STAGE-DISCHARGE MODELING FOR MEANDERING CHANNELS SAINE SIKTA DASH

STAGE-DISCHARGE MODELING FOR MEANDERING CHANNELS  SAINE SIKTA DASH
STAGE-DISCHARGE MODELING FOR
MEANDERING CHANNELS
A Thesis Submitted in Partial Fulfillment of the Requirements for the
Degree of
Master of Technology
In
Civil Engineering
SAINE SIKTA DASH
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
2013
STAGE-DISCHARGE MODELLING FOR
MEANDERING CHANNELS
A thesis
Submitted by
Saine Sikta Dash
(211CE4251)
In partial fulfillment of the requirements
for the award of the degree of
Master of Technology
In
Civil Engineering
(Water Resources Engineering)
Under The Guidance of
Dr. K.K Khatua
Department of Civil Engineering
National Institute of Technology Rourkela
Orissa -769008, India
May 2013
NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA,
ORISSA -769008, INDIA
This is to certify that the thesis entitled, “STAGE-DISCHARGE MODELLING FOR
MEANDERING CHANNELS” submitted by Saine Saikta Dash in partial fulfillment of the
requirement for the award of Master of Technology degree in Civil Engineering with
specialization in Water Resources Engineering at the National Institute of Technology
Rourkela is an authentic work carried out by her under our supervision and guidance. To the best
of our knowledge, the matter embodied in the thesis has not been submitted to any other
University/Institute for the award of any degree or diploma.
Research Guide
Place: Rourkela
Dr. K. K Khatua
Associate Professor
National Institute Of Technology
Date:
2
ACKNOWLEDGEMENTS
First and foremost, praise and thanks goes to my God for the blessing that has bestowed upon me
in all my endeavors.
I am deeply indebted to Dr. K.K Khatua, Associate Professor of Water Resources Engineering
Division, my advisor and guide, for the motivation, guidance, tutelage and patience throughout
the research work. I appreciate his broad range of expertise and attention to detail, as well as the
constant encouragement he has given me over the years. There is no need to mention that a big
part of this thesis is the result of joint work with him, without which the completion of the work
would have been impossible.
I am grateful to Prof. N Roy, Head of the Department of Civil Engineering for his valuable
suggestions during the synopsis meeting and necessary facilities for the research work. And also
i am sincerely thankful to Prof. K.C. Patra, Prof. Ramakar Jha, and Prof. A. Kumar for their
kind cooperation and necessary advice.
I extend my sincere thanks to Mr. Prabir K. Mohanty & Mr. Alok Adhikari the senior
research scholar of Water Resources Engineering Division for their helpful comments and
encouragement for this work. I am grateful for friendly atmosphere of the Water Resources
Engineering Division and all kind and helpful professors that I have met during my course.
I would like thank my parents, Kirtikanta Sahoo and family members. Without their love,
patience and support, I could not have completed this work. Finally, I wish to thank many friends
for the encouragement during these difficult years, especially, Sonia, Mona, Roma, Gedi, Dutta
Roy, Shraddha, Arpan, Bhabani.
Saine Sikta Dash
i
ABSTRACT
Flow in meandering channel is quite ubiquitous for natural flow systems such as in rives. Rivers
generally follow this pattern for minimization of energy loss. However, several factors such as
environmental condition, roughness are responsible for generation of this path for rivers.
Selection of proper value of Roughness coefficient is essential for evaluating the actual carrying
capacity of Natural channel. An excessive value underestimates the discharge and a low value
can over estimates. Suggested values for Manning's n are found tabulated in many standard
articles. The resistance to the flow in a river is dependent on a number of flow and surface and
geometrical parameters. The usual practice in one dimensional analysis is to select a value of n
depending on the channel surface roughness and take it as uniform for the entire surface for all
depths of flow. The influences of all the parameters are assumed to be lumped into a single value
of Manning’s n .It is seen that Manning’s coefficient n not only denotes the roughness
characteristics of a channel but also the energy loss in the flow. The larger the value of n, the
higher is the loss of energy within the flow. Experimental investigations concerning the variation
of roughness coefficients for a highly meandering channel for different flow condition, geometry
are presented. The flow properties are found to be affected by different geometric, surface and
flow parameters. In the present work, an attempt has been made to analyses the important
parameters affecting the flow behavior and flow resistance in term of Manning’ n in a
meandering channel. The factors influencing for predicting the roughness coefficient of a
meandering channel are non-dimensionlised and its dependency with different parameters are
studied. Further, a regression analysis has been done to formulate a mathematical model to
predict the roughness coefficient. The equation can be suitably applied to predict the stage-
ii
discharge relationships of a meandering channel. The efficacy of the proposed equation has also
been tested not only to the present data sets but also the data of other investigators.
KEYWORDS: Meandering channel, Manning’s n, Flow variables, Stage, Discharge.
iii
TABLE OF CONTENTS
Title
Page No.
ACKNOWLEDGEMENTS .......................................................................................... i
ABSTRACT ................................................................................................................. ii
TABLES OF CONTENTS ........................................................................................ .iv
LIST OF FIGURES ................................................................................................... vii
LIST OF TABLES .................................................................................................... viii
NOTATIONS .............................................................................................................. ix
CHAPTER 1
INTRODUCTION
1.1. Overview ................................................................................................................1
1.2. Stage Discharge Relationships ...............................................................................3
1.3. Types of Flow channels .........................................................................................4
1.3.1. Straight Channels ................................................................................................5
1.3.2. Meandering Channels .........................................................................................6
1.3.3. Braided Channels ................................................................................................8
1.4. Estimation of Stage discharge ................................................................................8
CHAPTER 2
LITERATURE REVIEW
2.1. Overview ..............................................................................................................10
2.2. Literature on roughness, discharge and velocity..................................................11
CHAPTER 3
EXPERIMANTAL SETUP AND METHODOLOGY
3.1. Overview ..............................................................................................................26
3.2. Design & Construction of Channel ......................................................................26
3.3. Apparatus and Equipments used ..........................................................................29
3.4. Experimental Procedure .......................................................................................30
iv
3.5. Experimental Channel ..........................................................................................31
3.6. Measurement of Bed Slope ..................................................................................31
3.7. Calibration of Notch ............................................................................................32
3.8. Measurement of Depth of Flow and Discharge ...................................................33
3.9. Measurement of Longitudinal Velocity ...............................................................33
CHAPTER 4
EXPERIMENTAL RESULTS
4.1. Overview ..............................................................................................................35
4.2. Stage Discharge Results.......................................................................................35
4.3. Distribution of Longitudinal Velocity Results .....................................................37
4.4. Variation of Roughness Coefficient in Meandering Channels ............................40
CHAPTER 5
THEORITICAL ANALYSIS AND MODEL DEVELOPMENT
5.1.
Summary .........................................................................................................47
5.2.
Problem Statement ..........................................................................................47
5.3.
Background .....................................................................................................49
5.4 Description ...........................................................................................................52
5.4.1 Regression Analysis ........................................................................................52
5.4.2. Linear Regression Analysis ............................................................................53
5.4.3. Regression Calculations ..................................................................................55
5.5.
Source of Data and Selection of Parameters ...................................................59
5.5.1 Source of Data.................................................................................................59
5.5.2. Selection of Hydraulic, Geometric, and Surface Parameters ..........................59
5.6.
Model Development........................................................................................60
5.6.1. Introduction .....................................................................................................60
v
5.6.2. Modelling of the Meandering Channel at N.I.T Rourkela ..............................62
5.7
Validation ........................................................................................................68
CHAPTER 6
CONCLUSION
6.1. Conclusion ...........................................................................................................71
6.2. Scope for Future Work.........................................................................................73
ACKNOWLEDGEMENT ........................................................................................73
REFERENCES .........................................................................................................74
vi
LIST OF FIGURES
Title
Page No
Fig.1.1: Different types of straight channel ..................................................................5
Fig.1.2: Different types of meandering channel ...........................................................6
Fig.1.3: A simple illustration of Meandering Channel formation ................................7
Fig.1.4: Geometry of Meandering Channel ..................................................................8
Fig.1.5: Different types of braided channel ..................................................................8
Fig.3.1. Designed plan of the experimental meandering channel (Sr=4.11)…….. ....28
Fig.3.2: Schematic diagram of Experimental meandering channels with setup .........28
Fig.3.3: Longitudinal & Cross sectional dimension of the meandering channels ......28
Fig.3.4 (i to iv): Apparatus used in experimentation in the meandering channels .....29
Fig.3.5: Typical grid showing the arrangement of velocity measurement points
along horizontal and vertical direction at the test section for meandering channel. ..30
Fig.3.6: meandering channel inside the flume with measuring equipment ................31
Fig.4.1: Stage-discharge curve for meandering Channel ............................................36
Fig.4.2: Grid Points of longitudinal velocity measurement ........................................38
Fig.4.3 (i to v): Longitudinal velocity contours for simple meandering channels ....38
Fig.4.4: Variation of Manning’s n with Depth of Flow ..............................................42
Fig.4.5: Variation of Chezy’s C with Depth of Flow .................................................42
Fig.4.6: Variation of Friction factor f with Depth of Flow .........................................43
Fig 4.7: Variation of Manning’s n with Aspect Ratio ................................................44
Fig 4.8 Variation of Manning’s n with Reynolds’s Number ......................................45
vii
Fig 4.9 Variation of Manning’s n with Froude’s Number ..........................................45
Fig 5-(1-5) Comparison of Manning’s n with different flow parameters ...................64
Fig 5-(6 to 9) Validation occurs in between actual vs. Predicted ...............................68
LIST OF TABLES
Table 2.1: Degree of meandering................................................................................11
Table 3.1: Details of Geometrical parameters of the experimental runs…………....31
Table.4.1 Details of Hydraulic parameters of the experimental runs…………….…41
Table 5.1: Details of Hydraulic parameter for all data collected from Globe……....60
Table 5.2: Unstandardized Coefficient by Linear Regression Analysis……………67
viii
LIST OF NOTATIONS
A
Area of channel cross section
a
Intercept
B
Meander channel top width
b
Bottom width of channel
b/h
Width depth ratio or aspect ratio
C
Chezy’s channel coefficient
Cd
Coefficient of discharge
d84
Size of the intermediate particles
e
Error
f
Darcy-Weisbach Friction factor
Fr
Froude’s Number
g
Acceleration due to gravity
Hn
Height of water above the notch
Hw
Height of water in the volumetric tank
h
Height of channel
k
Von Karman’s constant
Lc
Wave Length
Lm
Length of Meander
n
Manning’s roughness coefficient
Q
Discharge
Qa
Actual discharge
Qth
Theoretical discharge
R2
Coefficient of Determination
R
Hydraulic radius of the channel cross section
Re
Reynolds Number
So
Bed Slope of the Channel
Sr
Sinuosity of the Channel
ix
SSy
Sum of Square Error
V
Mean flow velocity
X
Independent Variable
Y
Dependent Variable
P
Dynamic Velocity

Density of flowing fluid
μ
Dynamic viscosity of water
α
Aspect ratio
2
Variance
ν
Kinematic viscosity

