An efficient algorithm for ogee spillway discharge with partiallyopened radial gates by the method of Design of Small Dams and comparison of current and previous methods

The International Journal Of Engineering And Science (IJES)
|| Volume || 5 || Issue || 7 || Pages || PP -13-26 || 2016 ||
ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805
www.theijes.com The IJES Page 13
An efficient algorithm for ogee spillway discharge with partially-
opened radial gates by the method of Design of Small Dams and
comparison of current and previous methods
Tefaruk Haktanir1
, Hatice Citakoglu2
, Hurmet Kucukgoncu3
1,2,3
Department of Civil Engineering, Erciyes University, Kayseri, Turkey
--------------------------------------------------------ABSTRACT-----------------------------------------------------------
Ogee profile flood spillways equipped with radial gates are common, and accurate computation of spilled
discharge through partially-opened radial gates is an important problem. A new algorithm is developed for the
method given in the latest edition of the book: Design of Small Dams for computation of discharge over ogee
spillways equipped with radial gates for the partial opening case. This algorithm is more efficient with less
computational load than the one presented in ‘Hydraulic Design Criteria, Sheets 311-1 to 311-5’ by US Army
Corps of Engineers which is the method by ‘Design of Small Dams’. For a wide range of partial gate openings
on a few existing dams, discharges are computed by this method and are compared with those given by the
previous method comprised in the former editions of ‘Design of Small Dams’. As both yield close values for
small gate openings, the current method gives spillway discharges about 10 % to 30 % greater than the
previous method for large gate openings. Next, discharge coefficients are computed using the measured data
taken on 1:50 scale laboratory model of the spillway of Kavsak Dam and are compared with those given by the
charts in ‘Design of Small Dams’, which are found to be deviant as much as 10 %.
Keywords: discharge of radial-gated spillways
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Date of Submission: 17 May 2016 Date of Accepted: 15 July 2016
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I. INTRODUCTION
The subsections 194 of the first and 201 of the second editions of the book: Design of Small Dams [1, 2] are
exactly the same, which both give the equation to compute the discharge over a radial-gated ogee-crested
spillway while the gates are partially opened, which is a dimensionally homogeneous equation, as:
Q = (2/3)∙(√2g)∙C∙Le∙(H1
1.5
– H2
1.5
) (1)
where, g is the acceleration of gravity, C is a coefficient, Le is the effective length of crest, H1 and H2 are defined
as: “H1 and H2 are the total heads (including the velocity head of approach) to the bottom and top of the orifice,
respectively.”, and Q is the discharge. A copy of Fig. 197 of the first edition, which is Fig. 257 of the second
edition, of Design of Small Dams, is given here as Fig. 1. C in equation (1) is given as a function of the ratio
(d/H1) in this figure, where d is the vertical gate opening, which equals H1 – H2. In two other relevant books by
the Bureau of Reclamation, also, which are Design of Gravity Dams [3] and Design of Arch Dams [4], equation
(1) is given along with Fig. 1 for computation of the discharge under partially-opened radial gates.
Figure 1. Copy of Fig. 197 in the „Spillways‟ chapter of the first edition of Design of Small Dams [1], which is
the same as Fig. 257 in the „Spillways‟ chapter of the second edition of Design of Small Dams [2].

An efficient algorithm for ogee spillway discharge with partially-opened radial gates by the method..
The effective length, Le, is equal to the net length of spillway crest minus a reduction due to the contraction of
the flow while passing from the lake first to the approach channel and next into the spillway bays separated by
the gate piers, and is expressed as [1, 2]:
Le = L – 2∙(Np∙kp + ka)∙H (1a)
where, L is the net length of the spillway crest excluding the sum of widths of the existing piers, Np is the number
of piers on the crest, kp is the pier contraction coefficient, ka is the approach abutments contraction coefficient,
and H is the actual total head with respect to the spillway apex elevation. Three values for kp and ka are
suggested as 0, 0.01, 0.02, and 0, 0.1, 0.2, respectively, depending on geometrical shapes of pier noses and of
abutment headwalls [1, 2]. In all three Design of Small Dams [1, 2, 5], there are numerical examples for the
uncontrolled (free flow) ogee spillway discharges, but, there are no examples for the gate-controlled flows.
In Fig. 1, the radial gate looks as if its gate seat is right on the apex in closed position. However, for radial gates
whose gate seats are a little further downstream from the apex, which is a common application, the „bottom of
the orifice‟ should be the point on the spillway crest curve closest to the lip of the partially-opened gate. For
computational simplicity, the bottom of the orifice may be taken constantly as the spillway apex elevation. For
example, in the final projects of two dams in Turkey [6, 7], the designers used equation (1) and took the spillway
apex elevation consistently as the bottom of the orifice, although their gate seats are a little downstream from the
apex. The vertical gate opening (d) with respect to the spillway apex is slightly smaller than that with respect to
the point closest to the gate lip; but, H1 with respect to the spillway apex is also just a little smaller than H1 with
respect to the point closest to the gate lip. Therefore, the magnitude of d/H1 does not change appreciably from
both viewpoints, and the magnitudes of the discharge coefficient (C) are close. However, the magnitude of
discharge computed with d and H1 for the more realistic case may be non-negligibly greater than with d and H1
with respect to the spillway apex.
In subsection 9.16 of the third edition of Design of Small Dams [5] however, a different equation is given for
discharge over a radial-gated ogee spillway while the gates are partially opened, which is:
Q = C∙D∙L∙(2g∙H)1/2
(2)
where, C is a dimensionless coefficient, D is the shortest distance between the gate lip and the spillway crest
curve, L is the net length (not the effective length) of the spillway crest, g is the acceleration of gravity, and H is
the vertical difference between the total head just upstream of the gate and the center of the gate opening.
Equation (2) also is dimensionally homogeneous. Because it is in the updated edition of Design of Small Dams
[5], equation (2) should be used, and for example, according to [8], the Spanish Committee on Large Dams
recommends usage of this equation.
Fig. 2, giving the C coefficient of equation (2) as a function of the angle between the tangent to the crest curve at
the point closest to the gate lip and the tangent to the gate at its lip, is a copy of Fig. 9-31 of the third edition of
Design of Small Dams [5]. This figure is a replica of the figure captioned as: 'Hydraulic Design Chart 311-1'
given in Sheets 311-1 to 311-5 of Volume 2 of Hydraulic Design Criteria [9], which was developed by the
Waterways Experimentation Station of US Army Corps of Engineers based on measurements on three laboratory
models and on three prototype structures; and, it can be downloaded from:
chl.erdc.usace.army.mil/Media/2/8/1/300.pdf. The angle symbolized by β in Hydraulic Design Chart 311-1 [9]
is denoted by θ in Fig. 9-31 of the third edition of Design of Small Dams [5]. And, Hydraulic Design Chart 311-
1 [9] has a high-resolution background for better visual reading of the magnitude of C. Actually, equation (2)
and the new method is originally proposed by the Waterways Experimentation Station of US Army Corps of
Engineers in the publication: Sheets 311-1 to 311-5 of Volume 2 of Hydraulic Design Criteria [9], and, the third
edition of Design of Small Dams [5] directly refers to this method. Equation (2) and the figure for C as „Plate 6-
1‟, are given in Chapter 6 of EM-1110-2-1603 [9], also. Although there are detailed examples for the free-flow
(un-gated ogee spillway) case in Design of Small Dams [5], the case for the partially-opened gates is given
briefly without any examples, where it is stated: “For additional information and geometric computations see
[20].” (Reference number 20 being the publication: Sheets 311-1 to 311-5 of Volume 2 of Hydraulic Design
Criteria [9]. Available at the above-mentioned web site, this is a technical report comprising seven pages, which
articulately presents the details of a method for computing the C coefficient of equation (2) and finally the
discharge for any partial opening. Investigation of this report will reveal that (1) there is a long table having 20
columns which lead to the magnitudes of the angle β and the orifice opening D, (2) there is another table having
15 columns leading to the magnitude of the spilled discharge versus a definite gate opening, and (3) there are
two charts for the quantitative relationships of (i) coordinates of point on crest curve closest to the gate lip and of
(ii) derivative of crest curve with respect to the x variable. These two charts need to be prepared beforehand in a

