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Hydraulic Structures II - Lecture Note Page 28

1.3.1.1 Incipient Motion

Particle movement will occur when the instantaneous fluid force on a particle is just

larger than the instantaneous resisting force related to the submerged particle weight

and the friction coefficient.

The driving forces are strongly related to the local near-bed velocities. In turbulent flow

conditions the velocities are fluctuating in space and time, which make together with the

randomness of particle size, shape and position that initiation of motion is not merely a

deterministic phenomenon but a stochastic process as well.

Incipient motion is important in the study of sediment transport, channel degradation,

and stable channel design. Due to the stochastic nature of sediment movement along

an alluvial bed, it is difficult to define precisely at what flow condition a sediment particle

will begin to move.

Let us consider the steady flow over the bed composed of cohesionless grains. The

forces acting on the grain is shown in Fig.1.10.

Figure 1.10 Forces acting on a grain resting on the bed.

The driving force is the flow drag force on the grain,

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where the friction velocity u* is the flow velocity close to the bed. α is a coefficient, used

to modify u* so that αu* forms the characteristic flow velocity past the grain. The

stabilizing force can be modeled as the friction force acting on the grain.

If u*, c, critical friction velocity, denotes the situation where the grain is about to move,

then the drag force is equal to the friction force, i.e. FD = f (W’ – FL),

which can be re-arranged into

Shields parameter is then defined as

dgs

u

1

2

**

(1.15)

We say that sediment starts to move if

c*,c*,* uvelocityfrictioncriticaluu

where

𝒖∗ = 𝝉𝒃 𝝆

τb = the mean bed shear stress and r = fluid density

or

𝝉𝒃 > 𝝉𝒃,𝒄 critical bottom shear stress

where

𝝉𝒃 = 𝝆𝒈𝒉𝑺

S = bed slope, h = water depth

or dgs

uparameterShieldscritical

c

cc1

2

*,

*,*,*

Fig.1.11 shows Shields experimental results, which relate τ*,c to the grain Reynolds

number defined as

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(1.16)

The figure has 3 distinct zones corresponding to 3 flow situations

1) Hydraulically smooth flow; for 2du

R n*e

.

dn is much smaller than the thickness of viscous sublayer. Grains are embedded in

the viscous sublayer and hence, τ*,c is independent of the grain diameter. By

experiments it is found that τ*,c = 0.1/Re.

2) Hydraulically rough flow; for Re ≥ 500.

The viscous sublayer does not exist and hence, τ*,c is independent of the fluid

viscosity. τ*,c has a constant value of 0.06.

3) Hydraulically transitional flow; for 2 ≤ Re ≤ 500.

Grain size is the same order as the thickness of the viscous sublayer. There is a

minimum value of τ*,c of 0.032 corresponding to Re = 10.

Note that the flow classification is similar to that of the Nikurase pipe flow where the bed

roughness ks is applied instead of dn.

Figure 1.11. The Shields diagram giving τ*,c as a function of Re (uniform and cohesionless grain).

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Figure 1.12 Turbulent velocity distribution

1.3.1.2 Bed Load, Suspended Load, Wash Load and Total Load Transport

When the values of the bed shear velocity just exceeds the critical value for initiation of

motion, the bed material particles will be rolling and/or sliding in continuous contact with

the bed. For increasing values of the bed shear velocity the particles will be moving

along the bed by more or less regular jumps, which are called saltations.

When the value of the bed shear velocity begins to exceed fall velocity of the particles,

the sediment particles can be lifted to a level at which the upward turbulent forces will

be of comparable or higher order than the submerged weight of the particles and as a

result the particles may go into suspension.

Usually, the transport of particles by rolling, sliding and saltating is called bed load

transport, while the suspended particles are transported as suspended load transport.

The suspended load may also include the fine silt particles brought into suspension

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from the catchment area rather than from the streambed material (bed material load)

and is called the wash load. A grain size of 63 μm (dividing line between silt and sand)

is frequently used to separate between bed material and wash load.

Bed load and suspended load may occur simultaneously, but the transition zone

between both modes of transport is not well defined.

The following classification and definitions are used for the total sediment transported in

rivers.

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Bed Load Transport Usually, the transport of particles by rolling, sliding and saltating is called the bed load

transport. Saltation refers to the transport of sediment particles in a series of irregular

jumps and bounces along the bed (see Figure 1.13).

