HYDRAULICS OF ALLUVIAL CHANNELS

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 crosssection, slopes, planform 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 noncohesive 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 noncohesive 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 semitheoretical 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 KarmanPrandtl 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 1020° 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 regimedependent.
It is also evident that the stagedischarge 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 righthand 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 powerlaw 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 powerlaw 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, nonuniformity 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 powerlaw 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
stagehydraulic radius curve and stagearea curve of crosssection, the stagedischarge curve
for an alluvial channel can be computed as follows:
(i) Assume a stage and find hydraulic radius R and area of crosssection A from
stagehydraulic radius and stagearea 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 crosssection 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 crosssection. Assuming that the total area of
crosssection 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
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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 crosssectional 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
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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 stagedischarge 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 crosssection 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
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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 bedmaterial 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 bedmaterial 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 bedmaterial load in suspension is added to the bedmaterial load moving as
bed load, one gets the total bedmaterial load which may be a major or minor fraction of the
total load comprising bedmaterial load and wash load of the stream depending on the catchment
characteristics.
Irrigation channels carrying siltladen 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 bedmaterial
load are such that errors of the order of one magnitude are not uncommon. If the bedmaterial
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 semitheoretical – for the computation
of bed load. The most commonly used empirical relation is given by MeyerPeter 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 semitheoretical 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 sedimentwater
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 twodimensional 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 bedmaterial 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 bedmaterial 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 bedmaterial 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 bedmaterial transport rate for Example 7.2. 