Wave Length
x
CHAPTER 1
INTRODUCTION
Chapter 1
1.1
OVERVIEW
Water is perhaps the most fundamental and necessary resource available to mankind. It arrives
on land in the form of precipitation and returns to the sea by means of river channels. For the
most part, river channels adequately convey the water back to the sea but occasionally, under
conditions of high rainfall and large flow rates, the river channel may overtop its banks and flow
onto the flood plain with possible danger to life and property. Rivers are a natural aspect of
our landscape and form an integral part of the water cycle. By default rivers are the effect of
magnificence and the historic essence of a settlement. Also river are providing peace and
serenity to the human beings. People have lived near to rivers for centuries due to the reason of
mainly food, water, transport and protection. But sometimes, it may cause serious damage to
people and the places in which they live even if it is a small, slow-flowing stream or gentle river.
Normally river flow patterns are divided into two types such as (i) Straight river and (ii)
Meandering river and (iii) Braid ed river. Discharge through straight type and meandering type
are totally different from each other. According to the geometrical shape and other parameters
flow through meandering channel are more complicated than straight channel. So In a
meandering river, distribution of flow and velocity play a major role in relation to practical
problems such as flood protection, flood plain management, bank protection, navigation, water
intakes and sediment transport-depositional patterns. River flows in a meandering channel often
inundate the adjacent plains at high discharges. Due to this it generates a complicated complex
flow structure throughout the channel. There are two main issues involved in tackling such
complex flow problems. The first is to collect the fundamental data for an internal structure
under simple flow conditions as a bench mark. Without understanding the structure in a simple
flow case one cannot proceed to more complex flow situations. Secondly an advanced tool is
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Chapter 1
required to predict the accurate flow variables for complex flow structures. To predict such types
of complex flow, experimental facilities, instrumentation and computer models have been
gradually improved in the world. In fact, for the last 2 or 3decades, development of new velocity
measuring devices, data collection systems and numerical models has made possible
considerable advances in knowledge relating to water engineering problems. Extensive research
has already been conducted not only in simple straight channels but also in complex meandering
channels, such as the study was being carried out by Ikeda & Parker (1989) and Ozawa (1995),
and compound channel flows by Ishigaki (2001).
Meander migration is a significant river engineering problem, which interferes with many human
activities on and around the rivers, such as navigation, highway construction, flood control, and
farming. There is a well-documented need (U.S. Army Corps of Engineers 1981) for guidelines
for evaluating the problem, which is one of stability of river channel alignment. Guidelines are
particularly needed to determine the extent to which a given channel alignment is prone to future
changes. The meandering river system is characterized by recurrent river planform patterns,
repeated with little variation from one river to the next irrespective of their magnitude and from
one scale to another within each river. This consistency suggests that a higher level of processes
forms by self-organization from the physical processes of deposition and erosion operating in the
system. These physical processes may be described by continuum fluid mechanics. Although
meandering dynamics can be simulated from models based on continuum mechanics, such
models reveal little about the holistic, spatiotemporal properties of the meandering process, for
example, the hierarchical, fractal geometry of the river planform. Present study attempts to
justify the computational hydraulics analyses of meandering channel with formulation of
equation for discharge. The natural flow mechanism was complicated and hence the calculation
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Chapter 1
of discharge in meandering open channel flow is also complex and conventional methods cannot
predict discharge with sufficient accuracy. A simple but reliable prediction technique for
estimating roughness of a channel is highly necessary for field engineers, designers and
researchers. However, with the recent development of soft computing, data mining and artificial
intelligence, there is a choice of better techniques that can made easy to analysis the problems
and can solve these problems to get the satisfactory results efficiently. When relationship
between input and output is difficult to establish using numerical, analytical and mathematical
methods and it becomes complicated and consumes the valuable time, therefore an easily
implementable technique like a computing technique using a generalized Regression method can
be adopted. The main advantage of this technique, it is totally a mathematical regression process
Experiments are shown to examine the variation of roughness coefficients of a meandering
channel with different hydraulic and geometric parameters. In addition to fresh experimental
dataset, other data sets of meandering channels are collected to estimate the roughness
coefficients of a meandering channel by Statistical system. The results validated by Statistical are
compared with other well established and widely used traditional models Sellin (1960), Willet &
Hardwick (1993), Shino & knight (1999), Glasgow (1997), Abreeden (1997), Myers (1957),
Khatua(2007). Statistical error analysis is also carried to examine the performance of these
models.
1.2
STAGE-DISCHARGE RELATIONSHIPS:-
Normally in river, for hydrological analysis continuous data measurement was needed but in
practically data measurement in flowing condition was usually impractical. For stage we need
data for analysis so it was observed continuously or at time intervals with relative ease and
economy. Luckily, a relation occurs between stage and discharge at the river section. This
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Chapter 1
relation is named a stage-discharge relationship or stage-discharge rating curve. This relationship
occurs by making a number of concurrent observations of stage and discharge over a period of
time covering the estimated range of stages at the river gauging section. This is conventional
method for stage-discharge analysis.
Stage discharge rating curve is a very important device for open channel flow because the
consistency of discharge data is highly dependent on a suitable stage-discharge relationship at
the gauging station. Though the planning of rating curves appears to be an essentially task, a
wide theoretical background is needed to create a reliable tool to switch from measured water
height to discharge. The traditional and simple way to gather information on current discharge is
then to measure the water level with gauges and to use the stage-discharge relationship to
estimate the flow discharge. It is well known, in fact, that direct measurements of discharge in
open channels is costly, time consuming, and sometimes impractical during floods. Prediction of
stage-discharge relationship in a meandering river section is required in several river hydraulic
problems such as river engineering, environmental engineering and intake designs. In a
meandering river distribution of flow and velocity play a major role in relation to practical
problems such as flood protection, Flood plain management, bank protection, navigation, water
intakes and sediment transport-depositional patterns.
1.3
TYPES OF FLOW CHANNELS:-
In rivers, phenomena which may vary considerably in time and space, involve mainly two
important subjects: open channel flow hydraulics and sediment transport. The boundaries in
rivers are generally loose, and the problems are therefore complex. For a basic understanding of
river morphology knowledge of sediment transport is obviously essential. Generally the natural
rivers are of various kinds. Typical patterns in the plan geometry of streams correspond to
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Chapter 1
characteristic of cross-sectional form, ability of bed load, downstream slope, and cross-valley
slope, inclination to cut or fill, or position within the system. They can be classified on the basis
of geometry of their flow path as follows,
1) STRAIGHT CHANNEL
2) MEANDERING CHANNEL
3) BRAIDED CHANNEL
1.3.1 Straight channels
If a channel no variation occurs it passage along its flow path. A straight channel, mainly
unstable in nature and it’s developing along the lines of faults and joints, on steep slopes where
hills closely follow the surface gradient. Flume experiments show that straight channels of
uniform cross section rapidly develop pool-and-riffle orders.
The Straight River in
Owatonna
The Straight River Township
Minnesota
Fig.1.1 Different types of straight channel
1.3.2 Meandering channels
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Chapter 1
If a channel deviates from its axial path and a curvature of reverse order is developed with short
straight reaches. Meandering represents a degree of adjustment of water and sediment load in the
river. Meandering channels are solo channels that are sinuous in geometry, but there is no
principle, the degree of sinuosity essential before a channel is called meandering. The design of
bends is measured by flow resistance. In meandering channel the flow reaches a minimum when
the radius of the bend is between two and three times the width of the bed. Meandering channel
is a bend in a sinuous river where single channels deviate and it becomes meander. It is basically
formed by sediment erosion from the outer wall of bend and depositing them on the inside as
results widens its valley. Natural channels never stop changing their geomorphic characteristics.
The channel geometry, slope, degree of curvature, roughness and other concerning parameters
are adjusted in such a way that the river fixes the least work in rotating, while carrying these
loads. Flow in a meandering channel significances in geomorphic changes.
Typical Meandering Channel
(River-Amazon)
Typical Meandering Channel
Northern Owens Valley
Fig.1.2 Different types of meandering channel
Topographic high points and low points are count as pools and riffles correspondingly (Watson,
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Chapter 1
et al. (2005)).The grain sizes found in riffles is greater than pools stated by Keller (1971). Fine
sediments are removed from riffles during low flow velocity and shear stress and which get
deposited in to pools. As a results the outer side of the bank became deeper than the inside bank
where a point bar formation occurs. In addition to this the following features are seen in natural
streams: point bars, middle bars, alternate bars and braiding. In meandering channel a helical
flow (corkscrew shaped flow) is occurs, which is the water surface being raised on the outer
bank of each curve, and return existing at depth directing the flow towards the opposite bank.
The outer bank is eroded for this type situation and its gives a result of the higher flow velocity,
for this deposition takes place on the inner side it forms a point bar.
Fig.1.3 A simple illustration of Meandering Channel formation
There are two methods available to analyze meander geometry (Knighton, 1998). The first
method focused on the individual bend statistics and the second method is a series approach
method that treats the stream trace as a differential change of flow direction (Knighton 1998).
An over view of geometry of meandering channel Watson et al. (2005) is shown below
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Chapter 1
Fig.1.4 Geometry of Meandering Channel
1.3.3 Braided Channel
A braided river has a channel that consists of a network of small channels. Water courses which
are divided by Small Island into multiple channel but term doesn’t describes the multiple channel
of an Ana-branching river. It is a section of a river or stream that diverts from the main channel
or stem of water course and rejoins the main stem downstream.
Typical braided Channel
(Waimakariri River)
Typical braided Channel
(white river)
Fig.1.5 Different types of braided channel
1.4
Estimation of Stage-Discharge:-
For estimating a suitable rating curve in a natural channel the flow is considered steady and
uniform and the discharge is estimated using traditional open channel flow formulations such as,
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Chapter 1
Manning’s formula, Chezy’s formula and Darcy’s Weisbach formula etc. are used. Normally
discharge is calculated by area and velocity.
Q=A.V
(1.1)
In previously our scientist such as Manning’s, Chezy’s, Darcy’s weisbach gives some formula to
calculate velocity for straight channel. These are given below:-
Manning’s equation
Chezy’s equation
1 2 1
v  R 3S 2
n
v  C RS
v
Darcy-Weisbach’s equation
8 gRS
f
(1.2)
(1.3)
(1.4)
In above equation were traditional formulas for calculating velocity. The equation fails to predict
the flow in meandering channels because of many variability and irregularities in geometry, flow
and surface conditions. For this type of situation we need to create a mathematical modeling for
calculating roughness and hence the stage-discharge relationships for meandering channel for a
generalised conditions of flow and geometry.
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CHAPTER 2
LITERATURE REVIEW
CHAPTER 2
2.1
OVERVIEW
An attempt has been made in this chapter to draw together various aspects of past research in
hydraulic engineering concerning the behavior of rivers and channels during in bank and
overbank flow. Prior to the early Sixties, very little was known of the complex flow patterns
which exist between a channel and its associated flood plains, but more recent developments
have led to a clearer understanding of the hydraulic mechanisms involved, at least at the level of
model studies. An important step in receiving a better understanding of river systems is to study
its velocity distribution with maximum accuracy. The flow prediction of river flows is vital
information for flood control, channel design, channel stabilization and restoration projects and it
affects the transport of pollutants and sediments. However, natural river channels are neither
straight nor uniform somewhat typical curved or meandering channel forms. Flow in meandering
channels is increasing interest because this type of channel is common for natural rivers, and
research work regarding flood control, discharge estimation and stream restoration need to be
conducted for this type of channel. It has exposed from investigators that the flow structure of
meandering channels is unpredictably more complex than straight channels due to its velocity
distribution. There are limited studies available in literature concerning the flow in meandering
channels. Rivers are observed to meander, as may readily be seen from maps or aerial
photographs (such as the well-known view of the Thames through London). Meandering
effectively lengthens the channel path, within the existing valley or flood plain. The degree of
meandering may be measured by the term sinuosity, which is defined as the ratio of channel
length to valley length. Chow (1959) described the degree of meandering as follows:
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CHAPTER 2
Table 2.1:-Degree of meandering
Sinuosity ratio
Degree of meandering
1.0 - 1.2
Minor
1.2 - 1.5
Appreciable
1.5 and greater
Severe
In rivers, phenomena which may vary considerably in time and space, involve mainly two
important subjects: open channel flow hydraulics and sediment transport. The boundaries in
rivers are generally loose, and the problems are therefore complex. For a basic understanding of
river morphology knowledge of sediment transport is obviously essential. In this research,
experiments have been carried out in a meandering channel or compound channel and the results
from these different channel configurations compared. This chapter is therefore divided into two
sections: the hydraulics of open channel flow with either meandering or compound channels.
These two sections briefly explain the necessary information concerning flow structure, twophase flow and sediment transport. The prediction of the flow characteristics in meandering
channels is a challenging task for rivers engineers due to the nature of flow. The dominant
feature consists of the effect in the fast moving flow in meandering channel. Some of the
extension literature are studied and presented below.
2.2
LITERATURE ON ROUGHNESS ,DISCHARGE AND VELOCITY PROFILE
Thomson (1876) studied the characteristic spiral motion of the flow in a channel bend. He also
dedicated much research work to both clarifying the flow structure and explaining its
mechanism. It was observed that the source of this phenomenon was the centrifugal force
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CHAPTER 2
generated due to the curved flow path, and resulting spiral motions, i.e. secondary flows, have a
significant effect on engineering matters such as flow resistance, sediment transport, erosion and
deposition.
Bhowmik and Demissie (1932) studied data from two rivers in the United States and it is
observed the rating curves obtained from these two rivers. It can be seen that, for both rivers
there is a significant reduction in the main channel velocity during overbank flow.
Shukry (1950) observed regarding the entrance of the bend and gave the information about the
path deviates from its normal course towards the inner bank and called this phenomenon an