log-log graph paper for specific values of K and n coefficients of the analytical expression depicting the
downstream crest curve profile of the ogee spillway, which is equation (3) in the next section below. Besides,
instead of using the actual analytical expression, a few circular arc replacements are suggested and used. In
computing the angle β and the orifice opening D in the first table, various steps like using relationships of similar
triangles are applied. The method is correct, but it is too winding (This seven-page publication [9] can be
downloaded by anyone interested to witness for themselves this cumbersome and too long method.). Instead of
using arcs of circles for the crest curve, and instead of such a long procedure involving so many intermediate
steps, a new analytical method is developed here in this study, which provides all the terms of equation (2) and
finally the discharge as a result of fewer steps with a less computational load. Therefore, the main objective of
this study is to present this less complicated method for equation (2).
Figure 2. Copy of Fig. 9-31 in the third edition of Design of Small Dams [5].
Whether by using either equation (1) or equation (2), computation of discharge passing over a radial-gated ogee
flood spillway for the case of partially-opened gates is a common and significant problem, for which the studies
by Haktanir et al [10] and Zargar et al [11] are typical examples. Computation of discharge over ogee spillways
has always been a significant research topic investigated by various researchers [e.g., 12, 13, 14]. There may be
other methods for computing discharge through partially-opened gates. For example, Ansar and Chen [15]
presented generalized equations for discharge over ogee spillways with sharp-edged sluice gates using data
measured at many prototype canal control structures in South Florida. Bahajantri et al [16] proposed a numerical
method based on finite element approach. Saunders et al [17] developed a method using the Smoothed Particle
Hydrodynamics model. The objective of the study whose summary is presented here is not to compare the
method of Design of Small Dams [5] with the others, rather mainly to present an efficient algorithm for the
USBR, and hence the USACE, method.
The second objective is to compare the (spillway discharge)↔(lake water surface elevation) relationships over
wide ranges of (1) water heads from small magnitudes to the design head and of (2) partial gate openings from as
small as 10 cm to close to the full opening given by both equation (1) and equation (2) on radial-gated ogee
spillways of a few existing dams.

1. Computation of Spillway Discharge by Equation (2)
The relevant numerical data known from dimensions of the spillway and the radial gates, denoted by the symbols
used herein are:
Rg : radius of the gate,
xL0 : abscissa of the gate seat (of the gate lip when the gate is at closed position),
yL0 : ordinate of the gate seat,
xT : abscissa of the gate trunnion,
yT : ordinate of the gate trunnion,
yL : ordinate of the gate lip when it is partially opened (= vertical gate opening with respect to the spillway apex).
When the origin of a Cartesian coordinates system is located at the apex of an ogee spillway according to the
right-hand rule, the profile of its downstream face is expressed analytically as [1, 2, 5]:
y = – K∙Hd
(1–n)
∙xn
(3)
where, y is the ordinate of a point on the crest curve downstream from the apex whose abscissa is x, Hd is the
design head in the approach channel with respect to the spillway apex elevation, and K and n are “constants
whose values depend on upstream inclination and on the velocity of approach” [5, page: 365]. The K and n
coefficients are given in Fig. 9-21 of Design of Small Dams [5, page: 366] as a function of (1) the ratio: ha /Hd ,
where ha is the velocity head in the approach channel for the design discharge, and of (2) the inclination of the
upstream face of the spillway. K varies within the interval: 0.47 < K < 0.54, and n varies within: 1.74 < n < 1.87.
Case 1: The gate seat is some distance downstream from the apex of the spillway
Fig. 3 schematically depicts the geometrical configuration of a partially-opened radial gate for this case.
Figure 3. Tangents to the lip of the partially-opened radial gate and to the crest curve of the ogee spillway at the
point closest to the gate lip, the angles these tangents make with the x axis, and the angle θ between these
tangents.
In order to simplify the analytical expression of the downstream crest curve, equation (3), in Fig. 9-22 of Design
of Small Dams [5, page: 368] this profile is closely approximated by adjacent arcs of three circles, whose center
coordinates and radiuses are indicated in that figure. In this study, equation (3) is used throughout the analyses
and circle approximations are not applied. But, the crest curve of an ogee spillway upstream from its apex is
necessarily represented by arcs of two adjacent circles [5, page: 368] simply because the range of equation (3) is:
0 < x < +∞. As seen in Fig. 3, the shortest distance between the gate lip and the surface of the ogee spillway (D
in Fig. 2) is on the line which is perpendicular to the line tangent to the crest curve at the point having the
shortest distance to the lip. The line tangent to the radial gate at its lip and the angles these tangents make with
the x axis and with each other are also shown in Fig. 3.
According to the adapted Cartesian coordinates, the analytical expression of the circle of which the radial gate is
an arc is
y2
– 2∙yT ∙y + x2
– 2∙xT ∙x + xT
2
+ yT
2
– Rg
2
= 0 (4)