Bed load transport occurs when the bed shear stress, τ0 exceeds a critical value (τ0)c. In

dimensionless terms, the condition for bed-load motion is:

transportloadBedc**

where τ* is the Shields parameter (i.e. gd1s0

*

and (τ*)c is the critical Shields

parameter for initiation of bed load transport.

The sediment transport rate may be measured by weight (units: N/s), by mass (units:

kg/s) or by volume (units: m3/s). In practice the sediment transport rate is often

expressed per unit width and is measured either by mass or by volume.

Bed load, as one part of the bed material load, is often quantitatively small and hence

does not represent a severe problem of sedimentation. On the other hand, as the main

factor of the bed formation process, it is always of major importance. Roughness of

alluvial channels is to a great extent determined by the movement of the bed load.

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Figure 1.13 Bed load motion. (a) Sketch of saltation motion. (b) Definition sketch of the bed-load layer.

Bed Load Formulae

Various formulas are developed in the past for estimation of bed load discharge.

Estimates of bed load transport using different formula for the same set of given data

are also found to give widely different results. The most common formulae and

approaches are:

a) The discharge approach (bed load expressed in terms of discharge)

b) Shear stress approach

c) The probabilistic approach

Out of these methods the shear stress approach shall be discussed next.

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Shear Stress Approach

This approach is much more favored today, because of the importance accorded to the

shear stress in all aspects of the sediment movement in alluvial channels. Formulae of

this type are those of Straub-Du Boys, Shield, Kalinske, Meyer-Peter and Mueller, etc.

The best known of these, and probably the most widely used, is the Meyer-Peter and

Mueller formula; it also gives the best agreement with measured data.

Meyer-Peter and Mueller Formula

The original Meyer-Peter formula was the discharge type. The new type of formula was

arrived at in collaboration with R. Mueller. The formula is given as follows:

dShn

nq s

B

Gb

s

047.0125.0

23

32

32

31

(1.17)

where qb = dry weight of transported sediment (N/s/m width of channel)

nG = Manning’s grain roughness coef.

nB = Manning’s bed roughness coef.

h = depth of flow (m), S = energy slope,

d = dm = representative grain size of the bed material (m)

The roughness coefficients nB, which comprises of bottom roughness due to the

sediment and to form resistance should be estimated. The grain roughness coef. nG is

defined by nG =d901/6/26.

Equation (1.17) is valid for fully developed turbulence.

The representative grain diameter, d, should best be determined by dividing the grain

size distribution curve into several fractions and then computing the grain size by

100

pddm

(1.18)

where d = average size of grains in a size fraction

p = percentage of a given fraction in respect to the total

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Equation (1.17) gives fair agreement with measured quantities for coarse sediments, i.e.

for gravel or cobble-bed streams.

Shields Formula

The semi-empirical formula derived by Shields for a level bed is

dSq

10qss

cb

(1.19)

where d = d50 , and S = bed slope.

In this formula τ and τc can be calculated from

50sc gd056.0 and

SRg

Equation (1.19) is dimensionally homogeneous, and can be used for any system of

units. The critical shear stress can also be obtained from Shields diagram.

Suspended Load Transport

Suspended load refers to sediment that is supported by the upward components of

turbulent currents and stays in suspension for an appreciable length of time. In most

natural rivers, sediments are mainly transported as suspended load.

The suspended load transport can be defined mathematically as

h

a

sv dzcuq (1.20a)

h

a

ssw dzcuq (1.20b)

where qsv and qsw are suspended load transport rates in terms of volume and weight,

respectively; candu are time averaged velocity and sediment concentration, by

volume at a distance z above the bed, respectively; a is thickness of the bed load

transport; and h is the water depth.

Before eq. (1.20) is integrated, candu must be expressed mathematically as a function

of z. Under steady equilibrium conditions, the downward movement of sediment due to

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the fall velocity must be balanced by the net upward movement of sediment due to

turbulent fluctuations, i.e.,

0dz

dCC s (1.21)

where εs is the momentum diffusion coefficient for sediment, which is a function of z; ω

is fall velocity of sediment particles; and C is sediment concentration.

For turbulent flow, the turbulent shear stress can be expressed as

dz

dumz (1.22)

where εm is kinematic eddy viscosity of fluid or momentum diffusion coefficient for fluid.

It is generally assumed that

ms (1.23)

where β is a factor of proportionality.

