sluice openings than the ordinary weir formula for sharp-edged weirs. It should be remembered, however, that the friction on the sides and crest of the weir has been neglected, and that this tends to reduce a little the discharge. The formula is equivalent to the ordinary weir formula with c = 0.577.
Special Cases of Discharge from Orifices
§ 45. Cases in which the Velocity of Approach needs to be taken into Account. Rectangular Orifices and Notches.—In finding the velocity at the orifice in the preceding investigations, it has been assumed that the head h has been measured from the free surface of still water above the orifice. In many cases which occur in practice the channel of approach to an orifice or notch is not so large, relatively to the stream through the orifice or notch, that the velocity in it can be disregarded.
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| 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,
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
And for a rectangular notch for which h1 = 0, the discharge is
In cases where u can be directly determined, these formulae give the discharge quite simply. When, however, u is only known as a function of the section of the stream in the channel of approach, they become complicated. Let Ω be the sectional area of the channel where h1 and h2 are measured. Then u = Q/Ω and ɧ = Q2/2g Ω2.
This value introduced in the equations above would render them excessively cumbrous. In cases therefore where Ω only is known, it is best to proceed by approximation. Calculate an approximate value Q′ of Q by the equation
Then ɧ = Q′2/2gΩ2 nearly. This value of ɧ introduced in the equations above will give a second and much more approximate value of Q.
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| Fig. 49. |
§ 46. Partially Submerged Rectangular Orifices and Notches.—When the tail water is above the lower but below the upper edge of the orifice, the flow in the two parts of the orifice, into which it is divided by the surface of the tail water, takes place under different conditions. A filament M1m1 (fig. 49) in the upper part of the orifice issues with a head h′ which may have any value between h1 and h. But a filament M2m2 issuing in the lower part of the orifice has a velocity due to h″ − h″′, or h, simply. In the upper part of the orifice the head is variable, in the lower constant. If Q1, Q2 are the discharges from the upper and lower parts of the orifice, b the width of the orifice, then
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
where h is the difference of level of the head and tail water, and h2 is the head from the free surface above the weir to the weir crest (fig. 50).
From some experiments by Messrs A. Fteley and F.P. Stearns (Trans. Am. Soc. C.E., 1883, p. 102) some values of the coefficient c can be reduced
| h3/h2 | c | h3/h2 | c |
| 0.1 | 0.629 | 0.7 | 0.578 |
| 0.2 | 0.614 | 0.8 | 0.583 |
| 0.3 | 0.600 | 0.9 | 0.596 |
| 0.4 | 0.590 | 0.95 | 0.607 |
| 0.5 | 0.582 | 1.00 | 0.628 |
| 0.6 | 0.578 |
If velocity of approach is taken into account, let ɧ be the head due to that velocity; then, adding ɧ to each of the heads in the equations (3), and reducing, we get for a weir
an equation which may be useful in estimating flood discharges.
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| Fig. 50. |
Bridge Piers and other Obstructions in Streams.—When the piers of a bridge are erected in a stream they create an obstruction to the flow of the stream, which causes a difference of surface-level above and below the pier (fig. 51). If it is necessary to estimate this difference of level, the flow between the piers may be treated as if it occurred over a drowned weir. But the value of c in this case is imperfectly known.
§ 47. Bazin’s Researches on Weirs.—H. Bazin has executed a long series of researches on the flow over weirs, so systematic and complete that they almost supersede other observations. The account of them is contained in a series of papers in the Annales des Ponts et Chaussées (October 1888, January 1890, November 1891, February 1894, December 1896, 2nd trimestre 1898). Only a very abbreviated account can be given here. The general plan of the experiments was to establish first the coefficients of discharge for a standard weir without end contractions; next to establish weirs of other types in series with the standard weir on a channel with steady flow, to compare the observed heads on the different weirs and to determine their coefficients from the discharge computed at the standard weir. A channel was constructed parallel to the Canal de Bourgogne, taking water from it through three sluices 0.3 × 1.0 metres. The water enters a masonry chamber 15 metres long by 4 metres wide where it is stilled and passes into the canal at the end of which is the standard weir. The canal has a length of 15 metres, a width of 2 metres and a depth of 0.6 metres. From this extends a channel 200 metres in length with a slope of 1 mm. per metre. The channel is 2 metres wide with vertical sides. The channels were constructed of concrete rendered with cement. The water levels were taken in chambers constructed near the canal, by floats actuating an index on a dial. Hook gauges were used in determining the heads on the weirs.
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| Fig. 51. |
Standard Weir.—The weir crest was 3.72 ft. above the bottom of the canal and formed by a plate 14 in. thick. It was sharp-edged with free overfall. It was as wide as the canal so that end contractions were suppressed, and enlargements were formed below the crest to admit air under the water sheet. The channel below the weir was used as a gauging tank. Gaugings were made with the weir 2 metres in length and afterwards with the weir reduced to 1 metre and 0.5 metre in length, the end contractions being suppressed in all cases. Assuming the general formula