where, x and y are the coordinates of any point on the circle, xT and yT are the abscissa and the ordinate of the
trunnion pin of the gate, and Rg is the radius of the gate.
The derivative of y with respect to x in equation (3) gives the slope of the line tangent to the downstream crest
curve of the spillway, which is
y′ = – n∙K∙Hd
(1–n)
∙x(n–1)
(5)
The ordinate of the gate lip (yL) is equal to the vertical opening of the gate with respect to the spillway apex
elevation; and, the abscissa of the lip (xL) can be computed by inserting yL for y in equation (4), which can be
rewritten as
x2
– 2∙xT ∙x + yL
2
– 2∙yT ∙yL + xT
2
+ yT
2
– Rg
2
= 0 (6)
Out of the two roots of equation (6), a quadratic equation, the one below gives the abscissa of the gate lip (xL =
r1).
r1 (= xL) = xT – (2∙yT ∙yL – yL
2
– yT
2
+ Rg
2
)1/2
(7)
It can be easily shown that as long as the vertical gate opening with respect to the spillway apex elevation, yL, is
smaller than (yT + Rg), which corresponds to the point on top of the circle of the gate, the term in parentheses is
positive-valued, and so equation (7) yields real roots all the time. The second root gives the abscissa of the point
at the other side of the circle whose ordinate is the same as that of the gate lip, which has no physical meaning in
this problem.
The line of the shortest distance between the gate lip and the point on the surface of the spillway is perpendicular
to the line tangent to the spillway surface at that point closest to the lip. Denoting the slope of this line by S1 and
the slope of the line tangent to the spillway crest curve by S2, the following equation holds.
S1∙S2 = –1 (8)
Symbolizing the ordinates of the point on the surface of the spillway closest to the gate lip by xc and yc, S2 is
given by the right-hand side of equation (5) by inserting xc for x. And, from equation (8) S1 is depicted as
S1 = 1 / [n∙K∙Hd
(1–n)
∙xc
(n–1)
] (9)
From geometry, the analytical equation of the line passing through the point (xL, yL) with the slope S1 can be
written as
y = S1∙x + yL – S1∙xL (10)
Because this line passes through the point (xc, yc), equation (10) holds at this point also:
yc = S1∙xc + yL – S1∙xL (11)
The point (xc, yc) satisfies equation (3) also because it is on the surface of the spillway. Therefore, when xc is
inserted for x in equation (3), y at the left-hand side becomes yc, and when the right-hand side of equation (3) in
this form is inserted for yc in equation (11), the below equation results:
S1∙xc – S1∙xL + K∙Hd
(1–n)
∙xc
n
= – yL (12)
Now, inserting the right-hand side of equation (9) for S1 in equation (12), after arranging, the below equation is
obtained.
xc
(2–n)
– xL∙xc
(1–n)
+ n∙[K∙Hd
(1–n)
]2
∙xc
n
+ n∙K∙Hd
(1–n)
∙yL = 0 (13)
The root of this equation is the abscissa of the point on the spillway crest curve closest to the gate lip (xc). It is
obvious that an analytical solution for the root is not possible. However, it can be easily computed by the

