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 [h] be the
head due to that velocity; then, adding [h] to each of the
heads in the equations (3), and reducing, we get for a weir
Q = cb [root]{2g} [(h2 + [h]) (h + [h])^(½) - (1/3)(h + [h])^(3/2)
- (2/3)[h]^(3/2)]; (5)
an equation which may be useful in estimating flood discharges.
[Illustration: 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.
[Illustration: FIG. 51.]
_Standard Weir._--The weir crest was 3.72 ft. above the bottom of the
canal and formed by a plate ¼ 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