For fine sediments in suspension, it can be assumed that β = 1 without causing

significant error. Eq.(1.21) can also be written as

0dz

C

dC

s

(1.24)

and integration of eq.(1.24) yields

z

a sa

dzexpCC (1.25)

where C and Ca are sediment concentrations by weight at distance z and a above the

bed, respectively.

The shear stress at a distance z above the bed is

h

z1zhSz (1.26)

where τ and τz are shear stresses at channel bottom and a distance z above the bed,

respectively.

Assume the Prandtl – von Karman velocity distribution is valid, i.e.,

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zk

U

dz

du * (1.27)

From equations (1.22), (1.26) and (1.27),

)zh(h

zUk *m (1.28)

and )zh(h

zUk *s (1.29)

Equation (1.28) indicates that εm =0 at z = 0 and z =h. The maximum value of εm occurs

at z = ½ h.

On substituting eq. (1.29) into eq. (1.24) and integration from a to z, assuming β = 1,

yields

Z

a ah

a

z

zh

C

C

(1.30)

where *Uk

Z

is known as the Rouse constant and equation (1.30) is called the Rouse

equation. This equation gives the distribution of the suspended sediment concentration

over the vertical for various values of Z (see Fig. 1.14).

Figure 1.14 Suspended sediment distribution according to equation (1.30)

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Total Load Transport

Based on the mode of transportation, total load is the sum of bed load and suspended

load. Based on the source of material being transported, total load can also be defined

as the sum of bed material load and wash load. Wash load consists of fine materials

that are finer than those found in the bed. The amount of wash load depends mainly on

the supply from the watershed, not on the hydraulics of the river. Consequently, it is

difficult to predict the wash load based on the hydraulic characteristics of a river. Most

total load equations are, therefore, total bed material load equations.

General Approaches

There are two general approaches to the determination of total load: (1) computation of

bed load and suspended load separately and then adding them together to obtain total

load – indirect method, and (2) determination of total load function directly without

dividing it into bed load and suspended load – direct method.

Out of the number of methods available for total load computation, only the Engelund

and Hansen method (which is based on the second approach) is presented here.

Engelund and Hansen Method

The basic expression for this method is given by

2

5

10 .f (1.31)

where f = total friction factor, computed from Darcy- Weisbach equation for

friction losses,

Φ = dimensionless sediment discharge, given by

3501 dgs

qT

(1.32)

where qT = bed material discharge per unit width and time, s = specific gravity of

sediments, and

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5050 11 ds

Sh

ds

(1.33)

where θ = dimensionless form of the bed shear stress, τ

h = mean flow depth, S = hydraulic gradient.

By analogy with Darcy-Weisbach equation for friction losses, the total friction factor can

be expressed as

2

2

rF

Sf (1.34)

where Fr = Froude number of the stream flow.

1.4 Cross-sectional Index and Meandering Index

Alluvial stream channels, due to the continuous process of erosion and deposition, have

ever-changing cross-sections, now being aggraded (deposition), now being degraded

(erosion). In order to express these changes, a characteristic ratio, called cross-section

index, is often used.

(1.35)

in which d =A/B - hydraulic depth; B - water-surface width,. Alluvial streams rarely flow for any appreciable length along a straight line, but rather in

a series of curves, called meanders. Even those streams which at first sight appear not

to meander for longer stretches, will invariably prove to be also subject to natural

oscillations.

The geometric aspect of a meandering stream is expressed by a characteristic index

which denotes to what extent a given alluvial stream deviates from following the center-

line of the valley, (see Fig. 1.15).

The ratio of the actual stream channel alignment to the corresponding length of the

valley line is called the meandering index, denoted M,

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(1.36)

Figure 1.15 Meandering stream

According to the very nature of alluvial streams, meandering index is never a constant

for a given alluvial channel, but is rather continuously varying around a characteristic

value. Lack of constancy for both the cross-section index and the meandering index is

another expression of the fact that alluvial streams are generally in temporary and

precarious equilibrium only.

1.5 Development Process of a Stream

1.5.1 Stream channel formation

Streams exhibit a wide range of physical characteristics at different phases of their

formation and will react differently to management or restoration efforts by resource

managers. Stream managers or users must understand stream channel formation to

adequately address stream problems and restoration.

