Contents preface (VII) introduction 1—37

Fig. 17.12 Shaft spillway
Depending upon the type of crest, the shaft spillway can be either standard-crested or flat-crested (Fig. 17.13). In a standard-crested spillway, the water begins its free fall immediately upon leaving the crest whereas in the flat-crested spillway water approaches the crest on a flat slope before beginning its free fall. The standard-crested spillway would have a smaller diameter crest since its coefficient of discharge is greater than that of a flat crest. Therefore, if the shaft spillway is to be constructed in the form of a tower, it would be economical to have a standard-crested spillway. However, a flat-crest shaft spillway has a smaller funnel diameter and is, therefore, more advantageous when the spillway is to be excavated in rock. The design of a standard-crested shaft spillway has been discussed here.
The design of a standard-crested shaft spillway involves the determination of the funnel radius, R, and the head over the theoretical sharp crest, H, for known discharge, Q and the allowable maximum head, h on the spillway crest (Fig. 17.14). The method given by Creager, et al. (6) is valid for negligible velocity of approach and involves trial at two stages and is based on the following equations:

582 IRRIGATION AND WATER RESOURCES ENGINEERING

Funnel

Funnel	Crotch
Standard-
crested spillway
Free-falling		shaft	section
section		Vertical	Verticalshaft
	Crotch
section	Vertical	shaft
Vertical
shaft

H.W.S.

Weir section
Free-falling

section
Flat-crested

spillway

Fig. 17.13 Standard-crested and flat-crested profiles for shaft spillway


^{H h} Spillway crest

	x
Theoretical
sharp crest
R
Top of boil	spillway
	L
	C
Crotch
y

Fig. 17.14 Jet profile over standard-crested shaft spillway

SPILLWAYS				583
	Q = 2 π RC1	g H 3 / 2	(17.9)
	r		H
		= 0.11 – 0.10			(17.10)
	H	R
	H = h + r					(17.11)

In these equations, C₁ is the coefficient of discharge which is related to H/R as shown in Fig. 17.15, and r is the rise of the lower nappe above the theoretical sharp crest. The different steps involved in the trial method are as follows:

0.61

0.60

0.59

0.58
^C1
0.57

0.56

0.55

Boil

^C1

25
20
15boil
10	oftopof
	y/H
5	Ordinate
0
–5

0

0.1 0.2 0.3 H/R

0.4

0.5

Fig. 17.15 Variations of C₁ and the ordinate y/H of the top of boil for standard-crested shaft spillway (6)
(i) Assume some suitable values of H and R and, hence, H/R, and obtain the value of C₁ from Fig. 17.15.
(ii) Determine the discharge intensity, q per unit length of the crest, i.e.,

q = C g H ^3 /2	(17.12)
1

(iii) Obtain the required radius R from

R =	Q	(17.13)
2πq

Compare this value of R with the assumed value of R in step (i). If these two values do not match, assume another value of R for the same assumed value of H and re-peat the procedure until the value of R obtained in step (iii) matches with the as-sumed value of R.

(iv) Determine r from Eq. (17.10) and then obtain h from Eq. (17.11). If this value of h does not agree with the given value of h, one has to assume another value of H, and repeat steps (i) to (iv) until the agreement is reached.

584 IRRIGATION AND WATER RESOURCES ENGINEERING
Vittal (7) has obtained a direct solution for R and H by rewriting Eqs. (17.9) to (17.11) as follows:
From Eq. (17.9),

Q* =

Q

= 2π

R

C

F

H

I 3 /2

(17.14)

1/ 2

5 /2

g

h

h

1G

J

H

h K

Using Eq. (17.11), Eq. (17.10) can be rewritten in the following two forms:

r

=

H − h

= 1 −

h

= 0.11 − 0.10

H

H

H

H

R

∴

h

= 0.89 + 0.10

H

(17.15)

H

R

r

H − h

H

−

h

H

= 0.11 − 0.10

and

=

=

R

R

H

H

H

R

F H I ^2

R

H

h

∴

G

J

+ 8.9

− 10

= 0

(17.16)

