Contents preface (VII) introduction 1—37

Fig. 7.2 Yalin and Karahan curve for incipient motion condition (4)

256 IRRIGATION AND WATER RESOURCES ENGINEERING
For given values of d, ρ_s, ρ, and ν, the value of τ_c can be obtained from Fig. 7.1 or Fig. 7.2
only by trial as τ_c appears in both parameters τ_c* and R_c*. However, the ratio of R_c* and τ^*_c yields a parameter R₀* which does not contain τ_c and is uniquely related to τ_c*.

R0* =	Rc^*	^u*c	d F	∆ ρs	gd I ^1/ 2	F	∆ ρs gd^3 I ^1/ 2
	* ^=	ν	G	τ c	J	^= G	ρν	2	J
	^τ c	H	K	H		K

Since R_c* is uniquely related to τ_c* (Fig. 7.2), another relationship between can be obtained using Fig. 7.2 and Eq. (7.4).
(7.4)
R₀* and τ_c*

The relationship between R₀* and τ_c* is as shown in Fig. 7.3 and can be used to obtain direct solution for τ_c for given values of d, ρ _s, ρ and ν.

0.2

0.1

_c*

0.01

1.0 10.0 100.0 1000 R₀*
Fig. 7.3 Variation of R₀* and τ_c* based on Fig. 7.2
Example 7.1 Water flows at a depth of 0.3 m in a wide stream having a slope of 1 × 10^–3. The median diameter of the sand on the bed is 1.0 mm. Determine whether the grains are stationary or moving (ν = 10^–6 m²/s).
Solution:

F	∆ ρs gd^3 I ^1/ 2	F	165.	× 9.81 × (1 × 10^−^3 )^3
R0* = G			J	= G
ρ ν	2		(10		−6	)	2
	H		K	H

I

J

K
1/ 2
= 127.23

From Fig. 7.3,	τc* = 0.035 =	^τ c	=	^τ c
∆ ρs gd	1650 × 9.81 × ( 1.0 × 10^−^3 )
∴	τ = 0.5665 N/m^2
	c

Shear stress on the bed, τ₀ = ρ ghS = 9810 × 0.3 × 10^–3 = 2.943 N/m²

Since τ₀ > τ_c, the grains would move. τ_c can also be computed from Eq. (7.3).

0.409 (10.)<sup>2

τ</sup>c ^{= 0.155 +} _{[1 + 0.177 (10.)}2 _]1/ 2 = 0.532 N/m².

HYDRAULICS OF ALLUVIAL CHANNELS

257

7.3. REGIMES OF FLOW
When the average shear stress on the bed of an alluvial channel exceeds the critical shear stress, the bed particles are set in motion and thus disturb the plane bed condition. Depending upon the prevailing flow conditions and other influencing parameters, the bed and the water surface attain different forms. The features that form on the bed of an alluvial channel due to the flow of water are called ‘bed forms’, ‘bed irregularities’ or ‘sand waves’. Garde and Albertson
(5) introduced another term ‘regimes of flow’ defined in the following manner:
‘As the sediment characteristics, the flow characteristics and/or fluid characteristics are changed in alluvial channel, the nature of the bed surface and the water surface changes accordingly. These types of the bed and water surfaces are classified according to their characteristics and are called regimes of flow.’
Regimes of flow will affect considerably the velocity distribution, resistance relations, and the transport of sediment. The regimes of flow can be divided into the following four categories:
(i) Plane bed with no motion of sediment particles, (ii) Ripples and dunes,

(iii) Transition, and (iv) Antidunes.