"adjusting process". Due to this the maximum primary velocity occurs along the inner bank in
the first half of bend. After advance proceeding into bend, secondary flows begin to act on the
distribution of the primary flow, driving the faster moving fluid to the outer bank. From the exit
of the bend onward the maximum forward velocity lies near the outer side of the bend. Also
Francis & Asfari (1971), Kalkwijk & De Vriend (1980) and Johannesson & Parker (1989b)
analysed about the distribution of the primary velocity in channel bends theoretically, taking into
account the advective effect of secondary flows.
Shukry (1950), Rozovskii (1961) and Onishi et at. (1976) conducted experimental work to
identify both the main sources of extra energy losses in channel bends and the parameters which
will affect such energy losses. Also it is concluded that the major sources of energy loss in
channel bends can be attributed to: a) skin friction along the channel boundaries; b) increased
bed friction caused by secondary flows and c) internal fluid friction due to secondary flows. The
parameters on which energy loss in a channel bend is deemed to depend are: a) geometrical
conditions, such as bend radius r c, bend angle 8 and the channel cross section shape; b)hydraulic
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CHAPTER 2
conditions, such as flow depth h, Reynolds number Re and Froude number Fr; and c) roughness
condition, i.e. friction factor j, n, C etc.
Prandtl (1952) divided secondary flows into three categories. According to him, the secondary
flows in a channel bend are skew induced and are of the first kind. On the other hand, secondary
flows of the second kind, stress induced ones; appear in straight ducts or channels. In this paper
the growth and decay of secondary flows are described. The analysis that, after entering a bend,
secondary flows start to develop until they become fully developed at some middle point on the
bend. It is seen that the rotation of the flows is that the currents at the water surface move
towards the outside of the bend, whereas they move inwards near the bed. At the fully developed
point single cell type circulation of the flow can be observed and then secondary flows maintain
their conditions uniformly until the exit of the bend. They initiate to decay when they leave the
bend and enter the downstream straight reach. A much longer distance is necessary to weaken
the flow than that for the development. The results show that both geometrical and hydraulic
conditions are essential parameters.
The U.S. Army Corps of Engineers (Hydraulic 1956) studied a series experiments in
meandering channels at the Waterways Experiments Station in Vicksburg. This paper
investigates the stage-discharge and the effect of geometric parameters like radius of curvature of
the bends, sinuosity of the channel, depth of flow, channel roughness on conveyance capacity in
meandering channels.
Cowan (1956) discusses a procedure for estimating the effects of some factors to determine the
value of n for a channel. This paper provides much more flexibility and accuracy than can be
achieved using Chow (1959) in isolation.
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CHAPTER 2
Chow (1959) shows the tables specifying roughness coefficients for natural channels with
constant roughness characteristics along a full river reach. However in any one reach these
characteristics may vary considerably.
Sellin (1960) also presented the similar work at the University of Bristol. This paper investigates
in a 6.1 meters long, 0.457 meters wide flume with symmetrical flood plains constructed from
fiber glass. The consequential bank full depth was 0.0445 m. Preliminary tests indicated
depressions on the water surface caused by large scale vortices transporting momentum from
regions of high velocity to regions of slower flow on the flood plain. Based on the experiments
it is attempted to quantify the extent of these vortices by using two similar photographic
techniques. Thus the Schlieren principle can be used as a method of photographing the
depressions on the water surface, generated by the vortices mechanism.
Spitsin (1962) explained about the behaviour of a trapezoidal channel with a channel bed width
of 1.66 meters during overbank flow. To compare the flow in the main channel under interacting
and isolated conditions, a glass wall at the channel/flood plain junction is inserted. Also he was
able to calculate the energy existing in the channel and flood plain under isolated and interacting
conditions.
The original Soil Conservation Service (SCS) (1963) method is also used for selecting
roughness coefficient values for meandering channels. It is consisted of an empirically-based
model which incorporates the extra flow resistance resulting from the influence of a channel
sinuosity by adjusting the roughness coefficients that are used in the standard resistance
formulae.
Page | 14
CHAPTER 2
Sellin (1964) discusses about the existence of vertical vortices at the junction using a flow
visualization technique. He also explained that momentum is exchanged between the main
channel and the flood plain through these vortices.
Cruff (1965) studied the use of the Preston tube technique as well as the Karman - Prandtl
logarithmic velocity-law to estimate the boundary shear stress resulting from uniform flow in a
rectangular channel. A Preston tube is traversed around the boundary of a rectangular channel
and an estimation of the boundary shear stress distribution obtained. From considerations of the
longitudinal force equilibrium Equation, an apparent shear force, which is essentially an "out of
balance" force, could be calculated to act on any vertical plane in the flow. Although he did not
measure boundary shear stresses in a channel with overbank flow, his work recognized a method
to enable investigators to calculate the apparent shear stress and hence momentum transfer
between a channel and its flood plain. Also Wright and Carstens used the Preston tube technique
to measure boundary shear stresses in a closed conduit aerodynamic model 6 meters long.
Posey (1967) shows the problems associated with the use of the hydraulic radius in estimating
discharges in rivers with overbank flow. The hydraulic radius is distinct as the cross-sectional
area of a channel divided by its wetted perimeter. As a flood inundates the flood plain, there is a
sudden increase in the wetted perimeter with only a small increase in the total channel crosssectional area. It shows that at just above bank full level, the hydraulic radius, as commonly
calculated, is suddenly reduced and if conventional relationships are used to estimate discharge
(such as Chezy’s or Manning) then the predicted discharge will also be reduced, since the
discharge is a function of the hydraulic radius. Since the actual discharge is not reduced, better
methods of estimating the discharge in compound channels are required. He suggested four
possible methods which might be used in situations described above: (i) consider the whole
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CHAPTER 2
cross-sectional area of the compound channel and divide it by the total wetted perimeter. (ii)
Divide the channel and flood plains by imaginary walls at the channel/flood plain junction and
compute the discharge for each section including the vertical imaginary walls for the calculation
of the hydraulic radius for each section. The disadvantage of this method is that no allowance is
made for the turbulent shear interaction and momentum transfer which occurs across each
division line. Neglecting of this channel/flood plain interaction will lead to overestimation of
discharges at low flood plain depths. (iii) Method (iii) is similar to method two except that the
imaginary walls are exceeded in the calculations of the hydraulic radius for each section. (iv)
Method (iv) is perhaps the most complicated approach and involves the introduction of
imaginary walls inclined towards the center of the channel from the channel bank. The hydraulic
radius is then weighted by considering the area of the section it represents, against the total
cross-sectional area. It is concluded that method (ii) was the most accurate method at low flood
plain depths, whereas at greater depths, method (i) became more accurate.
Toebes and Sooky(1967) conducted an experiment from which the roughness, slope and
channel depth on the discharge capacity of a meandering channel was investigated. A sinuosity
of 1.09 was set for all the models which meant that the key parameters of these models were
dissimilar to the key parameters in natural river channels. They observed the insight into general
flow behavior and the dependency of meandering channels on longitudinal slope and channel
aspect ratio.
Townsend (1967) conducted an experimental programmer using a 9-10 meter long Perspex
flume of width 0.61 meters. This paper discussed the measurement of turbulence intensities in
the longitudinal and transverse directions. The results of two tests illustrate a distinct lateral
distortion of the maximum velocity, filament away from the interaction region.
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CHAPTER 2
Limerions’s (1970) discusses an equation for natural alluvial channels. It is only valid for a
particular range of flow volume. The vary from 6 – 430 m3/s, and n/R0.17 ratios up to 300
although it is reported that little change occurs over R > 30.limits of discharges
Muramoto & Endo (1970) presented the experimental work and measured longitudinal velocity
in 1800 curved open channels by means of a propeller current-meter and studied turbulent
intensity, autocorrelation coefficients and energy spectra.
Ghosh and Jena (1972) shows the distribution of boundary shear stress for rough and smooth
walls in a compound channel. The experiment is conducted in an 8*5 meter long flume with a
main channel width of 0.203 meters flanked by two flood plains, each of width 76 mm. Also
they obtained the boundary shear distribution along the wetted perimeter of the total channel for
various depths of flow using the Preston tube technique combined with the Patel calibration. It is
observed that the maximum shear stress on the channel bed occurs approximately midway
between the center line and corner, and the maximum shear in the flood plain always occurs at
the channel/flood plain Junction. Also they prepared no direct reference to the interaction
between &-channel and its flood plain, but results obtained can be used to calculate the extent of
any interaction which was taking place during their tests. From the experimental results of the
shear distribution it is possible to calculate Tc' the average shear stress in the- channel during
interaction. It is observed that by roughening the total periphery of the channel and flood plain
the boundary shear in the channel could be redistributed with the maximum shear in the channel
bed now occurring at the channel center line.
Yen and Overton (1973) suggested the inadequacies of the methods proposed for discharge
estimation in rivers with overbank flow. They projected a method which involved the selection
of division lines across which; the net momentum transfer was zero. These lines would therefore
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CHAPTER 2
be excluded in the calculation of the hydraulic radius since by definition, no shear stress exists
on division lines through which no momentum is transferred.
Myers and Elsawy (1975) studied the effects of the existence of a flood plain on the boundary
shear distribution of a channel. They used the Preston tube technique and Patel’s calibration to
determine the magnitude and distribution of boundary shear in a compound channel. It is seen
that the average shear in the main channel is reduced during channel/flood plain interaction. It is
found that this average shear stress could be reduced by as much as 20% at very low flood plain
depths. At greater flood plain depths the reduction in shear stress decreased.
Leeder & Bridges (1975) give the information about the flow separation in meander bends and
it is expressed as a function of bend tightness r clB and Froude number Fr. This paper shows that
even at modest Froude numbers, say, 0.27-0.42, flow separation is likely to occur. This implies
that many bends in rivers could be expected to induce separation zones.
James and Brown (1977) conducted an extensive study into the nature of the turbulent shear
interaction between a channel and its flood plain. The experiment is carried out in a flume 26.82
meters long, 1-52 meters wide and 0.457 meters deep. Tests were conceded out on asymmetric
and symmetric cross sections of varying channel and flood plain widths. Since the main channel
was trapezoidal in shape, there was a less rapid change in the depth of flow across the channel.
However the investigators did note some interaction between the channel and flood plain and
suggested an empirical adjustment to the Manning resistance Equation applied to the total cross
section.
Bathurst et al. (1979) presented the field measurements for the bed shear stress in a curved river
and it is reported that the distribution of bed shear stress is affected by both the position of the
core of the main velocity and the structure of secondary flows.
Page | 18
CHAPTER 2
Rajaratnam and Ahmadi (1979) conducted experimental work in a channel 18.29 meters long,
1.22 meters wide and 0.9 meters deep. A main channel 0.2032 meters wide, flanked by two flood
plains, each 0.508 meters wide is used to exhibit the Interaction mechanism in a symmetrical
compound channel. Velocity traverses and boundary shear stresses were recorded. Analysis of
velocity profiles revealed that the lateral velocity profiles at different depths in the main channel
exhibited similarity.
Crory (1980) studied extensive tests in a flume described in Section 2-3i2. The analysis consist a
model of an asymmetric compound channel, i. e. a channel flanked by only one flood plain. By
inserting a moveable Perspex wall, it is able to test 4 different main channel widths and use was
made of a laser Doppler anemometry system to give instantaneous point velocities and
turbulence levels throughout the channel/flood plain cross-section. The point velocity
measurements are integrated over the whole cross-sectional area giving a mean total channel
discharge within 0.7% of the measured discharge, thus demonstrating the usefulness of the laser
system. He also plotted isovel contours of the cross-sections and found that the maximum
velocity filament in the channel was depressed below the water surface and away from the centre
line of the channel, towards the non-interacting side of the main channel. It is observed at low
flood plain depths, the maximum velocity filament occurred at the channel/flood plain.
Donald W. Knight, et. al. (1983) conducted experiments on flood plain and main channel flow
interaction.The discharge characteristics, boundary shear stress and boundary shear force
distributions in a compound section comprising of one rectangular main channel and two
symmetrically disposed flood plains are obtained from experimental results. Equations are
formed giving the shear force on the flood plains as a percentage of the total shear force in terms
Page | 19
CHAPTER 2
of two dimensionless parameters. The l Shear force results from experiments are used to derive
ancillary equations for the lateral and vertical transfer of momentum within the cross section.
The apparent shear force acting on the vertical interface between one flood plain and the main
channel is shown to increase rapidly for low relative depths and high flood plain widths.
Equations are modeled also giving the proportion of the total flow which occurs in the various
sub areas. The division of flow based on linear proportion of the areas is shown to be inadequate
on account of the interaction between the flood plain and main channel flows.
Chang (1984) conducted an experiment on the meander curvature and other geometric features
of the channel using energy approach. It directly accounts for variations in bend radius along the
length of a channel. The modified Chang (1984) method is based on the assumption that the
channel is widecompared to its depth. This paper shows that it is difficult to apply this method to
natural channels because of their variable configuration. In some of the instances the modified
Chang method will give physically justifiable results however in most of the circumstances the
simple LSCS method will be more appropriate than this method.
Jarrett (1984) studied a model to determine Manning’s n for natural high gradient channels
having stable bed flow without meandering coefficient. This paper shows an mathematical
equation which gives the value of Manning`s n is developed. The equation is meant for natural
channels having stable bed and bank materials (boulders) without bed rock. It is intended for
channel gradients from 0.002 – 0.04 and hydraulic radii from 0.15 – 2.1m.
Booij (1985) presented the experimental work and measured of the various shear stress
components in a mildly curved flume. The analysis considered a 2-component LOA which was
set up a unique configuration of the laser beams to obtain lateral and vertical components. This
paper calculated the eddy viscosity coefficients in three directions: 1'yx -'-I ( a.---uv -8yx -J+ b-
Page | 20
CHAPTER 2
Tl) -8yax.Jp 0/ & 0/ and so on. It is shown that the assumption of isotropic eddy viscosities was
not justified in the curved channel.
Anwar (1986) carried out a field measurement of three velocity components in a river bend
using a two-component electro-magnetic current meter. The analysis considered
(a) The velocity profile in the mean flow direction in the bend did not obey the logarithmic law
(b) The Reynolds shear stress and normal stresses had their maxima near the outer bank.
Arcement and Schneider (1989) studied the Cowan method discussed above. This paper relates