1.5.2 Dynamic equilibrium

All streams try to move towards a state of dynamic equilibrium. One way to describe this

equilibrium is the amount of sediment delivered to the channel from the watershed is in

long-term balance with the capacity of the stream to transport and discharge that

sediment. Sediment suspended in water eventually equals sediment settling out of the

water column or being deposited. Sediment load is the total amount of sediment,

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including that in the bed of the stream, being moved by flowing water. The streams

dynamic equilibrium can be expressed with the “stream power proportionality” equation

developed by Lane (see Figure 1.16).

Figure 1.16 Lane’s stream power proportionality equation that expresses the stream dynamic

equilibrium

According to Lane’s equation, the products of Q S and Qs

D50

are proportional to each

other although not equal to each other. The equilibrium occurs when all four variables

are in balance. The bottom line is that a given amount of water with a certain velocity

can only move so much sediment of a given size.

For example, if slope is increased and streamflow remains the same, either the

sediment load or the size of the particles must also increase. Likewise, if flow is

increased (e.g., by an interbasin transfer) and the slope stays the same, sediment load

or sediment particle size has to increase to maintain channel equilibrium. A stream

seeking a new equilibrium tends to erode more sediment and of larger particle size.

Stable streams are in dynamic equilibrium and called graded (poised). The slope of a

graded stream, over a period of years, has delicately adjusted to provide, with the

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available discharge and prevailing channel characteristics, the velocity required for the

transportation of the sediment load supplied from the drainage basin. A graded stream

can have depositional and erosional events but overall the sediment transported and

supplied to the stream is balanced over long periods. Disturbance of the equilibrium

leads to unstable streams that are degrading (eroding) or aggrading (depositing).

Degrading streams have a deficit of sediment supply, while aggrading streams have an

excess of sediment supply. In both degrading and aggrading streams, the stream is

trying to adjust its slope based on the sediment supply. A stream can typically exhibit all

three equilibrium states in various reaches along the same stream.

1.5.3 The channel evolution model

In addition to channel stability (dynamic equilibrium), sediment transport and channel

dimensions (width and depth of the channel) are very important characteristics for

describing streams. These characteristics are incorporated in a conceptual model called

the incised channel evolution model (CEM). This model builds upon the dynamic

equilibrium theory and describes the stages a stream goes through to reach a new

dynamic equilibrium following a disturbance. It also describes the stream bank erosion

processes (downcutting, headcutting, or lateral erosion) that are dominant during the

different stages.

There are five different stages (Figure 1.17):

Stage I (Stable): The stream flow discharge of Q2

will spill in the floodplain and

deposit sediment and organic matter. Q2

is a discharge that has a probability of

occurring every two years and is associated with bankfull discharge of undisturbed

streams in normal to wet environments. The stream bank height h is below the

critical height hc. Critical height h

c is that height above which the banks have high

potential of collapsing by gravitational forces.

Stage II (Incision): This stage starts after disequilibrium conditions occur. These

conditions occur as a result of higher Q (stream flow discharge) or S (slope), which

lead to an increase in Qs

(sediment discharge) capacity in order to maintain the

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dynamic equilibrium. The increased Qs

capacity causes downcutting of the stream

bed. The height of the stream banks increases to higher than critical. As a result the

banks now can hold stream flow discharge of Q10

. Q10

is the discharge that has the

probability to occur every ten years. A knick point can indicate the movement of

incision upstream and in the tributaries. A knick point is a point in the stream profile

where the slope abruptly changes.

Stage III (Widening): The extensive increase in bank heights (higher than the

critical height) of the channel leads to excessive stream bank instability. The banks

start collapsing and the stream starts widening. These streams are extremely deep

and wide. Most of the sediment is still moving downstream.

Stage IV (Stabilizing): Excessive sediment deposition from the stream banks in the

channel makes it impossible for the stream flow discharge to remove all of it. The

stream bank height starts decreasing (typically equals the critical height). Vegetation

starts growing on the sloughing material that is not removed. A new lower capacity

stream channel is formed.

Stage V (Stable with terraces): A new channel develops and the new banks have

heights shorter than the critical bank height. The new floodplain is connected with

the stream. Terraces are the remnants of the original floodplain.

Within each of the five stages of channel development described by the Channel

Evolution Model, channel adjustment is dominated by one of the several processes. For

example, in Stage II, downcutting yields the majority of the stream sediment while in

Stage III lateral (stream bank) erosion is the primary mechanism of channel adjustment.

