MEANDERING EFFECT FOR EVALUATION OF ROUGHNESS COEFFICIENTS AND BOUNDARY SHEAR

MEANDERING EFFECT FOR EVALUATION OF ROUGHNESS COEFFICIENTS AND BOUNDARY SHEAR
MEANDERING EFFECT FOR EVALUATION OF
ROUGHNESS COEFFICIENTS AND BOUNDARY SHEAR
DISTRIBUTION IN OPEN CHANNEL FLOW
A Thesis Submitted in Partial Fulfillment of the Requirements for the
Degree of
Master of Technology
in
Civil Engineering
PINAKI PRASANNA NAYAK
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
2010
MEANDERING EFFECT FOR EVALUATION OF ROUGHNESS
COEFFICIENTS AND BOUNDARY SHEAR DISTRIBUTION IN
OPEN CHANNEL FLOW
A Thesis Submitted in Partial Fulfillment of the Requirements for the
Degree of
Master of Technology
in
Civil Engineering
Under the guidance and supervision of
Prof. K.C.Patra
&
Prof. K.K.Khatua
Submitted By:
PINAKI PRASANNA NAYAK
ROLL.NO: 607CE003
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
2010
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled “Meandering Effect for Evaluation of Roughness
Coefficients and Boundary Shear Distribution in Open Channel Flow” being submitted
by Pinaki Prasanna Nayak in partial fulfillment of the requirements for the award of Master
of Technology in Civil Engineering at National Institute of Technology Rourkela, is a
bonafide research carried out by her under our guidance and supervision.
The work incorporated in this thesis has not been, to the best of our knowledge, submitted to
any other University or Institute for the award of any degree or diploma.
Prof. K.C. Patra
Prof. K. K. Khatua
(Supervisor)
(Co-Supervisor)
i
ACKNOWLEDGEMENTS
I sincerely express my deep sense of indebtedness and gratitude to Prof. K.C.Patra and
Prof. K.K.Khatua for providing me an opportunity to work under their supervision and
guidance. Their continuous encouragement, invaluable guidance and immense help have
inspired me for the successful completion of the thesis work. I sincerely thank them for
their intellectual support and creative criticism, which led me to generate my own ideas and
made my work interesting as far as possible.
I would also like to express my sincere thanks to Prof. M. Panda, Head, Civil
Engineering Department, Prof. A.K. Pradhan, Prof. P.K.Ray and Prof. R.Jha in providing
me with all sorts of help and paving me with their precious comments and ideas. I am
indebted to all of them.
I am also thankful to staff members and students associated with the Fluid Mechanics
and Hydraulics Laboratory of Civil Engineering Department, especially Mr. P. Rout and
Mr. K. M. Patra for their useful assistance and cooperation during the entire course of the
experimentation and helping me in all possible ways.
Friendly environment and cooperative company made I had from my stay at NIT
Rourkela, memorable and pleasant. The affection received from my batchmates and juniors
will always remind me of my days as a student here. I wish to thank all of them heartily.
Their support and suggestions were indeed of great help whenever needed.
I am thankful to my parents and my in-laws for their emotional support and being
patient during the completion of my dissertation. Last but not the least; I thank the
ALMIGHTY for blessing me and supporting me throughout.
Pinaki Prasanna Nayak
ii
ABSTRACT
Almost all the natural channels meander. During uniform flow in an open channel the
resistance is dependent on a number of flow and geometrical parameters. The usual practice
in one dimensional flow analysis is to select an appropriate value of roughness coefficient
for evaluating the carrying capacity of natural channel. This value of roughness is taken as
uniform for the entire surface and for all depths of flow. The resistance coefficients for
meandering channels are found to vary with flow depth, aspect ratio, slope and sinuosity and
are all linked to the stage-discharge relationships. Although much research has been done on
Manning's n for straight channels, less works are reported concerning the roughness values
for meandering channels. An investigation concerning the variation of roughness
coefficients for meandering channels with slope, sinuosity and geometry are presented. The
loss of energy in terms of Manning’s n, Chezy’s C, and Darcy-Weisbach coefficient f are
evaluated. A simple equation for roughness coefficient based on dimensional analysis is
modeled and tested with the recent experimental data. The method gives discharge results
that are comparable to that of the observed values as well as to other published data.
Knowledge on wall shear stress distribution in open channel flow is required to solve a
variety of engineering and river hydraulic problems so as to understand the mechanism of
sediment transport, and to design stable channels etc. The study on boundary shear and its
distribution also give the basic idea on the resistance relationship. Good many works have
been reported on the distribution of wall shear for straight channels, but only a few studies
are reported on the works involving meandering channels. Experiments are conducted to
evaluate wall shear from point to point along the wetted perimeter of the meandering open
channel flow. The wall shear distributions in meandering channel is found to be dependent
on the dimensionless parameters such as sinuosity, aspect ratio, and channel roughness.
Equations are developed to predict the wall shear distribution in meandering channel which
is adequate for the present data and other published data.