specifically for selection of resistance factor in natural meandering channels. This is mainly
performed for the U.S. Department of Transportation. In the modified equation each variable
values are selected from tables in Arcement and Schneider (1989).The equation is verified with
flow depths from 0.8-1.5 m.
James and Wark (1992) studied the step function defined above with a linear function to avoid
the discontinuity at the certain limits of the defined sinuosity ranges with consequent ambiguity.
To overcome from this difficulty the existing equation was further linearised known as the
Linearised SCS (LSCS) Method [1992] and this method was easy to apply and yields a
significant result.
D. A. Ervine, et.al. (1993) shows the factors affecting conveyance in meandering compound
flows. This paper accumulates together recent data from the large-scale flood channel facility,
Wallingford, England, with other model studies to study the main parameters affecting
conveyance in meandering compound river channels. Parameters measured include sinuosity,
boundary roughness, main channel aspect ratio, width of meander belt, and flow depth above
bank-full level as well as the cross-sectional shape of the main channel. The effect of each
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CHAPTER 2
parameter is quantified through a non-dimensional discharge coefficient F*. Possible scale
effects in modeling such flows are also investigated.
Willets and Hardwick (1993) conducted an experiment to study flow in a small laboratory
flume where meandering channels of different sinusitis and geometry were utilized. It was found
that the conveyance of channel vary with sinuosity. In other words, the flow resistance increases
substantially with an increase in channel sinuosity. The flow interaction responsible for the flow
resistance was also found to be dependent on channel cross section geometries.
Genadii A. Atanov, et.al. (1999) conducted an experimental work to measure estimation of
roughness profile in trapezoidal open channels. In de Saint Venant equations, it is difficult to
measure the bed roughness-coefficient directly and therefore needs to be estimated. The
estimation process is referred to as ‘‘parameter identification,’’ which is a mathematical process
depending on the difference between the solution of the model equations and the measured
system response. This paper shows an approach for solving the parameter identification problem
in the de Saint Venant equations. The method proposed herein is widely used in gas dynamics;
but it is not used before for unsteady problem identification of open channel flow parameters.
The assumed solution is also used for other parameters, e.g., cross-sectional area, bed width, etc.
Starting with an initial guess of the roughness coefficient, the algorithm iteratively improves the
guesses in the direction of the gradient of the least square criterion. The gradient is obtained by
means of a variation approach, while the conditions of the criterion minimum are identified by
the general method of indefinite Lagrangian multiplier.
Maria and Silva (1999) studied friction factor of meandering channel by conducting
experiments in two meandering channels of different sinuosity and expressed that it is a function
Page | 22
CHAPTER 2
of sinuosity and position. This expression was found to compute vertically averaged flows that
were in agreement with the flow pictures measured for both large and small values of sinuosity.
Shiono, et.al. (1999) studied the effect of bed slope and sinuosity on discharge estimation of a
meandering channel. Conveyance capacity of a meandering channel was derived using
dimensional analysis and consequently helped in finding the stage-discharge relationship for
meandering channels. Ta paper shows that discharge increased with an increase in bed slope and
decreased with increase in sinuosity for the same channel.
Kahnu C. Patra and Srijiv k. kar (2000) studied on the flow interaction of meandering river
with floodplains. A series of laboratory test results are conducted about the boundary shear
stress, shear force, and discharge characteristics of compound meandering river sections
composed of a rectangular main channel and one or two floodplains disposed of to its sides. Five
dimensionless parameters are used to form equations representing the total shear force
percentage carried by floodplains. A set of smooth and rough sections is studied with an aspect
ratio varying from 2 to 5. Apparent shear forces on the assumed vertical, diagonal, and horizontal
Interfacial plains are found to be different from zero at low depths of flow and change sign with
an increase in depth over the floodplain. Here a variable-inclined interface is proposed for which
apparent shear force is calculated as zero. This paper shows equations related to proportion of
discharge carried by the main channel and floodplain. The equations concur well with
experimental and river discharge data. Using the variable-inclined interface, the error between
the measured and calculated discharges for the meandering compound sections is found to be the
minimum when compared with that using other interfaces.
Patra, Kar and Bhattacharya (2004) shows that the flow and velocity distributions in
meandering channels are strongly governed by flow interaction. By taking adequate care of the
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CHAPTER 2
interaction affect, they proposed equations that are found to be in good agreement with natural
rivers and also the experimental meandering channel data obtained from a series of symmetrical
and unsymmetrical test channels with smooth and rough sections.
Patra and Khatua (2006) observed that Manning’s roughness coefficient not only denotes the
roughness characteristics of a channel but also the energy loss in the flow.
Jana (2007) assumed the stage-discharge relationship in meandering channels of low sinuosity
using dimensional analysis. For meandering channel Manning’s roughness co-efficient is
described as a function of aspect ratio, sinuosity and bed slope.
Khatua (2008) studied that distribution of energy is an important aspect needs addressed
adequately. It is resulted from the variation of the resistance factors Manning’s n, Chezy’s C, and
Darcy –Weisbach’s f with flow depths. He found out Stage-discharge relationship ranging from
in-bank to the over-bank flow, channel resistance coefficients for meandering channel. It is
stated that Flow distribution becomes more complicated due to interaction mechanism as well as
with sinuosity.
Lai Sat Hin et. al. (2008) measured the estimation of discharge capacity in river channels with
variations in geometry and boundary roughness. It is shown that the findings afield study
including the stage-discharge relationships and surface roughness in term of the Darcy-Weisbach
friction factor for several frequently flooded equatorial natural rivers. The resulted friction factor
was found to increase rapidly for low flow depth.
Paarlberg et. al. (2008) presented the experimental work to determine the channel roughness of
rivers by predicting dune dimensions during a flood wave by using an idealized mathematical
model. It was a new physically-based method and the result of model predicts the shape of a
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CHAPTER 2
flood wave which influences dune dynamics and thus roughness coefficient development in
hydraulic models which can serve as starting point for future model calibrations.
Pinaki (2010) also analysed a series of laboratory tests for smooth and rigid meandering
channels and developed mathematical equation using dimension analysis to evaluate roughness
coefficients of smooth meandering channels of less width ratio and sinuosity.
Khatua and Patra (2012) conducted a series of laboratory tests for smooth and rigid
meandering channels and developed mathematical equation using dimension analysis to evaluate
roughness coefficients.The important variables considered in affecting the stage-discharge
relationship were velocity, hydraulic radius, viscosity, gravitational acceleration, bed slope,
sinuosity, and aspect ratio.
Moharana (2012) also presented the effect of geometry and sinuosity on the roughness of a
meandering channel. Using a large data set applied a soft computing technique (ANFIS) to
predict the roughness of a meandering channel.
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CHAPTER 3
EXPERIMENTAL AND
METHODOLOGY
CHAPTER 3
3.1
OVERVIEW
Normally, experimental work based on river is somewhat difficult to present in a laboratory in
practical point of view. For this cause a designed model of a river in a laboratory is required.
Rivers are generally meandering in nature but maintaining different sinuosity related to flow of
water is difficult. To predict Stage-Discharge relationships in straight channel is easier than
meandering channel. For a meandering channel a generalised model is required to be formulated
to estimate the stage discharge relationships in meandering rivers. Many researchers have
developed a number of models which partially satisfied the condition of meandering channels.
So no proper model has been developed so far to predict stage-discharge relationship of a
meandering channel taking all important parameters into account. Our present study is related to
development of a model which predicts a suitable model which gives more accurate stage
discharge prediction for meandering channel than previous. In our National Institute of
Technology Rourkela, by the help of DST, there was a R&D project titled “Sinuosity
dependency in stage-discharge and boundary shear distribution modeling for meandering
compound channels”, here different meandering channels for different sinuosity river based
model have been created for study of flow variables due to change in sinuosity and flow
conditions. According to the requirement of the project the construction of the channel is
maintained as follows.
3.2
DESIGN AND CONSTRUCTION OF CHANNEL
For carrying out study in meandering channels, experimental setup was built in Fluid mechanics
and Hydraulics Laboratory of NIT, Rourkela. Here the main channel is constructed with Perspex
sheet of 6mm/10mm thickness. The main channel is formed in trapezoidal shape having bottom
width 0.33m, depth 0.065m and side slope 1:1 was built inside a steel tilting flume. The total
Page | 26
CHAPTER 3
length of the flume is 15m and the flumes can be tilted by hydraulic jack arrangement. (Fig
shows the overall view of the channel.) The main channel is a sine generated curve of one and
half wave length (wavelength, λ=2.162m) and is preceded and followed by a straight portion
jointed with a transitional curved portion in order to have proper flow field developed in the test
reach which is at the second bend apex of the central curve (Fig 3.1 shows Meandering channel
of cross over angle 400, Fig.3.2 shows the schematic diagram of experimental setup and fig 3.3
represents the dimensions of channel with test section respectively). By the help of centrifugal
pump (15Hp) the required amount of water is supplied to the flume from an underground sump
via an overhead tank. This water is re circulated through the downstream volumetric tank fitted
with closure valves for calibration purpose. Water entered the channel through bell mouth
section via an upstream rectangular notch specifically built to measure discharge in such a wide
laboratory channel. An adjustable vertical gate along with flow straighteners was constructed in
upstream section. It is provided to reduce turbulence and velocity of approach in the flow near
the notch section. At the downstream end another adjustable tail gate was also provided to
control the flow depth and maintain a uniform flow in the channel. A movable bridge (approx.
1.2m width and 4m long) was provided across the flume in both axes over the channel area so
that each location on the plan of meandering channel could be accessed for taking measurements.
The broad parameters of this channel such as aspect ratio of main channel (δ), width ratio (α)
were kept constant for all different sinuous channels. In all the experimental channels, the flow
has been maintained uniform i.e. the water surface is parallel to bed of channel. This simplified
approach has been tried to achieve which is also in line with the experimental work of Shino AlRomaih and Knight (1999). This stage of flow is considered at normal depth, which can carry a
particular flow only under steady and uniform conditions.
Page | 27
CHAPTER 3
Fig.3.1. Designed plan of the experimental meandering channel (Sr=4.11)
Fig.3.2. Schematic diagram of Experimental meandering channels with setup
Fig.3.3 Longitudinal & Cross sectional dimension of the meandering channel
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CHAPTER 3
3.3
APPARATUS & EQUIPMENTS USED:-
In this present study Measuring devices like a pointer gauge having least count of 0.1 mm,
rectangular notch, five micro-Pitot tubes each of them having 4.6 mm external diameter and five
manometers were used in the experiments. These are used to measure velocity and its direction
of flow in the channels. In the experiments structures like baffle walls, stilling chamber, head
gate, travelling bridge, sump, tail gate, volumetric tank, overhead tank arrangement, water
supply devices, two parallel pumps etc. are used. The measuring equipment and the devices were
arranged properly to carry out experiments in the channels.
i. Pointer Gauge
iii. Micro Pitot tubes
ii. Rectangular Notch
iv. Series of Manometers
Fig.3.4 (i to iv) Apparatus used in experimentation in the meandering channel
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CHAPTER 3
3.4
EXPERIMENTAL PROCEDURE
All the observations are recorded at the central bend apex of the meandering channel. Point
velocities were measured along verticals spread across the main channel so as to cover the width
of entire cross section. Also at a number of horizontal layers in each vertical, point velocities
were measured. Measurements were thus taken from left edge point to the right edge of the main
channel bed and side slope. The lateral spacing of grid points over which measurements were
taken was kept 4cm inside the main channel and also Pitot tube is moved from the bottom of the
channel to upwards by 0.4H, 0.6H, 0.8H (H=total depth of flow of water)(Fig.3.5 shows the grid
diagram used for experiments). Velocity measurements are taken by Pitot static tube (outside
diameter 4.77mm) and two piezometers fitted inside a transparent fiber block fixed to a wooden
board and hung vertically at the edge of flume. The ends of which were open to atmosphere at
one end and the other end connected to total pressure hole and static hole of Pitot tube by long
transparent PVC tubes. Before taking the readings the Pitot tube along with the long tubes
measuring about 5m were to be properly immersed in water and caution was exercised for
complete expulsion of any air bubble present inside the Pitot tube or the PVC tube. Even the
presence of a small air bubble inside the static limb or total pressure limb could give erroneous
readings in piezometers used for recording the pressure. Steady uniform discharge was
maintained in each run of the experiment and altogether five runs were conducted.
Fig.3.5 Typical grid showing the arrangement of velocity measurement points along horizontal
and vertical direction at the test section for meandering channel.
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CHAPTER 3
3.5
EXPERIMENTAL CHANNELS:
The meandering channel is constructed of 330 mm wide at bottom, 460 mm at top having full
depth of 65 mm, and side slopes of 1:1. The channel has wavelength L = 2162 mm and double
amplitude 2A’ = 1555 mm. Sinuosity of the channel is maintained 4.11.For better information
the details of geometrical parameters for both the experimental channels are tabulated below and
also fig.3.6 shows details overview of meandering channels.
Fig. 3.6 meandering channel inside the flume with measuring equipment
Table 3.1: Details of Geometrical parameters of the experimental runs
3.6
Sl
No
Item description
Present Experimental
Channels
1
2
3
Channel Type
Flume size
Geometry of Main channel section
Meandering 1
4.0m×15m×0.5m long
Trapezoidal (side slope 1:1)
4
5
6
7
8
9
10
Nature of surface of bed
Channel width
Bank full depth of channel
Bed Slope of the channel
Sinuosity
Amplitude
Wave length
smooth and rigid bed
33cm at bottom and 46 cm at top
6.5cm
0.00165
4.11
1555 mm
2162 mm
MEASUREMENT OF BED SLOPE
Measuring the bed slope of the flume, there are several methods exists which are used according
to the practical conditions and researcher’s interest. Here in our present study we measured the
bed slope through water level piezometric tube. So first of all we took the water level with
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CHAPTER 3
reference to the bed of the channel at the upstream side and then downstream side of the
meandering channel which is 15m apart. Here the level is taken from the bottom of the bed
excluding the Perspex sheet thickness. After taking the level at the two points, the difference in
the corresponding level was measured. The bed slope of the channel is calculated by dividing
this with the length of the channel (15m). For more accuracy this procedure was continuing for
three times and the average was taken as the bed slope of 0.00165.
3.7
CALIBRATION OF NOTCH
For getting accurate and continuous discharge estimations of the channels, a rectangular notch is
fitted at the upstream side of the flume. For calculating discharge more accurately notch
calibration is required. For this, area of volumetric tank located at the downstream of the channel
was measured properly. The height of water in the volumetric tank was recorded from the
corresponding measuring scale connected to it. A constant time or a constant height of the water