The difference between these types of erosion has implications for determining the type

of restoration efforts. In Stage II success of restoration depends upon stopping stream

downcutting by what is called “bed stabilization.” Bed stabilization is usually done by

installing grade control structures such as gabions or check dams. In Stage III, where

lateral erosion dominates, restoration efforts should focus on the stream banks.

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Figure 1.17 The five stages of channel evolution model

1.6 Bed forms and alluvial roughness

In alluvial channels the movable bed will take on different and changing forms,

depending on the interaction between the sediment and the flow of water. A general

picture of bed forms and their relationship with flow regimes is essential for engineering

purposes, as the resistance to flow in alluvial channels is largely determined by bed

configuration.

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Flowing over an alluvial bed, water exerts a shear stress on individual sediment

particles, given as τ = g RS. If Manning equation for uniform flow is used, this shear

stress can be expressed as

(1.37)

in which n is the Manning roughness coefficient, R the hydraulic radius and V the mean

velocity. Assuming further that n and R are constant; this is a simple quadratic relation

between τ and V. If, on the other hand, roughness coefficient n changes as a result of

the shear stresses on the loose bed, the above simple relationship generally assumes a

form similar to the one shown in Figure 1.17.

Figure 1.17 Schematic relationship τ = f (V) in alluvial channels.

1.6.1 Bed forms

Many types of bedforms can be observed in nature. When the bed form crest is

perpendicular (transverse) to the main flow direction, the bedforms are called transverse

bedforms, such as ripples, dunes and anti-dunes (see Fig. 1.18). Ripples have a length

scale smaller than the water depth, whereas dunes have a length scale much larger

than the water depth.

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Ripples and dunes travel downstream by erosion at the upstream face (stoss side) and

deposition at the downstream face (lee side). Antidunes travel upstream by lee side

erosion and stoss side deposition (see Figure 1.19).

Bedforms with their crest parallel to the flow are called longitudinal bedforms such as

ribbons and ridges. In laboratory flumes the sequence of bedforms with increasing flow

intensity is

Flat bed → Ripples →Dunes → High stage flat bed → Antidunes

Plane (flat) bed: is a plane bed surface without elevations or depressions larger than

the largest grain of the bed material.

Ripples: Ripples are formed at relatively weak flow intensity and are linked with fine

materials, with d50 less than 0.7 mm. The size of ripples is mainly controlled by grain

size. By observations the typical height and length of ripples are

At low flow intensity the ripples have a fairly regular form with an upstream slope 6° and

downstream slope 32°. Ripple profiles are approximately triangular, with long gentle

upstream slopes and short, steep downstream slopes.

Dunes: The shape of dunes is very similar to that of ripples, but it is much larger. The

size of dunes is mainly controlled by flow depth. Dunes are linked with coarse grains,

with d50 bigger than 0.6 mm. With the increase of flow intensity, dunes grow up, and the

water depth at the crest of dunes becomes smaller. It means a fairly high velocity at the

crest, dunes will be washed-out and the high stage flat (plane) bed is formed.

Transition: This bed configuration is generated by flow conditions intermediate

between those producing dunes and plane bed. In many cases, part of the bed is

covered with dunes while a plane bed covers the remainder.

Antidunes: These are also called standing waves. When Froude number exceeds unity

antidunes occur. The wave height on the water surface is the same order as the

antidune height. The surface wave is unstable and can grow and break in an upstream

direction, which moves the antidunes upstream.

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Chutes and Pools: These occur at relatively large slopes with high velocities and

sediment concentrations.

Figure 1.19 Bed form types in rivers

Figure 1.20 Bed form migration in lower and upper regimes

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Bars: These are bed forms having lengths of the same order as the channel width, or

greater, and heights comparable to the mean depth of the generating flow. Bars have

generally elongated shapes, usually reaching lengths equal to channel width or more.

These are point bars, alternate bars, middle bars, and tributary bars.

Point bars are formed on the convex side of channel bends or meandering alluvial

streams.

Alternate bars are generally a characteristic feature of crossings, i.e. straight stretches

between successive meanders. They appear alternately along both banks of the

stream, and as a rule occupy much less than the width of the channel.

Tributary bars are formed at confluence of tributaries with the main stream, and they

extend downstream. Tributary bars, developed during high flows may appear as

detached small islands during low water.

Figure 1.21 Types of bars

Alluvial Cones and Fans

At locations in which an alluvial stream suddenly changes its slope from relatively steep

to mild, as for instance when leaving mountainous area and entering alluvial plain, or

where a steep tributary meets a flat stream, an alluvial fan may develop.

