Transcript Chapter 2: Design of Overflow Structures

Chapter 2:
Design of Overflow Structures
2.1 Overflow Structures
2.1.1
Overflow Gates:
• The overflow structure has a hydraulic behavior that the
discharge increases significantly with the head on the
overflow crest. Remember that Q=CLH3/2.
• The height of the overflow is usually a small portion of the
dam height.
• Further, gates may be positioned on the crest for “overflow
regulation”.
• During the floods, if the reservoir is full, the gates are
completely open to promote the overflow.
• A large number of reservoirs with a relatively small design
discharges are ungated.
Bottom pressure profile
Gated
Ungated
Bottom pressure
profile
• Currently most large dams are equipped with gates
to allow for a flexible operation.
• The cost of the gates increases mainly the
magnitude of the flood, i.e.: with the overflow area.
• Improper operation and malfunction of the gates is
the major concern which may lead to serious
overtopping of the dam.
• In order to inhibit floods in the tailwater, gates are to
moved according to gate regulation.
• Gates should be checked against vibrations.
The Advantages and Disadvantages of Gates
The advantages of gates at overflow structure are:
• Variation of reservoir level,
• Flood control,
• Benefit from higher storage level.
The disadvantages are:
• Potential danger of malfunction,
• Additional cost, and maintenance.
Depending on the size of the dam and its location, one would prefer
the gates for:
• Large dams,
• Large floods, and
• Easy access for gate operation.
Three types of gates are currently favored:
• Hinged flap gates,
• Vertical lift gates,
• Radial gates.
Flap Gate
Vertical Gate
Radial Gate
The flaps are used for a small head of some meters, and
may span over a considerable length.
The vertical gate can be very high but requires substantial
slots, a heavy lifting device, and unappealing
superstructure.
The radial gates are most frequently used for medium or
large overflow structures because of
• their simple construction,
• the modest force required for operation and
• absence of gate slots.
They may be up to 20m X 20m, or also 12 m high and 40 m
wide. The radial gate is limited by the strength of the
trunnion bearings.
• The risk of gate jamming in seismic sites is relatively
small, if setting the gate inside a stiff one-piece frame.
• For safety reasons, there should be a number of
moderately sized gates rather than a few large gates.
• For the overflow design, it is customary to assume
that the largest gate is out of operation.
• The regulation is ensured by hoist or by hydraulic jacks
driven by electric motors.
• Stand-by diesel-electric generators should be provided
if power failures are likely.
2.1.2 Overflow Types
Depending on the site conditions and hydraulic
particularities an overflow structure can be of various
designs:
• Frontal overflow,
• Side-channel overflow, and
• Shaft overflow.
Other types of structures such s labyrinth spillway use a
frontal overflow but with a crest consisting of successive
triangles or trapezoids in plan view.
Still another type is the orifice spillway in the arch dam.
Main Types of Overflow Structures
Frontal Overflow
Side Overflow
Shaft Overflow
•
•
1.
2.
3.
The non-frontal overflow type of spillways are used for small
and intermediate discharges, typically up to design floods of
1000 m3/s.
The shaft type spillway was developed in 1930’s and has proved
to be especially economical, provided the diversion tunnel can be
used as a tailrace. The structure consists of three main elements:
The intake,
The vertical shaft with a 90o bend, and
The almost horizontal spillway tunnel.
Air by aeration conduits is provided in order to prevent cavitation
damage at the transition between shaft and tunnel.
Also, to account for flood safety, only non-submerged flow is
allowed such that free surface flow occurs along the entire
structure, from the intake to the dissipator.
The hydraulic capacity of both the shaft and the tunnel is thus
larger than that of the intake structure.
The system intake-shaft is also referred to as “morning glory
overflow” due to similarity with a flower having a cup shape.
Morning Glory Spillway
The Monticello Dam
Morning Glory Spillway
• The side channel overflow was successively used at
• the Hoover dam (USA) in the late 1930’s.
• The arrangement is advantageous at locations where a
frontal overflow is not feasible, such as earth dams, or
when a different location at the dam site yields a better
and simpler connection to the stilling basin.
• Side channels consist of a frontal type of overflow
structure and a spillway with axis parallel to the overflow
crest.
• The specific discharge of overflow structure is
normally limited to 10 m3/s/m, but for lengths of over
100 m.
• This type of overflow is restricted to small and medium
discharges.
Side-Channel Spillway
• The frontal type of overflow is a standard overflow structure, both
due to simplicity and direct connection of reservoir to tailwater. It can
normally be used in both arch and gravity dams. Also, earth dams
and frontal overflows can be combined, with particular attention
against overtopping. (eg: Hasan Ugurlu Dam)
• The frontal overflow can easily be extended with gates and piers to
regulate the reservoir level, and to improve the approach flow to
spillway.
• Gated overflows of 20 m gate height and more have been
constructed, with a capacity of 200 m3/s per unit width. Such
overflows are thus suited for medium and large dams, with large
floods to be conveyedto the tailwater.
• Particular attention has to be paid to cavitation due to immense
heads that may generate pressure below the vapor pressure in the
crest domain.
• Also, the gate piers have to be carefully shaped in order to obtain a
symmetric approach flow.
• The downstream of frontal overflow may have various shapes.
Usually, a spillway is connected to the overfall crest as a transition
between overflow and energy dissipator.
• Also, the crest may abruptly end in arch dams to
include a falling nappe that impinges on the tailwater.
Another design uses a cascade spillway to dissipate
energy right away from the crest end to the tailwater,
such that a reduced stilling basin is needed.
• The standard design involves a smooth spillway that
convey flow with a high velocity either directly to the
stilling basin, or to a trajectory bucket where it is
ejected in the air to promote jet dispersion and
reduce the impact action.
Significance of Overflow Structure
•
•
•
•
According to ICOLD* (1967),the overflow structure and the
design discharge have a strong impact on the dam safety.
Scale models of overflow structures are currently needed in
cases where:
The valley is narrow and approach velocity is large such that
asymmetric flow pattern develops.