Key words: Aspect ratio(α), Bed slope(S), Boundary shear, Chezy’s C, Darcy-Weisbach
coefficient f, Manning’s n, Dimensional analysis, Meandering channel, Open channel, Flow
resistance, Sinuosity(S r), Stage-discharge relationship, Wall shear distribution.
iii
CONTENTS
CERTIFICATE
i
ACKNOWLEDGEMENTS
ii
ABSTRACT
iii
LIST OF TABLES
vii
LIST OF FIGURES AND PHOTOGRAPHS
viii
LIST OF ACRONYMS AND SYMBOLS
xi
CHAPTER 1: INTRODUCTION
1
1.1 General
1
1.2 Meandering – Its Effects and Influence
2
1.3 Essence of Proper Value of Roughness Coefficient and Boundary
Shear Distribution in Meandering Channel Flow
4
1.4 Scope and Approach of Present Research Work
5
1.5 Organisation of the Thesis
6
CHAPTER 2: LITERATURE REVIEW
8
2.1 General
8
2.2 Previous Work on Roughness Coefficients and
Discharge Estimation
8
2.3 Overview of the Research Work on Boundary Shear
14
2.4 Objectives of the Present Research Work
20
CHAPTER 3: EXPERIMENTAL WORK
22
3.1 General
22
iv
3.2 Experimental Setup
22
3.3 Experimental Procedure
27
3.3.1 Apparatus and Methodology
27
3.3.2 Determination of Channel Slope
29
3.3.3 Measurement of Discharge and Water Surface Elevation
29
3.3.4 Measurement of Velocity and its Direction
33
CHAPTER 4: RESULTS AND DISCUSSIONS
36
4.1 General
36
4.2 Stage-Discharge Curves for Meandering Channels
39
4.3 Variation of Reach Averaged Longitudinal Velocity U With
Depth of Flow
39
4.4 Distribution of Tangential (Longitudinal Velocity)
40
4.4.1 Straight Channel
47
4.4.2 Meandering Channel
47
4.5 Measurement of Boundary Shear Stress
48
4.5.1 Velocity Profile Method
48
4.6 Distribution of Boundary Shear Stress
50
4.6.1 Straight Channel
51
4.6.2 Meandering Channel
51
4.7 Roughness Coefficients in Open Channel Flow
52
4.7.1 Variation of Manning’s n with Depth of Flow
55
4.7.2 Variation of Chezy’s C with Depth of Flow
55
4.7.3 Variation of Darcy-Weisbach Friction Factor f with
Depth of Flow
56
v
4.8 Dimensional Analysis
57
4.8.1 Evaluation Of Roughness Coefficients
4.8.1.1 Derivation of Roughness Coefficients using Dimensional Analysis
57
58
4.9 Discharge Estimation Using the Present Approach
61
4.10 Application of Other Methods to the Present Channel
62
4.11 Boundary Shear Distribution in Meandering Channel
64
4.11.1 Analysis of Boundary Shear in Meandering Channels
CHAPTER 5: CONCLUSIONS AND FURTHER WORK
68
72
5.1 Conclusions
72
5.2 Scope for Future Work
73
References
74
Dissemination of Work
79
Brief Bio-Data
80
vi
LIST OF TABLES
Sl.No.
Details of Tables
Page No.
Table 3.1
Details of Geometrical Parameters of the Experimental Channels
24
Table 3.2
Hydraulics Details of the Experimental Runs
35
Table 4.1(a)
Experimental Runs for Type-I and Type-II Straight Channels
37
Table 4.1(b)
Experimental Runs for Type-III Meandering Channels
37
Table 4.1(c)
Experimental Runs for Type-IV Meandering Channels
38
Table 4.2
Comparison of the Boundary Shear Results for Straight
Channel and Meandering Channels at Bend Apex-AA
50
Table 4.3(a)
Experimental Results for Straight Channels of Type-I and
Type-II Showing n, C and f (Sinuosity=1)
53
Table 4.3(b)
Experimental Results of Meandering Channel of Type-III
Showing n, C and f (Sinuosity=1.44)
53
Table 4.3(c)
Experimental Runs for Meandering Channel of Type-IV
Showing n, C and f (Sinuosity=1.91)
54
Table 4.4(a)
Summary of Experimental Results Showing Overall Shear
Stress in Straight Channels (Type-I and II) (Sinuosity=1)
66
Table 4.4(b)
Summary of Experimental Results Showing Overall Shear
Stress in Meandering Channel of Type-III (Sinuosity=1.44)
66
Table 4.4(c)
Summary of Experimental Results Showing Overall Shear
Stress in Meandering Channel of Type-IV (Sinuosity=1.91)
67
Table 4.5
Results of Experimental Data for Boundary Shear Distribution
in Type-III and Type-IV Meandering Channels
70
Table 4.6
Results of Experimental Data of Kar (1977) and Das (1984)
(from Patra, 1999)
70
vii
LIST OF FIGURES AND PHOTOGRAPHS
Sl.No.
Details of Figures
Page No.