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Several chronological stages of the alluvial-fan development are schematically shown

on Fig. 1.22.

Stage (a) - A relatively unobstructed or recently regulated stream channel carries the

water safely within its banks;

Stage (b) - Beginning of sediment deposition, part of the available cross section is

blocked;

Stage (c) - Sediment deposition continues, stream channel fills up and the water starts

to overflow the banks, flooding part of the adjacent area, the flood-plain;

Stage (d) - The water which has overflown the banks moves with a very low velocity

over the flood plain and fine sediment settles down on both sides of the stream-bed.

This is already a highly undesirable situation, since the flood-waters cannot be drained

back into the main stream channel;

Stage (e) - At this stage the channel bed is higher than the surrounding area, and this

situation is justly known as an "elevated stream-bed". It consists mainly of recently

deposited fine sediment. The area of sediment deposition outside the stream channel

proper grows continuously from this stage on.

This elevated area is called the alluvial cone. When the water flowing over the cone

fans out in the form of branching gullies, it is known by the name of alluvial fan (see

Figure 1.23).

As mentioned before, the main cause for the formation of an alluvial cone is probably

the abrupt changes of the channel slope, but recent studies and field observations seem

to indicate that the deposition is also the result of change in channel width and the

corresponding reduction of flow volume as the water fans out over an ever larger area.

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Figure 1.22. Alluvial cone and fan formation stages.

Figure 1.23 Alluvial fan.

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The apex of the cone is located at the head of the mountain front, and from this point

deposits spread out fan-like into the plain and finally merge with it. At the apex the

sediment is generally a mixture of gravel and sand, becoming progressively finer toward

the margins. Slope of the cone is the steepest at the apex, diminishing toward the

periphery.

Stream Delta

When a stream finally reaches the sea, or any other expanse of water, it loses most of

its tractive power and deposits all of its sediments, including the finest silt and clay

fractions. The deposited sediment generally takes the form similar to an alluvial fan, but

its formation is much more complex, because of additional parameters that are of

considerable influence, such as sea waves and breakers, offshore currents and tidal

motion.

From the engineering point of view, stream mouths may generally be of three main

types: 1) estuaries, 2) lagoons, and 3) deltas. In the following only delta-type stream

mouths will be reviewed, not only because they are by large the most widespread, but

also because they have many features in common with all the other forms.

Delta is a highly dynamic natural phenomenon, since it is actually the result of a

continuous contest and interaction between the stream and the sea or other water

expanse. The stream manages to deposit its sediments more quickly and efficiently

than the dispersive action of sea waves and currents to carry them off into the sea.

Climatic conditions of the drainage basin determine to a great extent the shape of the

delta. In temperate climates, where the flow and the sediment load generally are more

or less evenly distributed throughout the year, stream channels of the delta tend to be

stable and well adapted to the whole range of discharges conveyed by the stream. In

arid climates, on the other hand, flood conditions tend to be erratic, and large

quantities of water and sediment are carried by the stream during relatively short

periods of time. As a consequence, distributary channels never fully adjust themselves

to such large influx of water and sediment. Many new channels are rapidly formed

during the flood wave, to be filled up and abandoned as floodwaters recede. This

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process, often repeated, causes instability and migratory tendency of the whole deltaic

system.

Within the delta proper, sediment transportation is in form of bed load and suspended

load, but mainly in the latter form. Stability of distributary channels depends to some

degree on the form of sediment transportation:

Channels carrying larger volumes of bed load generally tend to be wide and

shallow, and are subject to rapid lateral migration;

High concentrations of suspended load tend to cause narrow and deep channels

that are relatively much more stable.

A typical cross-section through a stream delta generally shows several sets of bed

layers superimposed one on top of the other. In the subaerial part (see Fig. 1.24)

mostly channel sands and natural silts are found. Deposits at the delta front are laid

down in a subaqueous environment immediately seaward of the delta coastline, and

are of much finer gradation. In general, it can be said that all deposits tend to grade

from coarser to finer in the offshore direction, and from finer to coarser in a vertical

section, starting upward from the bottom.

Figure 1.24 Schematic drawing of a river delta.

