Open channel hydraulics for engineers. Chapter 5 spillways
Spillways are familiar hydraulic structures built across a stream to control the water level. This chapter emphasizes the classification of weirs and spillways as well as the application of hydraulic formulas for designing their shape and dimensions.
OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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Chapter SPILLWAYS
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5.1. Introduction
5.2. General formula
5.3. Sharp-crested weir
5.4. The overflow spillway
5.5. Broad-crested weir
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Summary
Spillways are familiar hydraulic structures built across a stream to control the water level.
This chapter emphasizes the classification of weirs and spillways as well as the application
of hydraulic formulas for designing their shape and dimensions.
Key words
Spillway; weir; crest; design head
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5.1. INTRODUCTION
Spillways are used at both large and small dams for letting flood flows pass,
thereby preventing overtopping and failure of the dam. A spillway as sketched in Fig. 5.1
is the most common type. Three zones can be distinguished: the crest, the face and the toe
– each with its separate problems.
crest
face
toe
Fig. 5.1. General view of a spillway
A weir is a notch of regular form through which water flows. The term is also applied to
the structure containing such a notch. Thus a weir may be a depression in the side of a
tank, a reservoir, or a channel, or it may be an overflow dam or other similar structure.
Classified in accordance with the shape of the notch, there are rectangular weirs, triangular
or V-notch weirs, trapezoidal weirs, and parabolic weirs. Weirs are usually designed to
control water levels in rice fields or wetlands. They are commonly used as a means of flow
measurement.
Of primary importance for hydraulic structures considered in this chapter is the magnitude
of backwater level they cause upstream of the structure for the given discharge; that is, the
head-discharge relationship for the structure. Both gradually varied and rapidly varied
flows are possible through these structures, but one-dimensional methods of analysis
usually are sufficient and well-developed in this branch of hydraulics. Essential to the
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“hydraulic approach” is the specification of empirical discharge coefficients that have been
well established by laboratory experiments and verified in the field. The determination of
controls in hydraulic analysis also is important, and critical depth often is the control of
interest. The energy equation and the specific-energy head diagram are useful tools in the
hydraulic analyses of this chapter.
5.2. GENERAL FORMULA
The equation for discharge over a weir cannot be derived exactly, because not only
the flow pattern of one weir differs from that of another, but also the flow pattern for a
given weir varies with the discharge. Furthermore, the number of variables involved is too
large to warrant a rigorous analytical approach. Approximate derivations are presented in
most texts. These derivations show effects of gravitational forces in an approximate
manner, but do not include the effects of viscosity, surface tension, the ratios of the
dimensions of the weir to the dimensions of the approach channel, the nature of the weir
crest, and the velocity distribution in the approach channel. A simplified derivation will be
made here to show the general character of the relationship between the discharge and the
most important variables and to demonstrate the nature of the effect of some of the
variables. The derivation will be made for sharp-crested weirs, but as will be shown later, a
similar derivation would apply to weirs that are not sharp-crested. Now, we consider a
rectangular weir, over which the water is flowing as shown in Fig. 5.2.
water surface dh
h
H
L
Fig. 5.2. Rectangular weir
Let, H = height of the water above the crest of the weir,
L = length of the weir, and
Cd = coefficient of discharge (see Chapter 1).
Let us consider a horizontal strip of water of thickness dh at a depth h from the water
surface as shown in Fig. 5.2.
Area of the strip = L.dh (i)
Assuming that the flow does not contract when passing over the weir (i.e. neglecting
streamline curvature), and that the pressure is atmospheric across the vertical section above
the weir and the upstream velocities are small, the velocity of the water through the strip
can theoretically be derived to be:
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theoretical velocity = 2gh (ii)
The discharge per unit width, q, flowing over a weir is generally expressed as:
dq Cd area of strip theoretical velocity
dq Cd .L.dh. 2gh (iii)
The total discharge over the weir may be found by integrating the above equation within
the limits 0 and H:
3
H
h 2
Q Cd .L.dh. 2gh Cd .L. 2g.
