**Post: #1**

HYDRAULICS OF ALLUVIAL CHANNELS

GENERAL

Civilisation prospered in agricultural lands by the side of rivers. From the beginning of

civilisation, mankind has given attention to the problems of rivers. The boundaries of many

rivers consist of loose material, which may be carried by the water flowing in these rivers.

Depending upon the prevailing conditions, the loose material may either get deposited or

scoured. Thus, the boundaries of such a river channel are mobile and not rigid. A change in

discharge of water flowing in a rigid boundary channel will cause a change only in the depth of

flow. But, in case of mobile (or loose) boundary channels, a change in discharge may cause

changes in cross-section, slopes, plan-form of the channel, bed forms and roughness coefficient.

The application of the theory of rigid boundary channels to loose boundary channels is, therefore,

not correct. Evidently, the problem of mobile boundary channels is more complicated.

The bed of a river channel generally consists of non-cohesive sediment (i.e., silt, sand,

and gravel) and such rivers are called alluvial rivers.

Sediment (also known as alluvium) is defined as the loose and noncohesive material

through which a river or channel flows. Sediment is also defined as fragmental material

transported by, suspended in, or deposited by water or air, or accumulated in the beds by other

natural agents. Ice, logs of wood, and organic materials flowing with water are excluded from

the definition of sediment.

A channel (or river) flowing through sediment and transporting some of it along with

the flowing water is called an alluvial channel (or river). The complex nature of alluvial channel

problems stands in the way of obtaining analytical solutions, and experimental methods are

generally adopted for obtaining solutions of problems to alluvial channels.

7.2. INCIPIENT MOTION OF SEDIMENT

Consider the case of flow of clear water in an open channel of a given slope with a movable bed

of non-cohesive material. At low discharges, the bed material remains stationary and, hence,

the channel can be treated as rigid. With the increase in discharge, a stage will come when the

shear force exerted by the flowing water on a particle will just exceed the force opposing the

movement of the particle. At this stage, a few particles on the bed move intermittently. This

condition is called the incipient motion condition or, simply, the critical condition.

A knowledge of flow at the incipient motion condition is useful in fixing slope or depth

for clear water flow in an alluvial channel. Knowledge of the incipient motion condition is also

HYDRAULICS OF ALLUVIAL CHANNELS

HYDRAULICS OF ALLUVIAL CHANNELS 253

required in some methods of calculation of sediment load. Hence, there is a need to understand

the phenomenon which initiates motion of sediment particles.

The experimental data on incipient motion condition have been analysed by different

investigators using one of the following three approaches (1):

(i) Competent velocity approach,

(ii) Lift force approach, and

(iii) Critical tractive force approach.

Competent velocity is the mean velocity of flow which just causes a particle to move. A

relationship among the size of the bed material, its relative density, and the competent velocity

is generally developed and used.

Investigators using the lift force approach assume that the incipient motion condition is

established when the lift force exerted by the flow on a particle just exceeds submerged weight

of the particle.

The critical tractive force approach is based on the premise that it is the drag (and not

lift) force exerted by the flowing water on the channel bed which is responsible for the motion

of the bed particles.

Of these three approaches, the critical tractive force approach is considered most logical

and is most often used by hydraulic engineers. Hence, only this approach has been discussed

here.

The critical tractive (or shear) stress is the average shear stress acting on the bed of a

channel at which the sediment particles just begin to move. Shields (2) was the first investigator

to give a semi-theoretical analysis of the problem of incipient motion. According to him, a

particle begins to move when the fluid drag F1 on the particle overcomes the particle resistance

F2. The fluid drag F1 is given as

F1 = k1 CD d ud

and the particle resistance F2 is expressed as

F2 = k2 [d3 (ρs – ρ) g]

where, CD = the drag coefficient,

d = the size of the particle,

ρ = the mass density of the flowing fluid,

ud = the velocity of flow at the top of the particle,

ρs = 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 ud can be

expressed as

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

τ0 / ρ and τ0 is the shear stress acting on the boundary of the channel.

254 IRRIGATION AND WATER RESOURCES ENGINEERING

Similarly,

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

Here, the subscript c has been used to indicate the critical condition (or the incipient

motion condition). The above equation can be rewritten as

On plotting the experimental data collected by different investigators, a unique

relationship between τc* and Rc* 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

parameter Rc

is, obviously, the ratio of the particle size d and ν/u*c. The parameter

is a measure of thickness of laminar sublayer, i.e., δ′. Hence, Rc* can be taken as a measure

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

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

This value of Rc* (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

Likewise, the constant value of τc* 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 τc (3) :

HYDRAULICS OF ALLUVIAL CHANNELS 255

— Data from different source

No particle movement

Particle movement

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 τc simply reduces to

in which τc is in N/m2 and d is in mm. Equations (7.2) and (7.3) are expected to give the value

of τc within about ± 5% of the value obtained from the Shields curve (3).