iterative Newton-Raphson algorithm because, symbolizing its left-hand side by f(xc), the analytical derivative of
f(xc) with respect to xc can be taken easily, which is:
f ′(xc) = (2–n)∙xc
(1–n)
– (1–n)∙xL∙xc
–n
+ n2
∙[K∙Hd(1–n)
]2
∙xc
(n–1)
(14)
The recursion formula of the iterative algorithm is
xc,1 = xc,0 – f (xc,0) / f ′(xc,0) (15)
where, f (xc,0) is the value of the left-hand side of equation (13) with the numerical value xc,0 inserted for xc, and
xc,1 is a value closer the root than xc,0. The absolute relative difference between xc,1 and xc,0 is computed by: RD =
|[f (xc,0) / f ′(xc,0)] / xc,1| . The iterations stop when RD ≤ 10–6
, which means the last xc,1 is the root correct to 6
significant digits. Otherwise, xc,1 is assigned as the new xc,0, and iterations continue.
For case 1, the gate seat being 20 ~30 ~ 40 cm or so below and a couple of meters or so downstream from the
spillway apex, the point on the crest curve closest to the gate lip must be between the apex, origin of the x–y
coordinates (0, 0), and the gate seat (xL0, yL0). Therefore, initially, the interval (0, xL0) is subdivided into ten
equal subintervals, and that one in which f (xc) changes sign is determined. Next, the bisection algorithm is
applied for the root of equation (13) (x1,3 = (x1,1 + x1,2) / 2) until the relative difference between the last two
iterations is less than or equal to 110–3
(|(x1,3 – x1,2) / x1,3| ≤ 110–3
). The last value of x1,3 is taken as the initial
estimate of the Newton-Raphson algorithm (xc,0 = x1,3) and in just a few iterations the root to 6 significant digits is
computed. This scheme yields convergent solutions always.
Next, the ordinate of the point on the spillway crest curve closest to the gate lip, yc, is computed by equation (3)
by inserting the value of xc for x.
Next, the closest distance between the gate lip and the surface of the spillway, D, is computed by:
D = [(xL – xc)2
+ (yL – yc)2
]1/2
(16)
Derivative of y in equation (4) with respect to x is
y′ = (xT – x) / (y – yT) (17)
which gives the slope of the tangent to the circle of the gate at the point whose ordinates are x and y. By inserting
xL and yL for x and y, y′ equals the tangent of the angle of the line tangent to the gate at its lip with respect to the
x axis, denoted by α in Fig. 3. Hence, the angle α in Fig. 3 is computed by:
α = arctan[(xT – xL) / (yL – yT)] (18)
Next, the angle of the line tangent to the crest curve at the point (xc, yc) with respect to the x axis, denoted by 
in Fig. 3, is computed by equation (5), which becomes
 = arctan[–n∙K∙Hd
(1–n)
∙xc
(n–1)
] (19)
Next, the angle symbolized by θ in Fig. 9-31 of Design of Small Dams [5], and in Fig. 3 here, is computed by:
θ =  – α (20)
Next, the magnitude of the C coefficient in equation (2) is taken from either Fig. 9-31 of Design of Small Dams
[5] or from Hydraulic Design Chart 311-1 of Hydraulic Design Criteria, Volume 2 [9], preferably from the
latter because it has a higher resolution, as a function of the angle θ.
Next, the vertical distance between the water surface elevation just upstream of the partially-opened radial gate
and the center of the gate opening is computed by:
H = H2 + (yL + │yL0│) / 2 = H2 + (yL – yL0) / 2 (21)

Here, H2 is the vertical distance between the water surface elevation plus the velocity head in the approach
channel just upstream of the gate and the gate lip.
Finally, the discharge through the partially-opened radial gates is computed by equation (2). Obviously, the last
three steps are the same as those of the method of Hydraulic Design Chart 311-1 of Hydraulic Design Criteria,
Volume 2 [9].
Case 2: The gate seat is on the apex of the spillway
The upstream side of the crest curve of the ogee spillway is expressed as arcs of two adjacent circles in Fig. 9-
21(A) of Design of Small Dams [5, page: 366]. The first circle is upstream of the apex of the spillway with a
radius of R1, and its center is on the ordinate axis. According to Fig. 9-21(f) of Design of Small Dams [5, page:
367], R1 varies within the interval: (0.34)∙Hd ≤ R1 ≤ (0.56)∙Hd. Taking R1 as a positive real number, the analytical
expression of this circle is
y = (R1
2
– x2
)1/2
– R1 (22)
The analytical manipulations for the desired quantities in case 2 are made such that for all positions of the
partially-opened radial gate, the point on the spillway crest curve closest to the gate lip is on the first circle,
because it is geometrically impossible for this point to be far upstream on the second circle. In this case, the
profile of the upstream part of the spillway surface is the arc of the circle number 1 of Fig. 9-21(A) of Design of
Small Dams [5, page: 366]. In Fig. 4, the geometrical configuration of the partially-opened radial gate and of the
spillway of the Yellowtail Afterbay Dam is given as an example for definition of the relevant geometrical
peculiarities for a spillway whose radial gate sits on the apex in closed form. This figure is drawn to scale for a
vertical gate opening of 6 feet with respect to the spillway apex (yL = 6.0 ft). The Yellowtail Dam is in Montana,
USA [18, 19].
Figure 4. Profile of the gate and the spillway of the Yellowtail Afterbay Dam for a partial opening of d = 6.0 ft
and the tangents to the gate lip and to the crest curve at the point closest to the gate lip.
First, the abscissa of the lip of the partially-opened gate (xL) is computed by equation (7) for this case also. If xL
is negative (xL < 0), which occurs more commonly for this case, then using equation (22) instead of equation (3)
and performing the same algebraic analyses of the previous section between equation (8) through equation (12)
ultimately results in equation (23) below for the abscissa of the point on the spillway crest curve closest to the
gate lip (xc).
xc = – R1 / {[(yL + R1) / xL]2
+ 1}1/2
(23)
Here, R1 is the radius of the first circle forming the spillway crest upstream of the apex. Notice that for this case
xc is directly computed whereas it is the root of equation (13) for the first case. Rarely, for very large gate
openings, the point on the spillway crest curve having the shortest distance to the lip may be somewhere just a
little downstream from the apex, for case 2 also. Then, xc is computed as the root of equation (13), instead,
similar to case 1.