R

R

H R K

From Fig. 17.15, and Eqs. (17.15) and (17.16), one obtains the following functional

relationships:

F H I

C = ^f1G

J

1

H

R K

h

F

H I

=

f2 G

J

H

H

R K

H

=

f

F

h

I

R

3 G

J

H

RK

Using the above functional relationships, Vittal (7) obtained the following functional relation:

F

H I

3 /2

F

h I

C1G

J

=

f4 G

J

(17.17)

H

h K

H

RK

Actual relationship of Eq. (17.17) can be obtained by obtaining C1 from Fig. 17.15, the

values of

h

from Eq. (17.15), and

h

from Eq. (17.16) for different values of

H

ranging from

R

H

R

F

H I 3 /2

F

h I

0 to 0.5. One can, therefore, prepare a curve of C1G

J

versus G

J . Vittal (7) obtained the

H

h K

H

RK

following equation for this curve:
C	F	H	I 3 /2	= 0.6988 − 0.0882 ^F	h	I	− 0.296 ^F	h	I ^2	(17.18)
1G		J	G		J	G		J
	H	h K	H	RK	H	RK

On combining Eq. (17.18) with Eq. (17.14), and solving the resulting quadratic equation, one obtains,

SPILLWAYS

F	RI	2	R
G		J	− (0.2280 Q* + 0.1263)		− 0.3861	= 0
	h
H		h K

	R	L			14432.
∴		= (0.1140 Q* + 0.0632) M1	+	1 +	(0.2280 Q + 0.1263)^2
h
		M				*
		N

O
P
P_Q

585

(17.19)

For large values of Q_* (say, greater than 25), Eq. (17.19) can be approximated to

R

= 0.2280 Q + 0.1264

(17.20)

h

*

For known Q and h and, hence, Q , one can easily determine

R

(and, hence, R) from one

*

h

H

of the Eqs. (17.19) and (17.20). Using Eq. (17.16) one can determine

and, hence, H. For

R

example, values of Q and h equal to 851.2 m^3/s and 3.05 m, respectively, yield

Q* = 16.73

R

= 4.03

∴

R = 12.29 m

h

H

= 0.27

∴

H = 3.325 m

R

The profile of the underside of the nappe over the circular sharp-crested weir can be determined from Fig. 17.16 which enables computation of x for a given value of y , and already computed R and H. If R₀ is the radius of the lower nappe at any given elevation y, then

R₀ = R – x
And x₀, representing the value of x for the upper side of the nappe at a given value of y, is obtained from (6)

x		= R –	R ^2	−		Q	(17.21)
0
			0		π	2 g(y + 1269. H)

Here, y + 1.269 H is the head available for vertical velocity. Proceeding in this manner, one can compute the value of x and x₀ for different values of y until x₀ equals R at which value of y the horizontal velocity ceases and its energy gets converted into a ‘boil’ as shown in Fig.

14. 
    The ordinate _H^y of the top of the boil can be computed from the curve shown in Fig.
    

14. 
    The point at which x₀ becomes equal to R is usually known as ‘crotch’. The above analysis does not include the friction loss as it is difficult to be considered and is within permissible limits for the accuracy desired (6).
    

The diameter of the vertical shaft below the crotch continues to decrease until the size becomes such that the discharge, Q can be carried according to the head available. The radius of the transition shaft, R′ at a given elevation y is obtained from

Q = π R ′ ^2 2gh	(17.22)
v

in which, h_v = y + 1.269 H – h_L1 – h_L2
where, h_{L 1} and h_L2 are the head losses due to friction, respectively, from the crest to the crotch and from the crotch to the elevation under consideration.

586
0.0
Crest of

theoretical weir

	1
	2
	3
Valuesofy/H	4
	5

6

7

8

9

10

0

IRRIGATION AND WATER RESOURCES ENGINEERING

50.0			Values
45.0
			of
40.0			H/R
				=
				0
				.
				0
35.0			.	0
		.
		0		0	10
	30.0	15
			.
	20

1 2 3 4 5

Values of x/H