was maintained to record the increase the level of the water in the measuring scale or the time
elapsed. Time variation depends on the rate of flow from the channel. Finally change in the mean
water level in the tank over the time interval was recorded. By getting the volume of the water
discharged and the corresponding time elapsed, the actual discharge was calculated for each run
of the channel. The height of water flowing above the rectangular notch is measured by point
gauge attached to the notch. Then form calibration the coefficient of discharge ‘Cd’ for each run
is calculated as for the equation given below.
Vw  A  h
(3.1)
Qa 
Vw
t
(3.2)
Qth 
2
32
L 2g H n
3
(3.3)
Page | 32
CHAPTER 3
Cd 
Qa
Q th
(3.4)
Where, Qa is the actual discharge, Qth is theoretical discharge, A is the area of volumetric tank, Vw
volume of water, t time in sec, Cd is the coefficient of discharge calculated from notch
calibration, h is the height of water in the volumetric tank, L is the length of the notch, Hn is the
height of water above the notch and g is the acceleration due to gravity.
From the notch calibration, coefficient of discharge ‘Cd’ of rectangular notch was found to be
0.66.
3.8
MEASUREMENT OF DEPTH OF FLOW AND DISCHARGE
Depth of flow for all the series of experimental are measured by pointer gauge above the bed of
the channel. The point gauge with least count of 0.1 mm was fitted to the movable bridge and
operated manually. The construction of rectangular notch is provided at the upstream side for
measuring the discharge in the channel. A volumetric tank located at the downstream side of
channel to receive the incoming water flow through the channels. The discharge ‘Qactual’ for each
run is calculated as for the equation given below.
Qactual  Cd
2
23
L 2g H n
3
(3.5)
Where, Qactual is the actual discharge, Cd is the coefficient of discharge calculated from notch
calibration, L is the length of the notch, Hn is the height of water above the notch and g is the
acceleration due to gravity
3.9
MEASUREMENT OF LONGITUDINAL VELOCITY
Generally in a meandering channel high energy loss occurs at the bend apex. So it is required to
measure the Velocity at this place due to minimum curvature effect and to study the flow
parameters covering half of the meander wave length. In the present study, by using Pitot tube
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CHAPTER 3
and manometer pressure head readings were taken. From these data corresponding velocities
were calculated. Normally Pitot tube was placed at the apex bend in the direction of flow and
then allowed to move along a plane parallel to the bed and until unless a relatively maximum
head difference obtained in manometer. The total head h reading by the Pitot tube at the
predefined points of the flow-grid in the channel is used to measure the magnitude of point
velocity vector as v=2gh, where g is the acceleration due to gravity. Here the tube coefficient is
taken as unit and the error due to turbulence considered negligible while measuring velocity.
Each experimental runs of the channel are carried out by maintaining the water surface slope
parallel to the valley slope to achieve the steady and uniform flow conditions.
Page | 34
CHAPTER 4
EXPERIMENTAL
RESULTS
CHAPTER 4
4.1
OVERVIEW
In chapter 3 the experimental procedures has been described with the outlines are given for
the experimental procedure carried out on the series of the tests. This chapter will now present
the results of these tests in terms of the local velocity distributions and stage-discharge
relationships and also the laboratory measurements are presented concerning the channel
geometry, the stage discharge data, and the calibration of bed roughness, velocity and flow
measurements.
It can be appreciated that before the effects of the meandering channel can be quantified,
an understanding regarding the hydraulic behavior, of the flow must be obtained when
the flow is confined to the channel alone. Therefore it is necessary to establish a proper
equation to evaluate the friction factors for the channels. A series of such tests were carried
out and their description and results are already presented in the previous chapters. Further
analysis and discussion of the results, new model development and comparisons with the work of
other researchers in the field are left until the following observations and comments on them are
presented in this chapter. The stage-discharge curves of the meandering channel, as well as
velocity contours for different flow depths are presented and discussed.
4.2
STAGE-DISCHARGE RESULTS:-
One of the most important relationships for a River Engineer is the stage-discharge
relationship, which is essential for design and flood management purposes. From examining
the stage-discharge data, shown in figure 4.1, it is possible to compare the efficiency of the
meandering channel. In order to investigate the influence of the momentum transport
mechanism on the discharges of meandering channel flow for different Froude numbers,
stage-discharge relationships were found. The first procedure is to obtain uniform flow for a
particular discharge. In order to achieve uniform flow, the tailgate/s was/were adjusted to give
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CHAPTER 4
several MI and M2 profiles. The mean water surface slopes and related depths were then
plotted versus tailgate position in a computer program and the tailgate setting which gave a
mean water surface slope equal to the floodplain bed slope of 1.650x 10-3 was interpolated
from the graphs .The depth related to this tailgate setting was then accepted as the normal depth.
This procedure was repeated for every single experiment in order to obtain accurate stagedischarge
for
the
meandering
channels.
Stage in cm
relationships
Discharge in cm3/s
Fig 4.1 Stage-discharge curve for meandering Channel
In the present work it was difficult to succeed the steady and uniform flow condition in
meandering channels due to the many effect such as curvature and the influence of a number of
geometrical and hydraulic parameters. However, it is tried to achieve the water surface slope
parallel to the valley slope so as to get an overall steady and uniform flow in the experimental
channels. In all the experimental runs this easy methodology has been tried to achieve which is
also in line with the experimental work of Shino, Al-Romaih and Knight (1999), Khatua (2008).
This stage of flow is considered as normal depth, which can carry a particular flow only steady
and uniform condition. The stage discharge curves plotted for meandering channels of sinuosity
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CHAPTER 4
4.11 are shown in Fig. 4.1. From the figures it is seen that the discharge increases with an
increase in stage in meandering channels.
4.3
DISTRIBUTION OF LONGITUDINAL VELOCITY RESULTS :-
In order to determine point velocities and to obtain the discharge, a device commonly used is the
Pitot static tube. A Pitot tube is a device which when used in conjunction with a manometer,
determines the difference between the total and static pressures, (i.e. the dynamic pressure), at
any point in a moving fluid. The dynamic pressure is related to the velocity via:
√
(4.1)
Where, V is the flow velocity
is the dynamic pressure,
g is gravitational acceleration
C is a non-dimensional constant
The coefficient C is dependent upon the degree of imperfection in the Pitot tube, and on its
method of use. The exact dependency of such factors is outlined in BS1042, part 2A(1983), and
as a consequence is not reiterated.The Pitot tube can also be used to evaluate the discharge in the
channel provided sufficient velocity measurements have been made. The number of
measurements of local velocity depends upon the nature of velocity distribution across the
channel, i.e. more measurements are needed for complex flow patterns. Averaging each velocity
over a small area of the channel, allows the evaluation of discharge to be obtained via numerical
integration. In order to check the discharges, the cross-section was divided into a grid. The mean
velocity of a sub area formed by the grid was assumed to be equal to the average of the four
point velocities at the corresponding nodes. Due to the curvature of the wall near the channel
boundaries, it was necessary to approximate some of the sub areas formed by the grid.
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CHAPTER 4
The Pitot tube used throughout this series of experiments had an outside diameter of 4.6mm, and
was connected to water manometer which was adjustable to the horizontal. In order to precisely
position the Pitot tube in the flow, it was placed in a transverse carriage, which allowed vertical,
horizontal and rotational motion.
Longitudinal velocity is noted by Pitot-tube in the experimentation. In these channels the
subsequent observations are carried out at the bend apex which is normal to flow direction. The
detailed velocity distribution is carried out for meandering channels of sinuosity 4.11
respectively. The radial distribution of longitudinal velocity in contour form for the runs of the
meandering channels is shown below. It gives the point form velocity measurement at different
points is shown below. The contours of velocity distribution are shown in Fig. 4.3(i to v).
Fig.4.2 Grid Points of longitudinal velocity measurement
(i)
(ii)
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CHAPTER 4
(iii)
(iv)
(V)
Fig.4.3 (i to v) -Longitudinal velocity contours for simple meandering channels
There were some points noted down from the counters of longitudinal velocity distribution in
meandering channel cross-sections at location of bend apex. These are given below
A. Longitudinal velocity distribution in the form of contours points that are skewed with
curvature. Contours with more velocity are getting gradually increasingly at the inner bank to
outer bank at the bend apex of meandering channel.
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CHAPTER 4
B. It is observed that in all cases of depth, the maximum velocity occurs at the inner wall in the
bend entrance where the radius of curvature is the minimum and negative pressure gradient
occurs from outer bank.
C. It was clearly proved that distribution of longitudinal velocity affected significantly by
sinuosity of meandering channel. The results of meandering channel with sinuosity 4.11
shows irregular longitudinal velocity distribution. Similar reports are also seen for deep
channels of Das (1984) and Khatua (2008) the distribution of longitudinal velocity as erratic.
4.4
VARIATION
OF
ROUGHNESS
COEFFICIENTS
IN
MEANDERING
CHANNELS
A major area of uncertainty in river channel analysis is the accuracy in predicting the discharge
carrying capability of river. The discharge calculation for channel is based mainly on refined one
dimensional analysis using the conventional Manning’s, Chezy’s or Darcy-Weischbach equation
already given in upper.
Sellin et al. (1993), Pang (1998), and Willetts and Hardwick (1993) reported that the Manning’s
roughness coefficient not only denotes the characteristics of channel roughness but also
influences the energy loss of the flow. For sinuous channels the values of n become large
indicating that the energy loss is more for such channels. The experimental results for Manning’s
n with depth of flow for simple meander channels are plotted. The plot indicates that the value of
n increases as the flow depth increases. An increase in the value of n can be mainly due to the
increase in resistance to flow for wider channel with shallow depth consuming more energy than
narrower and deep channel. It can also be seen from figure is that steeper channels consume
more energy than the flatter channels. The detailed calculation of roughness coefficients are
tabulated in Table.4.1.
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CHAPTER 4
Table.4.1 Details of Hydraulic parameters of the experimental runs.
Depth
of
flow
Manning`s
Length
(cm)
Discharge
3
(cm )
Area
2
(cm )
Perimeter
Slope
(cm)
Roughness
Darcy
Chezy`s
Weisbac
Reynold
Froude’
C
h Friction
s no Re
s no Fr
n
(cm)
Factor f
1.33
339
1486.9777
64.57143
34.8809
0.00165
0.014081
41.7941
0.04492
5483.18
0.5235
2.43
339
2128.8676
121.7565
36.4365
0.00165
0.014317
23.6187
0.1406
6251.53
0.5172
2.5
339
2197.1703
125.5114
36.5355
0.00165
0.014406
23.3224
0.1442
8054.92
0.4981
3.53
339
3462.3575
182.3641
37.9921
0.00165
0.014511
21.3987
0.1713
8230.69
0.4845
3.55
339
3542.5646
183.4977
38.0204
0.00165
0.014624
21.6999
0.1666
9490.98
0.4775
4
339
4205.8081
209.3036
38.6568
0.00165
0.014706
21.3243
0.1725
10205.6
0.4718
4.55
339
5361.4314
241.6219
39.4346
0.00165
0.014707
22.1357
0.1601
11688.4
0.4634
4.9
339
5830.2022
262.6336
39.9296
0.00165
0.014708
21.3739
0.1717
12441.9
0.4591
5
339
6214.5362
268.7005
40.0710
0.00165
0.01471
22.0546
0.1613
13182.5
0.4546
5.4
339
7210.4218
293.2513
40.6367
0.00165
0.014728
22.6015
0.1536
14936.6
0.4456
5.6
339
7622.3854
305.6964
40.9195
0.00165
0.014713
22.5267
0.1546
15607.1
0.4310
a.Variation
of Manning’s n with Depth of Flow
The experimental results for Manning’s n with depth of flow for the present highly meandering
channels investigated and plotted in Fig 5.1. Manning’s n is found to decrease with increase of
aspect ratio (ratio of width of the channel to the depth of flow) indicating that highly meander
channel consumes more energy as the depth of flow increases. So, with increase in aspect ratio
Manning’s n decreases. Manning’s n also varies with aspect ratio for different depth.
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CHAPTER 4
Fig.4.4 Variation of Manning’s n with Depth of Flow
b.Variation of Chezy`s C with Depth of Flow
The variation of Chezy’s C with depth of flow for the highly meandering channels investigated
for different depth of flows is shown in Fig 5.2. It can be seen from the figure that the
meandering channels, exhibits a steady decrease in the value of C with depth of flow. Chezy’s C
is found to be constant at higher depth of flow. So it can be suggested that the Chezy’s formula
can be applicable to predict stage-discharge relationship more correctly as compared to other
formulas mainly for highly meandering channels at higher depths of flow only.
Fig.4.5 Variation of Chezy’s C with Depth of Flow
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CHAPTER 4
c.Variation of Darcy-Weisbach f with Depth of Flow
The variations of friction factor f with depth of flow for the present meandering channels are
shown in Fig. 5.3. The behavioral trend of friction factor f is also increasing with flow depth.
From the Figs. 5.1 it is seen that the roughness coefficients n and f are behaving in similar
manner because, the relationship between the coefficients with hydraulic radius (R) can be
expressed as f 
8g
R
1
n2 .
3
Fig 4.6 Variation of Friction factor f with Depth of Flow
Here it can be suggested that the Darcy-weisbach formula can also be applicable to predict stagedischarge relationship more correctly as compared to Manning’s n formulas mainly for highly
meandering channels at higher depths of flow only. The behavior may change for other slope
conditions. Now it is required for further processing to see the behavioral trends of the roughness
coefficients of a highly meandering channel due to higher and lower slopes.
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CHAPTER 4
d.Variation of Manning’s n with Aspect Ratio
The variation of roughness co-efficient in terms of Manning’s n are plotted with aspect ratio in
Fig 5.4. It is seen that, when depth of flow increase the Manning’s n also increased, the flow
resistance are found to be increased. Further it is seen that for aspect ratio value up to 8 to 10,
increases sharply and Manning’s n also increases, after that Manning’s n value is found to
remain constant (in between 8 to 6). It indicates that for higher depth of flow for Manning’s n
value tends to be constant. For the higher depth of flow which is inter mingling of the flow with
the fluid particles for this turbulence is more as compared to in lower depth of flow the
turbulence in flow decreases
Manning's n
Manning's n vs aspect ratio
Aspect Ratio
Fig 4.7 Variation of Manning’s n with Aspect Ratio
e.Variation of Manning’s n with Reynolds’s no
The variation of roughness co-efficient in terms of Manning’s n are plotted with the variation of
Reynolds’s number in Fig 5.5. It is seen that, when Reynolds’s number is decreases Manning’s n
also decreases; it means Manning’s n is directly proportional to Reynolds’s number. When
Manning’s n increase Reynolds’s number is also increases. Because in higher sinuosity loss of
energy is low for this when Manning’s n increase Reynolds’s number are also increases by
Saine, et.al. (2013).
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CHAPTER 4
n
Manning's n vs Reynold's no
Re
Fig 4.8 Variation of Manning’s n with Reynolds’s Number
f.Variation of Manning’s n with Froude’s no
The variation of roughness co-efficient in terms of Manning’s n are plotted with the variation of
Froude’s number in Fig 5.6. In this case, we tried to find the effect of gravity on evaluation of
resistance of a meandering channel. It is seen than Manning’s n decreases with Froude’s no. This
may be due to that Froude’s number is directly proportional to mean velocity and at the same
time Manning’s n is inversely proportional to mean velocity. Due to this reason Manning’s n
decreases with successive increment of Froude’s number.
Manning's n vs Froude's no
0.015
0.0145
n
0.014
0.0135
0.013
0.0125
0.012
0
0.1
0.2
0.3
Fr
0.4
0.5
0.6
Fig 4.9 Variation of Manning’s n with Froude’s Number
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CHAPTER 4
Experiments are carried out to examine the effect of channel sinuosity, and cross section
geometry and flow depths on the prediction of roughness coefficients in a highly meandering
channel (Sr = 4.11). Based on analysis and discussions of the experimental investigations certain
conclusions can be drawn.

