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7 I
If the specific energy is a minimum dH,/dy = O , we may write
( 1 -29)
Equation 1-29 is valid only for steady flow with parallel streamlines in a channel of small slope. If the velocity distribution coefficient, U, is assumed to be unity, the crite- rion for critical flow becomes
Vc--- or V = i, = (g A,/B,)O.SO (1-30)
AC
2g 2Bc
- 2
Provided that the tailwater level of the measuring structure is low enough to enable
the depth of flow at the channel contraction to reach critical depth, Equations 1-2, 1-23, and 1-30 allow the development of a discharge equation for each measuring device, in which the upstream total energy head (HI) is the only independent variable.
Equation 1-30 states that at critical flow the average flow velocity V, = (g A,/B,)n.5n
It can be proved that this flow velocity equals the velocity with which the smallest
disturbance moves in an open channel, as measured relative to the flow. Because of this feature, a disturbance or change in a downstream level cannot influence an up- stream water level if critical flow occurs in between the two cross-sections considered. The 'control section' of a measuring structure is located where critical flow occurs and subcritical, tranquil, or streaming flow passes into supercritical, rapid, or shooting flow. Thus, if critical flow occurs at the control section of a measuring structure, the upstream water level is independent of the tailwater level; the flow over the structure is then called 'modular'.
1.9 The broad-crested weir
A broad-crested weir is an overflow structure with a horizontal crest above which
the deviation from a hydrostatic pressure distribution because of centripetal acce-
leration may be neglected. In other words, the streamlines are practically straight
and parallel. To obtain this situation the length of the weir crest in the direction
of flow (L) should be related to the total energy head over the weir crest as
u 7 < H , / L < 0.50. H,/L I 0.07 because otherwise the energy losses above the
weir crest c a p b e m a y occur on the-creqt; i I , / L _ > 0.50,
so tha; onlylight curvature-lGs occurTabove the crest and a hydrostatic
pressure distribution may be assumed. If the measuring structure is so designed that there are no significant energy losses in the zone of acceleration upstream of the control section, according to Bernoulli's equation (1-23):
- = c
~ H , = h, + ~V,'/2g= H = y + aV2/2g
(1-31)
where H, equals the total upstream energy head over the weir crest as shown in Figure
c c o z s ‘s, - -._ 5k c^ O
Figure 1.10 IhStrdtiOn of terminology
1.10. Substituting Q = VA and putting c1 = 1 .O gives
Q = A (2g(Hl -Y)}~.’~ (1-32)
Provided that critical flow occurs at the control section (y = y,), a head-discharge
equation for various throat geometries can now be derived from
Q = Ac {2idHi -YJ)~.’~ ( 1^ -33)
For a rectangular control section in which the flow is critical, we may write A, =
b,y, and A,/B, = y, so that Equation 1-30 may be written as Y2/2g = ‘I2 y,. Hence:
Broad-crestedweir with rectangular control section
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Substitution of this relation and A, = b, into Equation 1-33 gives, after simplification
This formula is based on idealized assumptions such as: absence of centripetal forces
Figure I. I 1 Dimensions of a rectangular control section
Figure 1.13 Dimensions of a parabolic control section
As explained in Section 1.9.1, correction coefficients have to be introduced to obtain
a practical head-discharge equation. Thus
Q = Cd C V J T fg hI2.O (1-43)
1.9.3 Broad-crested weir with triangular control section
For a triangular control section with A, = y: tan- and B, = 2yc tan- (see Figure
1.14), we may write Equation 1-30 as:
Hence
Substituting those relations and A, = y: tan- into Equation 1-33 gives
I
Figure 1.14 Dimensions of a triangular control section
It also shows that, if both b, and z, are known the ratio y,/Hl is a function of H I.
Values of yc/HIas a function of z, and the ratio Hl/bcare shown in Table I. I.
Substitution of A, = bcy, + zcy,2into Equation 1-33 and introduction of a discharge
coefficient gives as a head-discharge equation
Q = G {bCyc+ z,Y,~){%(Hi - YC))O.~O (^) \ ( 1^ -55)
Since for each combination of b,, zo and H,/b,, the ratio y,/H, is given in Table 1.1,
the discharge Q can be computed because the discharge coefficient has a predictable
value. In this way a Q-HI curve can be obtained. If the approach velocity head vI2/2g
is negligible, this curve may be used to measure the discharge. If the approach velocity
has a significant value, v12/2gshould be estimated and h, = H I -v12/2gmay be obtained
in one or two steps.