If xL is negative (xL < 0), then the ordinate of the point on the spillway crest curve closest to the gate lip (yc) is
computed by equation (22) by inserting xc for x. If xL is positive (xL > 0), then the ordinate of the point on the
spillway crest curve closest to the gate lip (yc) is computed by equation (3) by inserting the value of xc for x.
Next, the closest distance between the gate lip and the surface of the spillway (D) and the angle of the line
tangent to the gate at its lip with respect to the x axis (α) are computed by equations (16) and (18), respectively,
for case 2 also.
Next, the angle of the line tangent to the crest curve at the point (xc, yc) with respect to the x axis, denoted by 
here, is computed using the derivative of y in equation (22) with respect to x and inserting the value of xc for x,
which results in
 = arctan[–xc / (R1
2
– xc
2
)1/2
] (24)
For case 2, from geometrical relationships similar to those in Fig. 4, the angle symbolized by θ in Fig. 9-31 of
Design of Small Dams [5] is computed by equation (25) below.
θ =  + (π – α ) (25)
The rest of the procedure for computing the discharge is the same as that for case 1. Notice that neither in Design
of Small Dams [5] nor in Hydraulic Design Criteria, Volume 2, Sheets 311-1 to 311-5 [9] there is any method
suggested for computation of the angle θ and the orifice gate opening D, for case 2.
II. Application of the Developed Algorithm
The algorithm summarized above is coded as a computer program and applied to a few dams. In order to verify
its accuracy, this method is first applied to the example given in Sheets 311-1 to 311-5 Tainter Gates on
Spillway Crests, Discharge Coefficients [9], which is John Doe Dam in America. The relevant properties of this
dam and those of Yellowtail Afterbay Dam in America, Kavsak Dam in Turkey, and Bakhra Dam in India are
given in Table 1.
Table 1. Geometrical data of the flood spillways and their radial gates and some pertinent data of the dams
analyzed in this study as examples
Peculiarity John Doe(1) Yellowtail Afterbay(2) Kavsak(3) Bhakra(4)
Maximum WSE(*) : 325.00 ft 3192.0 ft 321.20 m 1685.00 ft
Top of active pool : 315.00 ft 3189.5 ft 320.00 m 1678.00 ft
Gate trunnion elevation : 300.00 ft 3189.5 ft 308.23 m 1662.25 ft
Spillway apex elevation : 288.00 ft 3179.5 ft 300.00 m 1645.00 ft
Gate seat elevation : 287.46 ft 3179.5 ft 299.86 m 1642.57 ft
Gate top elevation at
closed position
: 316.00 ft 3193.0 ft 326.27 m 1680.00 ft
Radius of gate : 30.75 ft 16.25 ft 18.00 m 38.0 ft
K and n coefficients of : 0.50 0.47 0.50 0.48
downstream crest curve :1.85 1.77 1.85 1.83
Radius of upstream
circle
: 18.0 ft 5.4 ft 11.24 m 18.0 ft
Number of gates : 4 5 3 4
Net spillway length : 42.0 ft 150.0 ft 37.8 m 230.0 ft
Angle of upstream face : 90º 2h:3v 2h:3v 90º
Design head, Hd : 37.0 ft 12.5 ft 21.2 m 38.5 ft
Spillway discharge at Hd : 23330 ft3
/s 25270 ft3
/s 7640 m3
/s 210000 ft3
/s
*: WSE: water surface elevation
(1): [9]
(2): [18, 19]
(3): [20]
(4): [21]

In the John Doe Dam example of the mentioned report [9], the pertinent quantities and spillway discharges are
computed for 12 different combinations of lake water surface elevations and partial gate openings. The
numerical values of some salient properties for all these 12 cases given in that report and those computed by the
method of this study are presented in Table 2.
Next, the spillway discharges have been computed by both equations (1) and (2) for a few partial gate openings
from very small values up to the full opening for John Doe and Yellowtail Afterbay Dams in America, Kavsak
Dam in Turkey, and Bhakra Dam in India for a lake water surface elevation (WSE) half way between the
maximum water surface elevation (Hmax) and top of active pool, and the results are given in Tables 3, 4, 5, and
6. As seen in these tables, the previous (1973) method mostly gives discharges smaller than the present (1987)
method, the difference being less than 10 % for gate openings smaller than 40 % of the total head, but being as
high as 30 % for large openings. The present method gives greater discharges for gate openings of about 70 ~ 80
% of the total head than the free flow discharge at the same head, which is a contradiction.
Table 2. Comparison of values given in Hydraulic Design Charts 311-2 and 311-5 (abbreviated as HDC below)
[9] for the example of John Doe Dam with the values computed by the algorithm developed in this study (lengths
are in feet, discharges are in cfs)
Gate
opening,
dsa
Angle θ D of Eqn.(2) C coefficient Water
surface
elevtn
Head on
spillway
Discharge, Q
HDC this
study
HDC this
study
HDC this
study
HDC this
study
(ft) (degrees) (ft) (ft) (cfs)
3.70 67.2 67.3 3.96 3.96 0.676 0.676 300 10.27 2900 2892
315 25.27 4500 4538
325 35.27 5400 5361
7.40 76.1 76.1 7.59 7.55 0.683 0.684 310 18.36 7500 7459
315 23.36 8400 8413
325 33.36 10100 10054
11.10 84.0 84.0 11.25 11.21 0.694 0.693 310 16.49 10700 10638
315 21.49 12200 12144
325 31.49 14800 14700
14.80 91.2 91.1 14.91 14.91 0.707 0.707 315 19.63 15800 15743
320 24.63 17600 17634
325 29.63 19300 19341
Table 3. Results of spillway discharge computations for partial gate openings by the 1973 and 1987 methods of
Design of Small Dams and for free flow at John Doe Dam in America.
All lengths are in ft and discharges are in cfs
Gate trunnion coordinates (y, x) : 12.00 33.55
Gate seat coordinates (y, x) : -0.54 5.48
Lake water surface elevation : 320.0
Variations of angle theta, orifice coefficient, and discharges by 1973 & 1987 editions of
Design of Small Dams versus vertical gate opening:
dsa(1) dbo(2) theta Corifice Q-1973 Q-1987 Rel.Diff.
5.0 5.18 71 0.679 5351 6486 -17 %
6.0 6.15 73 0.681 6259 7646 -18 %
7.0 7.13 75 0.683 7157 8801 -18 %
8.0 8.11 77 0.685 8031 9948 -19 %
9.0 9.10 80 0.688 8881 11092 -19 %
10.0 10.09 82 0.690 9707 12226 -20 %
11.0 11.08 84 0.693 10507 13351 -21 %
12.0 12.08 86 0.696 11280 14473 -22 %
13.0 13.07 88 0.700 12027 15598 -22 %
14.0 14.07 90 0.704 12746 16707 -23 %