The flow resistance in terms of Manning’s n, Chezy’s C and Darcy-Weisbach friction
factors f changes with flow depth for a meandering channel. The resistance coefficient not only
denotes the roughness characteristics of a channel but also the energy loss of the flow. The
assumption of an average value of flow resistance coefficient in terms of Manning’s n for all
depths of flow results in significant errors in discharge estimation.

The Manning’s n, Chezy’s C and Darcy-Weisbach friction factors f are found to very
significantly for low aspect ratio. The variation is less for higher depth of flow.

The variation of Chezy’s C and Darcy-Weisbach friction factors f are found to be less as
compared to Manning’s n for the present highly sinuous meandering channel.

The variation of Manning’s n is found to be mostly dependent upon the non-dimensional
parameters such as (i) geometric parameters like aspect ratio, slope and sinuosity (ii) flow
parameters like Reynolds no, Froude’s no.

Manning’s n is found to increase with aspect ratio for the lower depth of flow but at
higher depth of flow Manning’s n tends to be constant.

It increases with Reynolds no when Manning’s n increases because in higher sinuosity
loss of energy is low. Manning’s n decreases with Froude’s no. This may be due to that Froude’s
number is directly proportional to mean velocity and at the same time Manning’s n is inversely
proportional to mean velocity.
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CHAPTER 6
CONCLUSION
CHAPTER 5
I.
THERORETICAL STUDY
5.1
SUMMARY
In the previous chapter the experimental results for meandering channel having stage discharge,
velocity contour, and roughness variation are described. To formulate a generalized model to
predict stage discharge relationships in a meandering channel, the data of series of meandering
channels of same geometry but with different sinuosities are used for the present study. The other
meandering channels which are now added for the study consist of a meandering channels with
different sinuous are 1.00, 1.11, 1.41, 2.03
respectively and are similar to the present
meandering channel of different sinuosity of 4.11 but same dimension of meandering channels
i.e. width =330mm and depth =65mm. The details of the analysis of the other four channels can
be found from Mohanty et.al 2012, 2013, Patnaik et.al 2012 and Mohanty et.al 2012. This
chapter focuses on modeling the stage discharge distributions in meandering channels, since
these features in many engineering studies involving conveyance, sediment transport, bank
erosion, habitats and geomorphology.
5.2
Problem statement
Manning’s roughness coefficient, Chezy’s coefficients and Darcy;s coefficients are mainly
depending upon different independent dimensionless geometric, hydraulic and surface
parameters. Many previous researchers have presented models for evaluations of roughness
calculation only taking some limited dimensionless parameters and found to fit mainly for
straight channels. But for meandering channels the equations do not satisfy. If we consider all
dimensionless parameters, then in previous formulation given by many researchers cannot be
used in meandering channel for velocity and hence the discharge calculations. On behalf of this
we need to modified or generate a new mathematical model for discharge calculation for
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CHAPTER 5
meandering channel. In meandering channel there are many types of factors are present but in
every cases it is impossible to take all dimensionless parameters for analysis. Mainly some
dependent and also independent dimensionless parameters are present. Previous analysis for
stage discharge many models are modeled but they did not considered every dependent and
independent dimensionless parameters for analysis in meandering channels. In these cases
crucial conditions occurs for this we need a mathematical model for calculating stage discharge
in meandering channels. The stage-discharge relationship is of particular importance in flood
alleviation schemes and is often extrapolated when dealing with extreme flood events. The
stage discharge calculation for meandering channel is typically more difficult to model than the
velocity. Due to this many authors decide to ignore it altogether or gave little attention to it,
despite its vital role in river hydraulics. From the literature study and experimental analysis it is
found that velocity varies with some influencing dimensionless parameters. Such as
 Sinuosity
 Aspect ratio
 Froude number
 Reynolds's number
 Slope
Sinuosity:Geoscientists use the sinuosity ratio to determine whether a channel is straight or meandering.
The sinuosity ratio is the distance between two points on the stream measured along the channel
divided by the straight line distance between the two points. If the sinuosity ratio is 1.05 or
greater the channel is considered to be a meandering one.
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CHAPTER 5
Aspect ratio
The aspect ratio of a channel describes the proportional relation between its width and its height.
Froude number
It is a dimensionless number defined as the ratio of inertia force to gravity force.
(5.1)
√
Reynolds's number
It is a dimensionless number defined as the ratio of inertia force to viscous force.
(5.2)
Slope
It is the Sin of the angel of longitudinal axis at the surface of the channel with horizontal.
5.3
Background
From the knowledge of literatures there are some mathematical formulations given by different
investigators for evaluation of roughness coefficient in terms of n, friction factor f and Chezy’s
Care described below.
Cowan (1959) developed a procedure for estimating the effects of some factors to determine the
value of n for a channel. This equation provides much more flexibility and accuracy than can be
achieved using Chow (1959) in isolation. The value of n may be computed by
n  nb  n1  n2  n3  n4 m
(5.3)
Where, nb is the base value of n for a uniform, smooth and straight channel in natural materials,
n1is the correction factor for the effect of surface irregularities, n2 is the value for variations in
shape and size of the channel cross section, n3is the value for obstructions, n4is the value for
vegetation and flow conditions, m is the a correction factor for meandering of the channel.
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CHAPTER 5
The original Soil Conservation Service (SCS) (1963) method is also used for selecting
roughness coefficient values for meandering channels as shown below.
n
f
 ( )1 / 2  1.0 For s<1.2
n
f
(5.4a)
n
f
 ( )1 / 2  1.15 For 1.2=< s< 1.5
n
f
(5.4b)
n
f
 ( )1 / 2  1.3 For s >=1.5
n
f
(5.4c)
Toebes and Sooky (1967) carried out extensive series of experiments in laboratory flumes from
which the influence of channel roughness, slope and channel depth on the discharge capacity of a
meandering channel was investigated. To determine the overall conveyance of meandering
channels as a function of stage is given by
f
 1.0  6.89 R For s=1.1
f
(5.5)
Where, R = Hydraulic radius in meters.
Limerions’s (1970) proposed an equation which is meant for calculation of roughness
coefficient in terms of Manning’s in natural alluvial channels. This equation was developed for
discharges from 6 – 430 m3/s, and n/R0.17 ratios up to 300 although it is reported that little
change occurs over R > 30.It is given by
n
0.0926 R 0.17
1.16  2 log( R / d 84 )
(5.6)
Where, R is the hydraulic radius and d84 the size of the intermediate particles of diameter that
equals or exceeds that of 84% of the streambed particles, with both variables in feet.
Jarrett (1984) developed a model to determine Manning’s n for natural high gradient channels
having stable bed flow without meandering coefficient. He proposed, value of Manning`s n as
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CHAPTER 5
n
0.32S 0.38
R 0.16
(5.7)
Where, S is the channel gradient, R is the hydraulic radius in meters.
The simplest and most widely used method is the Soil Conservation Service (SCS) method for
selecting roughness coefficient values. The relationship was linearized, known as the Linearized
SCS (LSCS) Method (James1992). He proposed the value of Manning’s n using two cases of
sinuosity (Sr), i.e. LSCS method for meandering channel flows is given by
ns
 0.43s  0.5
n
For s < 1.7
And
(5.8a)
for s > 1.7
(5.8b)
Shino, Al-Romaih and Knight (1999) reported the effect of bed slope and sinuosity on
discharge of meandering channel. Conveyance capacity of a meandering channel was derived
using dimensional analysis is stated as
f 
S  10 
8
1
2
(5.9)
Where, S=Sinuosity of the Channel
Nayak (2010) developed mathematical equations by using dimension analysis to evaluate
roughness coefficients in terms of Manning’s n, friction factor f and Chezy’s C .These
mathematical formulations are given below.
C k
g 0.95 R1.35
S 0.02 0.9 S r
8 1.8 S r S 0.04
0.012  2 R 2.7 g 0.9
(5.10a)
2
f 
(5.10b)
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CHAPTER 5
Where, R is the hydraulic radius,is the viscosity, g is the gravitational acceleration, S is the bed
slope, Sr is the sinuosity, is the aspect ratio , k is a constant of value 0.001.
Khatua and Patra (2012) carried out a series of laboratory tests for smooth and rigid meandering
channels and developed numerical equation using dimension analysis to evaluate Manning’s n.
n
S r 0.72S 0.07m0.29
7kg 0.86 R1.2
(5.11)
Where, R is the hydraulic radius, is the viscosity, g is the gravitational acceleration, S is the
bed slope, Sr is the sinuosity, is the aspect ratio, k is a constant of value 0.001
5.4
DESCRIPTION
As previously describe that we want to produce a mathematical equation, by using which we can
easily calculate the stage discharge for meandering channels. In this present work, an efficient
technique is used for forming a mathematical equation. So mathematics like Statistics Data
Editor System (SPSS) is proposed for prediction of roughness coefficients in terms of Manning’s
n, Darcy-Weisbach f and Chezy’s C.
5.4.1 Regression Analysis
The term “regression” originates from the 14th century, where it had a biological mean in gas
“the act of going back”. It was first adapted to a more general statistical context by the wellknown statisticians Udny Yule and Karl Pearson. However, its first statistical form was
published by Legendre (1805) and by Gauss (1809) in the field of astronomy, where they applied
the method of least squares to the problem of determining orbits of bodies about the Sun. Since
then, regression analysis has been widely applied to the study of biology, behavioral and social
sciences and more recently in finance, industry and many other practical aspects of real life.
Regression is a generic term for all methods attempting to fit a model to observed data in order to
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CHAPTER 5
quantify the relationship between two groups of variables. The fitted model may then be used
either to merely describe the relationship between the two groups of variables, or to predict new
values. The first widely studied form of regression analysis has been linear regression, due to the
simplicity of the model and the statistical properties of the estimators. Linear regression is
usually used for the purpose of hypothesis testing or for the purpose of prediction and
forecasting. Many statistical methods and techniques have emerged from its study and one of
them is the simultaneous confidence band. This chapter provides a general review of linear
regression and presents some preliminary results necessary for the construction and comparison
of simultaneous confidence bands throughout the thesis.
5.4.2 Linear Regression Analysis
A common problem in experimental science is to observe how some sets of variables affect
others. Some relations are deterministic and easy to interpret, others are too complicated to
understanding or describe in simple terms, that possibly having a random component. In the
present study, by using relatively simple empirical methods we estimated these actual
relationships by simple functions or random processes. Among all the methods used for
calculating such complex relationships, linear regression possibly is the most useful. In this
methodology, it is necessary to assume a functional, parametric relationship between the
variables in question, and also unknown parameters which are to be estimated from the available
data. Two sets of variables can be well-known at this stage such as Predictor variables and
response variables. Predictors variables are those that can either be set to controlled or else take
values that can be observed without any error. Our objective is to discover that how changes
occurs in the predictor variables that affect the values of the response variables. Extra names
frequently attached to these variables in different books by different authors are the following:
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CHAPTER 5
Predictor variables / Input variables/X-variables/Regressors/Independent variables and
Response variable / Output variable/ Y-variable/Dependent variable.
Here some relation was give below,
Response variable = Linear model function in terms of input variables
+ Random error
(5.12)
In the simplest case when we have data (y1; x1), (y2; x2),..., (yn; xn), the linear function of the
form
yi = β0 + β1xi + ϵi ; i = 1; 2; …. ; n
(5.13)
can be used to relate y to x. We will also write down this model in generic terms as
y = β 0 + β1 x + ϵ
(5.14)
Here, ϵ is a calculating error of any individual function.
Linear regression analysis means it is a statistical technique used to model data comprising of a
dependent random variable and one or more independent variables, so as to evaluate the
relationship between the dependent variable and the independent variables. In statistics, simple
linear regression is the least squares estimator of a linear regression model with a single helpful
variable. In other words, simple linear regression fits a straight line through the set of n points in
such a way that makes the sum of squared residuals of the model (that is, vertical distances
between the points of the data set and the fitted line) as small as possible. Specifically, the
dependent variable y is expressed as a function of the independent variables x 1,x2, . . . , xm, the
corresponding parameters β0, β1, . . . , βk and an error term e as in
y = β0 + β1x1 + β2x2 + . . . + βmxm + e.
(5.15)
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CHAPTER 5
The error term e is a random variable that represents the unexplained variation in the dependent
variable y. If a sample of n observations are available with the ith observation given by (yi, xi1,
xi2, . . . , xim) for i = 1, . . . , n, the ith observation is assumed to satisfy the relationship
yi = β0 + β1xi1 + β2xi2 + . . . + βmxim + ei
(5.16)
Where b0, b1, . . . , bk are the same for all observations. The linear regression model can also be
represented in the matrix form
Y = Xβ + e
Where
(5.17)
Y=
β=
X=
( )
(
)
(
e=
)
( )
The matrix X is called the design matrix as its components can be suitably chosen via design.
Moreover, the linear regression model is subject to the following assumptions:
• The errors follow a normal distribution with the mean zero and constant variance σ2> 0 and
they are independent.
• The independent variables x1, . . . , xm are error-free and the design matrix X has full column
rank k + 1.
5.4.3 Regression Calculations

