air, so that p1= pa,h + pa/G = v12/2g + pa/G + (v12/2g) (1/cc− 1)2;(v1/2g) {1 + (1/cc− 1)2} = h;∴ v1= √(2gh) / √ {1 + (1/cc− 1)2};
(9)
If H1varies from 0 to ∞, σ( = H1/H2) varies from 0 to 1. The following table gives values of the two estimates of the discharge for different values of σ:—
Hence it is obvious that, except for very small values of σ, the simpler equation (5) gives values sensibly identical with those of (8). When σ < 0.5 it is better to use equation (8) with values of c determined experimentally for the particular proportions of orifice which are in question.
§ 40.Large Jets having a Circular Section from Orifices in a Vertical Plane Surface.—Let fig. 44 represent the section of the jet, OO being the free surface level in the reservoir. The discharge through the horizontal strip aabb, of breadth aa = b, between the depths h1+ y and h1+ y + dy, is
dQ = b √ {2g (h1+ y) } dy.
The whole discharge of the jet is
Q =∫d0b √ { 2g (h1+ y) } dy.
But b = d sin φ; y =1⁄2d (1 − cos φ); dy =1⁄2d sin φ dφ. Let ε = d/(2h1+ d), then
Q =1⁄2d2√ { 2g (h1+ d/2) }∫π0sin2φ √1 − ε cos φdφ.
From eq. (5), putting ω = πd2/4, h = h1+ d/2, c = 1 when d is the diameter of the jet and not that of the orifice,
Q1=1⁄4πd2√ {2g (h1+ d/2) },
Q/Q1= 2/π∫π0sin2φ √ {1 − ε cos φ} dφ.
For
h1= ∞, ε = 0 and Q/Q1= 1;
and for
h1= 0, ε = 1 and Q/Q1= 0.96.
So that in this case also the difference between the simple formula (5) and the formula above, in which the variation of head at different parts of the orifice is taken into account, is very small.
Notches and Weirs
§ 41.Notches, Weirs and Byewashes.—A notch is an orifice extending up to the free surface level in the reservoir from which the discharge takes place. A weir is a structure over which the water flows, the discharge being in the same conditions as for a notch. The formula of discharge for an orifice of this kind is ordinarily deduced by putting H1= 0 in the formula for the corresponding orifice, obtained as in the preceding section. Thus for a rectangular notch, put H1= 0 in (8). Then
Q =2⁄3cB √(2g) H3/2,
(11)
where H is put for the depth to the crest of the weir or the bottom of the notch. Fig. 45 shows the mode in which the discharge occurs in the case of a rectangular notch or weir with a level crest. As, the free surface level falls very sensibly near the notch, the head H should be measured at some distance back from the notch, at a point where the velocity of the water is very small.
Since the area of the notch opening is BH, the above formula is of the form
Q = c × BH × k √(2gH),
where k is a factor depending on the form of the notch and expressing the ratio of the mean velocity of discharge to the velocity due to the depth H.
§ 42.Francis’s Formula for Rectangular Notches.—The jet discharged through a rectangular notch has a section smaller than BH, (a) because of the fall of the water surface from the point where His measured towards the weir, (b) in consequence of the crest contraction, (c) in consequence of the end contractions. It may be pointed out that while the diminution of the section of the jet due to the surface fall and to the crest contraction is proportional to the length of the weir, the end contractions have nearly the same effect whether the weir is wide or narrow.
J. B. Francis’s experiments showed that a perfect end contraction, when the heads varied from 3 to 24 in., and the length of the weir was not less than three times the head, diminished the effective length of the weir by an amount approximately equal to one-tenth of the head. Hence, if l is the length of the notch or weir, and H the head measured behind the weir where the water is nearly still, then the width of the jet passing through the notch would be l − 0.2H, allowing for two end contractions. In a weir divided by posts there may be more than two end contractions. Hence, generally, the width of the jet is l − 0.1nH, where n is the number of end contractions of the stream. The contractions due to the fall of surface and to the crest contraction are proportional to the width of the jet. Hence, if cH is the thickness of the stream over the weir, measured at the contracted section, the section of the jet will be c(l − 0.1nH)H and (§ 41) the mean velocity will be2⁄3√(2gH). Consequently the discharge will be given by an equation of the form
Q =2⁄3c (l − 0.1nH) H √2gH
= 5.35c (l − 0.1nH) H3/2.
