Contents preface (VII) introduction 1—37

F₂ = k₂ [d³ (ρ_s – ρ) g] where, C_D = the drag coefficient,
d = the size of the particle,
ρ = the mass density of the flowing fluid,
u_d = the velocity of flow at the top of the particle, ρ_s = the mass density of the particle,
g = acceleration due to gravity,
k₁ = a factor dependent on the shape of the particle, and
k₂ = a factor dependent on the shape of the particle and angle of internal friction.
Using the Karman-Prandtl equation for the velocity distribution, the velocity u_d can be expressed as

^ud =	f	1	F u* dI	= f (R*)
^u*		G	J	1
			H	ν K

Here, ν is the kinematic viscosity of the flowing fluid, u_* the shear velocity equal to

_{τ 0} / _ρ and τ₀ is the shear stress acting on the boundary of the channel.

254

IRRIGATION AND WATER RESOURCES ENGINEERING

Similarly,

F

I

C

= f ′

ud

d

D

G

ν

J

H

K

F u* d I

*

or

CD

= f2

G

J = f 2

( R

)

ν

H

K

Thus,	F	= k f	(R^*) d^2	1	ρu*^2	[f1 (R*)]^2
	1	1 2	2

At the incipient motion condition, the two forces F₁ and F₂ will be equal. Hence,

k₁f₂ (R_c^*) d² ₂¹ ρ u_*c² [f₁ (R_c^* )]² = k₂ [d ³ (ρ_s − ρ) g]

Here, the subscript c has been used to indicate the critical condition (or the incipient motion condition). The above equation can be rewritten as

Alternatively,

where,

and
ρ u_*²_c

(ρ _s − ρ) gd

τ_c^*
τ_c^*
τ_c ∆ ρ_s

=	2 k2	f (Rc^* )
	k
	1
= f (R^*)	(7.1)
	c

• 
  ^τ c
  
  • 
    ρ_s gd
    
  • 
    ρ u_*²_c
    

• 
  ρ_s – ρ
  

On plotting the experimental data collected by different investigators, a unique relationship between τ_c* and R_c* was obtained by Shields (2) and is as shown in Fig. 7.1. The curve shown in the figure is known as the Shields curve for the incipient condition. The

*	F u cdI
parameter Rc	= G	*	J	is, obviously, the ratio of the particle size d and ν/u*c. The parameter
ν
		H	K

ν	is a measure of thickness of laminar sublayer, i.e., δ′. Hence, R* can be taken as a measure
^u*c	c

of the roughness of the boundary surface. The boundary surface is rough at large values of R_c*

and, hence, τ * attains a constant value of 0.06 and becomes independent of R* at R* ≥ 400.

c c c

This value of R_c* (i.e., 400), indicating that the boundary has become rough, is much higher than the value of 70 at which the boundary becomes rough from the established criterion

_δ′^d > 6.0. Likewise, the constant value of τ_c* equal to 0.06 is also on the higher side.

Alternatively, one may use the following equation of the Shields’ curve for the direct computation of τ_c (3) :

τ

c

0.06 d^2

= 0.243 +

*

(7.2)

F

ρ ν^2

I 1/ 3

(3600 + d*^2 )^1/ 2

∆ ρs g G

J

H ^∆ ρs

gK

in which,

d* =

d

(ρ ν

^2 / ∆ ρs g)^1/ 3

HYDRAULICS OF ALLUVIAL CHANNELS
1.0

c	gd	0.10
s
τ	∆ρ

0.01

0.4 1.0


255




— Data from different source

Particle movement

No particle movement

10	100	1000
	R*c

Fig. 7.1 Shields curve for incipient motion condition (2)
For specific case of water (at 20°C) and the sediment of specific gravity 2.65 the above relation for τ_c simply reduces to

τc = 0.155 +	0.409 d^2	(7.3)
(1 + 0.177 d^2 )^1/ 2

in which τ_c is in N/m² and d is in mm. Equations (7.2) and (7.3) are expected to give the value of τ_c within about ± 5% of the value obtained from the Shields curve (3).
Yalin and Karahan (4) developed a similar relationship (Fig. 7.2) between τ_c* and R_c* using a large amount of experimental data collected in recent years. It is noted that at higher values of R_c* (> 70) the constant value of τ_c* is 0.045. This relation (Fig. 7.2) is considered better than the more commonly used Shields’ relation (1).

c

0

10

— DataDatafromfromdifferentdifferentsourcesource

–1
10
gd
s
–2
10	0	1	2	3
–1
10	10	10	10		10

R*

c