sluice openings than the ordinary weir formula for sharpedged
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 e =o577.
SPECIAL Cases or DISCHARGE FROM ORIFICES
§ 45. Cases in which the Velocity of Approach needs to be taken
into Account. Rectangular Oriflces 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 hi, ho (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
E?E;Zq?¥sE, ........... . .. ........., .... 4
" “;i;ii: F2
TI'4'3' ' '“?§ ?f'F="; i, ':,
 7 Ti if; 
2 41
w he
))»; ... .> . I
I
I

... lf, . .
FIG. 48.,
channel of approach where the velocity is u. It is obvious that a fall of the free surface,
f>= M”/2g
has been somewhere expended in producing the velocity u, and hence the true heads measured in still water would have been lz1+f) and h2+b. Consequently the discharge, allowing for the velocity of approach, is
Q=§ ¢b~/2§ l(hfz+f))*(711+fJ>3l (I) And for a rectangular notch Q which hi=0, the discharge is Q=§ ¢b/ 2§ {(ll2 +f))if>3}~ (2) In cases where u can be directly determined, these formulae give the discharge quite simply. Wl1en, however, u is only known as a function of the section of the stream in the channel of approach, they become complicated. Let SZ be the sectional area of the channel where hx and la are measured. Then u=Q/ S2 and f)=Q2/2g $22. This value introduced in the equations above would render them excessively cumbrous. In cases therefore where SZ only is known, it is best to proceed by approximation. Calculate an approximate value Q' of Q by the equation
Q' = § cb/ Zglhgg h13}.
Then b= Q'2/2gQ” nearly. This value of Y) introduced in the equations above will give a second and much more approximate value of § 46. Partially Submerged Rectangular Orifices and NotchesfWhen 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 Mlml (fig. 49) in the upper part of the orikice issues with a head h' which may have any value between ir;Tf;;:Ef»;;;= § ¥ "i'i*"" "°"""" ?:€ E§ ;€gZZ;tsl;
lr, lim. it
E § ' if i,
1 1 tl 5 t
 n  1
5 5 "L " 'hm
M' ' t . UJ i.
I 4, L e ~= .= r'E
I mg
, %.—.. — FIG.
49.
hi and h. But a filament Mmzz issuing in the lower part of the orifice has a velocity due to h”lt”', 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
Q1=§ cb/ @{h3hi3} ()
Q.=@1»<1»2h>~/it ' 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/ 2£h(h2h/3), (4)
where h is the difference of level of the head and tail water, and hz the head from the free surface above the weir to the weir crest g. 50 .
From some experiments by Messrs A. Fteley and F. P. Stearns can be reduced
(Trans. Arn. Soc. C.E., 1883, p. 102) some values of the coefficient c ha/hz c' ha/hz c
01 o629 o7 0578
02 0614 o8 0583
o3 0600 09 0596
0'4 0'590 0'95 0'607
05 0582 1oo 0628
06 0578
If velocity of approach isitaken into account, let B be the head due to that velocity; then, adding f) to each of the heads in the equations (3), and reducing, we get for a weir W Q=cbJ3§ l(h2+5)(hIYJ);%(h+f>)%iflgl; (5) an eq uation 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, :;“:;'~j 3 i;~ t ' ' ' “g level above and below the ' " ' ' , pier(fig. 51). If it is neces /7 sary to estimate this difference
of level, the flow
between the piers may be
treated as if it occurred over,
a drowned weir. But the Q1
value of c in this case is V/l0?QJZ//Q/}7J, U%AL//A/Z///Q/A517//*W 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 Pants 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 0'3XI'O 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 16 metres. From I '~;;1
—L — FIG.
50.
"§ if
fi. %¢ * 1 17
 V
FIG. 51.11
this extends a channel 200 metres in length with a slope of I 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 fioats actuating an index on a dial. Hook gauges were used in determining the heads on the weirs. Standard Weir.The weir crest was 372 ft. above the bottom of the canal and formed by a plate i in. thick. It was sharpedged 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. Caugings were made with the weir 2 metres in length and afterwards with the weir reduced to 1 metre and 05 metre in length, the end contractions being suppressed in all eases. Assuming the general formula
Q = mlhwl (2gh), (I)