Contents preface (VII) introduction 1—37

Table 7.1 Variation of x with d65/δ′ (15)
d65/δ′	0.2	0.3	0.5			0.7		1.0			2.0	4.0	6.0	10
x	0.7	1.0	1.38		1.56		1.61			1.38	1.10	1.03	1.0

1.8
1.6

1.4
1.2

x
1.0
0.8
0.6
0.4

0.1

Smooth Wall

Rough wall

1.0 10 100

d₆₅/d¢

Fig. 7.8 Correction x in Eq. (7.28) (15)
Einstein and Barbarossa (15) recommended that one of the equations, Eq. (7.26) or Eq.

(7.27) may be used for practical problems. The resistance (or shear) due to bed forms τ ″ is

ob

computed by considering that there are N undulations of cross-sectional area a in a length of channel L with total wetted perimeter P. Total form drag F on these undulation is given by

F	1I	2
F = CD a G		J ρ U		N	(7.29)
H	2K

Here, C_D is the average drag coefficient of the undulations. Since this drag force acts on area LP, the average shear stress τ^″_ob will be given as

	τob^″		F		C D aN	U ^2
		=		=	LP ^ρ
	LP	2
∴	τ^″ob		= U ″ ^2	= C D aN U ^2
	ρ		*		LP	2
or	U		=	2LP / (C D aN)	(7.30)
U^″
		*

268 IRRIGATION AND WATER RESOURCES ENGINEERING
Here, U_*^″ is the shear velocity corresponding to bed undulations. According to Einstein and Barbarossa, the parameters on the right hand side of Eq. (7.30) would primarily depend

on sediment transport rate which is a function of Einstein’s parameter Ψ′ = ∆ρ	s	d	/ρ R′ S.
		35	b

Therefore, they obtained an empirical relation, Fig. 7.9., between	U	and Ψ′ using field data
U^″
	*

natural streams. The relationship proposed by Einstein and Barbarossa can be used to compute mean velocity of flow for a given stage (i.e., depth of flow) of the river and also to prepare stage– discharge relationship.

100
80
60
40
U 20
U*^²
10
8
6
4
0.4 0.6 0.8 1	2	4	6	8 10	20	40

Y¢
Fig. 7.9 Einstein and Barbarossa relation between U/U″ and Ψ′ (15)

*
The computation of mean velocity of flow for a given stage requires a trial procedure. From the known channel characteristics, the hydraulic radius R of the flow area can be determined for a given stage (or depth of flow) of the river for which the mean velocity of flow

is to be predicted. For a wide alluvial river, this hydraulic radius R approximately equals Rb. A
value of R′ smaller than R	b	is assumed and a trial value of the mean velocity U is calculated
b

from Eq. (7.25) or Eq. (7.26) or Eq. (7.27). The value of ^U is read from Fig. 7.9 for Ψ′

U_*^″

corresponding to the assumed value of R′ . From known values of U (trial value) and U/U″, U″
and, hence, R″	b	and R″ equals	*	*
can be computed. If the sum of R′	R the assumed value of R′
b	b		b	b	b
and, hence, the corresponding mean velocity of flow U computed from Eq. (7.25) or Eq. (7.26)

or Eq. (7.27) are okay. Otherwise, repeat the procedure for another trial value of R ′ till the

sum of R ′ and R″ equals R . The computations can be carried out easily in a tabular^b form as

b b b

illustrated in the following example:

Example 7.5 Solve Example 7.3 using Einstein and Barbarossa method.
Solution: For given d = 0.6 mm and bed slopes S = 3 × 10^–4

U^′ =

gR^′ S =

9.81 × R^′

× 3 × 10^−^4

= 0.054

R′

From Eq. (7.25)

*

b

b

b

F

I 1/ 6

U = 7.66 U′(R′ /d)^1/6

= 7.66 × 0.054

R^′

R^′

b

−3

J

*

b

b G

0.6

× 10

H

K

∴

U = 1.4243 R′ ^2/3

b

HYDRAULICS OF ALLUVIAL CHANNELS

	ψ′ =	∆ρ sd35
		ρR ^′	S
		b
From	U″ =	gR ^″ S
	*	b
		(u″ ) ^2
	R″ =	*		=
	b	gS

269

165.	× 0.6	× 10^−^3	3.3
=					=
R^′	(3 × 10^−^4 )
			R^′
			b				b
			(U ″ )^2		= 339.79 (U ^″ )^2
			*
	9.81 × 3 × 10^−^4
			*

The trial procedure for computation of mean velocity can now be carried out in a tabular form. It is assumed that the alluvial river is wide and, therefore,
R_b ≅ R

Trial	R′	U′	U	ψ′	U/U″	U″	R″	R	b	Comments
No.	b	*	(m/s)		*	*	b
(m)	(m/s)				(m/s)	(m)	(m)
1	1.2	0.059		1.6084	2.75	16.0	0.1005	3.432	4.632	higher than 1.4
2	0.5	0.038	0.8973	6.60	10.5	0.0855	2.484	2.984	higher than 1.4
3	0.2	0.024	0.4871	16.50	7.0	0.0696	1.646	1.846	higher than 1.4
4	0.1	0.017	0.3069	33.00	5.0	0.0614	1.281	1.381	close to 1.4
5	0.11	0.018	0.3270	30.00	5.2	0.0629	1.344	1.444	higher than 1.4
6	0.105	0.175	0.3170	31.43	5.1	0.0622	1.315	1.420	close to 1.4

Values of R _b (≅ R) in row nos. 4 and 6 are reasonably close to the given value of 1.4 m. Thus, the velocity of flow is taken as the average of 0.3069 m/s and 0.3170 m/s i.e., 0.312 m/s. The difference in the value of mean velocity obtained by Einstein and Barbarossa method compared with that obtained by Garde and Ranga Raju method (Example 7.3) should be noted.

For preparing a stage-discharge curve, one needs to obtain discharges corresponding to different stages of the river. If one neglects bank friction (i.e., R = R_b), the procedure, requiring no trial, is as follows:

For an assumed value of R′ , the mean velocity of flow U is computed from Eq. (7.26) and
	b
U/U" is read from Fig. 7.9 for Ψ′ corresponding to the assumed value of R′ . From known values
*				b	gives R which
of U and U/U" one can determine U″ and, hence, R″	. The sum of R′ and R″
*	*	b	b	b	b

equals R (if bank friction is neglected). Corresponding to this value of R, one can determine the
stage and, hence, the area of flow cross-section A. The product of U and A gives the discharge,

Q corresponding to the stage. Likewise, for another value of R′ , one can determine stage and

b

the corresponding discharge.