Contents preface (VII) introduction 1—37

f	R
(8.24)
	5	1	1

The factor f₁ has been named as the silt factor. For natural sediment of relative density equal to 2.65, the silt factor f₁ can be obtained by Lacey’s relation
f₁ = 1.76 d (8.25) where, d is the median size of sediment in millimetre.

On plotting Lindley’s data and other data of regime channels, Lacey obtained

	R^1/2 S = C″
and			C′ = 10.8C″^1/3
		U
∴				= 10.8 (R^1/2	S)1/3
	R	1/ 2
			U = 10.8R^2/3 S^1/3
which gives					(8.26)

Equation (8.26) is known as Lacey’s regime equation and is of considerable use in evaluating flood discharges.

	On the basis of the data of regime channels, Lacey also obtained
		Af1^2 = 140U^5	(8.27)
i.e.,		Qf ^2	= 140U^6	(8.28)
					1
on substituting the value of f1 from Eq. (8.24),
	QGF 5U ^2	JI ^2	= 140U^6
	H	2R	K
or	25 ×			Q		= 140U^2
	R^2
		4
or		25	Q	P2	= 140U^2
4	A^2
or					P2 = ^560 Q
								25
i.e.,					P = 4.733 Q ≅ 4.75 Q	(8.29)

296 IRRIGATION AND WATER RESOURCES ENGINEERING
Equation (8.29) with multiplying constant modified to 4.75 has been verified by a large amount of data and is very useful for fixing clear waterways for structures, such as bridges on rivers. Again, substituting the value of f₁ in Eq. (8.27),

1. 
   ^{F 5U 2 I 2} _{= 140U}5
   

^GH 2R ^JK

∴		A	560
			=	25 ^U
	R^2
or		P = 22.4 RU
or	4.75 Q = 22.4RU
or	RU = 0.212 Q^1/2	(8.30)
For wide channels, RU equals the discharge per unit width. Hence,
		q = 0.212 Q^1/2	(8.31)

Equation (8.31) relates the discharge per unit width of a regime channel with the total discharge flowing in the channel.
On substituting the value of U from Eq. (8.24) in Eq. (8.30), one gets

R	^2 f R = 0.212 Q^1/2
	5	1
		F	Q I	1/ 3
∴		R = 0.48 G		J	(8.32)
		H f1 K

For wide channels, the hydraulic radius is almost equal to the depth of flow. Equation (8.32), therefore, gives the depth of scour below high flood level. Hence, Eq. (8.32) can be utilised to estimate the depth of flow in a river during flood. This information forms the basis for the determination of the levels of foundations, vertical cutoffs, and lengths of launching aprons of a structure constructed along or across a river.
On combining Eqs. (8.31) and (8.32),

F q2 I ^{1/ 3}

R = 1.35G J (8.33)

H f₁ K

Further, eliminating U from Eqs. (8.24) and (8.26), one can obtain a relationship for the slope of a regime channel. Thus,

	^2 f 1R = 10.8R^2/3	S1/3
	5
			f	3 /2
∴	S = 0.0002		1				(8.34)
	R1/ 2
On substituting the value of R from either Eq. (8.32) or Eq. (8.33), in Eq. (8.34), one
obtains,
			f 5/3
	S = 0.0003		1				(8.35)
	Q1/6
						f 5/3
and	S = 0.000178		1			(8.36)
	q1/3

DESIGN OF STABLE CHANNELS

297

Lacey’s regime relations, Eqns. (8.23) to (8.36), are valid for regime channels and can be used suitably to design a regime channel for a given discharge and sediment size.

The following flow equation was also obtained by Lacey (11) :

	U =	1	R3/4 S1/2	(8.37)
					Na
where,	N	a	= 0.0225f ^1/4		(8.38)
				1

This means that the absolute roughness coefficient N_a is dependent only on the sediment size. On examining the data from various channels, the value of N_a was, however, not found to be constant. Lacey introduced the concept of ‘shock’ to explain the variation in N_a. He contended that a non-regime channel requires a larger slope (i.e. , large value of N_a) to overcome what he termed as ‘ shock resistance’ or the resistance due to bed irregularities. The shock resistance can, therefore, be considered similar to the form resistance of the bed undulations. This concept of Lacey leads one to conclude that a regime channel is free from shock. It is, however, known that the geometry of bed undulations can change even for the same sediment size and, hence, Lacey’s contention that a regime channel is free from shock, is unacceptable (12).
Lacey’s equations, commonly used for the design of alluvial channels, are summarised below :

f1 = 1.76		d	(8.25)
U = 10.8	R2/3 S1/3	(8.26)
P = 4.75		Q	(8.29)
	F	Q I 1/3
R = 0.48G			J	(8.32)
	H f1 K
S = 0.0003	f1^5 /3		(8.35)
Q1/ 6