15.0 15.07 91 0.708 13437 17800 -24 %
16.0 16.07 93 0.712 14099 18886 -25 %
17.0 17.08 95 0.716 14733 19971 -26 %
18.0 18.08 97 0.721 15338 21046 -27 %
19.0 19.09 99 0.725 15913 22104 -28 %
20.0 20.10 100 0.729 16457 23139 -28 %
21.0 21.11 102 0.735 16971 24206 -29 %
22.0 22.12 104 0.735 17454 25068 -30 %
24.0 24.15 107 0.735 18322 26713 -31 %
26.0 26.20 111 0.735 19053 28252 -32 %
28.0 28.25 114 0.735 19634 29687 -33 %
Fully open --- ----- 19279 19279 ----
(1): vertical gate opening with respect to spillway apex elevation
(2): vertical gate opening with respect to bottom of orifice
Design of Small Dams and for free flow at Yellowtail Afterbay Dam in America.
Gate seat coordinates (y, x) : 0.0 0.0
Variations of angle theta, orifice coefficient, and discharges by 1973 & 1987 editions of
Design of Small Dams versus vertical gate opening:
1.0 1.03 63 0.674 2854 2769 4 %
2.0 2.09 71 0.674 5521 5515 1 %
3.0 3.13 77 0.674 7960 8136 -2 %
4.0 4.15 82 0.674 10153 10588 -4 %
5.0 5.17 86 0.680 12102 12959 -6 %
6.0 6.17 90 0.686 13809 15185 -9 %
7.0 7.17 94 0.692 15275 17227 -11 %
8.0 8.16 97 0.698 16495 19083 -13 %
9.0 9.15 100 0.704 17454 20744 -15 %
10.0 10.13 103 0.709 18122 22207 -18 %
Fully open --- ----- 21362 21362 ----
(1): vertical gate opening w.r.t. spillway apex elevation
(2): vertical gate opening w.r.t. bottom of orifice
Design of Small Dams and for free flow at Kavsak Dam in Turkey.
All lengths are in m and discharges are in m3
/s
Variations of angle theta, orifice coefficient, and discharges by 1973 & 1987 editions of Design of
Small Dams versus vertical gate opening:
0.2 0.31 57 0.672 173 161 8 %
0.5 0.60 59 0.672 322 304 6 %
1.0 1.07 61 0.673 565 543 5 %
1.5 1.55 64 0.675 804 782 3 %
2.0 2.03 66 0.676 1040 1022 2 %
2.5 2.52 68 0.677 1274 1261 2 %
3.0 3.02 70 0.679 1505 1500 1 %

3.5 3.51 73 0.680 1733 1737 0 %
4.0 4.01 75 0.682 1954 1974 -1 %
4.5 4.50 77 0.684 2175 2210 -1 %
5.0 5.00 79 0.686 2389 2445 -2 %
6.0 6.00 82 0.691 2804 2911 -3 %
7.0 7.00 86 0.697 3197 3375 -5 %
8.0 8.00 89 0.704 3569 3834 -6 %
9.0 9.00 92 0.710 3920 4285 -8 %
10.0 10.00 95 0.717 4248 4732 -10 %
12.0 12.00 101 0.733 4838 5607 -13 %
14.0 14.01 107 0.735 5336 6333 -15 %
16.0 16.03 112 0.735 5738 6974 -17 %
18.0 18.06 118 0.735 6035 7550 -20 %
19.0 19.09 121 0.735 6137 7816 -21 %
Fully open --- ----- 7208 7208 ----
Design of Small Dams and for free flow at Bhakra Dam in India.
Variations of angle theta, orifice coefficient, and discharges by 1973 & 1987 editions of Design of
Small Dams versus vertical gate opening:
1.0 2.60 49 0.669 19871 20059 0 %
2.0 3.42 51 0.669 25696 26073 -1 %
3.0 4.26 53 0.670 31569 32155 -1 %
4.0 5.12 56 0.671 37479 38277 -2 %
5.0 6.00 58 0.672 43382 44422 -2 %
6.0 6.90 60 0.673 49196 50577 -2 %
7.0 7.81 62 0.674 55004 56728 -3 %
8.0 8.73 64 0.675 60741 62874 -3 %
9.0 9.66 66 0.676 66392 69017 -3 %
10.0 10.60 68 0.678 71944 75142 -4 %
12.0 12.51 72 0.680 82711 87290 -5 %
14.0 14.44 76 0.683 92971 99311 -6 %
16.0 16.40 79 0.687 102669 111211 -7 %
18.0 18.36 83 0.692 111759 122950 -9 %
20.0 20.35 86 0.697 120203 134566 -10 %
22.0 22.34 89 0.703 127966 146136 -12 %
24.0 24.34 92 0.709 135013 157349 -14 %
26.0 26.35 95 0.716 141305 168463 -16 %
28.0 28.37 98 0.724 146793 179367 -18 %
30.0 30.40 101 0.731 151411 189989 -20 %
32.0 32.44 104 0.735 155052 199111 -22 %
Fully open --- ----- 190195 190195 ----

III. Analysis of Laboratory Model Data of Kavsak Dam
Since early 1960s experiments are done on physical models of such appurtenant structures as spillways and
intake units of some dams in the hydraulics laboratory facility of the Technical Research and Quality Control
Department (known in Turkey as TAKK), which is somewhat similar to the hydraulics laboratory of Bureau of
Reclamation at Denver CO, of the General Directorate of State Water Works of Turkey (known in Turkey as
DSI) (www.dsi.gov.tr). The final report of such a recent work has been provided to the authors with permission
to be used in academic studies by the authorities of TAKK [20]. This is about the experiments on the 1:50 scale
model of the radial-gated flood spillway of Kavsak Dam, which is a dam on the Goksu River in Seyhan Basin in
Turkey. The relevant peculiarities of this spillway are given in Table 1 along with those of the other four dams.
The measured data of the gate openings, spillway discharges, and total heads with respect to the spillway apex
elevation are given in Table-3.5 of the report about spillway model of Kavsak Dam [20, pages: 36-38].
The discharge coefficients computed using the relevant quantities observed experimentally for the free-flow
(fully open gates) case with the Kavsak Dam laboratory model data turned out to be very close to those given by
the pertinent charts in Design of Small Dams [1, 2, 5], the difference being mostly less than ±5 %. Although the
empirical discharge coefficient are computed by multiplying four coefficients: (1) Co for the design head, (2)
correction for a head other than the design head, (3) correction for the angle of the upstream spillway face, and
(4) correction for the downstream effect, while the experimental discharge coefficient is computed from Q = Cnet
∙Le∙H1.5
only with the measured values inserted in it, both coefficients turn out to be very close to each other for
the Kavsak Dam laboratory model data. The discharge coefficients for the partially-opened cases are computed
by equation (1), the previous method [2], and by equation (2), the present method [5], with the experimentally
measured values inserted in them. The results are presented in Figs. 5 and 6.
Figure 5. Chart for discharge coefficients given in Figure 257 of the second edition of Design of Small Dams [2]
plotted together with the discharge coefficients computed by equation (1) with values measured experimentally
on the laboratory model of the Kavsak Dam [20].
Figure 6. Chart for discharge coefficients given in Figure-9.31 of the third edition of Design of Small Dams [5]
plotted together with the discharge coefficients computed by equation (2) with values measured experimentally
on the laboratory model of the Kavsak Dam [20].