Least Squares
In this section we will deal with datasets which are correlated and in which one variable, x, is
classed as an independent variable and the other variable, y, and is called a dependent variable as
the value of y depends on x.
We saw that correlation implies a linear relationship. Well a line is described by the equation
y = a +bx
(5.18)
Page | 55
CHAPTER 5
Here b = the slope of the line and a = the intercept.
Here we use the principal of least square and we draw a line through the data set so that sum of
square deviations of all points from the line is minimized.
Using some calculation, we can find the equation of this least square line:
Y=β0+β1x

(5.19)
The Simple Linear Regression Model
If there is a linear connection between x and y in the population the model will be as below. We
find that for a particular value of x, when an observation of y is made we get:
Y=β0+β1x+ϵ
(5.20)
Where ϵ is a random error which measures how far above or below the true regression line that
the actual observation of y lies. The mean of ϵ is zero. And this model is called probabilistic
model. It contains 3 unknown parameters,
β0 =the intercept of the line
β1= the slope of the line
And σ2 is the variance of ϵ
We need to estimate these parameters using data in our sample. In fact we have already seen the
sample statistics that we will use to estimate parameters. Remember σ2 measures how spread out
the points are from the true regression line.
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CHAPTER 5
According to our Linear Regression Model most of the variation in y is caused by its relationship
with x. Except in the case where all the points lie exactly on a straight line (i.e. where r = +1 or r
= -1) the model does not explain all the variation in y. The amount that is left unexplained by the
model is SSE. Suppose that the variation in y was not caused by a relationship between x and y,
then the best estimate for y would be y the sample mean. And the Sum of Squared Deviations of
the actual y’s from this prediction ȳ would be
SSyy=∑
(5.21)
If little or none of the variation in y is explained by the contribution of x then SSyy will be almost
equal to SSE (sum of squared errors). If all of the variation in y is explained by its relationship
with x then SSE will be zero.
The coefficient of determination is
(5.22)
This represents the proportion of the total sample variability in y i.e. explained by a linear
relationship between x & y.
R-Squared measures how well the model fits the data. Values of R-Squared close to 1fit well.
Values of R-Squared close to 0 fit badly.
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CHAPTER 5

Assumptions behind linear regression
The assumptions that must be met for linear regression to be valid depend on the purposes for
which it will be used. Any application of linear regression makes two assumptions:
(A) The data used in fitting the model are representative of the population.
(B) The true underlying relationship between X and Y is linear. All you need to assume to
predict Y from X are (A) and (B). To estimate the standard error of the prediction Sy, you
Also must assume that:
(C) The variance of the residuals is constant (homoscedastic, not heteroscedastic).
For linear regression to provide the best linear unbiased estimator of the true Y, (A) through (C)
must be true, and you must also assume that:
(D) The residuals must be independent.
To make probabilistic statements, such as hypothesis tests involving b or r, or to construct
confidence intervals, (A) through (D) must be true, and you must also assume that:
(E) The residuals are normally distributed.
Contrary to common mythology, linear regression does not assume anything about the
distributions of either X or Y; it only makes assumptions about the distribution of the residuals.
As with many other statistical techniques, it is not necessary for the data themselves to be
normally distributed, only for the errors (residuals) to be normally distributed. And this is only
required for the statistical significance tests (and other probabilistic statements) to be valid;
regression can be applied for many other purposes even if the errors are non-normally
distributed.
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CHAPTER 5
5.5
Source of data and Selection of Parameters
5.5.1 Source of data
The currently carried out experimental data set and also along with an extensive literature related
to analysis of meandering channels are studied. The regular data set was collected from
numerous references such as: Channel of Willets and Hardwick (Willets and Hardwick(1993));
University of Bradford (Shino, Al-Romaih and Knight (1999)); University of Glasgow,
(MacLeod AB. (1997)); United Kingdom (Sellin et. al (1964)) River Main (Myers (1957), N.I.T
Rourkela( Khatua (2007)), N.I.T Rourkela Experiment (Mohanty (2012)) , N.I.T Rourkela (
Patnaik & Mohanty(2013)); etc. are prepared in Table 5.1.
5.5.2 Selection of hydraulic, geometric and surface parameters
From the experimental results and extensive literature study for meandering channels, it is seen
that the investigators such as Acrement and Schneider (1989), Shino, Al-Romaih and Knight
(1999), Khatua et.al (2012) proposed models to predict roughness coefficients and justify the
dependency of flow variable on different hydraulic, geometrical and surface parameters. In the
present study we considered taking these parameters into our consideration that influencing nondimensional parameters such as bed slope of the channel (S), sinuosity (Sr), aspect ratio (α),
Reynolds Number (Re), Froude’s number (Fr), and side slope. These parameters are taken as
input for the model and roughness coefficient is taken as the output. Sinuosity (Sr) is the ratio of
channel length to valley length, refers to sinuous path of a channel. Aspect ratio (α) is the ratio of
channel bottom width (b) to flow depth (h) in the channel. To include the effect of roughness of
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CHAPTER 5
rough channels, the Manning’s n of a rough bed needs to be compared with a reference bed
surface.
Table 5.1:-Details of Hydraulic parameter for all types of data collected from Globe
Verified
test
channels
Sl
no
Present
Very High
MC NIT rkl
data
High MC
NIT rkl data
Mildly MC
NIT rkl data
Low MC NIT
rkl data
Straight
Channel
NIT rkl data
Glasgow
MC
Abreeden
MC
Shino&Knig
ht MC
Longitud
inal
slope(s)
Main
channel
width
(b) m
Main
channel
depth
(h) m
Sinuosity
(Sr)
1
0.00165
0.33
0.065
4.11
2
0.00165
0.33
0.065
2.03
3
0.0011
0.33
0.065
1.3
4
0.0011
0.33
0.065
1.12
5
0.0011
0.33
0.065
1
6
0.001
0.2
0.075
1.374
7
0.001
0.2
0.0525
1.374
0.152
0.052
1.372
0.9
0.15
8
0.00050.002
0.000996
0.001021
Observed
discharge
(Q)
range in
m3/sec
Flow
Aspect
Ratio
0.0021288
-0.007622
24.725.89
0.0120.0147
533913805
0.420.52
0.0013090.006508
0.0025000.008780
0.0005120.012894
19.416.6
14.045.29
23.25.32
0.0110.0139
0.00830.0123
0.00820.017
346517202
630717202
138525515
0.520.57
0.440.64
0.20.73
0.0033860.015950
11-5.07
0.00720.0092
816231040
0.60.83
0.01070.0133
0.01020.0134
0.0050.021
1079338017
26609691
27009710
0.250.36
0.350.5
0.170.9
Manning's
n
Reynolds
no
Froude
no
:
0.00050.0007
0.00040.0023
0.00060.03
3.82-8.6
1.3752.043
0.0230.067
6.114.09
0.012-0.06
2026154021
0.150.59
2.4-9.7
12-2.6
Sellin MC
9
Willets &
Hardwick
MC
10
0.001
0.139
0.05
1.2-2.06
0.00040.0015
6.712.85
0.0140.049
230-5509
0.10.34
Khatua
11
0.00190.0053
0.12
0.0800.120
1-1.91
0.0001920.007050
11-1.05
0-.0080.018
140018000
0.40.82
5.6
MODEL DEVELOPMENT
5.6.1 Introduction
The stage-discharge curve is defined by a relationship between the discharge Q and the water
depth H. According to Schmidt & Yen (2001), stage-discharge relationships for rivers are
traditionally based on empirical power law equations fitted to measure the discharge and the
corresponding stage. These measurements do not always form a unique relationship, either
because of scatter in the measured data that depends on the site conditions and river geometry, or
because of unsteady flow effects. For this data sets are collected and compared with each other
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CHAPTER 5
by linear regression analysis. So in the present work for regression analysis (Linear Regression
System) IBMSPSS software is used. In this analysis the variables are divided into two categories
such as dependent and independent variables. By considering these variables a mathematical
formulation is formulated. At the beginning all the variables are analysed through the
experiments with keeping different sinuosity. But the variation of variables in comparison to
Manning’s n and the relation between them are analysed through the IBMSPSS. From the
analysis part a mathematical equation is formulated here.
To represent the stage-discharge relationship, a number of empirical equations are already
formulated. But these empirical equations have some limitations about the range of data over
which they were obtained. This can lead to inaccuracies if they are extrapolated, as is often
required in flood prediction. Therefore a model is required for meandering channel which should
consider the physical phenomena and the variable geometric and roughness parameters. Only
then can a stage-discharge curve be 'safely' extrapolated. This chapter explores the application of
the IBMSPSS to meandering channel, not only to test the calibration philosophy developed in
Chapter 6, but also to test the general modeling approach to more complex channels shapes and
where the roughness is more likely to be heterogeneous than homogeneous. Data from different
channels such as, Channel of Willets and Hardwick (Willets and Hardwick(1993)); University of
Bradford (Shino, Al-Romaih and Knight (1999)); University of Glasgow, (MacLeod AB.
(1997)); United Kingdom (Sellin et. al (1964)) River Main (Myers (1957), N.I.T Rourkela(
Khatua (2007)), N.I.T Rourkela Experiment (Mohanty (2012)) , N.I.T Rourkela ( Patnaik &
Mohanty (2013)) are used for these purposes.
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CHAPTER 5
5.6.2 Modelling of the meandering channel at NIT Rourkela
One of the major factors affecting the stage-discharge curve is the geometry; hence particular
care needs to be taken in predicting the bankfull level in the simplified or analytical crosssection. These could have been made to be different without affecting the results by using
Regression analysis of solution using the IBMSPSS.
From the literature study, it is seen that the roughness coefficient of a meandering channel varies
from channel to channel and flow depth to flow depths. They are found to be function of
geometric parameter, flow parameter and surface parameters. All are non-dimensionalised.

