Contents preface (VII) introduction 1—37

Fig. 17.6 Ogee spillway with an overhang
The profile of an ogee spillway is designed for a given design discharge (or corresponding design surcharge head or, simply, design head). When the flowing discharge equals the design discharge, the flow adheres to the spillway surface with minimum interference from the boundary surface and air has no access to the underside of the water sheet. The discharge efficiency is maximum under such condition and the pressure along the spillway surface is atmospheric. If the flowing discharge exceeds the design discharge, the water sheet tends to pull away from the spillway surface and thus produces sub-atmospheric pressure along the surface of the spillway. While negative pressure may cause cavitation and other problems, it increases the effective head and increases the discharge. On the other hand, positive hydrostatic pressure will occur on the spillway surface, if the flowing discharge is less than the design discharge.

Model tests have indicated that the design head may be safely exceeded by about 50% beyond which cavitation may develop. Therefore, spillway profle may be designed for 75% of the peak head for the maximum design flood.

An upstream overhang, known as corbel, is added to the upstream face of the spillway as shown in Fig. 17.6. The effect of the corbel is to shift the nappe (and, hence, the spillway profile) backward which results in saving of concrete. If the height of the vertical face of the corbel is kept more than 0.3 times the head over the crest, the discharge coefficient of the spillway will be practically the same as it would be if the vertical face of the corbel were to extend to the full height of the spillway.

SPILLWAYS

573

The shape of an ogee crest, approximating the profile of the underside of a water jet flowing over a sharp-crested weir, depends on: (i) the head, (ii) the inclination of the upstream face of the overflow section, and (iii) the height of the overflow section above the floor of the entrance channel which affects the velocity of approach to the crest. A simple shape of an ogee crest suitable for dams with vertical upstream face is shown in Fig. 17.7. It consists of an upstream surface shaped as an arc of a circle upto the apex of the crest followed by a parabolic downstream surface. This type of ogee crest is suitable for preliminary estimate and for final designs when a more refined shape is not required (2).

^ho

ho

Origin of

coordinates

4

^ho

Y

8

5

x

R =

h

o

16

Parabola
2

X = 2h_oY

Fig. 17.7 Ogee crest for spillway with vertical upstream face
An extensive study of crest shapes has been made by USBR (3). These crest shapes can be represented by the profile shown in Fig. 17.8 (a) and defined with respect to the coordinate axes at the apex of the crest. The part of the profile upstream of the apex of the crest consists of either a single curve and a tangent or a compound circular curve. The profile downstream of the apex of the crest is defined by the equation

y	F	x	I ^n
	= − k G		J	(17.4)
H0	H0
		H	K

in which, H₀ is the total head on the crest including the velocity of approach and the constant k and n depend on: (i) the inclination of the upstream face, and (ii) the velocity of approach as has been shown in Fig. 17.8 (b) and 17.8 (c). Similarly, the value of X_c, Y_c, R₁, and R₂, shown as elements of the crest profile in Fig. 17.8 (a), can be obtained from Fig. 17.8(d), 17.8 (e), and 17.8 (f).

574 IRRIGATION AND WATER RESOURCES ENGINEERING

ha

h

a

Ho

Origin

xc

and apex of crest

yc

x

y

x

R1

X

Y

n

^R2

H

o

= –K ( H )

P

o

Upstream face

(a) Elements of ogee - shaped crest profiles

ha/H0

0

0.04

0.08

0.12

0.16

0.56

3:3

0.52

2:3

K

Vertical

and 1:3

0.48

0.44

(b) Values of K

1.88

Vertical

(c) Values of n

1.84

1:3

n

1.80

Slo

pin

g u

pstream f

1.76

ace2(H): 3(V)

3:3

1.72

0.04

0.08

0.12 ha/HD

0.16

0.28

Apex of crest

Centre of curvature

xc

0.26

for R2 is located at inter- ^yc

^R1 b

0

Slopin

g

u/s face

section of arcs ab and cd

d

1(

R2 c

/H

0.24

3

U/S face

c

H) :

(

a

R

2

X

0.22

Vertical)

location of centre

R

–1

2:3

for R2

(d) Values of xc

0.20

3:3

0.12

0

0.10

Vertical

/H

c

y

0.08

1:3

(e) Values of yc

0.06

2:3

0.04

3:3

0.55

R1

for

1:3

R

1 for

0.50

vertical

for

2:3

R1 for 3:3

R1

0.45

R2 for 3:3 =

	0.40		R
0	with	1	for	2:3
R/H		R
					=
0.35			2

0.30
0.25

0.20
0.15
0
2:3 for R2

R	for
2
	vertical
R	for 1:3	(f) Values of R1 and R2
2
0.04	0.08	0.12	0.16	0.20
		ha/H0

Fig. 17.8 Parameters for ogee-shaped crest profile

SPILLWAYS

575

The discharge Q flowing over an ogee crest of effective length L can be obtained from the formula

Q = CLH ^3/2	(17.5)
o

in which, C is a variable coefficient of discharge which depends on: (i) depth of approach [Fig. 17.9 (a)], (ii) slope of the upstream face [Fig. 17.9 (b)], (iii) position of the downstream apron [Fig. 17.9 (c )], (iv) the downstream tail-water level, [Fig. 17.9 (d)], and (v) the relation between the actual crest shape (corresponding to the design head) and the ideal nappe shape (corresponding to the actual head of flow) [Fig. 17.9 (e)]. The effect of the downstream apron and tail-water level would generally be felt when ogee crest is being used as control structure for a side-channel spillway and the tail-water level is high enough to affect the discharge, i.e., the crest is submerged. For usual ogee spillways, this situation would not arise.
The effective length of the crest L will be less than the total length of the crest on account of side contractions whenever crest piers are provided. The effective length L and total length L′ of a crest are related as follows (1):

L = L′ – 2 (N′Kp + Ka) Ho

(17.6)

Here, N′ is the number of piers, and K_p and K_a are, respectively, pier and abutment coefficients. Values of K_p vary between zero (for pointed nose piers) and 0.02 (for square-nosed pier with suitable rounded corners). Values of K_a vary between zero and 0.2.

ValuesofC

2.2

2.1

2.0

1.9

1.8
1.7

0	2	4	6	8	10	12
			P/HO

(a) Variation of coefficient of discharge for ogee-shaped crest with vertical uptream face

Inclined

C

1.04
1.02

Vertical
C	1.00
	0.98

0
Slope of the upstream face

3(H):1(V)
:3
:3

0.5 1.0 1.5 P/H_o

(b)- Coefficient of discharge for ogee-shaped crest with sloping uptream face