The overflow and pier geometry is not of standard shape, and
Structures at either side of the overflow may disturb the spilling
process.
For all other cases, the design of the overflow is so much
standardized that no model study is needed, except for details
departing from the recommendations.
• *
ICOLD= International Commisson for Large Dams, Paris
2.2 Design Discharge of Spillway
• 2.2.1
Concept of Crest Height
• A spillway is a safety structure against overflow. It should inhibit the
overflow of water at locations which were not considered.
• The spillway is the main element for overflow safety and
especially the safety against dam overtopping.
• A structure is known to be safe against damage if the load is
smaller than the resistance.
• What are the determining loads, and the resulting resistance with
respect to the overflow safety? Are these
• Crest heights
• Discharges
• Water volumes
??
•
•
•
•
In concept of crest height, the load is composed of the sum of:
Initial depth, h
A depth increase, r, due to a flood
A wave depth, b
load=h+r+b ……........................................(1)
Accordingly, the resistance of the crest height, k, (i.e.: the lowest
point) may be defined as:
(h+r+b)-k ≤ 0 ……………………………. (2)
Overtopping occurs provided that (h+r+b)-k >0.
The corresponding probability of overtopping is the complementary
value of the security, and also coupled to the probabilities of the
parameters h, r, b, and k.
Dam with reservoir
Probability of overtopping, p
Initial depth, h, before flood flow,
Depth, r, of flood level
Maximum reservoir depth, hmax
Maximum run-up height, bmax
Wave height, b
freeboard, f
Height of crest, k
• The probability of the initial depth h follows from the policy of
reservoir operation. For an existing dam, it can be obtained
from reservoir level records. For a future dam, it must be
estimated from the planned policy.
• The probability of the flood depth increase, r, can be
determined from flood routing. The result depends on the
probabilities of the approach flood and reservoir outflow. The
probability of the approach flood depends on general flood
parameters such as:
• Maximum discharge,
• Time to peak,
• Time of flood, and Flood volume.
Regarding the reservoir outflow, the degree of aperture of
the outflow structure has to be accounted for.
•
•
The probability of wave depth b, depends mainly on:
The character of the wind,its direction, and
The run-up slope of the dam relative to the water surface.
In particular cases, ship waves and surge waves due to shore
instabilities ( including rock and snow avalanches, and land slides)
are also included.
normally, the parameter k is considered as a fixed number and not a
stochastic value, which is an acceptable assumption for concrete
dams. For high earth dams, the probability of settlement could be
introduced, but this is often omitted and a maximum settlement is
accounted for.
The probability of combination of the parameters h, b, r, and k thus
mainly results in a variation of six basic variables:
1.
2.
3.
4.
5.
Initial flow depth, h
Peak flood discharge,
Time to peak,
Aperture degree
velocity and direction of wind
Assuming that these basic
variables are stochastically
independent, the result is
straight forward.
• As an example, one could apply the Monte Carlo Method* to obtain a
representative number of combinations, that is let
• s= number of combinations of (h+r+b-k)
• n= number of those sums with s≤0,
• Then the safety q against overflow is:
• q=n/s …………………………………(3)
• If m=s-n is the number of combinations where s>0, then the
probability of overflow is:
• p=m/s……………………………………(4)
The concept described is obvious, as it accounts for the relevant
parameters: depth of reservoir, and depth of crest. It can be used for
sensitivity analysis and answer questions such as:
What is the change of overtopping safety for changes in reservoir
operation? (relating to parameter h)
What is the effect of time to peak for a given peak discharge?
(relating to parameter r)
(*Monte Carlo method is a risk and decision analysis tool)
•
•
•
•
A set of stochastically independent parameters can
normally not be assumed:
In certain regions, large floods are combined with
thunderstorms and flood waves are thus related to wind
waves.
In other regions, the floods have their origin, far upstream
from the reservoir and accompaniying winds may hardly
reach the dam.
Normally, dams are erected to store water during the rain
period for the dry season.
Accordingly, the reservoirs have usually reached a high
level at the end of the rain period. The determining floods
occur often at the end of the rain period, such that the initial
depth is stochastically related to the flood depth.
• The design of spillways is often based on a fixed value of kfix for the
crest height. Further, the wave run-up height is determined more or
less independently from the season and a maximum value of bmax of
undetermined probability is considered. Then the modified equation
will become:
h+r+bmax-kfix ≤ 0 ............................. (5)
Further, it is assumed that the initial reservoir level is equal to the
maximum reservoir level, i.e.: at the maximum reservoir height, hmax.
Therefore above equation will be:
hmax +r+bmax-kfix ≤ 0 ............................. (6)
The only remaining stochastic parameter is thus r.
It is impossible to determine the security against overflow along this
model, because extreme values and stochastic values can not simply
be superposed.
Another popular approach uses instead of parameters kfix and hmax,
the free board height f = kfix-hmax, and requires that:
r ≤ (f-hmax)
2.2.2 Concept of Water Volumes
• The heights h, r, b, and k are related to particular volumes of
reservoir such as
• Vh= resrvoir volume at initial reservoir level,
• Vr= flood storage volume,
• Vb= wave run-up volume,
• Vk= maximum reservoir volume up to the dam crest level.
• (h+r+b-k) ≤ 0 equation may be written analogously with regard to
volume V as:
• (Vh+Vr+Vb-Vk) ≤ 0 ..................................... (9)
• And all concepts presented earlier can similtaneously be transposed.
• Again the result would be the same: the security against overtopping
can not be predicted by using stochastic parameters which are not
indepent to each other.
2.2.3 Concept of Discharges
• The effectively needed storage volume Vr can be determined from the
mass balance as:
v r   Qa - Qz  dt  Qr dt
T
T
• Where Qa= reservoir inflow during the filling time T,
Qz= reservoir outflow during the same time T,
Qr= reservoir storage,
The relation with the reservoir height h is:
dh
Qr = A
dt
Where A=A(h) is the reservoir surface. The filling time ends when Qz=Qa,
i.e.: when Qr=0
• Therefore it can be written that:
Vr ≤ VR
• Needed storage volume ≤ Available storage volume:
Q
dt