H
Cd .L. 2g.H
2 3
2
3 3
2 0
o
or the discharge per unit width, q, is:
q C d 2g.H 2
2 3
(5-1)
3
where H is to be conceived of as the total upstream specific-energy head on the weir crest
supposing that the upstream velocities are negligibly small; Cd is a discharge coefficient,
which can be approximated by Rehbock’s experimental formula (1929):
H
Cd = 0.611 + 0.08 (5-2)
P
where P is measured from top of the crest of the weir to the bottom of the reservoir; P is
called the weir height. Assuming P very large, Cd becomes equal to 0.611. In this case, Eq.
(5-1) can be written as:
q 1.80 H [1.80] = m½s-1
3
2
(5-3)
Experiments show that the rise from the sharp weir crest to the highest point of the nappe
(i.e. the “spillway crest”) is 0.11H (see Figs. 5.4 and 5.6). Using this fact we can express
Eq. (5-3) in terms of HD, the head over the spillway crest. We obtain:
q 2.14 H D [2.14] = m½s-1
3
2 (5-4)
where HD may be termed the design head.
Example 5.1: A rectangular weir, 4.5 m long, has a head of water 30 cm. Determine the
discharge over the weir, if the coefficient of discharge is 0.6.
Solution:
Given: length of weir: L = 4.5 m
head of water: H = 30 cm = 0.3 m
coefficient of discharge: Cd = 0.6
Discharge over the weir Q?
Using the relation:
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Q
2 3
Cd .L. 2g.H 2 = 1.31 m3/s Ans.
3
Example 5.2: The daily record of rainfall over a catchment area is 0.2 million m3. It has
been found that 80% of the rainfall reaches the storage reservoir and then passes over a
rectangular weir. What should be the length of the weir, if the water is not to rise more than
1 m above the crest? Assume a suitable value of the coefficient of discharge for the weir.
Solution:
Given: rainfall = 0.2 x 106 m3 per day
discharge into the reservoir: Q = 80% of rainfall
Q = 0.8 x 0.2 x 10 6 m3/day = 1.85 m3/s
head of water: H = 1 m
Let, L = length of the weir
Take: coefficient of discharge: Cd = 0.6
Using the relation:
Q Cd .L. 2g.H 2
2 3
3
1.85 = 1.77 L
L = 1.045 m Ans.
5.3. SHARP-CRESTED WEIR
5.3.1. Experiments on sharp-crested rectangular weirs
All tests on weirs of this type were made with the nappe fully aerated. When the
crest length L of a horizontal weir (see Fig. 5.3) is shorter than the width of the channel b,
as well as in the case of V-notch weirs, aeration is automatic. However, for horizontal
weirs extending over the full width of the channel, i.e. L/b = 1, air at atmospheric pressure
must be provided by vents. Otherwise the air beneath the nappe will be exhausted, causing
a reduction of pressure beneath the nappe, with a corresponding increase in the discharge
for a given head.
b
V2
L 2 V2
2g
2g
crest
H H
V P crest crest P
section through horizontal- V-notch weir round-crested
sharp-crested weir crested weir weir
L/b < 1
Fig. 5.3. Weirs, definition sketch
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The sharp-crested weir is of fundamental interest, because its associated theory forms a
basis for the design of spillways. Because the edge is sharp, opportunities for boundary-
layer development are limited to the vertical face of the weir, where velocities are low; we
may therefore expect the flow to be substantially free from viscous effects and the resultant
energy dissipation.
Vo 2
2g B total energy-head line
h
H C
p
V2
Vo 2g
A
P p
V2
2g
45
Fig. 5.4. The sharp-crested weir
Fig. 5.4 shows a longitudinal section of flow over such a weir. An elementary analysis can
be made by assuming that the flow does not contract as it passes over the weir (i.e.
neglecting streamline curvature), and that the pressure is atmospheric across the whole
section AB. Under these assumptions the velocity at any point such as C is equal to 2gh
(Henderson, 1966), and the discharge q per unit width accordingly equal to:
H o
V2
V 2 2 Vo2 2
3 3
2g
2ghdh H
2g
2 o
(5-5)
Vo
2g 2g
2
2g
3
the depth h being measured downwards form the total energy-head line, and not from the
upstream water surface. Vo is the approach velocity to the weir.