Fig. 1.1
Simple Illustration of Formation of Meandering Channel from
Straight Reach
02
Fig. 1.2
Planform of a Meandering River Showing Meander Wavelength,
Radius of Curvature, Amplitude and Bankfull Channel Width
03
Fig. 1.3
Classification of Channels According to Sinuosity Ratio
04
Fig. 3.1(a)
Plan View of Type-I Channel inside the Experimental Flume
24
Fig. 3.1(b)
Details of Geometrical Parameter of Type- I Channel
25
Fig. 3.2(a)
Plan View of Type-II Channel inside in the Experimental Flume
25
Fig. 3.2(b)
Details of Geometrical Parameter of Type- II Channel
25
Fig. 3.3(a)
Plan View of Type-III Channel inside in the Experimental Flume
26
Fig. 3.3(b)
Details of One Wavelength of Type- III Channel
26
Fig. 3.4(a)
Plan View of Type-IV Channel inside in the Experimental Flume
27
Fig. 3.4(b)
Details of One Wavelength of Type- IV Channel
27
Fig. 3.5(b)
Standard Features of a MicroADV
33
Fig. 3.5(b)
Details of Probe of a MicroADV
33
Fig. 4.1
Cross Section of Channel for Experimentation
36
Fig. 4.2
Stage Discharge Variations in Open Channel Flow
39
Fig. 4.3
Variation of Reach Averaged Longitudinal Velocity U with
Depth of Flow
40
Fig. 4.4 (a)
Location of Bend Apex (A-A) of Type-III Meandering Channel
41
Fig. 4.4 (b)
Location of Bend Apex (A-A) of Type-IV Meandering Channel
41
Figs. 4.5.14.5.4
Contours Showing the Distribution of Tangential Velocity and
Boundary Shear Distribution of Type-II Straight Channel
42-43
viii
Figs. 4.6.14.6.6
Contours Showing the Distribution of Tangential Velocity and
Boundary Shear Distribution of Type-III Meandering
Channel at Bend Apex (Section A-A)
43-45
Figs. 4.7.1 4.7.6
Contours Showing the Distribution of Tangential Velocity and
Boundary Shear Distribution of Type-IV Meandering
Channel at Bend Apex (Section AA)
45-46
Fig. 4.8
Variation of Manning`s n with Flow Depth
55
Fig. 4.9
Variation of Chezy`s C with Flow Depth
56
Fig. 4.10
Variation of Darcy - Weisbach f with Flow Depth
56
Fig. 4.11
Calibration Equation between  gR10 3 S 12

1
ν 2 m 3 

and (URSr να) for
59
Manning`s n
Fig. 4.12
Calibration Equation for between
 gR3S 12 ν 2  and


(URSr να) for
60
Derivation of Chezy`s C and Darcy Weisbach f
Fig. 4.13 (a)
Variation of Observed and Modeled Discharge Using Manning`s n
61
Fig. 4.13 (b)
Variation of Observed and Modeled Discharge Using Chezy`s C
62
Fig. 4.13 (c)
Variation of Observed and Modeled Discharge Using
Darcy Weisbach f
62
Fig. 4.14 (a)
Standard Error of Estimation of Discharge using Proposed
Manning`s Equation and other established Methods
63
Fig. 4.14(b)
Standard Error of Estimation of Discharge using Proposed
Chezy`s Equation other established Methods
63
Fig. 4.14(c)
Standard Error of Estimation of Discharge using Proposed
Darcy Weisbach equation and other established Methods
64
Fig. 4.15 (a)
Variation of Percentage of Wall Shear with Aspect Ratio
68
Fig. 4.15 (b)
Variation of Percentage of Wall Shear with Sinuosity
69
Fig. 4.16
Variation of Observed and Modeled Value of Wall
Shear in Meandering Channel
71
Fig. 4.17
Percentage of Error in Calculating %SFW with
Values of Aspect Ratios
71
ix
Sl.No.
Details of Photographs
Page No.
Photo 3.1
Plan form of Type-II Channel from Down Stream
23
Photo 3.2
Plan form of Type-III Channel from Down Stream
23
Photo 3.3
Plan form of Type-IV Channel from Down Stream
23
Photo 3.4
Mechanism of Re-circulation of Flow of Water
30
Photo 3.4(a)
Two Parallel Pumps for Re-circulation of Flow of Water
30
Photo 3.4(b)
Overhead Tank with a Glass Pipe for Storage of Water
30
Photo 3.4(c)
Stilling Tank with Baffles for Reduction of Turbulence of Water
30
Photo 3.4(d)
Channel Showing Flow of Water from Upstream to Downstream
30
Photo 3.4(e)
Tailgate end of the Channel at the Downstream
30
Photo 3.4(f)
Volumetric Tank with Glass Pipe at Tailgate of the Channel
30
Photo 3.5
Pointer Gauge used for Slope Measurements
31
Photo 3.6
Micro-ADV, Pointer Gauge and Micro- Pitot Tube Fitted to the
31
Traveling Bridge
Photo 3.7(a)
Micro-ADV with the Probes in Water
31
Photo 3.7(b)
The Processor and other Accessories fitted to Micro-ADV
31
Photo 3.8
Micro- Pitot Tube in Conjunction with Inclined Manometer
32
Photo 3.9(a)
Nozzle of the Pitot Tube in water
32
Photo 3.9(b)
A Manometer attached to the Pitot Tube
32
Photo 3.9(c)
Pitot Tube and Pointer Gauge attached to the travelling Gauge
32
Photo 3.10
One Wave Length of Type-III Meandering Channel
32
Photo 3.11
One Wave Length of Type-IV Meandering Channel
32
Photo 3.12
Slope Changing Lever
32
x
LIST OF ACRONYMS AND SYMBOLS
Abbreviations
ADV
Acoustics Doppler Velocity Meter
ENU
East, North and Upward
LSCS
Linear Soil Conservation Services
WBF
Bankfull channel width
WSD
West, South and Down ward
A
Area of channel cross section
b
Bottom width of channel
C
Chezy’s channel coefficient
f
Darcy-Weisbach Friction factor
h
Height of channel
n
Manning’s roughness coefficient
Symbols
%SFB
Percentage of shear force which is carried by the base