Q
dt

V
z
a
R


T
• or
T
Vz-Va ≤ VR
Where Vz= reservoir inflow volume during the filling time T,
Va= corresponding reservoir outflow volume
In fıgure below these relations are explained
Qz
Qr
Vz
Vr
Qa
r
Va
Q
V
Qa
Qz
a) Hydrograph of inflow flood, Qz(t)
b) Hydrograph of reservoir outflow, Qa(t)
c) Hydrograph of superposition and required
storage volume Vr until the end of time rise T,
d) Hydrograph of reservoir storage, Qr(t)
e) Hydrograph of increase of storage, r(t)
Maximum of functions
a) Hydrograph of inflow flood, Qz(t)
b) Hydrograph of reservoir outflow, Qa(t)
c) Hydrograph of superposition and required storage volume Vr
until the end of time rise T,
d) Hydrograph of reservoir storage, Qr(t)
e) Hydrograph of increase of storage, r(t)
Maximum of functions
The probabilities of (Vz-Va), and VR may be used to
determine the securities of overflow and relating
probabilities.
These computations become again useless if fixed values
instead of stochastic values are admitted.
A popular example involves the so-called (N-1) condition
for reservoir outflow. Instead of relating the availability of a
regulated outflow to probability, one is typically faced with
a situation such as: out of the N outlets, there is one
( and the one with largest capacity ) not available.
Other examples, such as fixing the maximum reservoir depth hmax
as the initial depth for floods have already been mentioned.
The most important parameter is the flood wave, such as shown
in figure below, and characterized with the peak discharge Qzmax
and time to peak tz.
The temporal wave profile is given as an empirical function Qz(t).
The determining flood for reservoir volume and spillway structure
is called design flood.
2.2.4 Design assumption
Numerous dam failures due to overtopping point to the significance
of the design flood.
Its resulting probability of failure has to be minimum, at least
infinitely small.
The lifespan of a usual dam is of the order of 100 years. Therefore,
the probability of failure has to be related also to this period.
If a value of 1%, or 0.1 % for the entire lifespan is assumed, then the
probability of failure per year is 10-4, and 10-5, respectively.
For a large potential of damage, one would choose one or two
orders of magnitude smaller .
How can such small values of probability be guaranteed?
It is not the purpose here to outline the corresponding philosophies of
each country worldwide.
Computational or Intuitive Approach
The probability of failure can be computed by combining the parameters
mentioned with their probability of occurrence such that the probability
of overtopping at the lowest point of the dam crest can be determined.
The difficulty of the procedure is in the estimation of probabilities of
some parameters.
For dams, which have existed for several decades and whose safety
against overflow is permanently checked, the wave run up has
eventually been observed and the availability of the outflow structures is
known. Also, information regarding the height of initial flow depth before
floods is probably available.
For new dams, estimations have to be advanced, however. It was
discussed that some parameters under consideration are stochastically
dependent, which adds further complications.
The probability of failure cannot be determined if some stochastic
parameters are assumed to have a certain probability, and others are
considered as fix values. The latter correspond normally to intuitively
chosen maxima. Such a mixed approach is currently the common
approach, however, and there is nothing to counter as long as no
probability computation is performed.
Normally, the mixed approach involves a rare design flood, for which
considerations of probability are still appropriate.
As an example, a 1000-year flood is chosen. Then, intuitive
security factor on parameters such as the initial reservoir
outlets are introduced.
As mentioned earlier, the maximum reservoir elevation is
often set equal to maximum reservoir level and the (N-1)
condition is added in relation to reservoir outlets
Further, an immobile power plant is considered, that is the
related power plant, the water supply station, or irrigation
works are switched off, and the free board is chosen higher
than the maximum wave run-up
With these additional safety measures, a probability of
overtopping well below 0.1 % within 100 years may be
achieved, i.e. a value of practically zero.
Design flood
The design flood is a reservoir inflow of extremely small probability, of
1000 or even 10 000 years of occurrence
To estimate these rare values, a data series is evidently not available.
There are conventions, however, by which extrapolations can be made
based on a data series of several decades. These extrapolations of the
reservoir inflow discharge, or the rainfalls are difficult to interpret. They
include
– knowledge of local particularities, and
– a detailed hydrological approach
The times when some flood discharge formula has been applied without
particular reference to a catchments area have definitely passed. As an
engineer would hardly transpose the geology of one dam site to the other,
it is impossible to use hydrologic data to cases other than considered.
Actually, two different design cases are used in many countries,
considering a smaller and a larger design flood. The smaller design
flood has a return period of the order of 100 years. It must be received
and diverted by the reservoir without damage. Often, a full reservoir
level is assumed and all intakes for power plants etc. are blocked, and
(N-1) spillway outlets are in operation.
Whether the bottom outlet can be accounted for diversion is a question,
but there is a tendency to include it in the approach. The freeboard is
specified and must be observed.
For the larger design flood, a return period of 10 000 years is
considered, for example
As such an extra ordinary event can be extrapolated from the limited
data available only as a rough estimation, other conventions are used.
• One approach increases the 1000 year flood by 50% both in peak
discharge and time to peak.
• Another approach is based on the concept of the possible maximum
flood (PMF). Accordingly, a rainfall-runoff model with the most
extreme combination of basic parameters is chosen, and no return
period is specified. This design flood has to be diverted without a
dam breaching. However, small damages at the dam and
surroundings may occur. Therefore, the conditions for wave run-ups
and the availability of (N-1) outlets, among others, are not entirely
satisfied.
In some countries, the definition of the design flood is also related to
the potential damage due to a dambreak. As prediction of such a
potential for the next century is difficult, hardly any figures are given.
A usual compromise is to increase the design quantity if high dams
and large reservoir volumes are involved.
Design Flood of Spillway Structure
• The design flood Qdmax of the spillway may be determined from Equation:
T Qz dt  TQadt VR
• that is from the inflow design flood, and includes the described effects of
initial reservoir depth, reservoir freeboard, and availability of outflow
structures.
• Where Qa= reservoir inflow during the filling time T,
Qz= reservoir outflow during thesame time T,
Va= available storage volume
2.3 Frontal Overflow
•
1.
2.3.1 Crest Shapes
Overflow structures of different shapes are:
Straight
(standard)
2.
Curved
3.
Polygonal
4.
Labyrinth
Plan view
The labyrinth structure has an increased overflow capacity with
respect to the width of the structure.
Labyrinth spillway
2.3.2 Standard Crest Shape
• When the flow over a structure involves curved streamlines with the
origin of curvature below the flow, the gravity component of a fluid
element is reduced by the centrifugal force.
If the curvature is sufficiently large, the internal pressure may drop
below the atmospheric pressure and even attain values below the
vapor pressure for large structures. Then cavitation may occur
with a potential cavitation damage. As discussed, the overflow
structure is very important for the dam safety. Therefore, such
conditions are unacceptable.
For medium and large overflow structures, the crest is shaped so as
to conform the lower surface of the nappe from a sharp-crested
weir.
•
•
The transverse section of an overflow structure may be rectangular,
trapezoidal, or triangular.
In order to have a symmetric downstream flow, and to
accommodate gates, the rectangular cross section is used almost
throughout.
•
1.
2.
3.
The longitudinal section of the overflow can be made either;
Broad-crested.
Circular crested, or
Standard crest shape (ogee-type)
•
•
For heads larger than 3 m, the standard overflow shape should be used.
Although its cost is higher than the other crest shapes, advantages result both in
capacity and safety against cavitation damage.
The crest shape should be knife sharp, with a 2 mm horizontal crest, and 45o downstream
bevelling.
H
2 mm
W
45o
•
•
•
In order to inhibit the scale effects due to viscosity and surface tension, the head on
the weir should be:
H≥ 100 mm, and the height of the weir, W ≥ 2Hmax
Then, the effects of approach velocity are insignificant.
Flow over a sharp-crested weir
Flow over a sharp-crested weir
• In order to inhibit the scale effects due to viscosity and surface
tension, the head on the weir should be:
• H≥ 100 mm, and the height of the weir,
• W ≥ 2Hmax
• Then, the effects of approach velocity are insignificant.
The shape of the crest is important regarding the bottom pressure
distribution. Slight modifications have a significant effect on the
bottom pressure, while the discharge characteristics remain
practically the same.
The geometry of the lower nappe cannot simply be expressed
analytically. The best known approximation is due to US Corps of
Engineers (USCE1970). They proposed a three arc profile for the
upstream quadrant and a power function for the downstream
quadrant, with the crest as origin of Cartesian coordinates (x,z).
USCE Crest Shape
The significant scaling length for standard overflow structure is the
so-called design head, HD.. All other lengths may be
nondimensionalized with HD. The radii of the upstream crest profile
are:
R3
R1
R2
 0.50,
 0.20,
 0.04
HD
HD
HD
The origins of curvature O1,O2, and O3, as well as the transition
points P1,P2, and P3, for the upstream quadrant are;
Point O1
x/HD 0.00
z/HD
O2
O3
P1
P2
P3
-0.105 -0.242 -0.175 -0.276 -0.2818
0.500 0.219
0.136
0.032
0.115
0.136
• The downstream quadrant crest shape was originally proposed by
Craeger as:
 x
z
 0.50
HD
 HD
1.85