Plane Bed with no Motion of Sediment Particles
When sediment and flow characteristics are such that the average shear stress on the bed is less than the critical shear stress, the sediment particles on the bed do not move. The bed remains plane and the channel boundary can be treated as a rigid boundary. The water surface remains fairly smooth if the Froude number is low. Resistance offered to the flow is on account of the grain roughness only, and Manning’s equation can be used for prediction of the mean velocity of flow with Manning’s n obtained from the Strickler’s equation, as discussed later in this chapter.
Ripples and Dunes
The sediment particles on the bed start moving when the average shear stress of the flow τ₀ exceeds the critical shear τ_c. As a result of this sediment motion, small triangular undulations known as ripples form on the bed [Fig. 7.4 (a)]. Ripples do not occur if the sediment is coarser than 0.6 mm. The length (between two adjacent troughs or crests) of the ripples is usually less than 0.4 m and the height (trough to crest) does not exceed 40 mm. The sediment motion is confined to the region near the bed and the sediment particles move either by sliding or taking a series of hops.
With the increase in discharge (and, hence, the average shear stress τ₀) the ripples grow into dunes [Fig. 7.4 (b)]. Dunes too are triangular undulations but of larger dimensions. These undulations are also unsymmetrical with a flat upstream face inclined at about 10-20° with the horizontal and steep downstream face whose angle of inclination with the horizontal is approximately equal to the angle of repose of the sediment material. Sometimes, ripples appear on the upstream face of a dune. The dunes in laboratory flumes may have length and height up to about 3 m and 0.4 m, respectively. But, in large rivers, the dunes may be several hundred metres long and up to about 15 m in height. The water surface falls over the crest of dunes and, hence, the water surface waves are out of phase with the bed waves. The flow conditions still correspond to the subcritical range. While most of the sediment particles move along the bed, some finer particles of the sediment may go in suspension.

258 IRRIGATION AND WATER RESOURCES ENGINEERING

(a) Ripples (d) Plane bed with

sediment motion

(b) Dunes	(e) Standing wave
	transition

(c) Washed out dunes (f) Antidunes
Fig. 7.4 Regimes of flow in alluvial channels
Ripples and dunes have many common features and, hence, are generally dealt with together as one regime of flow. Both ripples and dunes move downstream slowly. Kondap and Garde (6) have given an approximate equation for the advance velocity of ripples and dunes, U_w, as follows:

U	^w = 0.021	F	U	I ^3	(7.5)
	G		J
U	H	gh K

Here, U is the mean velocity of flow, and h is the average depth of flow. The average length (L) and height (H) of ripples and dunes can be estimated from the equations proposed by Ranga Raju and Soni (7):

L		O	3
H M	U	P		U	= 6500 (τ ′)^8/3	(7.6)
^d M	gR P		(∆ ρ s / ρ) gd	*
N		Q

L	U	O	3	^L = 1.8 × 10^8 (τ ′)^10/3	(7.7)
M	P	^U ×
M	∆ ρ s / ρ) gd P	gR D					*
N			Q				ρR ′ s
in which,					τ ′ =			(7.8)
				*		∆ ρ sd
Here, R′ (i.e., hydraulic radius corresponding to the grain roughness) is obtained from
the equation,
				U =		1	R′2 /3	S1/ 2	(7.9)
					n
						s

n_s (i.e., Manning’s roughness coefficient for the grains alone) is calculated from Strickler’s equation,

HYDRAULICS OF ALLUVIAL CHANNELS		259
n =	d1/ 6	(7.10)
25.6
	s

in which d is in metres. R is the hydraulic radius of the channel.
Transition
With further increase in the discharge over the duned bed, the ripples and dunes are washed away, and only some very small undulations are left [Fig. 7.4 (c)]. In some cases, however, the bed becomes plane but the sediment particles are in motion [Fig. 7.4 (d)]. With slight increase in discharge, the bed and water surfaces attain the shape of a sinusoidal wave form. Such waves, known as standing waves [Fig. 7.4 (e)], form and disappear and their size does not increase much. Thus, in this regime of transition, there is considerable variation in bed forms from washed out dunes to plane bed with sediment motion and then to standing waves. The Froude number is relatively high. Large amount of sediment particles move in suspension besides the particles moving along the bed. This regime is extremely unstable. The resistance to flow is relatively small.
Antidunes
When the discharge is further increased and flow becomes supercritical (i.e., the Froude number is greater than unity), the standing waves (i.e., symmetrical bed and water surface waves) move upstream and break intermittently. However, the sediment particles keep on moving downstream only. Since the direction of movement of bed forms in this regime is opposite to that of the dunes, the regime is termed antidunes, [Fig. 7.4 (f)]. The sediment transport rate is, obviously, very high. The resistance to flow is, however, small compared to that of the ripple and dune regime. In the case of canals and natural streams, antidunes rarely occur.