%SFW
Percentage of shear force which is carried by the walls
A`
Amplitude of a meander channel
b/h
Width depth ratio or aspect ratio
d`
The thickness of laminar sub-layer
ρ
Density of flowing fluid
fa
Apparent friction factor
fc
Composite friction
g
Acceleration due to gravity
h`
Distance from the channel wall
h`1 and h`2
Heights respectively from the wall
h0
Distance from the channel bottom at which logarithmic
law indicates zero velocity
k
Von Karman’s constant
Ks
The equivalent sand grain roughness of the surface
xi
M
Slope of the semi-log plot of velocity distributions
Nr
Reynolds number ratio
P
Wetted perimeter of the channel section
Q
Discharge
R
Hydraulic radius of the channel cross section
rc
Least centerline radius of curvature
S
Bed Slope of the Channel
SF
Total shear force shear force
Sr
Sinuosity of the Channel
U
Primary longitudinal velocity
u*
The velocity scale in the study of velocity distribution
close to the walls in open channels
u1 and u2
The time averaged velocities
V
The secondary flow velocities
Vx
Longitudinal component of velocity in ADV
Vy
Radial component of velocity in ADV
Vz
Vertical component of velocity in ADV
W
The secondary flow velocities
Dynamic viscosity of water
0
Mean Wall shear stress by Energy Gradient Method
w
Distribution of wall shear stress around wetted perimeter.
α
Αspect ratio
ν
Kinematic viscosity
xii
CHAPTER 1
IN TR OD U C TIO N
INTRODUCTION
1.1. GENERAL
Almost all the natural water resource conduits
are open channel that meander. In fact
straight river reaches longer than 10 to 12 times the channel widths are almost nonexistent in
nature. Reliable estimates of discharge capacity are essential for the design, operation, and
maintenance of open channels, more importantly for the prediction of flood, water level
management, and flood protection measures.
River meandering is a complicated process involving a large number of channel and
flow parameters. Inglis (1947) was probably the first to define meandering and it states
“where however, banks are not tough enough to withstand the excess turbulent energy
developed during floods, the banks erode and the river widens and shoals. In channels with
widely fluctuating discharges and silt charges, there is a tendency for silt to deposit at one
bank and for the river to move to the other bank. This is the origin of meandering…”
Leliavsky (1955) summarizes the concept of river meandering in his book which quotes
“The centrifugal effect, which causes the super elevation may possibly be visualised as the
fundamental principles of the meandering theory, for it represents the main cause of the
helicoidal cross currents which removes the soil from the concave banks, transport the
eroded material across the channel, and deposit it on the convex banks, thus intensifying the
tendency towards meandering. It follows therefore that the slightest accidental irregularity in
channel formation, turning as it does, the stream lines from their straight course may, under
certain circumstances constitute the focal point for the erosion process which leads
ultimately to meander”.
A river is the author of its own geometry. Distribution of flow and velocity in a
meandering river is an important topic in river hydraulics to be investigated from a practical
point of view in relation to the bank protection, navigation, water intakes, and sediment
transport-depositional patterns. This further helps to determine the energy expenditure and
bed shear stress distribution. The distribution of wall shear stress is influenced by many
factors, notably the shape of the cross-section, the longitudinal variation in plan form
geometry, the sediment concentration and the lateral and longitudinal distribution of wall
roughness. Accurate assessment of the discharge capacity of meandering channels are
1
Introduction
therefore essential in suggesting structures for flood control and in designing artificial
waterways etc.
1.2. MEANDERING - ITS EFFECTS AND INFLUENCE
The term meandering derives its meaning from the river known to the ancient Greeks as
Maiandros or Meander, characterised by a very convoluted path along the lower reach. As
such, even in Classical Greece the name of the river had become a common noun meaning
anything convoluted and winding, such as decorative patterns or speech and ideas, as well as
the geomorphological feature. Strabo said: "... its course is so exceedingly winding that
everything winding is called meandering." The Meander River is located in present-day
Turkey, south of Izmir, eastward the ancient Greek town of Miletus, now Turkish Milet. It
flows through a graben in the Menderes Massif, but has a flood plain much wider than the
meander zone in its lower reach. In the Turkish name, the Büyük Menderes River, Menderes
is from "Meander". Meanders are also formed as a result of deposition and erosion.