,
for x  0
• This shape is used up to so-called tangency point with a transition to
the straight-crested spillway.
• The disadvantage of USCE crest shape is the abrupt change of
curvature at locations P1 to P3 and at the origin. Such a crest
geometry can not be used for computational approaches due to the
curvature discontinuities.
• An alternative approach with a smooth curvature was provided by
Hager:
Z *  X *nX * ,
for X *  0.2818
• Where (X*,Z*) are transformed coordinates based on USCE shape
as:
• X*=1.3055(x’ +0.2818)
• Z*=2.7050(z’ +0.136) with x’=x/HD, and z’=z/HD
• This equation has the property that the second derivative is
d 2Z *
1
 *
*2
dX
X
• The inverse curvature varies linearly with x*.
• For design purposes, the difference between the two crest
geometries are usually negligible.
The crest shape given above for vertical spillways for which the velocity of
approach is zero, i.e.; for HD/P→0, where P is the height of the spillway.
•
In general, the shape of the crest depends on:
1. The design head HD,
2. The inclination of the upstream face,
3. The height of the overflow section above the floor of the entrance
channel (which influences the velocity of approach to the crest).
The crest shapes have been studied extensively in the Bureau of
Reclamation Hydraulic Laboratories. For most conditions, the data can
be summarized as:
q  CH D3 / 2  dischargeper unit width
q
Va 
 velocity of approach
p  h0
q2
ha 
 velocity of approachhead
2
2g p  h 0 
Elements of Nappe-Shaped Crest Profile
ha
h0
HD
xc
yc
P
R2
x
R1
y
• The portion upstream from the origin is defined as either a single
curve and a tangent, or as a compound circular curve. The portion
downstream defined by
n
 x 
y

 K 
H0
H0 
• Where K and n are constants whose values depend on the upstream
inclination and on the velocity approach head, ha.
Values of K
Values of n
Values of xc
Values of yc
Values of R1 and R2
2.3.3 Discharge Characteristics
•
•
•
•
•
•
1.
2.
3.
4.
5.
The discharge over an ogee crest is given by the formula:
3/2
e
Q  CLH
Where:
Q=discharge,
C=discharge coefficient,
L=effective length of crest,
He=total head on the crest, including the velocity of approach head, ha.
The discharge coefficient, C, is influenced by a number of factors:
The depth of approach,
Relation of actual crest shape to the ideal nappe shape,
Upstream face slope,
Downstream apron interface,
Downstream submergence.
Pier and Abutment Effects
• Where crest piers and abutments are shaped to cause side
contractions of the overflow, the effective length, L, will be less than
the net length of the crest. The effect of end contractions may be
taken into account by reducing the net crest length as follows:
• L=L’-2(NKP+Ka)He
• Where:
• L= effective length of crest,
• L’= net length of crest,
• N= number of piers,
• KP= pier contraction coefficient,
• Ka= abutment contraction coefficient,
• He= Existing total head on the crest.
2.3.4 Coefficient of Discharge for Ogee Crest
• i) The effect of depth of approach
• For a high sharp-crested weir placed in a channel, the velocity of
approach is small and the underside of the nappe flowing over the
weir attains the maximum contraction.
• As the approach depth (p+ho) decreaesed, the velocity of approach
increases, and the vertical contraction diminishes.
• If the sharp-crested weir coefficients are related to the head
measured from the point of maximum contraction instead of to the
head above the sharp crest, coefficients applicable to ogee crests
can be established.
• For an ideal nappe shape, i.e.: He
HD
1
2
q  C d 2g H 3 / 2
3
Cd  0.611 0.08
H
, (P  W)
P

yc
2

q  C d 2g 1 
3
 HD
2
 C  Cd
3
HD  H  y c




y
 H  HD  y cH 1  c
HD
D
3/ 2

y
2g 1  c
 HD
HD
3/ 2



Q  CLH D
3/ 2



Q
 q   CH D3 / 2
L
3/ 2
In the text book, the discharge coefficient C is given as function of
x=H/HD only:
2 
4 
C
1
for any angle  and   3