Yalin and Karahan (4) developed a similar relationship (Fig. 7.2) between τc* and Rc*

using a large amount of experimental data collected in recent years. It is noted that at higher

values of Rc* (> 70) the constant value of τc* is 0.045. This relation (Fig. 7.2) is considered better

than the more commonly used Shields’ relation (1).

— Data from different source

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

256 IRRIGATION AND WATER RESOURCES ENGINEERING

For given values of d, ρs, ρ, and ν, the value of τc can be obtained from Fig. 7.1 or Fig. 7.2

only by trial as τc appears in both parameters τc* and Rc*. However, the ratio of Rc* and τc *

yields a parameter R0* which does not contain τc and is uniquely related to τc*.

Since Rc* is uniquely related to τc* (Fig. 7.2), another relationship between R0* and τc*

can be obtained using Fig. 7.2 and Eq. (7.4).

The relationship between R0* and τc* is as shown in Fig. 7.3 and can be used to obtain

direct solution for τc for given values of d, ρs, ρ and ν.

Fig. 7.3 Variation of R0* and τc* 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:

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 τc. 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

(e) Standing wave

(b) Dunes transition

© 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,

Uw, as follows:

Here, R′ (i.e., hydraulic radius corresponding to the grain roughness) is obtained from

the equation,

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

equation,

HYDRAULICS OF ALLUVIAL CHANNELS 259

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 ©]. 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.

7.3.1. Importance of Regimes of Flow

In case of rigid boundary channels, the resistance to flow is on account of the surface roughness

(i.e., grain roughness) only except at very high Froude numbers when wave resistance may

also be present. But, in the case of alluvial channels, the total resistance to flow comprises the

form resistance (due to bed forms) and the grain resistance. In the ripple and dune regime, the

form resistance may be an appreciable fraction of the total resistance. Because of the varying

conditions of the bed of an alluvial channel, the form resistance is a highly varying quantity.

Any meaningful resistance relation for alluvial channels shall, therefore, be regime-dependent.

It is also evident that the stage-discharge relationship for an alluvial channel will also be

affected by regimes of flow.

The form resistance, which is on account of the difference in pressures on the upstream

and downstream side of the undulations, acts normal to the surface of the undulations. As

such, the form resistance is rather ineffective in the transport of sediment. Only grain shear

(i.e., the shear stress corresponding to grain resistance) affects the movement of sediment.

7.3.2. Prediction of Regimes of Flow

There are several methods for the prediction of regimes. The method described here has been

proposed by Garde and Ranga Raju (8).

The functional relationship for resistance of flow in alluvial channels was written,

following the principles of dimensional analysis, as follows:

260 IRRIGATION AND WATER RESOURCES ENGINEERING

Here, S is the slope of the channel bed. Since resistance to flow and the regime of flow are

closely related with each other, it was assumed that the parameters on the right-hand side of

Eq. (7.11) would predict the regime of flow. The third parameter (i.e., g1/2 d3/2/ν) was dropped

from the analysis on the plea that the influence of viscosity in the formation of bed waves is

rather small. The data from natural streams, canals, and laboratory flumes in which the regimes

had also been observed, were used to develop Fig. 7.5 on which lines demarcating the regimes

of flow have been drawn. The data used in developing Fig. 7.5 cover a wide range of depth of

flow, slope, sediment size, and the density of sediment.

It should be noted that the lines of 45° slope on Fig. 7.5 – such as the line demarcating

‘no motion’ and ‘ripples and dunes’ regimes – represent a line of constant value of τ* =FH G

This means that different regimes of flow can be obtained at the same shear stress by varying

suitably the individual values of R and S. Therefore, shear stress by itself cannot adequately

define regimes of flow.

Data from different sources

Ripples dunes

Transition

Antidunes

No motion

Transition

Ripples and dunes

No motion

Predictor for regimes of flow in alluvial channels (8)

The method of using Fig. 7.5 for prediction of regimes of flow consists of simply calculating

the parameters R/d and S/(Δ ρs/ρ) and then finding the region in which the corresponding

point falls. One obvious advantage of this method is that it does not require knowledge of the

mean velocity U and is, therefore, suitable for prediction of regimes for resistance problems.