This is Francis’s formula, in which the coefficient of discharge c is much more nearly constant for different values of l and h than in the ordinary formula. Francis found for c the mean value 0.622, the weir being sharp-edged.
§ 43.Triangular Notch(fig. 46).—Consider a lamina issuing between the depths h and h + dh. Its area, neglecting contraction, will be bdh, and the velocity at that depth is √(2gh). Hence the discharge for this lamina is
b√2ghdh.
But
B/b = H / (H − h); b = B (H − h) / H.
Hence discharge of lamina
= B(H − h) √(2gh) dh/H;
and total discharge of notch
= Q = B √(2g)∫H0(H − h) h1/2dh/H
=4⁄15B √(2g) H3/2.
or, introducing a coefficient to allow for contraction,
Q =4⁄15cB √(2g) H1/2,
When a notch is used to gauge a stream of varying flow, the ratio B/H varies if the notch is rectangular, but is constant if the notch is triangular. This led Professor James Thomson to suspect that the coefficient of discharge, c, would be much more constant with different values of H in a triangular than in a rectangular notch, and this has been experimentally shown to be the case. Hence a triangular notch is more suitable for accurate gaugings than a rectangular notch. For a sharp-edged triangular notch Professor J. Thomson found c = 0.617. It will be seen, as in § 41, that since1⁄2BH is the area of section of the stream through the notch, the formula is again of the form
Q = c ×1⁄2BH × k √(2gH),
where k =8⁄15is the ratio of the mean velocity in the notch to the velocity at the depth H. It may easily be shown that for all notches the discharge can be expressed in this form.
Coefficients for the Discharge over Weirs, derived from the Experiments of T. E. Blackwell. When more than one experiment was made with the same head, and the results were pretty uniform, the resulting coefficients are marked with an(*).The effect of the converging wing-boards is very strongly marked.
§ 44.Weir with a Broad Sloping Crest.—Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a movement sensibly rectilinear and uniform (fig. 47). Let the inner edge be so rounded as to prevent a crest contraction. Consider a filament aa′, the point a being so far back from the weir that the velocity of approach is negligible. Let OO be the surface level in the reservoir, and let a be at a height h″ below OO, and h′ above a′. Let h be the distance from OO to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at a is Gh″; at a′ it is Gz. If v be the velocity at a′,
v2/2g = h′ + h″ − z = h − e;
Q = be √2g (h − e).
Theory does not furnish a value for e, but Q = 0 for e = 0 and for e = h. Q has therefore a maximum for a value of e between 0 and h, obtained by equating dQ/de to zero. This gives e =2⁄3h, and, inserting this value,
Q = 0.385 bh √2gh,
as a maximum value of the discharge with the conditions assigned. Experiment shows that the actual discharge is very approximately equal to this maximum, and the formula is more legitimately applicable to the discharge over broad-crested weirs and to cases such as the discharge with free upper surface through large masonrysluice openings than the ordinary weir formula for sharp-edged weirs. It should be remembered, however, that the friction on the sides and crest of the weir has been neglected, and that this tends to reduce a little the discharge. The formula is equivalent to the ordinary weir formula with c = 0.577.
Special Cases of Discharge from Orifices
§ 45.Cases in which the Velocity of Approach needs to be taken into Account.Rectangular Orifices and Notches.—In finding the velocity at the orifice in the preceding investigations, it has been assumed that the head h has been measured from the free surface of still water above the orifice. In many cases which occur in practice the channel of approach to an orifice or notch is not so large, relatively to the stream through the orifice or notch, that the velocity in it can be disregarded.
Let h1, h2(fig. 48) be the heads measured from the free surface to the top and bottom edges of a rectangular orifice, at a point in the channel of approach where the velocity is u. It is obvious that a fall of the free surface,
ɧ = u2/2g
has been somewhere expended in producing the velocity u, and hence the true heads measured in still water would have been h1+ ɧ and h2+ ɧ. Consequently the discharge, allowing for the velocity of approach, is
Q =2⁄3cb √2g{ (h2+ ɧ)3/2− (h1+ ɧ)3/2}.