The effect of the flow contraction may be expressed by a contraction coefficient Cc leading
finally to the result:
V 2 2
3
Vo2 2 2
3
q C c 2g H Cd 2g.H
2 o
3
2
(5-6)
2g 2g 3
3
Vo2 2 Vo2 2
3 3
where the discharge coefficient: C d Cc 1 (5-7)
2gH 2gH
We should expect both Cc and the ratio (Vo2/2gH) to be dependent on the boundary
geometry alone, in particular on the ratio H/P; it follows that Cd should be a function of
H/P alone, which was indeed found by Rehbock (1929); see Eq. (5-2).
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In early experiments on weirs only small quantities of water were available. In most cases
results are given in the form of Eq. (5-1), with a discharge coefficient Cd.
Tests on weirs of this type were conducted by Kindsvater and Carter (1959). Their
tests cover a range of values of H/P from approximately 0.1 to 2.5, a range of heads from 3
cm to 22 cm, and weir heights from 9 cm to 44 cm. They also varied the weir length and
the channel width from 3 cm to 82 cm. In presenting their data they adopted the method
used by Rehbock of including the effect of H in the main body of the equation. Kindsvater
and Carter also introduced a method that includes the effect of the weir length L in the
main body of the equation. Their method is shown in the following three equations:
Q Ce .L e .H
3
2
(5-8)
Le L k L (5-9)
He H k H (5-10)
In these equations kL and kH are factors representing the effects of viscosity and surface
tension, and the subscript e indicates effective values, that is, He is the effective energy-
head. By treating the variables in this manner the authors were able to obtain a single linear
relationship between Ce and H/P for all values of H. The values of kL and kH were obtained
by trying successive values of kL and kH until the values of Ce were obtained that were the
most independent of H and L. They did this not only for their own data, but for several
other groups of experiments as well.
Their equations for Ce, with correcponding values of kH and kL are given hereafter.
The Kindsvater and Carter tests yielded
C e 1.78 0.22
H
(5-11)
P
kH = 0.001 m; kL = - 0.001 m
The Bazin tests yielded
C e 1.80 0.25
H
(5-12)
P
kH = 0.004 m; kL = 0 m
The Schoder and Turner tests yielded
C e 1.77 0.25
H
(5-13)
P
kH = 0.001 m; kL = 0 m
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The US Bureau of Reclamation (USBR) tests yielded
C e 1.78 0.24
H
(5-14)
P
kH = 0.001 m; kL = 0 m
5.3.2. Other types of sharp-crested weirs used for flow measurement
Of the many types of weirs developed in the last 100 years, only a few survived and
find practical use today. Excluding the suppressed (i.e. without lateral contraction effects)
rectangular weir, only the contracted rectangular weir and the triangular weir (see Fig. 5.5)
are employed with any frequency. Other types, such as the parabolic, the circular and the
compound-form weirs have been used from time to time for special applications.
The contracted rectangular weir
This type of weir was subject to considerable experimentation in the USA in the past
century. Most notable were the large scale tests by Francis, 1835, and Hamilton Smith,
1884. The Francis experiments were conducted with weirs between 2.44 m to 3.05 m in
width, with a crest made of a cast iron plate, 6.3 mm in width and carefully planed and
machined in the upstream corner. Francis suggested that the total discharge was diminished
with respect to the suppressed weir due to the contractions occurring at the sides. An
empirical correction was devised, that decreased the width of the weir by 5% of the head h
(see Fig. 5.5) for each lateral contraction. The Francis discharge equation was:
2 H
Q Cd 1 0.1 bH 2gH (5-15)
3 b
with Cd = 0.623
The more modern formulae of Hegly, 1921, and the Swiss Society of Engineers (SIA),
1924, for rectangular contracted weirs reproduce basically the same idea. These formulae
are quoted below:
Hegly:
b 0.004 H2
Cd 0.608 0.045 1 1 0.55
H
(5-16)
B (H P) 2
SIA:
b
2
3.615 3
b B 1 0.5 b
C d 0.578 0.037
2 4 2
2
H
1000H 1.6 B (H P)
(5-17)
B
In these relations, b is the width of the weir and B the width of the rectangular channel.