Example 7.2 An irrigation canal has been designed to have R = 2.5 m and S = 1.6 × 10–4.

The sediment on the bed has a median size of 0.30 mm. Find: (i) the bed condition that may be

HYDRAULICS OF ALLUVIAL CHANNELS 261

expected, (ii) the height and spacing of undulations, and (iii) the advance velocity of the

undulations. Assume depth of flow and mean velocity of flow to be 2.8 m and 0.95 m/s,

respectively.

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,

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,

Here, ks 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,

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 τc, 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

HYDRAULICS OF ALLUVIAL CHANNELS 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 = 10.8R2/3 S1/3 (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

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

between K1

was obtained for the prediction of

— Data from different sources

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

g1/2 d3/2/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, Δρs, 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 K2 (R/d)1/3

S

Δρs / ρ

and read the value of K1 U

(Δρs / ρ) g R

from Fig. 7.6.

(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.

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:

HYDRAULICS OF ALLUVIAL CHANNELS 265

7.4.2. Resistance Relationship Based on Division of Resistance

In dealing with open channel flows, hydraulic radius R of the flow cross-section is taken as the

characteristic depth parameter. The use of this parameter requires that the roughness over

the whole wetted perimeter is the same. Such a condition can be expected in a very wide

channel with alluvial bed and banks. However, laboratory flumes with glass walls and sand

bed would have different roughnesses on the bed and side walls. In such cases, therefore, the

hydraulic radius of the bed Rb is used instead of R in the resistance relations. The hydraulic

radius of the bed Rb can be computed using Einstein’s method (14) which assumes that the

velocity is uniformly distributed over the whole cross-section. Assuming that the total area of

cross-section of flow A can be divided into areas Ab and Aw corresponding to the bed and walls,

respectively, one can write

A = Aw + Ab

For rectangular channels, one can, therefore, write

(B + 2h) R = 2 hRw + B Rb

∴

= (B + 2h)(R/B) – 2hRw/B

= (PR/B) – 2hRw/B

= (A/B) – 2hRw/B

= h – 2hRw/B (7.20)

Using Manning’s equation for the walls, i.e.,

one can calculate the hydraulic radius of the wall Rw if the Manning’s coefficient for the walls,

nw is known. Using Eq. (7.20), the hydraulic radius of the bed Rb can be computed.

Example 7.4 A 0.40m wide laboratory flume with glass walls (nw = 0.01) and mobile bed

of 2.0 mm particles carries a discharge of 0.1 m3/s at a depth of 0.30m. The bed slope is

3 × 10–3. Determine whether the particles would move or not. Neglect viscous effects.

Solution:

Hydraulic radius, R =

= 0.0593 m

IRRIGATION AND WATER RESOURCES ENGINEERING

Using Eq. (7.20), Rb = 0 3

On neglecting viscous effects and using Yalin and Karahan’s curve,

τc

Δρs gd = 0.045

∴ Critical shear, τc = 0.045 × 1.65 × 9810 × 2 × 10–3

= 1.457 N/m2

Einstein and Barbarossa (15) obtained a rational solution to the problem of resistance

in alluvial channels by dividing the total bed resistance (or shear) τob into resistance (or shear)

due to sand grains τ′ob and resistance (or shear) due to the bed forms τob″, i.e.,

τob = τob′ + τob″ (7.22)

or ρg RbS = ρg Rb′S + ρg Rb″S

i.e., Rb = Rb′ + Rb″ (7.23)

where Rb′ and Rb″ are hydraulic radii of the bed corresponding to grain and form resistances (or

roughnesses).

For a hydrodynamically rough plane boundary, the Manning’s roughness coefficient for

the grain roughness ns is given by the Strickler’s equation i.e.,

Here, d65 (in metres) represents the sieve diameter through which 65 per cent of the

sediment will pass through, i.e., 65 per cent of the sediment is finer than d65. Therefore,

Manning’s equation can be written as

Einstein and Barbarossa (15) replaced this equation with the following logarithmic

relation having theoretical support.

HYDRAULICS OF ALLUVIAL CHANNELS 267

Equation (7.27) is valid for a hydrodynamically rough boundary. A viscous correction

factor x (which is dependent on d65/δ′, Table 7.1, Fig. 7.8) was introduced in this equation to

make it applicable to boundaries consisting of finer material (d65/δ′ 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 τob ″ is

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

Here, CD is the average drag coefficient of the undulations. Since this drag force acts on area

LP, the average shear stress τob

″ will be given as

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′ = Δρs d35/ρ R′bS.