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In the case of a radical reduction of sediment discharge, due to large-scale engineering

interference in the upstream course of the stream channel (such as the erection of a

large dam), the existing temporary equilibrium between the sediment transport and the

sea currents may be grossly disturbed. As a consequence, sea currents are likely to

attack and erode existing sediment deposits, thus actually shortening the delta. A

situation of this kind has been evolving at the Nile river delta after the construction of the

High Aswan Dam.

Stream Confluence and Bifurcation

Confluences are mainly present in the upper reach of a river whereas bifurcations are

usually present in the lower reach.

A few typical cases will be qualitatively analyzed in the following.

- It is assumed that the bottom elevation of the tributary at the confluence is

roughly the same as that of the main stream (case 1, 2, & 3).

- Liquid discharge, sediment discharge, mean grain size and the hydraulic gradient

in the main stream are Q1, Qs1, ds1, and S1, respectively; in the tributary they are

Q2, Qs2, ds2, and S2

- Both streams carry the maximum sediment discharge according to their

respective sediment transport capacity (STC) under the given flow conditions.

Case 1: The flood wave in both streams occurs roughly at the same time (Fig. 1.25a).

- Water stage in the main stream during the passage of the flood wave is usually

higher than in the tributary, and hence back-water curve will develop in the tributary

(Fig. 1.25a). As a result of this, hydraulic gradient in the tributary, in the reach

For a confluence, the continuity of equation for

water (Q) and sediment (QS) hold.

Q0 = Q1 + Q2

QS0 = QS1 + QS2

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upstream of the confluence, will decrease, causing a part of the sediment to be

deposited close to the stream-mouth.

- Downstream of the confluence liquid discharge in the main stream will increase from

Q1 to (Q1 + Q2), and the sediment discharge from QS1 to somewhat less than (Qs1 +

QS2), because a relatively small part of QS2 will already have been deposited.

- It can be assumed that there will be deposition, because STC will not be sufficiently

high. Sand bars downstream of the confluence will mainly consist of coarser

sediment fractions from the tributary, since the finer ones will probably be carried by

the increased water volume in the main stream.

Case 2: During the flood wave in the tributary, there is low water in the main stream

(Fig. 1.25b);

- The situation now will be the reverse: water level in the tributary will be higher

than in the main stream, and hence a drawdown curve will have to form in the

tributary upstream of the confluence.

- Due to flow velocities higher than for normal flow, STC of the tributary will be high

enough in the vicinity of the meeting point to carry the entire sediment load, QS2.

So the tributary is likely to stay clean.

- In the downstream main channel the combined discharge (Q1 + Q2) may well be

too low to carry the aggregate sediment load (Qs1 + QS2), and hence a

considerable part of it is likely to be deposited downstream of the confluence,

causing large sand bars.

Figure 1.25 Flow situation at confluence.

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Case 3: Low water in the tributary during the passage of a flood wave in the main

stream;

- Back-water curve extend much farther upstream into the tributary.

- In spite of the reduced flow velocities caused by the backed-up water, sediment

load in the tributary will be relatively low, and the tributary is likely to be capable

of handling it. Hence little deposition is expected to take place in the tributary.

- There is relatively modest addition of sediment from the tributary.

- High flow velocities in the main stream are likely to raise its STC just enough to

carry the additional load without much difficulty, and hence probably little or no

deposition in the main channel either (see Fig. 1.25a).

Case 4: water level in the tributary is higher than in the main stream and the bed

elevation of the tributary at the confluence is higher than in the main stream (Fig. 1.26);

- There will be a drawdown curve in the tributary upstream of the confluence,

accompanied by high velocities,

- Severe erosion is to be expected along the bed of the tributary.

- After some time, a part of channel bed may collapse, shifting the drop from 1 to

2; this process, generally known as back-erosion, may repeat itself several times

(points 3, 4, etc.), and thus endanger the stability of the channel.

- The eroded material will ultimately be carried into the main stream, settling

downstream of the confluence until entrained by high water during flood waves

Figure 1.26 Back-erosion at confluence

Although in both cases of confluence and bifurcation the main watercourse meets two

streams, there are some important differences. The geometry of the branching channels

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at the bifurcation and the available head determine the magnitude of discharges Q1 and

Q2. The sorting of the sediment at the bifurcation also depends mainly on the geometry.

There is generally no backwater effect at the bifurcation.

Meandering and Braided Stream Channels

Alluvial streams generally flow in a succession of clockwise and anti-clockwise bends,

interconnected by relatively short straight reaches called crossings. Such geometrical

alignment is generally known as a meandering river (Fig. 1.27).