Fig. 1.1 Simple Illustration of Formation of Meandering Channel from Straight
Reach
Meandering channels are equilibrium features that represent the most probable channel
plan geometry, where single channels deviate from straightness (Fig. 1.1). A meander, in
general, is a bend in a sinuous watercourse. It is formed when the moving water in a river
erodes the outer banks and widens its valley. A stream of any volume of water may assume a
meandering course, alternatively eroding sediments from the outside of a bend and
depositing them on the inside. The result is a snaking pattern as the stream meanders back
and forth across its down-valley axis. The degree of sinuosity required before a channel is
called meandering. As bed width is related to discharge, meander wavelength also is related
to discharge. The quasi-regular alternating bend of stream meanders are described in terms
of their wavelength
m,
their radius of curvature rm, and their amplitude αm and is bankfull
2
Introduction
channel width WBF. The radius of curvature of meander bends is not constant, so rm is
somewhat subjectively defined for the bend apex (Fig. 1.2).
Fig. 1.2 Planform of a Meandering River Showing Meander Wavelength, Radius of
Curvature, Amplitude and Bankfull Channel Width
It is an established fact that meandering represents a degree of adjustment of water and
sediment laden river with its size, shape, and slope. The curvature develops and adjusts itself
to transport the water and sediment load supplied from the watershed. The channel
geometry, side slope, degree of meandering, roughness and other allied parameters are so
adjusted that in course of time the river does the least work in turning, while carrying the
loads. With the exception of straight rivers, for most of the natural rivers, the channel slope
is usually less than the valley slope. The meander pattern represents a degree of channel
adjustment so that a river with flatter slope can exist on a steeper valley slope.
Sinuosity is one of the channel types that a stream may assume overall or part of its
course. All streams are sinuous at some time in their geologic history over some part of their
length. The meander ratio or sinuosity index is a means of quantifying how much a river or
stream meanders (how much its course deviates from the shortest possible path). It is
calculated as the length of the stream divided by the length of the valley. A perfectly straight
river would have a meander ratio of 1 (it would be the same length as its valley), while the
higher this ratio, the more the river meanders. Sinuosity indices are calculated from the map
or from an aerial photograph measured over a distance called the reach, which should be at
least 20 times the average fullbank channel width. The length of the stream is measured by
channel, or thalweg, length over the reach, while the bottom value of the ratio is the
downvalley length or air distance of the stream between two points on it defining the reach.
The sinuosity index plays a part in mathematical descriptions of streams (Fig. 1.3).
3
Introduction
Fig. 1.3 Classification of Channels According to Sinuosity Ratio
Two parameters are used to classify meandering channels into its categories. Rivers
with, is classified as straight when the sinuosity ratio is less than 1.05, and more are
classified as sinuous and meandering. The other parameter defining meandering plan form is
the ratio of the least centerline radius of curvature (rc) to the channel width (b) given as
(rc/b). Meandering channels are also classified as shallow or deep depending on the ratio of
the average channel width (b) to its depth (h). In shallow channels (b/h >5, Rozovskii, 1961)
the wall effects are limited to a small zone near the wall which may be called as "wall zone".
The central portion called "core zone" is free from the wall effects. In deep channels (b/h<5)
the influence of walls are felt throughout the channel width. Meandering and the flow
interaction between main channel and its adjoining floodplains are those natural processes
that have not been fully understood.
1.3. ESSENCE OF PROPER VALUE OF ROUGHNESS COEFFICIENTS
AND BOUNDARY SHEAR DISTRIBUTION IN MEANDERING
CHANNEL FLOW
The usual practice in one dimensional flow analysis is to select an appropriate value of
roughness coefficient for evaluating the actual stage-discharge relationship of natural
channel. The energy loss in a meandering channel is influenced by the channel flow
parameters and are assumed to be lumped into a single value manifested in the form of
resistance coefficients in terms of Manning`s n, Chezy`s C and Darcy Weisbach f. An
improper roughness value can either underestimates or overestimate the discharge. A
number of researchers have studied the phenomenon of flow mostly in straight channels and
4
Introduction
have proposed a number of methods for estimating the discharge. Sellin (1964) and Shino
and Knight (1999), Patra and Kar (2000), Patra and Khatua (2006), etc. have shown that the
structure of the flow is surprisingly complex, even for straight compound channels. It has
velocity in all the three directions. Consequently, the use of design methods based on
straight channels is inappropriate when applied to multiple stage meandering channels,
results in large errors when estimating the discharge. Determination of suitable value of
roughness coefficient for a given channel is therefore the most significant factor for
evaluating the actual carrying capacity of a natural channel.
Knowledge on wall shear stress distribution in open channel flow is required to solve a
variety of engineering and river hydraulics problems so as to give a basic idea on the
resistance relationship, to understand the mechanism of sediment transport, and to design
stable channels etc. The study also helps in the analysis of migration problems as well as
bank erosion in river channel. It is a great challenge to the river engineers and researchers to
predict the distribution of wall shear stress ( w) around the wetted perimeter of a given
channel for a certain flow-rate. By integrating the wall shear stress along the surface,
contribution to lift and drag on immersed objects can also be computed. Good many works
have been reported on the distribution of wall shear for straight channels, but only a few
studies are reported on the works involving meandering channels. The interaction between
the primary longitudinal velocity U, the secondary flow velocities, V and W are responsible
for non-uniform wall shear distribution in an open channel flow. So it is necessary for
researchers and engineers to predict the distribution of wall shear in open channel flow both
for straight and meandering reach in a sediment free condition.