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B
b
head
measuring
H
section
4 to 5 H
max. P
1 cm to 2 cm
/4 radians
upstream face of
weir plate
B
head
measuring
H
section
4 to 5 H
max. P
Fig.5.5. The contracted rectangular thin-plate weir (top)
and triangular thin-plate weir (bottom) and detail of crest and sides of notch (right)
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Triangular or V-Notch weir
Because the relative error in the measurement of the head becomes important for the
smaller heads (say less than 0.05 m) in a rectangular weir, it has been found advantageous
to replace this type of weir by triangular weirs or notches, when the discharges to be
measured are smaller than about 0.020 m3/s. This type of weir was devised by James
Thomson in England, 1858. Thomson experimented with 90 notches, but it was found
later that other angles were needed for particular applications, in the range from 15 to 120.
In this type of weir the crest should be finished as in the rectangular weir, with a sharp
upstream corner and a crest width of the order of 1 mm to 3 mm. As the weir nappe does
not span the whole width of the channel, it is not necessary to provide aeration ducts
beyond the weir plate.
For historical reasons (derived from Weisbach’s theory) it has become customary to define
a flow equation for a triangular weir with a notch angle as:
8
Q Cd tan H 2 2gH (5-18)
15 2
In many instances the product (8/15)Cd is replaced by a more convenient non-dimensional
coefficient m. This coefficient is expected to vary with the fluid properties and the weir
geometry in much the same way as in the case of the rectangular weir. The velocity-of-
approach effect is, however, much smaller than for rectangular weirs, as the cross-sectional
area of the notch is small in comparison with the channel cross-section. This condition
would ordinarily apply to weirs designed for accurate flow measurement. On the other
hand the surface tension effects are of the same order as in the case of the rectangular
weirs.
5.4. THE OVERFLOW SPILLWAY
5.4.1. The spillway crest
Normally the crest is shaped so as to conform to the lower surface of the nappe
from a sharp-crested weir, as shown in Fig. 5.6. The pressure on the crest will then be
atmospheric, provided that the resistance of the solid surface to flow does not induce a
material change in the pressure distribution. This could be happen only if the boundary
layer over the crest were very thick; and it is known (see Henderson, 1966) that the
boundary layer, which will grow effectively only from the neighbourhood of the point A, is
in fact a very small fraction of the head over the crest. Therefore we may expect the
pressure over the crest to be atmospheric, and in this fact lies the virtue of this crest shape;
pressures above atmospheric will reduce the discharge, and pressures below atmospheric
will increase the discharge, but at the risk of introducing instability and cavitation.
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Vo H HD
C
A
0.11H
P
Fig. 5.6. The crest and the equivalent weir of an overflow spillway
We consider the case of the high spillway. For the equivalent weir, H/P = 0 and the
substitution of the Eq. (5-2) into Eq. (5-1) then leads to the result of Eq. (5-3):
q 1.80 H [1.80] = m½s-1
3
2
(5-19)
Experiments show that the rise from the weir crest to the high point of the nappe (the
spillway crest) is 0.11 H, as in Fig. 5.6. Using this fact we can express Eq. (5-19) in terms
of HD, the head over the spillway crest. We obtain Eq. (5-4):
q 2.14 H D2 [2.14] = m½s-1
3
(5-20)
where HD may be termed the design head; as we have seen, operation at this head will
make the pressure over the crest atmospheric. However, the spillway will also have to
operate at lower heads, and possibly higher heads as well.
The former will evidently result in above-atmospheric pressures on the crest.