Therefore, they obtained an empirical relation, Fig. 7.9., between

″ and Ψ′ using field data

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.

0.4 0.60.81 2 4 6 8 10 20 40

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′b smaller than Rb is assumed and a trial value of the mean velocity U is calculated

from The value of

″ is read from Fig. 7.9 for Ψ′

corresponding to the assumed value of R′b. From known values of U (trial value) and U/U″*, U″*

and, hence, R″b can be computed. If the sum of R′b and R″b equals Rb the assumed value of R′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 Rb′ till the

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

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′* = gRbS Rb Rb

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,

Rb ≅ R

R″b Rb Comments

Values of Rb (≅ 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 = Rb), the procedure, requiring

no trial, is as follows:

For an assumed value of R′b, the mean velocity of flow U is computed from Eq. (7.26) and

U/U"* is read from Fig. 7.9 for Ψ′ corresponding to the assumed value of R′b. From known values

of U and U/U"* one can determine U″*

and, hence, R″b. The sum of R′b and R″b gives Rb which

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′b, one can determine stage and

the corresponding discharge.

TRANSPORT OF SEDIMENT

When the average shear stress τo on the bed of an alluvial channel exceeds the critical shear τc,

the sediment particles start moving in different ways depending on the flow condition, sediment

size, fluid and sediment densities, and the channel condition.

At relatively low shear stresses, the particles roll or slide along the bed. The particles

remain in continuous contact with the bed and the movement is generally discontinuous.

Sediment material transported in this manner is termed contact load.

On increasing the shear stress, some sediment particles lose contact with the bed for

some time, and ‘hop’ or ‘bounce’. The sediment particles moving in this manner fall into the

category of saltation load. This mode of transport is significant only in case of noncohesive

IRRIGATION AND WATER RESOURCES ENGINEERING

materials of relatively high fall velocities such as sand in air and, to a lesser extent, gravel in

water.

Since saltation load is insignificant in case of flow of water and also because it is difficult

to distinguish between saltation load and contact load, the two are grouped together and termed

bed load, which is transported on or near the bed.

With further increase in the shear stress, the particles may go in suspension and remain

so due to the turbulent fluctuations. The particles in suspension move downstream. Such

sediment material is included in the suspended load. Sediment particles move in suspension

when u*/wo > 0.5. Here, wo is the fall velocity for sediment particles of given size.

The material for bed load as well as a part of the suspended load originates from the bed

of the channel and, hence, both are grouped together and termed bed-material load.

Analysis of suspended load data from rivers and canals has shown that the suspended

load comprises the sediment particles originating from the bed and the sediment particles

which are not available in the bed. The former is the bed-material load in suspension and the

latter is the product of erosion in the catchment and is appropriately called wash load. The

wash load, having entered the stream, is unlikely to deposit unless the velocity (or the shear

stress) is greatly reduced or the concentration of such fine sediments is very high. The transport

rate of wash load is related to the availability of fine material in the catchment and its erodibility

and is, normally, independent of the hydraulic characteristics of the stream. As such, it is not

easy to make an estimate of wash load.

When the bed-material load in suspension is added to the bed-material load moving as

bed load, one gets the total bed-material load which may be a major or minor fraction of the

total load comprising bed-material load and wash load of the stream depending on the catchment

characteristics.

Irrigation channels carrying silt-laden water and flowing through alluvial bed are

designed to carry certain amounts of water and sediment discharges. This means that the

total sediment load transport will affect the design of an alluvial channel. Similarly, problems

related to reservoir sedimentation, aggradation, degradation, etc. can be solved only if the

total sediment load being transported by river (or channel) is known. One obvious method of

estimation of total load is to determine bed load, suspended load, and wash load individually

and then add these together. The wash load is usually carried without being deposited and is

also not easy to estimate. This load is, therefore, ignored while analysing channel stability.

It should, however, be noted that the available methods of computation of bed-material

load are such that errors of the order of one magnitude are not uncommon. If the bed-material

load is only a small fraction of the total load, the foregoing likely error would considerably

reduce the validity of the computations. This aspect of sediment load computations must always

be kept in mind while evaluating the result of the computations.

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

.

Here, qB is the rate of bed load transport in weight per unit width, i.e., N/m/s and da 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 qB and, hence, φB 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,

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 semitheoretical

analysis, a curve, Fig. 7.9, between the Einstein’s bed load parameter

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 φB from

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

Table 7.2 Relationship between φB and Ψ′ (17)

ψ′ 27.0 24.0 22.4 18.4 16.4 11.5 9.5 5.5 4.08 1.4 0.70

φB 10–4 5 × 10–4 10–3 5 × 10–3 10–2 5 × 10–2 10–1 5 × 10–1 1.0 5.0 10.0

Example 7.7 Determine the amount of bed load in Example 7.2 using Einstein’s method.