In the literature the trapezoïdal control section is sometimes described as the sum
of a rectangular and a triangular control section. Hence, along similar lines as will
be shown in Section I. 13 for sharp-crested weirs, a head-discharge equation is obtained
by superposing the head-discharge equations valid for a rectangular and a triangular
control section. For broad-crested weirs, however, this procedure results in a strongly
variable C,-value, since for a given H, the critical depth in the two superposed equa-
tions equals 2/3H, for a rectangular and 4/5Hcfor a triangular control section. This
difference of simultaneous y,-values is one of the reasons why superposition of various
head-discharge equations is not allowed. A second reason is the significant difference
in mean flow velocities through the rectangular and triangular portions of the control
section.
\ /
Figure I. 16 Dimensions of a trapezoidal control section
1.9.6 Broad-crested weir with circular control section
For a broad-crested weir with a circular control section we may write (see Figure
1
A, = -d,Z(O-sin€))
B, = d,sin l/,O and ( 1 -57)
d
y, = 2
(1 -cos ' / , O ) = d, sin2 1/
Substitution of values for A, and B, into Equation 1-30 gives
vc A, d, O-sin
2g 2B, 16 sin '/,O
and because H = H , = y, + v,2/2g we may write the total energy head over the
weir crest as
For each value of yJd, = sin2 1/40 a matching value of the ratios A,/d,Z and HJd,
can now be calculated with the above equations. These values, and the additional
values on the dimensionless ratios v,2/2gdCand yJH, are presented in Table 1.2.
For a circular control section we may use the general head-discharge relation 'given
earlier (Equation 1-33)
Q = C d Ac (2gWi - YJ}' 50 (1-61)
where the discharge coefficient Cd has been introduced for similar reasons to those
explained in Section 1.9.1. The latter equation may also be written in terms of dimen-
sionless ratios as
Figure 1.17 Dimensions of a circular control section
Table 1.2 Ratios for determining the discharge Q of a broad-crested weir and long-throated flume with
circular control section (Bos 1985)
- .o HdbC Vertical 0.25:1 0.5O:l 0.75:l 1 : l 1.5:1 2 1 2 5 :l 3.1 4 :l
- .o
- .o
- .O
- .O
- .O
- .O
- .O
- .O
- .O
- .I I O
- .I
- .I -. -. -. -. -. -. -. -. -. -. -. -. -. -. -. -. -. .so -. -. -.
- I. 1 .o
- I.
- I - , - , - , - , - , - , - , - , - , - , - , - , -. - , -. - , -. - , - , - , - , - , - , - , - , - , - , -. - , - , - , -. - , - , - , - , - , - , - , - , -. - , - , - , - , - , - , - , - , - , - , - , - , - , -. - , - , .67 I - , - , - , - , - , - , - , - , - , - , - , - , -. - , - , - , - , - , - , -. - , - , -. - , -. - , - , - .7 - , - , - , - , - , - , - , - , - , - , -. - , - , - , - , - , - , -. - , -. -. - .68 - , - , - , -. - , - , - , -. - , - , -. - , - , - , - , - , - , - , - , -. - , - , - , -. - , - , - , - , - , - , - , - , - , - , .67 I -. - , -. - , -. -. - , .68 I - , - , - , - , - , - , - , - , - , - , - , - , - , - , -. -. - , - , - , - , -. - , - , - , -. - , - , - , - , - , - , -. - , - , - , .67 I -. - , - , - , -. - , - , - , - , - , - , - , - , - , -. - , - , -. - , - , -. - , - , - , - .7 - , - , - , - , - , - , -. - , - , - , - , - , - , -. - , - , - , - , - , - , - , - , - , - .68 - , - , - , -. -. - , - , - , - , - , - , - .7I - , - , - , - , - , - , - , - , - , - , - , - , -. - , - , -. - , - , - , - , - , - .78 - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , - , -. - , - , - , -. .73 I - , - , - , - , - , -. - , - , - , - , -. - , .77 I - , - , - , - , - , -. -. - , -. - , - , -. - , - , - , - , - , - , - , - , - , -. - , - , - , - , - , - , - , - , - , - , - , - , -. -. - , - , - , - , - , -. - , -. - , - , .78 I - , - , - , - , - , - , -. .67I -. - , - , - , - , -. - , -. -. -. - , - , - , -. - , - , - , - , - , ,739’ - , -. -. - , - , - , - , - , - , -. - , -. - , -. - , - , - , -. - , - , -. -. - , - , -. - , - , - , - , - , - , - , -. - , - , - , - , -. - , - , -. - , - , -. - , - .75 - , - , - , -. -. - , -. - , -. - .77 -. - , -. - , - , - , - , - , - , - , - , -. - , - .O1 ,0033 ,0133 .O01 3 ,752 0.0001 .5 I .20 14 .7 I 14 ,4207 .7 I7 0. Y J ~ , v,2/2gdc ~ , / d , A& y c m 1 fw yc/dC v2/2& HiPC Acid: yc/Hi f(e) - .O2 ,0067 ,0267 ,0037 ,749 0.0004 .52 ,2065 .7265 ,4127 ,716 0. - .O3 .O101 ,0401 ,0069 ,749 0.0010 .53 .2 1 17 ,7417 ,4227 ,715 0. - .O4 .O1 34 ,0534 ,0105 ,749 0.001 7 .54 .2 I70 ,7570 ,4327 .7 13 0. - .O5 ,0168 ,0668 ,0147 ,748 0.0027 .55 ,2224 ,7724 ,4426 ,712 0. - .O6 ..O203 .O803 ,0192 ,748 0.0039 .56 ,2279 .7879 ,4526 ,711 0. - .O7 ,0237 ,0937 ,0242 ,747 0.0053 .57 .2335 ,8035 ,4625 ,709 0. - .O8 ,0271 ,1071 ,0294 ,747 0.0068 .58 ,2393 ,8193 ,4724 ,708 0. - .O9 ,0306 ,1206 .O350 ,746 0.0087 .59 ,2451 ,8351 ,4822 ,707 0. - I O ,0341 ,1341 ,0409 ,746 0.0107 .60 ,2511 ,8511 ,4920 .705 0. - I 1 ,0376 ,1476 .O470 ,745 0.0129 .61 ,2572 3672 ,5018 .703 0. - .I2 ,0411 ,1611 SO34 ,745 0.0153 .62 ,2635 ,8835 ,5115 ,702 0. - .I3 ,0446 ,1746 ,0600 ,745 0.0179 .63 ,2699 ,8999 ,5212 .700 0. - .I4 ,0482 ,1882 ,0688 ,744 0.0214 .64 ,2765 .9 165 ,5308 ,698 0. - .15 ,0517 ,2017 .O739 ,744 0.0238 .65 ,2833 .9333 ,5404 ,696 ,0. - .16 ,0553 ,2153 ,0811 .743 0.0270 .66 ,2902 ,9502 .5499 ,695 0. - .I7 ,0589 ,2289 ,0885 ,743 0.0304 .67 ,2974 ,9674 ,5594 ,693 0. - .I8 ,0626 ,2426 .O96^1 ,742 0.0340 .68 .3048 ,9848 S687 .69^1 0. - .I9 ,0662 ,2562 ,1039 ,742 0.0378 .69 .3125 1.0025 S780 ,688 0. - .20 ,0699 ,2699 I I I8 ,741 0.0418 .70 ,3204 1.0204 ,5872 ,686 0. - .21 .O736 ,2836 I I99 ,740 0.0460 .71 .3286 1.0386 S964 ,684 0. - .22 ,0773 .2973 .I281 ,740 0.0504 .72 ,3371 1.O571 ,6054 .681 0.497 - .23 .O8 1 I .3 1 1 1 .I365 ,739 0.0550 .73 ,3459 1.0759 ,6143 ,679 0. - .24 ,0848 ,3248 .I449 ,739 0.0597 .74 ,3552 1.0952 ,6231 .676 0. - .25 .O887 ,3387 .I535 ,738 0.0647 .75 .3648 1.1148 .6319 ,673 0. - .26 ,0925 ,3525 ,1623 ,738 0.0698 .76 ,3749 1.1349 ,6405 ,670 0. - .27 ,0963, ,3663 ,1711 ,737 0.0751 .77 ,3855 1.1555 ,6489 ,666 0. - .28 ,1002 ,3802 ,1800 ,736 0.0806 .78 .3967 1.1767 ,6573 ,663 0. - .29 .IO42 ,3942 ,1890 ,736 0.0863 .79 ,4085 1.1985 ,6655 ,659 0. - .30 ,1081 ,4081 ,1982 ,735 0.0922 3 0 .4210 1.2210 ,6735 ,655 0. - .3 1 I 12 I ,4221 .2074 .734 0.0982 .8 I ,4343 1.2443 .68 I5 ,651 0.635
