Contents preface (VII) introduction 1—37

11.7.2. UPIRI Method
For channels of constant depth, it was assumed (6) that the rate of change of velocity per unit length of the transition should be constant throughout the transition length. This means,

(vf – vi)/x = (vf – vc)/L

(11.16)

Here, suffixes f and c are, respectively, for the flumed section and normal section of the channel, and x is the distance of the i^th section of the transition from the flumed section. Thus, v_i is the velocity of flow at the chosen i^th section. L is the length of the transition which was arbitrarily assumed as 2 (B_c – B_f). Since the depth h is constant,

B_fv_f = B_iv_i = B_cv_c = Q/h

Hence,

vf =

Q

×

1

h

^Bf

v = ^Q

×

1

i

h

Bi

and

v

= ^Q

×

1

c

h

Bc

Therefore, using Eq. (11.16),

(Q / hBf ) − (Q / hBi )

=

(Q / hBf ) − (Q / hBc )

x

L

or

^Bi

=

Bc Bf L

(11.17)

LBc − x(Bc − Bf )

Equation (11.17) is the hyperbolic bed-line equation for constant depth and can be used for transition between rectangular flume and rectangular channel.
11.7.3. Vittal and Chiranjeevi’s Method
Vittal and Chiranjeevi (4) examined the above two methods and offered the following comments:
(i) Hinds’ method involves a trial procedure which can be easily avoided if the values assigned to m follow a smooth and continuous function [Eq. (11.6)] instead of an arbitrary set of values for m. The function to be chosen should be such that the side slope varies gradually in that part of the transition where the velocities are higher and it varies rather fast in other part of the transition where the velocities are lower.
(ii) A smooth and continuous bed width profile alone is not sufficient to avoid separation and consequently the high head loss.
(iii) The total head loss through transition can be obtained by summing up all the head losses in the sub-reaches of the transition. Using Eq. (11.10), the total head loss H_L is given by

CROSS-DRAINAGE STRUCTURES

395

^HL ^= ∑ ^hLi , i + 1 ^= ^Kh ∑	v 2	− v	2		v	^2 − v ^2
	i	i + 1	= Kh	f	c	(11.18)
	2g			2g

This means that the head loss in the transition depends only on the entrance and exit conditions, and is unaffected by the transition geometry. This, obviously, is not logical and is a limitation of Hinds’ method.
(iv) Hinds’ method first assumes free water surface and then computes the boundaries which would result in the assumed water surface. However, it is usually desirable to select the boundaries first and then compute the water surface profile.
(v) UPIRI method (6) would require lowering of the flume bed below the channel bed to achieve constant depth. This may not be always practical. For example, in case of a cross-drainage structure, the flume bed level may have to be lowered even below the drainage HFL. As a result, the cross-drainage structure, which otherwise can be an aqueduct, may have to be designed as a siphon aqueduct which is relatively more expensive.
On the basis of theoretical and experimental investigations, Vittal and Chiranjeevi (4) developed a method for the design of an expanding transition. The guiding principles for this method were minimisation of the energy loss and consideration of flow separation in the expanding flow. The design equations of Vittal and Chiranjeevi (4) for the bed width and side slope are as follows:
Transition bed width profile:

B − Bf

x ^L

F

x I ^n O

=

M1

− G

1 −

J

P

(11.19)

Bc − Bf

^L M

H

LK

P

N

Q

where,

n = 0.80 – 0.26 m

1/2

(11.20)

0

and length of transition,

L = 2.35 (Bc – Bf) + 1.65 m0 hc

(11.21)

Side slopes, varying according to the equation,

m

F

x I 1/ 2

= 1 − G 1 −

J

(11.22)

m0

H

LK

change gradually in the initial length of the transition where the flow velocity is high, and rapidly in the latter length of the transition where the velocity is low. For head loss computations, use of Hinds’ equation, viz., Eq. (11.10) with K_h = 0.3 has been suggested.
This method of design of expanding transition is applicable to all three types of conditions, viz., constant depth, constant specific energy, and variable depth-variable specific energy. The profiles of the bed line and the side slope will be the same for all the three conditions. The bed of the expanding transition would always rise in the direction of flow for the constant depth scheme. On the other hand, constant specific energy and variable depth-variable specific energy schemes would always result in the falling bed transition.
For the constant specific energy condition,

E = hc +	Q^2	= Ei = hi +	Q^2
2 g B	2 h 2		2g B	2 h 2
c
	c	c			i	i

Therefore, one can determine h_i and hence v_i. The transition loss between successive sections can be determined from Eq. (11.10) with K_h = 0.3. The bed has to be lowered at successive

396 IRRIGATION AND WATER RESOURCES ENGINEERING
sections just sufficient to provide for the transition loss [as can be seen from Eq. (11.9)] so that the specific energy remains constant throughout the transition. This means that,

^∆^zi, i + 1 ^= ^hLi, i + 1

(11.23)

Equations (11.19) to (11.23) along with Eq. (11.10) enable the design of an expanding transition as has been illustrated in Example 11.1.
Similarly, for the constant depth condition, h_f = h_c = h_i
Hence, one can calculate L, n, B_i and m_i for known value of x using Eqs. (11.19 – 11.22). Thereafter, one can compute the velocity v_i and the head loss h_{Li, i + 1} using Eqs. (11.7 and
11.10). Using Eq. (11.9), one can estimate the drop (or rise) ∆z_{i, i + 1} in the bed elevation at the known value of x from the flume end.
The method based on variable depth-variable specific energy scheme presupposes the value of ∆z₀ (i.e., difference in elevations of canal bed and flume bed) and the transition bed slope is assumed to be constant in the entire length of the transition. Length of transition L, the width of transition B_i and side slope m_i at known value of x (from the flume end) are calculated using the relevant equations discussed above. The depth of flow h_i is determined by trial using energy equation. To start with, the head loss term (unknown since the depth and, hence, velocity are unknown) is neglected and the value of depth of flow h_i is determined from the energy equation. Using this value of the depth of flow one can determine the velocity and the head loss between the adjacent sections and using energy equation, one can compute new value of h_i which is acceptable if it does not differ from the just previous trial value. Otherwise, one should repeat the trial.