Contents preface (VII) introduction 1—37

Table 11.3 Values of K in Eq. (11.1)
	Shape of pier	K
	Semicircular nose and tail	0.90
	Nose and tail formed of two circular curves each of radius	0.90
	equal to twice the pier width and each tangential to pier face
	Twin-cylinder piers with connecting diaphragm	0.95
	Twin-cylinder piers without diaphragm	1.05
	90°-triangular nose and tail	1.05
	Square nose and tail	1.25

The total head loss, h_L, for a flow through siphon or siphon aqueduct can be obtained as the sum of the losses at the inlet and outlet and the friction loss. If approach velocity V_a is also significant, the approach velocity head may also be taken into account. Assuming the downstream velocity head to be negligible, the head loss (or afflux) h_L is expressed as,

h

=

F

1

+ f

1

+ f

2

L

I V ^2

− Va^2

...(11.2)

L

G

J

2g

2g

H

RK

Here, L is the length of the barrel, R the hydraulic radius of the barrel section, V the velocity of flow through the barrel section, and V_a is the approach velocity which is generally neglected. f₁ is the entry loss coefficient whose value is 0.505 for an unshaped entrance and 0.08 for bell-mouthed entrance (1). f₂ is a coefficient similar to the friction factor and is equal to a [1 + (b/R)]. Here, R is expressed in metres. The values of a and b for different types of barrel surface are given in Table 11.4.
Table 11.4 Values of a and b for f₂ (3)

Barrel surface	a	b
Smooth iron pipe	0.00497	0.025
Incrusted iron pipe	0.00996	0.025
Smooth cement plaster	0.00316	0.030
Ashlar or brickwork or planks	0.00401	0.070
Rubble masonry or stone pitching	0.00507	0.250

In order to minimise the head loss and afflux, the barrel surface should be made smooth and the entrance of the barrel bell-mouthed.

11.7. DESIGN OF TRANSITIONS FOR CANAL WATERWAY
The cost of an aqueduct or siphon aqueduct will depend on its width and other factors. The width of the aqueduct is, therefore, reduced by contracting the canal waterway. However, the canal waterway is not contracted in case of earthen banks and, hence, contraction of the canal waterway is considered only in structures of type III [Fig. 11.4 (c)]. There is, however, a limit up to which a canal can be contracted or flumed. The fluming should be such that it should not result in supercritical velocity in the canal trough. The supercritical velocity in the canal trough may cause the formation of a hydraulic jump before the supercritical flow of the canal trough

CROSS-DRAINAGE STRUCTURES

391

meets the subcritical flow of the normal canal section. The jump formation would result in additional loss of energy and large forces on the structure. Also, the lengths of transitions increase with the amount of reduction in the canal width. The fluming should be such that the cost of additional length of transition is less than the savings in cost on account of the reduction in the width of the aqueduct (or siphon aqueduct). The canal should not be flumed to less than 75 per cent of bed width. If the velocity and the head loss permit, greater fluming may be allowed ensuring subcritical flow conditions in the flumed canal (1). Proper transitions need to be provided between the flumed portion and normal section of the canal. A minimum splay of 2 : 1 to 3 : 1 for the upstream contracting transition, and 3 : 1 to 5 : 1 for the downstream expanding transition is always provided (1).

An increase in the depth of flow in the transitions and also in the canal trough will result in deeper foundation and higher pressures on the roof as well as higher uplift on floor of the culvert. The canal trough and transitions are, therefore, generally designed keeping the depth of flow the same as in the normal section.
While constructing a cross-drainage structure, it is often required to connect two channels of different cross-sectional shapes. Usually, a trapezoidal channel is required to be connected to a rectangular channel or vice versa. A channel structure providing the cross-sectional change between two channels of different cross-sectional shapes is called channel transition or, simply, transition. Transition structure should be designed such that it minimises the energy loss, eliminates cross-waves, standing waves and turbulence, and provides safety for the transition as well as the waterway. Besides, it should be convenient to design and construct the structure. These transitions are usually gradual for large and important structures so that the transition losses are small. Abrupt transitions may, however, be provided on smaller structures.
For a contracting transition (Fig. 11.10), the flow is accelerating and as such any gradual contraction which is smooth and continuous should be satisfactory. A quadrant of an ellipse with its centre on bb can be chosen for determining the profile of the bed line. For a splay of 1 in m₁, the bed line profile can be written as

L

x

O^2

L

y

O

2

M

P

+ M

P

= 1

(11.3)

0.5m

(B

− B )

0.5 (B

− B )

M

1

c

f

P

M

c

f

P

N

Q

N

Q

Side slopes and the bed elevation may be linearly varied. The value of m₁ should be kept higher than 3. In the case of an expanding transition, however, more care must be exercised. Following discussions pertain to the design of expanding transitions.
Irrigation channels carry subcritical flow and the hydraulic design of gradual transitions for subcritical flow requires: (i ) prediction of flow conditions at section f-f (Fig. 11.10) for the given size and bed elevation of the flume and also the flow condition at the exit section c-c, and (ii) determination of the boundary shape and flow conditions within the transition. For given flow conditions in the exit channel, the depth and velocity of flow (and, hence, the energy loss) within the transition are governed by the following three boundary variations:

	B = f1	(x)	(11.4)
	∆z = f2	(x)	(11.5)
and	m = f3 (x)	(11.6)
where,	B = the bed width of the transition at distance x from the flume end of the transition
	(Fig. 11.10),

392

IRRIGATION AND WATER RESOURCES ENGINEERING

∆z = change in bed level (with respect to the flume bed) within the transition,

and

m = channel side slope (i.e., m(H) : 1 (V) within the transition).

a

m1

b

f

c

Water

1

2 i i+1

1

line

3

2

1

m = m0

m = 0

m = 0

m = m

Bf

B

0

BC

BC

X

0

Flume

Approach

Exit

3

2

1

1

2

channel

Contracting

channel

X

a transition b

f Expanding

c

Plan

transition

hf

Dh

^hC

Section

X

Dz

Dz0