(1)
And for a rectangular notch for which h1= 0, the discharge is
Q =2⁄3cb √2g{ (h2+ ɧ)3/2− ɧ3/2}.
(2)
In cases where u can be directly determined, these formulae give the discharge quite simply. When, however, u is only known as a function of the section of the stream in the channel of approach, they become complicated. Let Ω be the sectional area of the channel where h1and h2are measured. Then u = Q/Ω and ɧ = Q2/2g Ω2.
This value introduced in the equations above would render them excessively cumbrous. In cases therefore where Ω only is known, it is best to proceed by approximation. Calculate an approximate value Q′ of Q by the equation
Q′ =2⁄3cb √2g{h23/2− h13/2}.
Then ɧ = Q′2/2gΩ2nearly. This value of ɧ introduced in the equations above will give a second and much more approximate value of Q.
§ 46.Partially Submerged Rectangular Orifices and Notches.—When the tail water is above the lower but below the upper edge of the orifice, the flow in the two parts of the orifice, into which it is divided by the surface of the tail water, takes place under different conditions. A filament M1m1(fig. 49) in the upper part of the orifice issues with a head h′ which may have any value between h1and h. But a filament M2m2issuing in the lower part of the orifice has a velocity due to h″ − h″′, or h, simply. In the upper part of the orifice the head is variable, in the lower constant. If Q1, Q2are the discharges from the upper and lower parts of the orifice, b the width of the orifice, then
Q1=2⁄3cb √2g{ h3/2− h13/2}
Q2= cb (h2− h) √2gh.
(3)
In the case of a rectangular notch or weir, h1= 0. Inserting this value, and adding the two portions of the discharge together, we get for a drowned weir
Q = cb √2gh(h2− h/3),
(4)
where h is the difference of level of the head and tail water, and h2is the head from the free surface above the weir to the weir crest (fig. 50).
From some experiments by Messrs A. Fteley and F.P. Stearns (Trans. Am. Soc. C.E., 1883, p. 102) some values of the coefficient c can be reduced
If velocity of approach is taken into account, let ɧ be the head due to that velocity; then, adding ɧ to each of the heads in the equations (3), and reducing, we get for a weir
Q = cb √2g[ (h2+ ɧ) (h + ɧ)1/2−1⁄3(h + ɧ)3/2−2⁄3ɧ3/2];
(5)
an equation which may be useful in estimating flood discharges.
Bridge Piers and other Obstructions in Streams.—When the piers of a bridge are erected in a stream they create an obstruction to the flow of the stream, which causes a difference of surface-level above and below the pier (fig. 51). If it is necessary to estimate this difference of level, the flow between the piers may be treated as if it occurred over a drowned weir. But the value of c in this case is imperfectly known.
§ 47.Bazin’s Researches on Weirs.—H. Bazin has executed a long series of researches on the flow over weirs, so systematic and complete that they almost supersede other observations. The account of them is contained in a series of papers in theAnnales des Ponts et Chaussées(October 1888, January 1890, November 1891, February 1894, December 1896, 2nd trimestre 1898). Only a very abbreviated account can be given here. The general plan of the experiments was to establish first the coefficients of discharge for a standard weir without end contractions; next to establish weirs of other types in series with the standard weir on a channel with steady flow, to compare the observed heads on the different weirs and to determine their coefficients from the discharge computed at the standard weir. A channel was constructed parallel to the Canal de Bourgogne, taking water from it through three sluices 0.3 × 1.0 metres. The water enters a masonry chamber 15 metres long by 4 metres wide where it is stilled and passes into the canal at the end of which is the standard weir. The canal has a length of 15 metres, a width of 2 metres and a depth of 0.6 metres. From this extends a channel 200 metres in length with a slope of 1 mm. per metre. The channel is 2 metres wide with vertical sides. The channels were constructed of concrete rendered with cement. The water levels were taken in chambers constructed near the canal, by floats actuating an index on a dial. Hook gauges were used in determining the heads on the weirs.