1.4. SCOPE AND APPROACH OF PRESENT RESEARCH WORK
In general, the classical methods of discharge estimations are normally based on flow in a
straight channel. These are essentially flawed when applied to meandering channels. The
study undertaken in the present case is primarily a step in understanding the flow processes
and distribution of boundary shear in an idealised meander channel. It is supported by data
collected from three experimental channels of different geometry, slope, aspect ratio and
sinuosity. Experiments are conducted using fabricated channels at the Fluid Mechanics and
Hydraulics Engineering Laboratory at the National Institute of Technology, Rourkela, on
straight and meandering channels under sediment-free condition. A rigid boundary channel
5
Introduction
is considered to be more appropriate to study the basic nature of flow interaction and flow
parameters than going for movable bed. Most of the discharge calculation for channel is
based mainly on refined one dimensional (1D) method. However, both 2D and 3D
approaches are more complex to use in practice.
An investigation of energy losses resulting from boundary friction, secondary flow,
turbulence, and other factors in meandering channels has been presented. The energy loss is
manifested in the form of variation of resistance coefficients of the channel with depth of
flow. The variations of roughness coefficients with depths of flow are investigated. In the
present study, it is attempted to formulate the models for predicting discharge by selecting a
proper value of roughness coefficient in terms of Manning`s n, Chezy`s C and Darcy
Weisbach f. The results of the present model are compared well with other established
methods of Shino et al. (1999), LSCS method of James and Wark (1992), Toebes and Sooky
(1967) method and the conventional method for discharge estimation. Also, a modified
method has been proposed for evaluation of boundary shear in a meandering channel
considering sinuosity and aspect ratio as basic influencing parameter.
1.5. ORGANISATION OF THE THESIS
The thesis comprises of five chapters. General introduction is given in Chapter 1, literature
review is presented in Chapter 2, experimental work is outlined in Chapter 3, results and
discussions of the present work is Chapter 4 and finally the conclusions are in Chapter 5.
The first chapter provides us a glance about the contents of the research work along
with its related topics. It gives us an overview about the work done and presented in the
dissertation.
The literature review presented in Chapter 2 relates us to the work done by other
established authors in this field from the very beginning till date. The chapter emphasizes on
the research carried out in straight and meandering channels based on stage-discharge
relationships, roughness coefficients and boundary shear distribution.
Chapter 3 explains about the experimental work as a whole. This chapter discusses
about the experimental set up and the adopted procedures of flow parameters, observations
of channel. It also explains about the instruments used during the experimental processes.
6
Introduction
The experimental results concerning boundary shear force, stage-discharge, and energy
loss aspects in terms of Manning`s n, Chezy’s C and Darcy-Weisbach f with the discussion
of the results and theoretical considerations are depicted in Chapter 4.
Finally, Chapter 5 summarizes the important conclusions drawn from the present
work. Scope of future work that can be further carried out is also discussed in this chapter.
References that have been made in the subsequent chapters are given at the end of
the dissertation.
7
CHAPTER 2
L ITE R ATU R E
R E V IE W
LITERATURE REVIEW
2.1. GENERAL
A look into the plan form of all meandering channels show that there can be three distinct
type of geometry. They are straight, curved or meandering. Almost all natural rivers
meander. The name meander, which probably originated from the river Meanders in Turkey
is so frequent in river that it has attracted the interest of investigators from many disciplines.
Thomson (1876) was probably the first to point out the existence of spiral motion in curved
open channel. Since then, a lot of laboratory and theoretical studies are reported, more so, in
the last decade or so concerning the flow in straight or curved channel. It may be worth
while to understand the nature of flow in straight channels before knowing about the
meander channels. There are limited studies available concerning the flow in meandering
channels.
Discharge estimation is based on the most suitable value of roughness coefficient.
Knowledge on wall shear stress distribution in open channel flow is required to solve a
variety of engineering and river hydraulic problems so as to understand the mechanism of
sediment transport, and to design stable channels etc. The study on boundary shear and its
distribution also give the basic idea on the resistance relationship.
2.2. PREVIOUS WORK ON ROUGHNESS COEFFICIENTS AND DISCHARGE
ESTIMATION
The Meandering channel flow is considerably more complex than constant curvature bend
flow. Due to continuous stream wise variation of radius of curvature flow geometry in
meander channel is in the state of either development or decay or both. The following
important studies are reported concerning the flow in meander channels.
Chezy was the first to investigate the flow in open the channel of the new Paris water
supply. Chezy was the first to consider the wetted perimeter of channels as an analog of
boundary resistance (Rouse and Ince, 1957). Chezy’s equation has the form of V = C R1/2
Sc1/2 that uses a resistance coefficient C, the hydraulic radius (R=A/P) and the slope of the
channel (Sc).