3 3  9  5 
• For x→0, the overflow is shallow and almost hydrostatic pressure
distribution occurs. Then the overflow depth is equal to critical depth
and the discharge coefficient C= 2/3√3=0.385. For design flow X=1,
and Cd=0.495.
• The relationship of the ogee crest coefficient, C,, to various values of
P/H, is shown on Fig. 1. These coefficients are valid only when the
ogee is formed to the ideal nappe shape; that is, when He/H0 = 1.
Fig.1 Discharge Coefficients for vertical-faced
Ogee Crest
Effect of Heads Different from Design Head
When the ogee crest shape is different from the ideal shape or when the
crest has been shaped for a head larger or smaller than the one under
consideration, the discharge coefficient will differ from that shown on
Fig. 1. A wider shape will result in positive pressures along the crest
contact surface, thereby reducing the discharge. With a narrower crest
shape, negative pressures along the contact surface will occur, resulting
in an increased discharge. Fig. 2 shows the variation of the coefficient
as related to values of He/H0, where He, is the actual head being
considered. An approximate discharge coefficient for an irregularly
shaped crest whose profile has not been formed according to the
undernappe of the overflow jet can be estimated by finding the ideal
shape that most nearly matches it. The design head, HO, corresponding
to the matching shape can then be used as a basis for determining the
coefficients . The coefficients for partial heads on the crest, for preparing
a discharge-head relationship, can be determined from Fig. 2.
Fig.2 Discharge Coefficients for other than the
design head
Effect of Upstream Face Slope
For small ratios of the approach depth to the head on the crest, sloping
the upstream face of the overflow results in an increase in the discharge
coefficient. For large ratios the effect is a decrease in the coefficient.
Within the range considered in this text, the discharge coefficient is
reduced for large ratios of P/H, only for relatively flat upstream slopes.
Fig. 3 shows the ratio for the coefficient for an overflow ogee crest with
a sloping (inclined) face, Ci, to the coefficient for a crest with a vertical
upstream face, Cv, as obtained from Fig. 1 (and as adjusted by Fig. 2 if
appropriate), as related tovalues of P/H0,.
Fig.3 Discharge Coefficients for ogee-shaped
crest with sloping upstream face
Effect of Downstream Apron Interference and
Downstream Submergence
When the water level below an overflow weir is high enough to affect the
discharge, the weir is said to be submerged. The vertical distance from
the crest of the overflow to the downstream apron and the depth of flow
in the downstream channel, as it relates to the head pool level, are
factors that alter the discharge coefficient.
Five distinct characteristic flows can occur below an overflow crest,
depending on the relative positions of the apron and the downstream
water surface:
(1) flow can continue at supercritical stage;
(2) a partial or incomplete hydraulic jump can occur immediately
downstream from the crest;
(3) a true hydraulic jump can occur;
(4) a drowned jump can occur in which the high-velocity jet will follow
the face of the overflow and then continue in an erratic and
fluctuating path for a considerable distance under and through the
slower water; and
(5) no jump may occur-the jet will break away from the face of the
overflow and ride along the surface for a short distance and then
erratically intermingle with the slow moving water underneath. Fig. 4
shows the relationship of the floor positions and downstream
submergences that produce these distinctive flows.
• Where the downstream flow is at supercritical stage or where the
hydraulic jump occurs, the decrease in the discharge coefficient is
principally caused by the back-pressure effect of the downstream
apron and is independent of any submergence effect from the
tailwater.
Fig.4 Effect of downstream influences on the flow
over the weir crest
Downstream depths where jump occur
Downstream depth insufficient to form a good jump
depths sufficient to form a good jump
depths excessive to form a good jump
Drown jump with diving jet
No jump jet on surface
Fig.5 shows the effect of downstream apron conditions on the discharge
coefficient. It should be noted that this curve plots, in a slightly different
form, the same data represented by the vertical dashed lines on Fig.4.
As the downstream apron level nears the crest of the overflow, (hd +
d)/H, approaches 1.0, and the discharge coefficient is about 77 percent
of the coefficient for unretarded flow. On the basis of a coefficient of 4.0
for unretarded flow over a high weir, the coefficient when the weir is
submerged will be about 3.08, which is virtually the coefficient for a
broad-crested weir.
From Fig.4, it can be seen that when (hd + d)/H, exceeds about 1.7, the
downstream floor position has little effect on the coefficient, but there is
a decrease in the coefficient caused by tailwater submergence.
Fig.5 Ratio of discharge coefficients resulting
from apron effects
Fig.6 shows the ratio of the discharge coefficient where affected by
tailwater conditions to the coefficient for free flow conditions.
This curve plots, in a slightly different form, the data represented by the
horizontal dashed lines on Fig.4. Where the dashed lines on Fig.4 are
curved, the decrease in the coefficient is the result of a combination of
tailwater effects and downstream apron position.
Fig.6 Ratio of discharge coefficients caused by
tailwater effects
2.3.5 Uncontrolled Ogee Crest Design
Example on Design of a Spillway Crest
Design an uncontrolled overflow spillway crest, to
discharge 56 m3/s at 1.5-meter head. The upstream face
of the crest is sloped 1:1, and the entrance channel is 30
m. long. A bridge is to span the crest, and 50 cm- wide
bridge piers with rounded noses are to be provided. The
bridge spans are not to exceed 6m. The abutment walls
are rounded to a 1.5 m radius, and approach walls are to
be placed at 30o with the centerline of the spillway
entrance.
Procedure 1:
• First assume the position of the approacah anad
downstream apron level with respect to crest level, say
0.60 m below the crest level, i.e:
• Let P=0.60 m
• Then He+P≈ 2.1 m approximately
• To evaluate the approach channel losses, assume a
value of C to obtain an approximate approach velocity:
P
0.6

 0.4  C  2.07 (From Fig.1)
H e 1.5
Then the dischargeper unit length of the crestlength, q, is :
q  CH3/2  2.07( 1.5 )3 / 2  3.803m3 / s / m
The velocity of approach, Va , is then equal to :
q
3.803
Va 

 1.811 m/s
He  P
2.1
The approach velocity head, ha , is :
V a2 1.8112
ha 

 0.167 m
2g 2x 9.81
• In order to compute the frictional losses, we can use the
Manning formula:
V a
1
n
R 2 / 3 Sf 

V a n 2
S
,
f
for concreten  0.0225
1.811x 0.02252
Sf 
2.14 / 3
Sf 
h
L 
R
4/3
R  He  P 
 0.00062
Then the total approach channel friction loss,h , is :
h  S f L  0.00062x 30  0.0186m
• Assuming an entrance loss into the approach channel
equal to =0.1ha, the total head loss in the approach
channel is approximately:
h  total h  f h  m
h  total  0.0186 0.1x 0.167  0.0353m  0.04 m
• The effective head H0 is equal to: H0=1.5-0.4=1.46 m.
H 0 1.46 m
P
0.60

 0.411
H 0 1.46
From Fig.1, C0=2.08
Fig.3 is used to correct the discharge coefficient for the
inclined upstream slope:
1:1
Cinc
 1.018
C0
P
0.60