Standard Weir.—The weir crest was 3.72 ft. above the bottom of the canal and formed by a plate1⁄4in. thick. It was sharp-edged with free overfall. It was as wide as the canal so that end contractions were suppressed, and enlargements were formed below the crest to admit air under the water sheet. The channel below the weir was used as a gauging tank. Gaugings were made with the weir 2 metres in length and afterwards with the weir reduced to 1 metre and 0.5 metre in length, the end contractions being suppressed in all cases. Assuming the general formula
Q = mlh √(2gh),
(1)
Bazin arrives at the following values of m:—
Coefficients of Discharge of Standard Weir.
Bazin compares his results with those of Fteley and Stearns in 1877 and 1879, correcting for a different velocity of approach, and finds a close agreement.
Influence of Velocity of Approach.—To take account of the velocity of approach u it is usual to replace h in the formula by h + au2/2g where α is a coefficient not very well ascertained. Then
Q = μl (h + αu2/2g) √ { 2g (h + αu2/2g) }
= μlh √(2gh) (1 + αu2/2gh)3/2.
(2)
The original simple equation can be used if
m = μ (1 + αu2/2gh)3/2
or very approximately, since u2/2gh is small,
m = μ (1 +3⁄2αu2/2gh).
(3)
Now if p is the height of the weir crest above the bottom of the canal (fig. 52), u = Q/l(p + h). Replacing Q by its value in (1)
u2/2gh = Q2/ {2ghl2(p + h)2} = m2{h/(p + h) }2,
(4)
so that (3) may be written
m = μ [1 + k {h/(p + h)}2].
(5)
Gaugings were made with weirs of 0.75, 0.50, 0.35, and 0.24 metres height above the canal bottom and the results compared with those of the standard weir taken at the same time. The discussion of the results leads to the following values of m in the general equation (1):—
m = μ (1 + 2.5u2/2gh)
= μ [1 + 0.55 {h/(p + h)}2].
Values of μ—
An approximate formula for μ is:
μ = 0.405 + 0.003/h (h in metres)
μ = 0.405 + 0.01/h (h in feet).
Inclined Weirs.—-Experiments were made in which the plank weir was inclined up or down stream, the crest being sharp and the end contraction suppressed. The following are coefficients by which the discharge of a vertical weir should be multiplied to obtain the discharge of the inclined weir.
The coefficient varies appreciably, if h/p approaches unity, which case should be avoided.
In all the preceding cases the sheet passing over the weir is detached completely from the weir and its under-surface is subject to atmospheric pressure. These conditions permit the most exact determination of the coefficient of discharge. If the sides of the canal below the weir are not so arranged as to permit the access of air under the sheet, the phenomena are more complicated. So long as the head does not exceed a certain limit the sheet is detached from the weir, but encloses a volume of air which is at less than atmospheric pressure, and the tail water rises under the sheet. The discharge is a little greater than for free overfall. At greater head the air disappears from below the sheet and the sheet is said to be “drowned.” The drowned sheet may be independent of the tail water level or influenced by it. In the former case the fall is followed by a rapid, terminating in a standing wave. In the latter case when the foot of the sheet is drowned the level of the tail water influences the discharge even if it is below the weir crest.
Weirs with Flat Crests.—The water sheet may spring clear from the upstream edge or may adhere to the flat crest falling free beyond the down-stream edge. In the former case the condition is that of a sharp-edged weir and it is realized when the head is at least double the width of crest. It may arise if the head is at least 11⁄2the width of crest. Between these limits the condition of the sheet is unstable. When the sheet is adherent the coefficient m depends on the ratio of the head h to the width of crest c (fig. 53), and is given by the equation m = m1[0.70 + 0.185h/c], where m1is the coefficient for a sharp-edged weir in similar conditions. Rounding the upstream edge even to a small extent modifies the discharge. If R is the radius of the rounding the coefficient m is increased in the ratio 1 to 1 + R/h nearly. The results are limited to R less than1⁄2in.