IIII. RESULTS AND DISCUSSIONS
The algorithm developed in this study for computation of spillway discharge for the case of partially-opened
radial gates over an ogee spillway is more efficient than the one presented in Sheets 311-1 to 311-5 of Volume 2
of Hydraulic Design Criteria [9]. The method of Hydraulic Design Criteria [9] necessitates two separate tables
comprising sequences of computational steps in 20 columns in the first table followed by 15 columns in the
second table. Graphs of yc /Hd versus xc /Hd and dy/dx versus xc /Hd for a wide range of possible values of xc /Hd
pertaining to the geometrical dimensions of that spillway must be prepared initially before getting involved with
those tables. As would be appreciated by comparing the procedure outlined in the section: „Computation of
Spillway Discharge by Equation (2)‟ above with that in Sheets 311-1 to 311-5 of Volume 2 of Hydraulic Design
Criteria [9], the approach presented in the current study is more straightforward and algorithmically more
efficient. It is prepared in form of a Fortran code, which is freely available along with sample input data to
anybody interested. For some spillways whose gate seats are directly on the spillway apex, the point on the crest
curve closest to the gate lip is on the upstream crest. Although the method suggested in Hydraulic Design
Criteria, Volume 2, Sheets 311-1 to 311-5 Tainter Gates on Spillway Crests, Discharge Coefficients [9] does not
apply to this configuration, the algorithm developed in this study is valid for this case also.
As seen in Tables 3 – 6, discharges over ogee spillways having radial gates when the gates are partially opened
given by the method presented in the third edition of Design of Small Dams [5] are greater than those computed
by the method of the first and second editions of this book [1, 2]. The differences are very small for vertical gate
openings (d) up to 20 % of the design head (Hd), they are less than 10 % for 20 % of Hd < d < 50 % of Hd, but
they increase as much as 20 ~ 30 % for wider gate openings. The results of John Doe Dam seem to be more
pronounced than these values, and the discharge difference of the 1973 method is as much as –35 % from that of
the 1987 method. Table 3 of the model study report by USBR [19] presents actually measured discharges from
the spillway of Yellowtail Afterbay Dam and those computed by the 1987 method for five different d – WSE
combinations for fairly small gate openings between 12 % and 18 % of Hd. According to this table, the average
relative difference of the computed values from the observed ones is about +4 % which is affirmative of the
result of this study that the 1987 method has a tendency to over-estimate the discharges over partially-opened
radial-gated ogee spillways.
Further, the 1987 method gives greater discharges for gate openings of about 70 to 80 % of the total head than
the free flow discharge at the same head. On the contrary, the flow through a contracted gate opening should be
smaller than the free spilled flow under the same head. Fig. 9-31 of the third edition of Design of Small Dams
[5], which is a replica of the figure developed by the Waterways Experimentation Station of US Army Corps of
Engineers based on measurements on three laboratory models and on three prototype structures and is given as
'Hydraulic Design Chart 311-1' in Sheets 311-1 to 311-5 of Volume 2 of Hydraulic Design Criteria [9] is a
crucial figure as it yields the discharge coefficient (C) of equation (2), the 1987 method. It is believed that there
is still room for amendment of this C coefficient. Therefore, this figure, which is Fig. 9-31 of the third edition of
Design of Small Dams [5] also, needs to be revised based on many more laboratory and prototype data. The
laboratory data had better be obtained on models of scales ≤ 1:50.
Yet, as seen in Tables 4 and 5, at two of the four example dams, for very small gate openings of the order of 1 m,
the discharges computed by the 1973 method are slightly greater than those of the 1987 method. Such
differences are not as significant as the over-estimation of the 1987 method for discharges close to the design
flood however, because these are very small flowrates as compared to the real floods.
Although not given here, the 1973 and 1987 methods have been applied to a few more dams, like Bayramhacili,
Catalan, Aslantas, Bahcelik, Yamula Dams in Turkey, and Folsom Dam in America, and they all have revealed
results similar to those given in Tables 3 – 6.
The total head just upstream of the partially-opened gate (H1) must be with respect to the bottom of the orifice,
which is the point on the spillway crest curve closest to the gate lip, and not with respect to the spillway apex
elevation. This matter is not clear either in Design of Small Dams [5] or in HDC-311 [9]. Discharges computed
by taking H1 with respect to the spillway apex for gates whose seats are downstream from the apex will be a little
smaller than by taking the geometrically correct H1.
In the example of the spillway of Yellowtail Afterbay Dam whose gate seat is directly on the spillway apex, the
angle θ in Fig. 9-31 of Design of Small Dams (USBR 1987), which is symbolized by  in the figure: Hydraulic
Design Chart 311-1 [9] (Fig. 2 here), is found to vary in the interval: 63 º ≤ θ ≤ 105 º, while the bound values
are: 83 º ≤ θ ≤ 109 º in this figure. For the other three example dams whose gate seats are a little downstream
from the apex, the angle θ is found to vary in the interval: 49 º ≤ θ ≤ 124 º, while the same interval in Fig. 2 is
given as: 50 º ≤ θ ≤ 103 º. For the first example, the lower bound is a little too short, and for the next four
examples the upper bound is a little too short. In this study, the end values of the curves in that figure are taken
without extrapolation beyond their bounds.