 0.411
H 0 1.46
Then Cinc  1.018x 2.08  2.12
Next the relationships of (hd+d)/He and hd/He are
evaluated to determine the downstream effects.
The value of (hd+d)≈H0+P=1.46+0.60=2.06
hd  d 2.06

 1.41 from Fig.252
He
1.46
hd
 0.91 at supercritical flow
He
If supercritical flow prevails, hd should be equal to:
Vd2
hd  0.91 xHe  0.91x1.46  1.33 m 
2g
and d should be equal to : d  2.06 - 1.33  0.73 m
With the indicated unit discharge q=3.803 m3/s/m, the
downstream velocity will be approximately:
q 3.803
Vd  
 5.3 m/s
d
0.73
Vd2
hv 
 1.383m  1.33 m  hd
2g
The closeness of the values of hd and hv, verifies that flow is
supercritical . From Fig.252 it can be seen that the downstream
effect is due to apron influences only and the corrections shown
in Fig. 5 will apply.
To evaluate the effecet of submergence:
hd  d  P  H 0  0.60  1.46  2.06
hd  d 2.06
Cs

 1.41 from Fig.5
 0.966
H0
1.46
Cinc
Therefore C s  0.966x 2.12  2.05
C=2.05
This coefficient has now been corrected for
all influencing effects.
The next step is to determine the required crest length.
For the design head, H0, of 1.46 m, the required effective
crest length, L, is equal to:
Q
56
L

 15.485m.
3/ 2
3/ 2
CH 0
2.05x 1.46
To correct for the pier and abutment effects, the net
length is:
L  L  2NKP K a H e
If the bridge spans are not to exceed 6 m, two piers will
be required for the approximate 16 m total span and N
will be equal to 2. Then:
L  15.485  22x0.01 01.46  15.543m
This procedure establishes a coefficient of discharge for
the design head. For computing a rating curve , Q v.s H,
coefficient for lesser heads must be obtained. Since
variations of the different corrections are not consistent,
the procedure for correcting the coefficients must be
repeated for each lesser head.
Procedure 2:
• First assume an overall coefficient of discharge, say
2.00. The discharge per unit length, q, is then equal to:
q  CH3/2  2.00
( 1.5 )3 / 2  3.674m3 / s / m
• Then the required effective length of crest ,L, is equal to:
Q
56
L 
 15.24 m.
q 3.674
• Next, the approach depth is approximated by use of
Fig.1
P
C  2.00 
 0.26 (From Fig.1)
He
P  0.26H e  0.26x 1.50  0.39 m.
• Thus the approach depth P can not be less than 0.39 m.
To allow for other factors which may reduce the
coefficient, an approach depth of about 0.60 m might be
reasonably assumed.
• With P=0.60 m the computation for approach losseswill
be the same as in procedure 1 solution, and the effective
head H0 will become 1.46 m.
• Similarly, the value of Cinc=2.12.
• Since, the overall coefficient of 2.00 was assumed for
1.5 m gross head, the corresponding coefficient for 1.46
m effective head is:
q  C gross 1.53 / 2  C 0 1.463 / 2
2.001.5
3/ 2
 C 0 1.46
3/ 2
 1.46 
 C0  2.00

 1.50 
3/ 2
 2.08
• The submergence ratio:
Cs
2.08

 0.98 and from Fig.5 :
C inc 2.12
hd  d
 1.485, thus hd  d  1.485x 1.46  2.17 m.
He
Therefore the downstream apron should be placed
• 2.17-1.46=0.71 m below the crest level.
• Since it was demonstrated previously that the pier and
contraction effects are small, they can be neglected in
this example, and the crest length is therefore 15.25 m.
• This crest length and the downstream apron position can
be varied by altering the assumptions of overall
coefficient, and approach depth.
Crest Shape:
q  CH3/2  2.08
( 1.46)3 / 2  3.67 m3 / s / m
The velocity of approach, Va , is then equal to :
Va 
q
The approach velocity head, ha , is :
h P
V a2
q2
3.672
0.686
ha 



2
2
2g 19.62h  P 
h  P 2
19.62h  P 
h  H e  P  - ha h  P  2.06
ha 
( i 1 )

H
0.686
 P  - ha 
as an initial guess

( i) 2
e
h  P  2.06  ha 
( 0)
by iteration ha can be obtained
0.686
0.686


 0.19 m
2
2
h  P  2.06
ha 
( i 1 )

H
0.686
 P  - ha 
as an initial guess

( i) 2
e
h  P  2.06  ha 
( 0)
by iteration ha can be obtained
0.686
0.686


 0.162 m
2
2
h  P  2.06
h  P  2.06  ha
( h+P) (m)
ha (m)
2.06
0.162
1.898
0.19
1.87
0.196
1.864
0.1976
1.862
0.198
1.862
0.198
Therefore ha=0.198 m.
• Now we can determine the crest shape:
ha 0.198

 0.1356 & 3 : 3  K  0.52, and n  1.748
H 0 1.46
xc
 0.194  x c  0.194x 1.46  0.283m
H0
yc
 0.038  y c  0.038x 1.46  0.055 m
H0
R1 R 2