Drowned Weirs.—Let h (fig. 54) be the height of head water and h1that of tail water above the weir crest. Then Bazin obtains as the approximate formula for the coefficient of discharge
m = 1.05m1[1 +1⁄5h1/p]3√ { (h − h1) / h },
where as before m1is the coefficient for a sharp-edged weir in similar conditions, that is, when the sheet is free and the weir of the same height.
§ 48.Separating Weirs.—Many towns derive their water-supply from streams in high moorland districts, in which the flow is extremely variable. The water is collected in large storage reservoirs, from which an uniform supply can be sent to the town. In such cases it is desirable to separate the coloured water which comes down the streams in high floods from the purer water of ordinary flow. The latter is sent into the reservoirs; the former is allowedto flow away down the original stream channel, or is stored in separate reservoirs and used as compensation water. To accomplish the separation of the flood and ordinary water, advantage is taken of the different horizontal range of the parabolic path of the water falling over a weir, as the depth on the weir and, consequently, the velocity change. Fig. 55 shows one of these separating weirs in the form in which they were first introduced on the Manchester Waterworks; fig. 56 a more modern weir of the same kind designed by Sir A. Binnie for the Bradford Waterworks. When the quantity of water coming down the stream is not excessive, it drops over the weir into a transverse channel leading to the reservoirs. In flood, the water springs over the mouth of this channel and is led into a waste channel.
It may be assumed, probably with accuracy enough for practical purposes, that the particles describe the parabolas due to the mean velocity of the water passing over the weir, that is, to a velocity
2⁄3√(2gh),
where h is the head above the crest of the weir.
Let cb = x be the width of the orifice and ac = y the difference of level of its edges (fig. 57). Then, if a particle passes from a to b in t seconds,
y =1⁄2gt2, x =2⁄3√(2gh)t;
∴ y =9⁄16x2/h,
which gives the width x for any given difference of level y and head h, which the jet will just pass over the orifice. Set off ad vertically and equal to1⁄2g on any scale; af horizontally and equal to2⁄3√(gh). Divide af, fe into an equal number of equal parts. Join a with the divisions on ef. The intersections of these lines with verticals from the divisions on af give the parabolic path of the jet.
Mouthpieces—Head Constant
§ 49.Cylindrical Mouthpieces.—When water issues from a short cylindrical pipe or mouthpiece of a length at least equal to l1⁄2times its smallest transverse dimension, the stream, after contraction within the mouthpiece, expands to fill it and issues full bore, or without contraction, at the point of discharge. The discharge is found to be about one-third greater than that from a simple orifice of the same size. On the other hand, the energy of the fluid per unit of weight is less than that of the stream from a simple orifice with the same head, because part of the energy is wasted in eddies produced at the point where the stream expands to fill the mouthpiece, the action being something like that which occurs at an abrupt change of section.
Let fig. 58 represent a vessel discharging through a cylindrical mouthpiece at the depth h from the free surface, and let the axis of the jet XX be taken as the datum with reference to which the head is estimated. Let Ω be the area of the mouthpiece, ω the area of the stream at the contracted section EF. Let v, p be the velocity and pressure at EF, and v1, p1the same quantities at GH. If the discharge is into the air, p1is equal to the atmospheric pressure pa.
The total head of any filament which goes to form the jet, taken at a point where its velocity is sensibly zero, is h + pa/G; at EF the total head is v2/2g + p/G; at GH it is v12/2g + p1/G.
Between EF and GH there is a loss of head due to abrupt change of velocity, which from eq. (3), § 36, may have the value
(v − v1)2/2g.
Adding this head lost to the head at GH, before equating it to the heads at EF and at the point where the filaments start into motion,—
h + pa/G = v2/2g + p/G = v12/2g + p1/G + (v − v1)2/2g.
But ωv = Ωv1, and ω = ccΩ, if ccis the coefficient of contraction within the mouthpiece. Hence
v = Ωv1/ω = v1/cc.
Supposing the discharge into the air, so that p1= pa,
h + pa/G = v12/2g + pa/G + (v12/2g) (1/cc− 1)2;
(v1/2g) {1 + (1/cc− 1)2} = h;
∴ v1= √(2gh) / √ {1 + (1/cc− 1)2};