A watercourse is generally called a meandering stream when the ratio between its

actual length and the length of the valley is 1.5 or more (the ratio is rarely more than

about 2.5).

The actual shape of bends in a meandering stream is rarely symmetrical and

geometrically well-defined. Radius of curvature varies over a wide range, depending

upon the type of bend. Free bends in plain alluvial material, easily erodible and mobile,

generally have the ratio of the radii of curvature to the width of the stream in the range

of 4-5, while in case of more consolidated bank materials; the ratio may be as high as 7-

8. On the other hand, in forced bends, formed by a stream being deflected by a

practically non-eroding bank, the ratio may be as low as 2-3.

Figure 1.27. Schematic layout of a meandering stream.

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A general characteristic of all meandering watercourses is the migration of the bends

downstream and under certain circumstances even laterally. The migration velocity

changes from stream to stream, and there are slow-moving and fast-moving streams.

Short straight reaches connecting consecutive bends are known under the name of

"crossings", and they generally are relatively shallow compared to deep parts of the

bends that precede and follow them. A considerable part of the bed material eroded

from the concave bank of the bend is deposited in the crossings by the spiral cross

currents which do not decay as soon as they leave the bend, but extend downstream.

At lower discharges, sand bars also may be formed in the crossings. The main erosion

process is to be expected at the concave side of the flow channel.

Figure 1.27a

An alternative alignment of an oscillating alluvial watercourse is known as a braided

stream. The characteristic features of such a configuration are a wide channel, unstable

and poorly defined banks and shallow water. The watercourse consists of a number of

entwined channels divided by islands, which meet, cross and separate again.

The main causes which bring about the braiding of a stream seem to be

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1) Supply of more sediment than warranted by its STC, hence part of the load is

deposited,

2) Steep longitudinal slopes that tend to produce a wide and shallow channel in

which bars readily form, become stabilized by armouring and vegetation and

form islands, and

3) Easily erodible banks, allowing the widening of the stream channel at high flows.

It is generally assumed that a braided channel has a steep slope, a large bed load in

comparison with the suspended load, and usually small amounts of silt and clay

particles in both bed and banks. A decrease in longitudinal slope may often change a

channel from braided into meandering.

Figure 1.28. A braided stream

1.6.2 Bed Roughness

In open channel hydraulics with rigid-boundary, the roughness coefficient can be treated

as a constant. After the roughness coefficient has been determined, a resistance

formula can be applied directly for the computation of velocity, slope, or depth. In fluvial

hydraulics, the boundary is movable and the resistance to flow or the roughness

coefficient is variable. In this case, a resistance formula cannot be applied directly

without knowledge of how the resistance coefficient will change under different flow and

sediment conditions.

Resistance to flow with a movable boundary consists of two parts. The roughness that

is directly related to grain size is called GRAIN ROUGHNESS. The roughness that is

due to the existence of bedforms and that changes with changes of bedforms is called

FORM ROUGHNESS.

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If Manning’s roughness coefficient is used, the total coefficient n can be expressed as

nnn (1.38)

where n’ = Manning’s coefficient due to grain roughness, and n’’ = Manning’s roughness

due to form roughness.

The value of n’ is proportional to the sediment particle size to the sixth power. There is

no reliable method for the computation of n’’, which poses a major problem in the study

of alluvial hydraulics.

Manning’s Formula

One of the most commonly used resistance equations for open channel flows is

Manning’s equation, namely,

2

13

2

SRVn1 (1.39)

Strickler defined Manning’s n as a function of sediment particle size as:

1.21

dn

61

(1.40a)

where d = sediment size of uniform sand in m.

Meyer-Peter and Mueller, considering a sand mixture, transformed Strickler’s formula to

26

dn

61

90 (1.40b)

where d90 = sediment size (in m) for which 90% of the mixture is finer.

Similar to the division of total roughness into grain roughness and form roughness, the

shear stress or drag force acting along an alluvial bed can be divided into two parts, i.e.,

RRS (1.41)

where τ = total drag force acting along alluvial bed, τ´ and τ´´ = drag force due to grain

roughness and form roughness, respectively, γ= specific weight of water, S = energy

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slope, and R´ and R´´= hydraulic radii due to grain roughness and form roughness,

respectively.

aaitcivil.files.wordpress.comthe representative grain diameter, d, should best be determined by...

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