 0.455  R1  R 2  0.455x 1.46  0.66 m
H0 H0
1.748
 x 
y

 0.52
H0
H0 
Discharge rating curve
He/H
He
C/C0
Ci
Hd+
d
Hd+
d/He
Cs/C
Cs
q
He+
P
Va
ha
Sf
hm
hl
HG
Q
0.1
0.14
6
0.82
1.74
0.74
6
5.11
1.00
1.74
0.09
7
0.74
6
0.13
0.00
1
0.00
0.0
0.00
0.15
1.50
0.2
0.29
2
0.85
1.80
2
0.89
2
3.05
1.00
1.80
2
0.28
4
0.89
2
0.32
0.00
52
0.00
01
0.0
0.00
0.29
4.41
0.4
0.58
4
0.90
1.90
8
1.18
4
2.03
1.00
1.90
8
0.85
2
1.18
4
0.72
0.02
64
0.00
02
0.0
0.01
0.59
13.2
0
0.6
0.87
6
0.94
1.99
3
1.47
6
1.68
1.00
1.99
3
1.63
4
1.47
6
1.11
0.06
25
0.00
04
0.0
0.01
0.89
25.3
3
0.8
1.16
8
0.97
2.06
1.76
8
1.51
0.98
2
2.02
3
2.55
4
1.76
8
1.44
0.10
63
0.00
05
0.0
0.01
1.18
39.5
8
1.0
1.46
1.00
2.12
2.06
1.41
0.96
6
2.05
3.61
3
2.06
1.75
0.15
68
0.00
06
0.00
01
0.02
1.48
56.0
0
1.2
1.75
2
1.03
2.18
4
2.35
2
1.34
0.95
2.07
4.81
1
2.35
2
2.05
0.21
33
0.00
07
0.00
01
0.02
1.77
74.5
8
0
2.3.6 Bottom pressure characteristics
The bottom pressure distribution Pb(x) is important, because it yields:
• an index for the potential danger of cavitation damage, and
• the location where piers can end without inducing separation of flow.
The bottom pressure head Pb/g nondimensionalized by the design
head HD as a function of location X=x/HD for various c=H/HD is
shown in figure below.
• The minimum pressure Pm occurs on the upstream quadrant.
Pb g
0
HD
Pb g
0
HD
H
for c 
1
HD
H
for c 
1
HD
• The most severe pressure minima along the piers due to significant
streamline curvature effects.
Bottom pressure distribution
Free surface profile
H/HD
Pb/g
Plane flow
Plane flow
Between piers
Axial between piers
Pb/g
Pb/g
along piers
• A generalized analysis is given by Hager as
Pm
P m
vs c which may be approximated as :
gH
P m 1  c 
Accordingly the minimum bottom pressure is positive compared to
the atmospheric pressure when c<1.
Also, the minimum pressure head Pm/g
Pm
 H  the effective head plus 1  c 
g
• The location of the minimum pressure is:
• Xm = - 0.15 for c<1.5
• Xm = - 0.27 for c1.5 i.e.; just at the transition of the crest to the
vertical abutment.
The crest bottom pressure index Pc

Pc
gH
as a function of c is significantly
above the minimum pressure. Approximately:
2
Pc  Pm
3
• The figure below refers the location of zero bottom pressure, X0=x0/HD,
• i.e.; where atmospheric pressure occurs.
H
X0  c 
plus   chute angle
HD
• Hager gives this relation as
X 0  0.9 tan c  10.43 where
x0
X0 
 relative position of atmospheric pressure
H0
Minimum bottom pressure index
Crest pressure
Discharge coefficient in relation
to relative head c=H/HD
Location of atmospheric bottom
pressure c0( c )
2.3.7 Cavitation Design
Standart overflow with
c<1
underdesigned
c>1
over design and thus subatmospheric bottom pressures.
Initially over design of dam overflows was associated with advantages
in capacity.
However the increase in discharge coefficients C for c>1 is relatively
small, but the decrease of minimum pressure, Pm, is significant.
• Overdesigning, thus adds to the cavitation potential.
• Incipient cavitation is a statistical process depending greatly on the
water quality and the local turbulence pattern. Generally, one
assumes an incipient pressure head:
Pvi
 7.6 m.
g
• The limit head HL for incipient cavitation to occur is
HL  b 1  c 
-1
Pvi 
g 
 
• The constant b was introduced to account for additional effects, such
as the variability of Pvi with c.
2.3.8 Spillway Face
• The spillway face is a straight line between the point of tangency,
P.T.(xt,yt) and the point of curvature P.C. It has a length of L1, and a
slope of , which is determined from the stabilty analysis.
P.T
L1

P.C
Point of tangency
At the point of tangency:
 y 
 x 
dy 1

and    K  
dx 
 H0 
 H0 
1 dy
H 0 dx

1
H0
 x 

 H0 
nK 
 xt   1 
    

H

nK

 0 
1
n 1
n 1
n
x 
 nK  t 
 H0 
n 1

 yt   1 
    

H

nK

 0 
1

n
n 1
• Flow down the steep face of spillway, normally at about 45o to the
horizontal, has a rather special charactecter which makes the
methods of Gradually-Varied Flow unsuitable for its treatment.
• In this case acceleration and boundary layer development are both
taking place along the spillway face, as shown in figure below.
• 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, but the extreme steepness of the slope introduces
more complications, chiefly the phenomenon of air entrainment, or
“insufflation”.
Spillway Face
Recommended radius of toe: R=H0+0.25P
x
P
R
Air entrainment on the face of a spillway
Flow on a spillway face and air entrainment
Oldman river dam, Alberta
• 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-increasing proportion of
air, continues to accelerate until uniform flow occurs, or the base of
spillway is reached.
• Clearly, the designer will wish to know the velocity reached at the
base, or toe, of the spillway.
• The computation of this velocity can be obtained by using boundary
layer development over the spillway face. The boundary layer will
start to develop from point A where spillway crest starts.
• The thickness of the boundary layer, d, the displacement thickness,
d1, and the energy thickness d3, at the point of curvature is given by:
d
L 
 0.08 
L
k 
0.233
d 1 0.18d
d 3  0.22d
Spillway crest and boundary layer
• The boundary layer will start to develop from point A where spillway
crest starts.
• Where:
d= the boundary layer thickness at P.C.
• L=Lc+L1= total length of crest
• Lc= length of curved crest,
• L1= Length of face,
• k= roughness height of concrete= 0.061 cm
d1= the displacement thickness at P.C.
d3= the energy thickness at P.C.
The length of curved crest, Lc,can be obtained from Fig.7 as a
function of xt/H0.
The depth of flow at the point of curvature can be obtained from
energy equation by assuming that the head loss is zero. This depth
dp is called as potential-flow depth, because head loss is neglected.
• However, a boundary layer is developing along the spillway face.
Hence a head loss will occur. Therefore, the actual depth, d, and the
head loss,hl, can be computed by using the displacement and energy
thicknesses as follows:
d  dp  d 1  actual flow depth
d 3U p3
h 
 spillway energy loss
2gq
where
q
Up 
 potential - flow velocity
dp
Example on Spillway Energy Loss
Given:
H0= 10 m
H0=10 m
P=107 m
k=6.1x10-4 m
Face slope: 1:0.78
P=107 m
For a high spillway,
The crest shape is:
 x 
y

 K 
H0
H0 
x
yT
xT
x
Y2
(P.C)
n
K  0.5, n  1.85
(P.T)
y
R
COMPUTE THE ENERGY HEAD ENTERING
THE STILLING BASIN
1. Boundary Geometry
a) Length of curved crest, Lc
XT
 1.47 (from equation)
H0
Lc
 2.15(from figure)
H0
L c  2.15x 10  21.5 m
b)
Length of tangent, LT
YT
 1.02 (from equation)
H0
YT  10.2 m
Y 2 YT  107  10.2  96.8 m
1
Tan 
 1.2821
0.78
1
Y 2 YT
Sin   0.7885 