8
Literature Review
The U.S. Army Corps of Engineers (Hydraulic 1956) conducted a series of stagedischarge experiments in meandering channels at the Waterways Experiments Station in
Vicksburg. The main purpose of these experiments was to investigate the effect of the
following geometric parameters on the conveyance capacity of compound meandering
channels: radius of curvature of the bends, sinuosity of the main channel, depth of the
floodplain flow, ratio of the floodplain area to the main channel area, and floodplain
roughness. Although the study did not investigate the behaviour of the flow, its conclusions
were of importance and were outlined as follows: (1) The effect of the main channel
sinuosity on the channel conveyance was small when the main channel was narrow in
comparison to the floodplain width; (2) the channel conveyance decreased as the sinuosity of
the main channel increased; (3) the effect of the main channel sinuosity on the total channel
conveyance was small when the floodplain width was more than three times the width of the
meander belt; and (4) as the floodplain roughness increased, the conveyance decreased
accordingly. Sinuosity is defined as the ratio of the length along the channel centre line
between two points to the straight line distance between the points.
Cowan (1956) proposed an equation to evaluate n as
n = (n b+ n1+ n2+ n3+ n 4) m
where, nb = base n value; n1 = addition for surface irregularities; n2 = addition for
variation in channel cross section; n 3 = addition for obstructions; n 4 = addition for
vegetation; m = ratio for meandering. The above method was improved by Arcement and
Schneider (1989). Suggested values for Manning’s n are tabulated by Chow (1959).
The Soil Conservation Service (SCS) method for selecting roughness coefficient values
for channels (Guide 1963) was found to be the best, giving a better stage prediction than the
other methods. It proposed accounting for meander losses by adjusting the basic value of
Manning's n on the basis of sinuosity (s), as follows:
n`
= 1 .0
n
for s < 1.2
n`
= 1 . 15
n
for 1.2
n`
= 1 . 15
n
for s
s < 1.5
1.5
9
Literature Review
where, n` = the adjusted value; and n = the basic value. Because n is proportional to f l/2 the
adjustment should be squared when using the Darcy-Weisbach equation.
Toebes and Sooky (1967) carried out several experiments in small-scale flumes to
determine the overall conveyance of meandering channels as a function of stage for specific
geometries and to study the redistribution of the erosive forces acting on the boundaries
under overbank flow conditions. They concluded that the energy losses, caused partially by
the main channel and floodplain interaction and the sinuosity of the main channel, depend on
both the Reynolds and the Froude numbers. Velocity components were measured for the
different geometries to investigate the secondary flows and the formation of helical currents
in the main channel and floodplain. From experimental results in a small laboratory channel
with a sinuosity of 1.09, they proposed an adjustment to Darcy Weisbach`s f.
Chang (1983) investigated energy expenditure and proposed an analytical model for
obtaining the energy gradient, based on fully developed secondary circulation for wide
rectangular sections. The method was for fully developed secondary circulation. In fact, the
circulation took considerable distance to develop through a bend and begins to decay once
the channel straightens out. For meanders there was also a tendency for the circulation to
reverse between successive bends. The average energy gradient associated with secondary
circulation along the channel was therefore substantially less than predicted assuming full
development. Chang’s approach was subsequently adapted to meandering channels by James
and Wark (1992).
Later, Chang (1984) analysed the meander curvature and other geometric features of the
channel using energy approach. It established the maximum curvature for which the river
did the last work in turning, using the relations for flow continuity, sediment load, resistance
to flow, bank stability and transverse circulation in channel bends. The analysis
demonstrated how uniform utilisation of energy and continuity of sediment load was
maintained through meanders. He accounted for applying his full circulation loss model
together with non-uniform flow calculations and predicted the distribution of losses,
boundary shear stresses, and water levels through bends.
Jarrett (1984) developed a model to determine Manning’s n for natural high gradient
channels having stable bed and in bank flow without meandering coefficient. He proposed, a
10
Literature Review
0 . 38
value for Manning`s n as n = 0 . 32 0S. 16
where, S is the channel gradient, R the hydraulic
R
radius in meters. The equation was developed for natural main channels having stable bed
and bank materials (boulders) without bed rock. It was intended for channel gradients from
0.002 – 0.04 and hydraulic radii from 0.15 – 2.1m, although Jarrett noted that extrapolation
to large flows should not be too much.
Wormleaton and Hadjipanos (1985) measured the velocity in each subdivision of the
channel, and found that even if the errors in the calculation of the overall discharge were
small, the errors in the calculated discharges in the floodplain and main channel may be very
large when treated separately. They also observed that, typically, underestimating the
discharge on the floodplain was compensated by overestimating it for the main channel. The
failure of most subdivision methods is due to the complicated interaction between the main
channel and floodplain flows. The interaction phenomenon has been studied by many
researchers [e.g., Myers (1978), Knight and Demetriou (1983), Knight and Shiono (1990),
and Shiono and Knight (1991)] in an attempt to quantify the effect and hence to predict
correctly the flow in compound channels.
Arcement and Schneider (1989) was basically modified Cowan method discussed
above. It was designed specifically to account for selecting n values for natural channels and
flood plains resistance. This work was performed for the U.S. Department of Transportation.
In the modified equation the coefficients were taken as nb = a base value n for the
floodplain`s natural bare soil; n1 = a correction factor for the effect of surface irregularities
on floodplain (range0-0.02); n2 = a value for variation in channel shape and size of the
floodplain cross section; n3 = a value for obstructions on the floodplain (range 0-0.03); n 4 =
a value for vegetation on the floodplain (range 0.001-0.2); m = a correction factor for
sinuosity of the floodplain=1.0. Each variable values were selected from tables in Arcement
and Schneider (1989). The equation was verified for wooden floodplains with flow depths
from 0.8-1.5 m.