Geometric parameter:
(a) Aspect ratio
(b) Slope
(c) Sinuosity,

Flow parameters:
(a) Reynolds no
(b) Froude's no.
The dependency of roughness and the best functional relationships have been found out from
different plots of the global data sets exit in the literatures. So it is in the following form
n = f (δ, S0, Sr, Re, Fr)
(5.23)
The variation of roughness co-efficient has been found out for five meandering channels of
different sinuosity. The variation of roughness co-efficient in terms of Manning’s n are plotted
for different aspect ratio in Fig (5.1). It is seen that, as sinuosity increased, Manning’s n also
increased for the low sinuosity channel and straight channel (channels- 1 &2) the Manning’s n is
found to be decreased with flow depth but for meandering channel of higher sinuosity but the
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CHAPTER 5
flow resistance are found to be increase in aspect ratio. Further it is seen that for high sinuosity
channel, Manning’s n is found to remain constant. This may be due to the reason that at lower
depth the meandering channel exhibits the higher energy loss due to bend affect but in higher
depth of flow, the effect of bend loss diminishes. Next, the mean velocities of the meandering
channels are calculated for each depth of flow.
Then the Reynolds no vs. Manning’s n are plotted for all the channels and presented in Fig (5.2).
From the Fig (5.2) it is seen that Manning’s n decreases with Reynolds no for lower sinuosity
channel but for higher sinuous channel Manning’s increase with Reynolds number. Because for
straight channel and low sinuous channel the loss of energy is less for higher depth of flow but
higher sinuous channel, Manning’s n increase with Reynolds number. The reason of the results
may be considered as similar to the results of figure 1 i.e. Manning’s vs. Aspect ratio.
In the third case, we tried to find the effect of gravity on evaluation of resistance of a meandering
channel. Therefore, Manning’s, n values are plotted with different Froude’s no. Here in this fig
Froude’s no vs. Manning’s n are plotted for all the flow channels. From Fig (5.3) it is seen than
Manning’s n decreases with Froude’s no for both lower sinuous channel as well as higher
sinuous channel. This may be due to that Froude’s number is directly proportional to mean
velocity and at the same time Manning’s n is inversely proportional to mean velocity. Due to this
reason Manning’s n decreases with successive increment of Froude’s number.
Similar to the previous cases, the variation of Manning’s n is tested for channels of different
sinuosity. Therefore in Fig (5.4) the relationships between Manning are n and sinuosity for
different aspect ratios is plotted. In this case, it is clearly seen that when sinuosity increases,
Manning’s n substantially increases for a constant aspect ratio. The reason may be that velocity
of the flow gradually decreased with increased sinuosity. It can be stated that, increase of
Page | 63
CHAPTER 5
sinuosity is found to have direct effects to de-crease the values of Manning’s n, for a constant
geometry of a meandering channel. Finally we have tested the variation of Manning’s n with
longitudinal slope in Fig (5.5). It is a well-known fact that the conveyance is mainly affected by
longitudinal slope. Attempt has been made now to see the variation of roughness coefficient with
respect to the longitudinal slope. Due to absence of different slope data, in the present work data
of Shino (1999) has been analysed with our experimental data. From experimental data, it was
clearly noticed that when slope increases the gravity component increase, so when force increase
which subsequently reduces the roughness coefficient, therefore the Manning’s n found to bed
decrease. But in higher slope means greater than 0.001 from Fig (5.5), it is clearly show that
roughness value in-creases. This reason may be that the higher value of slope, the formation of
turbulence, eddies, starts producing more loss of energy, so increasing the value of Manning’s n.
Manning's n ~ Aspect Ratio
channel 1
channel 2
channel 3
channel 4
channel 5
n
Fig 5.1 Manning’s n vs. Aspect Ratio
n = 0.0161δ-0.116
R² = 0.9944
Aspect Ratio
Page | 64
CHAPTER 5
Manning's n vs Reynold's no
channel 1
channel 2
channel 3
channel 4
channel 5
n
Fig 5.2 Manning’s n vs. Reynolds’s no
n = 0.0053Re0.0924
R² = 0.9729
Re
Manning's n ~ Froude's number
0.016
channel 5
channel 1
0.014
channel 2
0.012
channel 3
channel 4
0.01
Fig 5.3 Manning’s n vs. Froude’s no
n
0.008
n = -0.003ln(Fr) + 0.0067
R² = 0.9787
0.006
0.004
0.002
0
0
0.2
0.4
0.6
0.8
1
Fr
Manning's n ~ Longotudinal Slope
0.02
0.018
0.016
0.014
0.012
0.01
n
Fig 5.4 Manning’s n vs. longitudinal Slope
aspect ratio 7.2
0.008
aspect ratio 3.25
n = 4600So2 - 12.3So + 0.023
R² = 1
0.006
0.004
aspect ratio 4
aspect ratio 4.5
aspect ratio 5.5
0.002
aspect ratio 6.5
0
0
0.0005
0.001
So
0.0015
0.002
0.0025
Page | 65
CHAPTER 5
Manning's n ~ Sinuosity
n
Fig 5.5 Manning’s n vs. Sinuosity
aspect ratio 6.5
n = 0.0041ln(Sr) + 0.0089
R² = 0.9734
aspect ratio 7
aspect ratio 8.25
aspect ratio 9
aspect ratio 13.75
Sr
Fig 5-(1 to 5) Comparison of Manning’s n with different subsequent flow parameters.
By analysed the above plots, corresponding functional relationships of n with different nondimensional geometric and hydraulic parameters are,
𝑛
𝑛
𝐴 𝛿
𝑛
𝑓 𝑆𝑜
𝑛
𝐴 𝑆𝑜
+𝑐
(5.24a)
𝐵𝑆 + 𝐶
(5.24b)
𝑛
𝑛
𝐴𝑙𝑛 𝑆
𝐴𝑙𝑛 𝑅
(5.24c)
𝐴
𝑓 𝑅
+𝑐
(5.24d)
𝑛
𝑛
𝑓 𝑆
+𝑐
𝑛
𝑛
𝑓 𝛿
𝑓
(5.24e)
From the above graphs it’s shown that R2 value is very high and varies from 0.97 to 1. By using
above relationships we compile to develop the n value by help of IBMSPSS software with
analysis data sets. Dependent variables are taken in y axis and actual Manning’s n value
Page | 66
CHAPTER 5
considered as independent variable in x axis. By using IBMSPSS software, mathematical model
is now formulated. From the regression analysis an empirical formulation is formulated. These
equations shows the relation between Manning’s n, with Aspect Ratio, Slope, Sinuosity,
Reynolds number, Froude number.
Table 5.2: Unstandardized Coefficient by Linear Regression Analysis
Unstandardized Coefficients
Model
1
B
Std. Error
(Constant)
-.015
.052
AR
-.156
.284
So
.801
3.389
Sr
.978
.130
Re
.364
.307
Fr
.124
.159
R2=0.91
Table 5.2 data represent the result of linear regression analysis. From the above table the
corresponding co-efficient are found and these values are used in the above equation 5.13. After
formulation, a mathematical empirical relation is created and it shows in (5.25) the
corresponding roughness co-efficient. After then by putting the different dependent variables
from the above graph (5.1 to 5.15) then it will give a relation in a modified form which shows in
(5.26) in accurate linear form.
n=-0.015-0.156δ+0.801So+0.978Sr+0.364Re+0.124Fr
(5.25)
After simplify this above equation,
n =-0.015-0.156*(0.0161δ-0.116) +0.801*(4600 So2 - 12.3So + 0.023) +0.978*(0.0041 ln (Sr)
+ 0.0089) +0.364*(0.0053 Re0.0924) + 0.124*(-0.003 ln (Fr) + 0.0067)
n=0.013(1-0.015 δ-0.116+0.3021 ln(Sr)+0.15Re0.0924-0.3 ln(Fr)-9.852So(1-374 So)
(5.26)
n=0.013(1-0.015 δ-0.116+0.3021 ln (Sr) +0.15Re0.0924-0.3 ln (Fr) -9.852 So (1-374 So)
Page | 67
CHAPTER 5
5.7 Validation
After formulation of the new mathematical model, an attempt has been taken to validate with the
data of other investigators and present experimental channel data sets. So in the present study the
necessary validation is carried out with previous known data sets. The results are as follows.
Experimental Datas (2013)
0.018
y = 1.0035x + 0.0003
R² = 0.9515
0.016
0.014
Predicted
0.012
Fig 5.5 Actual vs. Predicted in
Experimentation in Channel (2013)
0.01
0.008
Experimental datas
0.006
0.004
0.002
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Actual
Sellin Datas (1964)
0.012
0.0115
Predicted
Fig 5.6 Actual vs. Predicted in
Sellin (1964)
y = 0.9263x + 0.0009
R² = 0.9249
0.011
Sellin
0.0105
0.01
0.01
0.0105
0.011Actual
0.0115
0.012
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CHAPTER 5
Abreeden Datas (1997)
0.013
0.0125
y = 0.5538x + 0.0047
R² = 0.9331
Predicted
0.012
Fig 5.7 Actual vs. Predicted in
Abreeden (1997)
0.0115
0.011
Abreeden
0.0105
0.01
0.01
0.0105
0.011
0.0115
Actual
0.012
0.0125
0.013
Khatua (2012)
0.0108
0.0107
y = 1.0788x - 0.0008
R² = 0.988
0.0106
Fig 5.8 Actual vs. Predicted in
Khatua (2012)
Predicted
0.0105
0.0104
0.0103
Khatua
0.0102
0.0101
0.01
0.0099
0.0098
0.0098
0.01
0.0102
0.0104
0.0106
0.0108
Actual
Shino Datas (1999)
0.012
y = 0.8539x + 0.0016
R² = 0.9468
Predicted
0.0115
0.011
0.0105
Fig 5.9 Actual vs. Predicted in Shino
(1999)
shino
0.01
0.0095
0.009
0.009
0.0095
0.01
0.0105
Actual
0.011
0.0115
0.012
Fig 5-(6 to 9) Validation occurs in between actual vs. predicted
Page | 69
CHAPTER 5
It is clearly seen that from the graph, it’s plotted between the predicted Roughness coefficient
verses actual Roughness coefficient for meandering channels. In this case, it is found that in Fig
5.5 both predicted vs. actual roughness coefficient agrees perfectly and also gives a satisfactory
result along appropriately with determination/ Roughness coefficient is R2=0.95.
Similarly, the comparison occurs in between the predicted vs. actual has been shown in Fig 5.65.8 for data sets of Sellin (1964), Abreeden (1997), Khatua et.al. (2012), Shino (1999)
respectively the Regression coefficient R2 for all these data sets are found to be 0.92, 0.93, 0.98,
and 0.94.and these all are gives a satisfactory results. Therefore from the above figure it is
clearly shown that the present model gives a better Stage-Discharge result which proves to
adequacy of present developed model.
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CHAPTER 5
CONCLUSIONS
CHAPTER 6
CONCLUSION
Intensive literature survey has been carried out to study the stage-discharge relation in
meandering channels. From the literature, it is found that stage-discharge relation depends upon
the properties prediction of roughness coefficient in terms of Manning’s n. A method has been
proposed method for predicting roughness coefficients of meandering channels. The results from
the method are compared well with the other standard models. For the present experimental
work, flow in meandering channels has been investigated. The following are the results from
present experimentations.

There were some points noted down from the counters of longitudinal velocity
distribution in meandering channel cross-sections at location of bend apex and
longitudinal velocity distribution in the form of contours points that are skewed with
curvature. Contours with more velocity are getting gradually increasingly at the inner
bank to outer bank at the bend apex of meandering channel. It is observed that in all cases
of depth, the maximum velocity occurs at the inner wall in the bend entrance where the
radius of curvature is the minimum and negative pressure gradient occurs from outer
bank.

Manning’s n is found to be depending upon many non-dimensional hydraulic and
geometric parameters. The experimental Investigation has been carried out to found the
dependency of Manning’s n with respect to geometric parameters like aspect ratio, slope
and sinuosity and flow parameters like Reynolds no, Froude’s no.

The flow resistance in terms of Manning’s n, Chezy’s C and Darcy-Weisbach friction
factors f also changes with flow depth for the meandering channels. The resistance
coefficients not only denote the roughness characteristics of the channel but also the
Page | 71
CHAPTER 6
energy loss of the flow. The assumption of an average value of flow resistance coefficient
in terms of Manning’s n for all depths of flow results in significant errors in discharge
estimation.

The Manning’s n, Chezy’s C and Darcy-Weisbach friction factors f are found to very
significantly for low aspect ratio. The variation is less for higher depth of flow.
Manning’s n is found to increase with aspect ratio for the lower depth of flow but at
higher depth of flow Manning’s n tends to be constant. It increases with Reynolds no
when Manning’s n increases because in higher sinuosity loss of energy is low. Manning’s
n decreases with Froude’s no. This may be due to that Froude’s number is directly
proportional to mean velocity and at the same time Manning’s n is inversely proportional
to mean velocity.

Regression analysis has been carried out to formulate a mathematical model to predict
Manning’s n of a meandering channel. It is seen that, the model gives best result not only
to the present experimental data sets but also the data sets of other investigation such as,
Sellin (1964), Abreeden (1997), Khatua (2012) and Shiono (1999).

The accuracy of models has also been studied and the R2=0.91 are found for the presnt
data sets. By using this model in our other experimental data sets also gives a satisfactory
result along appropriately with determination/ Roughness coefficient is R2=0.95.

The method is also applied to the data of other investigators such as Sellin (1964), Abreeden
(1997), Khatua (2007), Shino (1999) respectively and the R2 value are found to be 0.92,
0.93, 0.98, and 0.94.respectively showing the satisfactory results of the present model.
This proves the adequacy of present model.
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CHAPTER 6
6.2
SCOPE FOR FUTURE WORK
The present work leaves a wide scope for future investigators to explore many other aspects of a
meandering channel analysis. The equations developed may be improved by incorporating more
data from channels of different roughness conditions. Further investigation is required to study
the flow properties and develop models to predict the boundary shear, zonal flow distribution,
and energy loss aspects of having different geometry, sinuosity and roughness conditions. The
channels here are rigid. Further investigation for the flow processes may also be carried out for
mobile and of meandering channels of different geometry and sinuosity. Numerical analysis can
also be applied to predict the flow variables of such channels for different geometry, flow and
surface conditions.
ACKNOWLEDGEMENT
The authors wish to acknowledge thankfully the support received by the second author from
Department
of
Science
and
Technology,
Government
of
India,
under
grant
no.SR/S3/MERC/066/2008 for the research project work on compound channels at NIT,
Rourkela.
Page | 73
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