- , .32 I 161 ,4361 ,2167 ,.734 0.1044 3 2 ,4485 1.2685 .6893 ,646 0. - .33 ,1202 ,4502 ,2260 ,733 0.1 108 .83 ,4638 1.2938 ,6969 ,641 0. - .34 ,1243 ,4643 ,2355 ,732 0.1 I74 .84 ,4803 1.3203 ,7043 ,636 0. - .35 .1284 ,4784 ,2450 ,732 0.1289 .85 ,4982 1.3482 ,7115 ,630 0. - .36 ,1326 ,4926 ,2546 ,731 0.1311 3 6 ,5177 1.3777 ,7186 ,624 0. - .37 ,1368 ,5068 ,2642 ,730 O 1382 .87 ,5392 1.4092 ,7254 ,617 0. - .38 .1411 ,5211 ,2739 ,729 0.1455 3 8 ,5632 1.4432 ,7320 ,610 0. - .39 ,1454 ,5354 .2836 ,728 0.1529 .89 S900 1.4800 .7384 ,601 0. - .40 ,1497 ,5497 .2934 ,728 0.1605 .90 ,6204 1.5204 .7445 ,592 0. - .41 .I541 ,5641 ,3032 ,727 0.1683 .91 .6555 1.5655 .7504 ,581 0. - .42 ,1586 .5786 ,3130 ,726 0.1763 .92 .6966 1.6166 ,7560 ,569 0. - .43 ,1631 ,5931 ' ,3229 ,725 0.1844 .93 ,7459 1.6759 ,7612 ,555 0.
- ,44 ,1676 ,6076 .3328 ,724 0.1927 .94 .8065 1.7465 .7662 ,538 0.
- .45 ,1723 ,6223 .3428 ,723 0.2012 .95 ,8841 1.8341 ,7707 .5 18 I .O
Thus experimental data are made to fit a head-discharge equation which is structurally
similar to that of a broad-crested weir b.utjn_which the dischape coefficient expresses- the influence ,of streamline curvature in addition to these factorssplaine_d in S s
*act, the same measuring structure can act as a broad-crested weir for low heads
(H,/L < 0.50), while with an increase of head (H,/L > 0.50) the influence of the
streamline curvature becomes significant, and the structure acts as a short-crested weir.
For practical purposes, a short-crested weir with rectangular control section has a
head-discharge equation similar to Equation 1-37, i.e.
m
____.
Q = C ' C - ['- g ]05°'b, h,l.SO
The head-discharge equations of short-crested weirs with non-rectangular throats are
structurally similar to those presented in Section 1.9. An exception to this rule is pro-
vided by those short-crested weirs which have basic characteristics in common with
sharp-crested weirs. As an example we mention the WES-spillway which is shaped
according to the lower nappe surface of an aerated sharp-crested weir and the triangu-
lar profile weir whose control section is situated- above a separation bubble down-
stream of a sharp weir crest.
Owing to the-gressure and velocity distributions above the wsa- indicated
i i F i s r e 1.19, the discharge coefficient (-cd) of a short-crested weir TsAgher than
T h a t of a broad-creste-d-xeir,The rate of departure from the hydrostatic pressure d i s t r
bution is defined by the local centripetal acceleration v2/r (see Equation 1-10).
-P-
Depending on the degree of curvature in the overflowing nappe, an underpressure
may develop near the weir crest, while under certain circumstances even vapour pres-
sure can be reached (see also Annex 1). If the overfalling nappe is not in contact with
the body of the weir, the air pocket beneath the nappe should be aeratedto avoid
an underpressure,wJhich increases the streamline curvature aroF-
mÓre details on this aeration demand the reader is referred to Section 1.14.
II (^) v; 129 II (^) p /
q E
Figure 1.19 Velocity and pressure distribution above a short-crested weir