Solution: From the solution of Example 7.2,

7.5.2. Suspended Load

At the advanced stage of bed load movement the average shear stress is relatively high and

finer particles may move into suspension. With the increase in the shear stress, coarser fractions

of the bed material will also move into suspension. The particles in suspension move with a

velocity almost equal to the flow velocity. It is also evident that the concentration of sediment

HYDRAULICS OF ALLUVIAL CHANNELS 273

particles will be maximum at or near the bed and that it would decrease as the distance from

the bed increases. The concentration of suspended sediment is generally expressed as follows:

(i) Volume concentration: The ratio of absolute volume of solids and the volume of sediment-

water mixture is termed the volume concentration and can be expressed as percentage

by volume. 1 % of volume concentration equals 10,000 ppm by volume.

(ii) Weight concentration: The ratio of weight of solids and the weight of sediment-water

mixture is termed the weight concentration and is usually expressed in parts per million (ppm).

Variation of Concentration of Suspended Load

Starting from the differential equation for the distribution of suspended material in the vertical

and using an appropriate diffusion equation, Rouse (18) obtained the following equation for

sediment distribution (i.e., variation of sediment concentration along a vertical):

where, C = the sediment concentration at a distance y from the bed,

Ca = the reference concentration at y = a,

h = the depth of flow,

and is the exponent in the sediment distribution equation,

wo = the fall velocity of the sediment particles,

and k = Karman’s constant.

Rouse’s equation, Eq. (7.34), assumes two-dimensional steady flow, constant fall velocity

and fixed Karman’s constant. However, it is known that the fall velocity as well as Karman’s

constant vary with concentration and turbulence. Further, a knowledge of some reference

concentration Ca at y = a is required for the use of Eq. (7.34).

Knowledge of the velocity distribution and the concentration variation (Fig. 7.11) would

enable one to compute the rate of transport of suspended load qs. Consider a strip of unit width

and thickness dy at an elevation y. The volume of suspended load transported past this strip in

a unit time is equal to

Cudy.

Velocity

profile

Concentration

profile

Variation of velocity of flow and sediment concentration in a vertical

274 IRRIGATION AND WATER RESOURCES ENGINEERING

Here, C is the volume concentration (expressed as percentage) at an elevation y where

the velocity of flow is u. Thus,

where, qs is the weight of suspended load transported per unit width per unit time. Since the

suspended sediment moves only on top of the bed layer, the lower limit of integration, a, can be

considered equal to the thickness of the bed layer, i.e., 2d.

Instead of using the curves of the type shown in Fig. 7.11, one may use a suitable velocity

distribution law and the sediment distribution equation, Eq. (7.34). For the estimation of the

reference concentration Ca appearing in Eq. (7.34), Einstein (17) assumed that the average

concentration of bed load in the bed layer equals the concentration of suspended load at y = 2d.

This assumption is based on the fact that there will be continuity in the distribution of suspended

load and bed load. Making use of suitable velocity distribution laws, the velocity of the bed

layer was determined as 11.6 u*′ and as such the concentration in the bed layer was obtained as

11.6 ′ 2 . Hence, the reference concentration Ca (in per cent) at y = 2d is given as

Equation (7.35) can now be integrated in a suitable manner.

Example 7.8 Prepare a table for the distribution of sediment concentration in the vertical

for Example 7.2. Assume fall velocity of the particles as 0.01 m/s.

Solution: From the solution of Example 7.2 and 7.3,

qB = 0.286 N/m/s

and R′ = 0.651 m

Using Eq. (7.36)

HYDRAULICS OF ALLUVIAL CHANNELS 275

The variation of C with y can now be computed as shown in the following table:

The total bed-material load can be determined by adding together the bed load and the

suspended load. There is, however, another category of methods too for the estimation of the

total bed-material load. The supporters of these methods argue that the process of suspension

is an advanced stage of tractive shear along the bed, and, therefore, the total load should be

related to the shear parameter. One such method is proposed by Engelund and Hansen (19)

who obtained a relationship for the total bed-material load qT (expressed as weight per unit

width per unit time) by relating the sediment transport to the shear stress and friction factor

The median size d50 is used for d in the above equation.

Example 7.9 Determine the total bed-material transport rate for Example 7.2.