2
2
LT
1  0.78
Y 2 YT
LT 
 122.8 m
Sin 
c) Total crest length, L
L=Lc+LT=21.5+122.8=144.3 m
2. Hydraulic Computation:
a) Boundary-layer thickness, d:
L
144.3
5


2.336x10
k 6.1x 104
-0.233
d
L
 
 0.08 
 0.08x(2.33 6x105 )-0.233  0.00448
L
k 
d  0.00448x 144.3  0.646 m
b) Energy thickness, d3:
d30.22d0.142 m.
c) Unit discharge, q:
q  CH 03 / 2
C  2.18 (
P
 10.7  Fig .1)
H0
q  2.18x 103 / 2  68.94 m 3 / s / m
d) Potential flow depth dp and velocity U at PC of toe curve
q2
HT  d p cos 
2gd p2
Cos  0.6150
HT  107  10  117 m
117  0.615d p 
d 
p
( i 1 )
68.942
2x 9.81d p
2
 0.615d p 
242.22
d p2
242.22

by iteration dp  1.44 m
( i)
117  0.615d p 
q
Up 
 47.73 m/s
dp
e) Spillway energy loss, hl:
d 3U p3 0.142x 47.733
h 

 11.41m
2gq
2x 9.81x 68.94
f) Energy head entering to the stilling basin:
Hb=117-11.41=105.59 m
g) Depth of flow, d, at PC of toe curve:
d=dp+d1
d1=0.18 d=0.116 m
d=1.44+0.12=1.56 m
U act
q 68.94
 
 44.2 m/s
d
1.56
Since d<d, no bulking of flow from air entrainment
2.3.9 Spillway Toe
• When the flow reaches the end of inclined face of spillway it is
deflected through a vertical curve into the horizontal or into an
upward direction. In the latter case we have the ski-jump and the
bucket-type energy dissipators, to be discussed later.
• In either case, centrifugal pressures will be developed which can set
up a severe thrust on the spillway side walls. These pressures
cannot be accurately calculated by elemantary means, but there are
certain approximations. One of them is:
• Assume that the depth yO at the center of curve is equal to the depth
y1 of approaching flow. Then the centrifugal pressure at point O will
be equal to:
2
V1 y 1
PO 
R
Where
V1= velocity of approaching flow, and R= radius of curvature of toe
Spillway toe and flip bucket
Spillway toe
Flip bucket
Spillway cross section
A typical cross section of a spillway with a ski-jump
• If pressure is increasing, velocity must be decreasing by the Bernoulli
Equation. Then the average velocity must be smaller than V1, and
the depth must be greater than yO, so that this equation is not
correct.
• A better approximation can be made by assuming that the
streamlines crossing OA form parts of concentric circles, and the
velocity distribution along this line is accordingly the same as that in
free, or irrotational vortex, i.e.:
C
v 
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 streamline at A, then
C=V1R1. The discharge q across AO is given by
R
R
R
V1R1
1
q V1 y 1  vdr  
dr V1R1  dr
r
R
R
R r
1
1
1
R 
y1

 log  and
R1
R1 
R 
y1 R1

log 
R R
R1 
Since y1 and R are known in advance, R1 can be obtained by trial
from the last equation. Given R1/R, we can obtain PO, the pressure
at O, from the condition :
1
1
2
PO  VO  V12 hence :
2
2
PO
1
V 1 2
2
VO
 1  
 V1
2

R
  1   1 
R 

2
• The “free vortex” method leads to results that are quite accurate
within a certain range, but it has a 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 natural logarithms. Applying this result to the
equation for y1/R ,we see that R/y1 has a minimum value of e, when
R/R1 =e, even though R/y1 is by nature of the problem an
independent variable, which may in practice assume any value at all.
• Therefore, the theory cannot be applied when R/y1 <e.
• 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, P=gy, to the
pressures obtained above.
Gate controlled ogee crest
Releases for partial gate openings for gated crests occur as orifice flow.
With full head on a gate that is opened a small amount, a free
discharging trajectory will follow the path of a jet issuing from an orifice.
For a vertical orifice the path of the jet can be expressed by the
parabolic equation
2
x
y 
4H
where H is the head on the center of the opening.
For an orifice inclined an angle q from the vertical, the equation is:
 y  x tanq 
x2
4H cos2 q
If subatmospheric pressures are to be avoided along the crest contact,
the shape of the ogee downstream from the gate sill must conform to
the trajectory profile.
Gates operated with small openings under high heads produce negative
pressures along the crest in the region immediately below the gate if the
ogee profile drops below the trajectory profile. Tests showed the
subatmospheric pressures would be equal to about one-tenth of the
design head when the gate is operated at small openings and the ogee
is shaped to the ideal nappe profile:
 x 
y

 K 
H0
H0 
n
For maximum head Ho. The force diagram for this condition is shown on
figure 8.
Subatmospheric crest pressures for undershot
gate flow
The adoption of a trajectory profile rather than a nappe profile
downstream from the gate sill will result in a wider ogee, and reduced
discharge efficiency for full gate opening. Where the discharge
efficiency is unimportant and where a wider ogee shape is needed for
structural stability, the trajectory profile may be adopted to avoid
subatmospheric pressure zones along the crest.
Where the ogee is shaped to the ideal nappe profile for maximum head,
the subatmospheric pressure area can be minimized by placing the gate
sill downstream from the crest of the ogee. This will provide an orifice
that is inclined downstream for small gate openings and will result in a
steeper trajectory closer to the nappe-shaped profile.
Discharge Over Gate-Controlled Ogee Crests
The discharge for a gated ogee crest at partial gate openings will be
similar to flow through an orifice and may be computed by the equation:
Q  CDL 2gH
where:
H = head to the center of the gate opening (including the velocity head
of approach),
D = shortest distance from the gate lip to the crest curve, and
L = crest width.
The coefficient, C, is primarily dependent upon the characteristics of the
flow lines approaching and leaving the orifice. In turn, these flow lines
are dependent on the shape of the crest and the type of gate. Figure 9,
which shows coefficients of discharge for orifice’ flows for different q
angles, can be used for leaf gates or radial gates located at the crest or
downstream of the crest. The q angle for a particular opening is that
angle formed by the tangent to the gate’lip and the tangent to the crest
curve at the nearest-point of the crest curve for radial gates. This angle
is affected by the gate radius and the location of the trunnion pin.
Definition sketch for gated crests
Discharge coefficient for flow under gates