IV. CONCLUSIONS
The methods to compute discharges over ogee spillways having radial gates when the gates are partially opened
presented in the 1973 and 1987 editions of the classical book: Design of Small Dams [2, 5] are different from
each other. Because the 1987 is a newer date than 1973, the 1987 method should annul the former one. Although
Design of Small Dams [5] presents the new method, it simply refers to the original source which developed it [9]
for the computational procedure. An analytical and numerical scheme for the 1987 method, which is different
from and algorithmically more efficient than the approach presented in Hydraulic Design Criteria, Volume 2,
Sheets 311-1 to 311-5 Tainter Gates on Spillway Crests, Discharge Coefficients [9] is developed in this study.
The spillway discharges over partially-opened radial-gated ogee spillways by both the 1973 and 1987 methods
are computed for vertical gate openings of: d = 0.1, 0.2, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, ..., 10, 11, …, fully open, in
meters, and compared with each other for a few existing dams having radial-gated spillways. And, it is seen that
while both methods yield very close discharges for small gate openings, the 1987 method gives discharges
greater than the 1973 method, the gap widening with increasing gate openings.
The essential part of the new method [5, 9] is the chart for the discharge coefficient (C). It is believed that this
chart should be revised based on more laboratory and prototype data because the present chart has been
developed using three laboratory and three prototype data only.
REFERENCES
[1]. USBR, Design of Small Dams, Spillways, 194. Discharge Over Gate-Controlled Ogee Crests, First edition, Second print, US
Department of The Interior, Bureau of Reclamation, (US Government Printing Office, 1961, Was
[2]. USBR, Design of Small Dams, Spillways, 201. Discharge Over Gate-Controlled Ogee Crests, Second edition, US Department
of The Interior, Bureau of Reclamation, (US Government Printing Office, 1973, Washington DC).
[3]. USBR, Design of Gravity Dams, Ch.9 Spillways, 9-14. Discharge Over Gate-Controlled Ogee Crests, A Water Resources
Technical Publication, US Department of The Interior, Bureau of Reclamation (US Government Printing Office, Washington
DC, 1976).
[4]. USBR, Design of Arch Dams, Ch.9 Spillways, 9-14. Discharge Over Gate-Controlled Ogee Crests, A Water Resources
Technical Publication, US Department of The Interior, Bureau of Reclamation, (US Government Printing Office, Washington
DC, 1977).
[5]. USBR, Design of Small Dams, Ch.9 Spillways, 9.16. Discharge Over Gate-Controlled Ogee Crests, A Water Resources
Technical Publication, Third edition, US Department of The Interior, Bureau of Reclamation, (US Government Printing Office,
Washington DC, 1987).
[6]. Sanko, Final Feasibility Report of Yedigoze Dam and HPP Volume 1, Appendix-4: Hydraulic Design of Flood Spillway (in
Turkish). Sanko Engineering and Consulting Inc., Cetin Emec Boulevard, 6th
Street, No: 61/7, 06520 Balgat, Ankara, Turkey,
2007.
[7]. Temelsu, Final Design of Bayramhacili Dam and HPP (Civil Works), 3.1 Hydraulic Computations, Flood Spillway (in
Turkish). Temelsu International Engineering Services Inc., Ankara, Turkey, 2007.
[8]. F Salazar, R Moran, R Rossi, E Onate, Analysis of discharge capacity of radial-gated spillways using CFD and ANN – Oliana
Dam case study, Journal of Hydraulic Research, 51(3), 2013, 244–252.
[9]. USACE, Hydraulic Design Criteria, Volume 2, Tainter Gates on Spillway Crests, Sheets 311-1 to 311-5, Department of the
Army, Corps of Engineers, Mississippi River Commission, Waterways Experimentation Station, Vicksburg, Mississippi, 1987.
[10]. T Haktanir, H Citakoglu, N Acanal, Fifteen-stage operation of gated spillways for flood routing management through artificial
reservoirs, Hydrological Sciences Journal, 58(5), 2013, 1013–1031.
[11]. M Zargar, H M V Samani, A Haghigih, Optimization of gated spillways operation for flood risk management in multi-reservoir
systems. Natural Hazards, 82(1), 2016, 299–320.
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Engineering, ASCE, 127(8), 2001, 640–649.
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CFD model. KSCE Journal of Civil Engineering, 9(2), 2005, 161–169.
[14]. Bagatur T and Onen F, 2016. Computation of design coefficients in ogee-crested spillway structure using GEP and regression
models. KSCE Journal of Civil Engineering, 20(2), 951–959. DOI 10.1007/s12205–015–0648–x
[15]. Ansar M and Chen Z, 2009. Generalized flow rating equations at prototype gated spillways. Journal of Hydraulic Engineering,
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[18]. USBR, 1965. Hydraulic Model Studies of the Sluiceway and Overflow Weir – Yellowtail Afterbay Dam, Missouri River Basin
Project, Montana, Report No. Hyd-523, US Department of The Interior, Bureau of Reclamation, Hydraulics Branch, Division of
Research, Office of Chief Engineer, Denver, Colorado.
[19]. USBR, 2013. Discharge Curves and Equations for Yellowtail Afterbay Dam, Missouri River Basin Project, Montana,
Hydraulic Laboratory Report HL-2013-01, US Department of The Interior, Bureau of Reclamation, Technical Service Center,
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An efficient algorithm for ogee spillway discharge with partiallyopened radial gates by the method of Design of Small Dams and comparison of current and previous methods

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An efficient algorithm for ogee spillway discharge with partiallyopened radial gates by the method of Design of Small Dams and comparison of current and previous methods