As to the details of the crest shape, extensive experiments by the U.S. Bureau of
Reclamation (USBR) have resulted in the development by the U.S. Army Corps of
Engineers of curves which can be described by simple equations, yet approximately close
to the nappe profiles measured in the USBR experiments. The profile for a vertical
upstream face is shown in Fig. 5.7; others were also developed for various angles of the
upstream face to the vertical.
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0.282HD
origin of coordinates
0.175HD
X
R= 0.2HD R= 0.5HD
X
1.85
2.0
Y
HD
Y
HD
Fig. 5.7. Standard spillway crest
(US. Army Engineers Waterways Experiment Station)
5.4.2. The spillway face
Flow down the steep face of the spillway is normally at about 45 to the horizontal.
In this case acceleration and boundary layer development are both taking place during
much of the journey down the spillway face, as shown in Fig. 5.8. Turbulence does not
become fully developed until the boundary layer fills the whole cross section of the flow,
at the point marked C. Downstream of this point the flow might be expected to conform to
the S2-profile (see Chapter 4) but the extreme steepness of the slope introduces more
complications, chiefly the phenomenon of air entrainment, or “insufflation”.
H
limit of boundary layer
C Z
Fig. 5.8. Boundary layer development on spillway face
It is now generally agreed that insufflation begins at this very point C, where the boundary
layer meets the water surface. The resulting mixture of air and water, containing an ever-
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increasing proportion of air, continues to accelerate until uniform flow occurs, or the base
of the spillway is reached.
Clearly the designer will wish to know the velocity reached at the base, or toe, of the
spillway, but the above remarks make it clear that the computation of this velocity would
be tedious and difficult, even if one were certain of the correct assumptions to adopt
concerning the nature of the flow. Considerable work have been done on this problem at
the U.S. Bureau of Reclamation.
The mechanism of air entrainment is not yet completely understood, and reliable field data
on the concentration of entrained air are extremely sparse.
5.4.3. The spillway toe
When the flow reaches the end of the inclined face of the spillway it is deflected
through a vertical curve into the horizontal or into an upward direction (Fig. 5.9). In the
latter case we have the ski-jump and the bucket-type energy dissipators, to be discussed in
Chapter 6.
In either case, centrifugal pressures will be developed which can set up a severe thrust on
the spillway sidewalls. These pressures cannot be accurately calculated by elementary
means, but certain approximations suggest themselves; e.g., if one assumed that the depth
ho at the center of the curve (Fig. 5.8.a) is equal to the depth h1 of the approaching flow,
then the centrifugal pressure at the point O will be equal to:
V12 h1
po (5-21)
R
where V1 and R are also defined in Fig. 5.8.a. This result can only be an approximation, for
a pressure rise along AO must, by the Bernoulli equation, be accompanied by a fall in
velocity, so that the velocity profile will be somewhat as shown in Fig. 5.8.a. The average
velocity will then be less than V1, and the depth ho greater than h1, so that Eq. (5-21) will
not be correct.
V = V1
R1
h1 R
A
h0
O
velocity distribution
(a) Spillway toe
(b) Flip bucket
Fig. 5.9. Flow at the spillway toe, after Henderson and Tierney (1962, 1963)
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A better approximation can be made by assuming that the streamlines crossing OA form
parts of concentric circles, and that the velocity distribution along this line is accordingly
the same as that in the free, or irrotational, vortex, i.e.,
V
C
(5-22)
r
where C is a constant and r is the radius of any streamline. Since the streamlines are
concentric circles, r is also a measure of distance along AO from A to O. If R1 is the radius
of the streamline at A, then C = V1R1. The discharge q across AO is given by:
q V1h1 Vdr V1R 1 r 1dr V1R 1 ln
R R R
(5-23)
R1 R1 R1
ln
h1 R h1 R 1 R
i.e., and ln (5-24)
R1 R1 R R R1
Since h1 and R are known in advance, R1 can be obtained by trial from this equation. Given
R1/R, we can obtain p o, the pressure at O, from the condition:
p o Vo2 V12
1 1
(5-25)
2 2
V R
2 2
1 o 1 1
po
i.e., (5-26)
1
V12 V1 R
2
assuming no energy dissipation between A and O; this assumption appears to be justified
by experiment.
