Contents preface (VII) introduction 1—37

262 IRRIGATION AND WATER RESOURCES ENGINEERING
7.4. RESISTANCE TO FLOW IN ALLUVIAL CHANNELS
The resistance equation expresses relationship among the mean velocity of flow U, the hydraulic radius R, and the characteristics of the channel boundary. For steady and uniform flow in rigid boundary channels, the Keulegan’s equations (logarithmic type) or power-law type of equations (like the Chezy’s and the Manning’s equations) are used. Keulegan (9) obtained the following logarithmic relations for rigid boundary channels:

For smooth boundaries,			F u* RI
		U
		u*	= 5.75 log G			J + 3.25	(7.12)
				H v		K
For rough boundaries,
		^U = 5.75 log (R/k ) + 6.25	(7.13)
		u*					s
For the range 5 <	R	< 700, the Manning’s equation,
k
	s
		U =	1	R2 /3 S1/ 2			(7.14)
				n

has been found (9) to be as satisfactory as the Keulegan’s equation [Eq. (7.13)] for rough boundaries. In Eq. (7.14), n is the Manning’s roughness coefficient which can be calculated using the Strickler’s equation,

	k 1/6
n =	25^s.6	(7.15)

Here, k_s is the equivalent sand grain roughness in metres. Another power-law type of equation is given by Chezy in the following form:

U = C RS

(7.16)

Comparing the Manning’s equations,

U	C	=	R1/ 6	F	R I 1/ 6	25.6	(7.17)
=	g	n g	= G	J	g
u*			H ks K

In case of an alluvial channel, so long as the average shear stress τ₀ on boundary of the channel is less than the critical shear τ_c, the channel boundary can be considered rigid and any of the resistance equations valid for rigid boundary channels would yield results for alluvial channels too. However, as soon as sediment movement starts, undulations develop on the bed, thereby increasing the boundary resistance. Besides, some energy is required to move the grains. Further, the sediment particles in suspension also affect the resistance of alluvial streams. The suspended sediment particles dampen the turbulence or interfere with the production of turbulence near the bed where the concentration of these particles as well as the rate of turbulence production are maximum. It is, therefore, obvious that the problem of resistance in alluvial channels is very complex and the complexity further increases if one includes the effects of channel shape, non-uniformity of sediment size, discharge variation, and other factors on channel resistance. None of the resistance equations developed so far takes all these factors into consideration.

The method for computing resistance in alluvial channels can be grouped into two broad categories. The first includes such methods which deal with the overall resistance and use

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263

either a logarithmic type relation or a power-law type relation for the mean velocity. The second category of methods separates the total resistance into grain resistance and form resistance (i.e., the resistance that develops on account of undulations on the channel bed). Both categories of methods generally deal with uniform steady flow.

7.4.1. Resistance Relationships based on Total Resistance Approach
The following equation, proposed by Lacey (10) on the basis of analysis of stable channel data from India, is the simplest relationship for alluvial channels:

U = 10.8R^2/3 S^1/3

(7.18)

However, this equation is applicable only under regime conditions (see Art. 8.5) and, hence, has only limited application.
Garde and Ranga Raju (11) analysed data from streams, canals, and laboratory flumes to obtain an empirical relation for prediction of mean velocity in an alluvial channel. The functional relation, [Eq. (7.11)] may be rewritten (11) as

U

= f M^L

R

,

S

,

g1/ 2 d3 /2

(∆ρ

/ ρ) g R

d

∆ρ

/ ρ

ν

s

M

s

N

O

P (7.19)

PQ

By employing usual graphical techniques and using alluvial channel data of canals, rivers, and laboratory flumes, covering a large range of d and depth of flow, a graphical relation

	1	U	2	G	R	J	1/ 3	S
between K		and K	F	I	, Fig. 7.6, was obtained for the prediction of
		(∆ρ s / ρ) g R
					H d K		∆ρs / ρ

U	∆ρs
	1
K

3.0



— Data from different sources
1.0

gR

ρ

0.10
0.03

–4	–3			–2	–1
10	10	1/3	S	10	10
	K (	^R ) (	)
	∆ρs/ρ
		2		d

Fig. 7.6 Resistance relationship for alluvial channels (12)

264 IRRIGATION AND WATER RESOURCES ENGINEERING
the mean velocity U. The coefficients K₁ and K₂ were related to the sediment size d by the graphical relations shown in Fig. 7.7. It should be noted that the dimensionless parameter g^1/2 d^3/2/ν has been replaced by the sediment size alone on the plea that the viscosity of the liquid for a majority of the data used in the analysis did not change much (12). This method is expected to yield results with an accuracy of ± 30 per cent (13). For given S, d, ∆ρ_s, ρ, and the stage-hydraulic radius curve and stage-area curve of cross-section, the stage-discharge curve for an alluvial channel can be computed as follows:
(i) Assume a stage and find hydraulic radius R and area of cross-section A from stage-hydraulic radius and stage-area curves, respectively.
(ii) Determine K₁ and K₂ for known value of d using Fig. 7.7.

(iii) Compute K		(R/d)^1/3	S	and read the value of K	U	from Fig. 7.6.
	2		∆ρ s / ρ		^1 (∆ρ s / ρ) g R

(iv) Calculate the value of the mean velocity U and, hence, the discharge. (v) Repeat the above steps for other values of stage.

Finally, a graphical relation between stage and discharge can be prepared.

2.0

K₂

K1	K
1.0	1
K2
2
,K
1
K
0.10

K₁

K₂

0.01 0.10 1.0 10.0 d in MM

Fig. 7.7 Variation of K₁ and K₂ with sediment size (12)
Example 7.3 An alluvial stream (d = 0.60 mm) has a bed slope of 3 × 10^–4 Find the mean velocity of flow when the hydraulic radius is 1.40 m.
Solution:
From Fig. 7.7, K₁ = 0.75 and K₂ = 0.70

S

F

1.40

I 1/3

3

× 10

−4

K2

(R/d)^1/3

= 0.70

G

J

×

= 1.69 × 10^−^3

∆ρs

/ ρ

−3

1.65

H 0.6 × 10

K

From Fig. 7.6,

K1

U

= 0.135

or

0.75 ×

U

= 0.135

(∆ρ s / ρ) g R

1.65

× 9.81

× 1.4

∴

U = 0.86 m/s.

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265