Contents preface (VII) introduction 1—37

7.5.1. Bed Load
The prediction of the bed load transport is not an easy task because it is interrelated with the resistance to flow which, in turn, is dependent on flow regime. Nevertheless, several attempts have been made to propose methods – empirical as well as semi-theoretical – for the computation of bed load. The most commonly used empirical relation is given by Meyer-Peter and Müller (16). Their relation is based on: (i) the division of total shear into grain shear and form shear, and (ii) the premise that the bed load transport is a function of only the grain shear. Their equation, written in dimensionless form, is as follows:

HYDRAULICS OF ALLUVIAL CHANNELS

271

L ns O^3 /2

ρRS

qB^2 /3

M

P

= 0.047 + 0.25

(7.31)

∆ρ s da

F

∆ρs I

1/ 3

N

n Q

2 /3

ρs

^gda G

J

H

ρ K

which is rewritten as

τ ′ = 0.047 + 0.25 φ ^2/3

(7.32)

*

B

or

φ

= 8 (τ ′ – 0.047)^3/2

(7.33)

B

*

where, φB is the bed load function and equals ρ

g 3 / 2

qB

∆ρ

/ρ ^,

d 3 /2

s

a

s

τ ′

τ ′ is the dimensionless grain shear and equals

o

,

*

∆ρ s gda

and τ ′ is the grain shear and equals^L ^n^s ^O^3 /2 ρgRS.

o

M

P

N n Q

Here, q_B is the rate of bed load transport in weight per unit width, i.e., N/m/s and d_a is the arithmetic mean size of the sediment particles which generally varies between d₅₀ and d₆₀(1).
From Eq. (7.33), it may be seen that the value of the dimensionless shear τ_*′ at the incipient motion condition (i.e., when q_B and, hence, φ_B is zero) is 0.047. Thus, (τ_*′ – 0.047) can be interpreted as the effective shear stress causing bed load movement.
The layer in which the bed load moves is called the bed layer and its thickness is generally taken as 2d.
Example 7.6 Determine the amount of bed load in Example 7.2
Solution:
From the solution of Example 7.2,
τ_*′ = 0.21

	From Eq. (7.33),
	∴	φ = 8 × (0.21 – 0.047)^3/2	= 0.5265
		B
i.e.,	qB	= 0.5265
ρ s g3 / 2 d3 /2
		∆ρs / ρ
		∴	q = 0.5265 × 2650 × (9.81 × 0.3 × 10^–3)^1.5	(1.65)^1/2	N/m/s
		B

= 0.286 N/m/s
A semi-theoretical analysis of the problem of the bed load transport was first attempted by Einstien (14) in 1942 when he did not consider the effect of bed forms on bed load transport. Later, he presented a modified solution (17) to the problem of bed load transport. Einstein’s solution does not use the concept of critical tractive stress but, instead, is based on the assumption that a sediment particle resting on the bed is set in motion when the instantaneous hydrodynamic lift force exceeds the submerged weight of the particle. Based on his semi-theoretical analysis, a curve, Fig. 7.9, between the Einstein’s bed load parameter

F	=			qB	I	and ψ′ (= ∆ρs d/ρR′S)
^φB G	3 / 2	d	3 /2	J
H	ρ s g				∆ρs / ρ K

272 IRRIGATION AND WATER RESOURCES ENGINEERING
can be used to compute the bed load transport in case of uniform sediment. The coordinates of the curve of Fig. 7.10 are given in Table 7.2. The method involves computation of Ψ′ for given sediment characteristics and flow conditions and reading the corresponding value of φ_B from Fig. 7.10 to obtain the value of q_B.

100
10
1.0
0.1
0.0001	0.001	0.01	0.1	1.0	10

B
Fig. 7.10 Einstein’s bed load transport relation (17)