The “free-vortex” method leads to results that are quite accurate within a certain range, but
it suffers from a curious limitation, arising from the fact that the function (lnx)/x has a
maximum value of 1/e, which occurs when x = e, the base of the natural logarithms.
Applying this result to Eq. (5-24), we see that the ratio R/h1 has a minimum value of e
when R/R1 = e, even though R/h1 is by the nature of the problem an independent variable,
which may in practice assume any value at all. The effect of this curious result is that the
theory cannot be applied when R/h1 < e, and a curve displaying the results of the theory
must, as in Fig. 5.10, terminate at the point where R/h1 = e, although lower values of R/h1
are quite possible.
The corresponding terminal value of p o will, from Eq. (5-26), be equal to 1 2 .
1
1 e
2
V1
2
A complete solution of the problem requires the use of the mathematical theory of
irrotational flow. This has been done by Herderson and Tierney (1962, 1963) for the case
where there is an open sluice through the spillway, as shown by broken lines in Fig. 5.9.a
and for the more usual case discussed above, where the toe is a curved solid surface.
Theoretical and experimental results for the latter case are shown in Fig. 5.10, which
displays the behavior of po, the pressure at O, for angles (in Fig. 5.9.a) of 45 and 90. It
is seen that the free vortex method gives results approximating closely to those of the
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complete theory when R/h1 > 6, as does the elementary result of Eq. (5-21). However, the
latter fails to predict any thickening of the jet and therefore seriously underestimates the
total thrust and bending moment on the sidewalls.
1.0
e 2.718
R
h1
0.9
po
1 2 0.865
1
1
V12 e
0.8 2
complete theory
0.7
free vortex approximation
Eq. (5-21)
0.6
po
experiment ( = 45)
V12
1 0.5
2 = 90
0.4 = 45
0.3
0.2
0.1
R
0.0 h1
1 2 3 4 5 6 7 8 9 10 11
Fig.5.10: Flow at spillway toe: Maximum pressure – theory and experiment
(after Henderson and Tierney, 1962, 1963)
All the above discussions imply the assumption that the flow is irrotational. This
assumption is a reasonable one, since losses must be small over the short length of spillway
involved, and the highly turbulent approaching flow must have a transverse velocity
distribution very close to the uniform distribution which is characteristic of irrotational
flow of a perfect fluid. Also, there is little risk of separation, when a solid boundary is
continually curving into the flow, as in this case. Pressure distributions should therefore be
close to those in the irrotational flow of a perfect fluid, and this conclusion is confirmed by
the good agreement between theory and experiment shown in Fig. 5.10.
Further, the effect of gravity has been ignored, so that the pressures derived are purely
those due to centrifugal action. We take gravity into account simply by adding hydrostatic
pressure to the pressure obtained above. This additional pressure may be substantial in the
case of the bucket-type energy dissipator, in which a structure like that of Fig. 5.9.b. is
deeply drowned under a turbulent but stationary eddy.
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5.5. BROAD-CRESTED WEIR
5.5.1. Introduction
Sharp-crested weirs have proved to be too fragile to be considered reliable gauging
structures in open channels, especially in irrigation channels. Thus, other types of
structures with a wider crest were developed. The intention was to prevent damage to the
crest section from floating debris, and to shape the structure such that deposition of
sediment on the upstream side would not seriously alter the coefficient of discharge of the
weir. The structure that resulted from these considerations was termed a broad-crested
weir. A broad-crested weir is a flat-crested structure with a crest length large compared
with the flow thickness (see Fig. 5.11).
total head line
HT H hc
P
Lcrest
(a)
energy-head line
HT H hmin
P
Lcrest
(b)
Fig. 5.11: Flow pattern above a broad-crested weir
(a) Broad-crested weir flow. (b) Undular weir flow
The ratio of crest length to upstream head over the crest must typically be greater than 1.5
to 3 (e.g. Chow, 1973; Henderson, 1966):
1.5 to 3
L crest
(5-27)
HT P
where Lcrest is the crest length in the flow direction;
HT is the upstream total head; and
P is the weir height above the channel bed.
When the crest is “broad” enough for the flow streamlines to be parallel to the crest, the
pressure distribution above the crest is hydrostatic and the critical flow depth is observed
on the weir crest. Broad-crested weirs are sometimes used as critical depth meters (i.e. to
measure stream discharges). The hydraulic characteristics of broad-crested weirs were
studied during the 19th and 20th century. Hager and Schwalt (1994) recently presented an
authoritative work on the topic.
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5.5.2. Broad-crested weir discharge formula
Consider the section on the weir crest in case of a critical flow condition At this
section, Vc gh c , where hc = (HT – P) is the critical depth and (HT –P) the specific-
2
3
energy head at the section with respect to the weir crest as indicated in Fig. 5.11. The
discharge above the weir, equals:
q Vc .h c Vc (H T P)
2
3
q g HT P
2 2 3
or (5-28)
3 3
The above equation may be conveniently rewritten as:
q 1.705 H T P [1.705] = m½s-1
3
2
(5-29)
These equations are used for ideal fluid flow calculation.
Notes:
In a horizontal rectangular channel, assuming a hydrostatic pressure distribution,
the critical flow depth equals:
hc E
2
(5-30)
3
where E is the specific-energy head. The critical depth and the discharge per unit width are
related by:
q2
hc 3 (5-31)
g
q gh 3
c (5-32)
At the crest of a broad-crested weir, the continuity equation and the Bernoulli
equation yield:
HT P h c
2
(5-33)
3
5.5.3. Undular weir flow and discharge coefficients
Undular weir flow
HP
For low discharge (i.e. 1 ), several researchers observed free surface
P
undulations above the crest of a broad-crested weir (Fig. 5.11.b). According to Hubert
Chanson (1999), model studies suggest that undular weir flow occurs for:
1.5
q
(5-34)
gh 3
min
where hmin is the minimum flow depth upstream of the first wave crest. Another criterion
is:
HT P
0.1 (5-35)
L crest
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Chapter 5: SPILLWAYS 105
OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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This equation is a practical criterion based on the ratio of energy head on crest to weir
length. In practice, design engineers should avoid flow conditions leading to undular weir
flow. In the presence of free surface undulations above the crest, the weir cannot be used as
a discharge meter, and waves may propagate in the downstream channel.
Discharge coefficients
Experiment measurements indicate that the discharge versus the total head
relationship departs slightly from the equation for ideal fluid flow:
2
3
q g HT P
2
(5-36)
3
For a real flow rate it holds:
2
3
q Cd g H T P
2
(5-37)
3
The above equation is usually rearranged as:
q Cd g HT P
2 2 3
(5-38)
3 3
where the discharge coefficient Cd is a function of the weir height, the crest length, the
crest width, the upstream corner shape and the upstream total head, as given Table 5.1
below:
Table 5.1. Discharge coefficient for broad-crested weir
Reference Discharge coefficient Cd Range Remarks
(1) (2) (3) (4)
HT P
0.1 1.5
Sharp- corner deduced from
weir 2 laboratory
9
L crest
Hager and 0.85 1 9
4
experiments
1 HT P
Schwalt 7
(1984)
L crest
HP
0.05
Rounde- corner based upon
weir L crest r L crest r laboratory and
1 0.01 1 0.01
Lcrest
P HP
field tests
H P 0.06 m
Bos (1976)
HP
1.5
P
Ackers et al. 0.95 * P 0.15 m
(1978) H P
0.15< T 0.6
P
Notes: * Re-analysis of experimental data presented by Ackers et al. (1978);
r = curvature radius of upstream corner.
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Chapter 5: SPILLWAYS 106