James and Wark’s comprehensive study (1992) compared different methods of
estimating the conveyance in meandering compound channels, and was summarized in Wark
and James (1994). The methods were compared with three sets of laboratory-based data.
The step nature of the SCS recommendation introduces discontinuities at the limits of
the defined sinuosity ranges, with consequent ambiguity and uncertainty. To overcome the
11
Literature Review
problem the relationship was linearised, known as the Linearised SCS (LSCS) Method. He
proposed the value of Manning`s n using two cases of sinuosity (Sr), i.e. Sr <1.7 and Sr
1.7.
Ervine et al. (1993) investigated the main parameters that affect the conveyance of
compound meandering channels with different floodplain roughness using the Flood
Channel Facility (FCF) at HR Wallingford, Oxfordshire, U.K. Three configurations were
examined: (1) A 607 angle to the crossover with a regular trapezoidal main channel cross
section; (2) a 607 angle to the crossover with a natural main channel cross section; and (3) a
1107 angle to the crossover with a natural main channel cross section. It was observed that
the sinuosity has a large effect on both smooth and rough floodplain channels. The effect of
the sinuosity decreases as the depth on the floodplain increases. The magnitude of this
decrease is noticeably higher for channels with roughened floodplains. The aspect ratio of
the main channel was also found to contribute significantly to the interaction between the
floodplain and the main channel and, hence, also to the flow structure mechanisms. Higher
meander belt ratios resulted in higher energy losses.
Greenhill and Sellin (1993) compared different subdivision methods applied to
meandering channels, using FCF stage discharge data from smooth meandering channels
with two sinuosities. It was found that the error in discharge prediction was drastically
reduced when a refined division method was used. The refined method was based on
dividing the channel into three sub-channels: the main river channel, using the main channel
bed slope; the floodplain within the meander belt, using the valley slope; and the floodplain
outside the meander belt, also using the valley slope. The refined method was found to
predict discharges with a very low error margin when checked against the experimental
smooth channel data, provided that the value of the Manning coefficient was accurately
chosen.
Willets and Hardwick (1993) investigated flow in meandering channels using a small
laboratory flume. Four sinuosities were examined with trapezoidal and natural cross sections
for the channel. The conveyance of the overall channel was found to be affected inversely by
the channel sinuosity. In other words, the flow resistance increases substantially with an
increase in the main channel sinuosity. The main channel/floodplain flow interaction
responsible for the flow resistance was also found to be dependent on the main channel cross
section geometries.
12
Literature Review
James (1994) reviewed the various methods for bend loss in meandering channel
proposed by different investigators. He tested the results of the methods using the data of
FCF, trapezoidal channel of Willets, at the University of Aberdeen, and the trapezoidal
channels measured by the US Army Corps of Engineers at the Waterways Experiment
Station, Vicksburg. The results were found considerably different. He proposed some new
methods accounting for additional resistance due to bend by suitable modifications of
previous methods. His modified methods predicted well the stage discharge relationships for
meandering channels.
Pang (1998) conducted experiments on compound channel in straight reaches under
isolated and interacting conditions. It was found that the distribution of discharge between the
main channel and floodplain was in accordance with the flow energy loss, which can be
expressed in the form of flow resistance coefficient. In general, Manning's roughness
coefficient n not only denoted the characteristics of channel roughness, but also influenced
the energy loss in the flow. The value of n with the same surface in the main channel and
floodplain possessed different values when the water depth in the section varied.
Shiono, Al-Romaih, and Knight (1999) reported the effect of bed slope and sinuosity on
discharge of two stage meandering channel. Basing on dimensional analysis, an equation for
the conveyance capacity was derived, which was subsequently used to obtain the stagedischarge relationship for meandering channel with over bank flow. It was found that the
channel discharge increased with an increase in bed slope and it decreased with increase in
sinuosity for the same channel. An error of 10% in discharge estimation was reported for
relative depths exceeding 0.01.
Maria and Silva (1999) expressed the friction factor of rough turbulent meandering
flows as the function of sinuosity and position (which is determined by, among other factors,
the local channel curvature). They validated the expression by the laboratory data for two
meandering channels of different sinuosity. The expression was found to yield the computed
vertically averaged flows that were in agreement with the flow pictures measured for both
large and small values of sinuosity.
Patra (1999), Patra and Kar (2000), Patra and Khatua (2006) and Khatua (2008) have
shown that Manning’s n not only denotes the roughness characteristics of a channel but also
13
Literature Review
the energy loss in the flow. Jana & Panda et.al (2006) have performed dimensional analysis
and predicted the stage-discharge relationship in meandering channels of low sinuosity.
Lai Sat Hin et al. (2008) expressed that estimation of discharge capacity in river
channels was complicated by variations in geometry and boundary roughness. Estimating
flood flows was particularly difficult because of compound cross-sectional geometries and
because of the difficulties of flow gauging. They presented results of a field study including
the stage-discharge relationships and surface roughness in term of the Darcy-Weisbach
friction factor, fa for several frequently flooded equatorial natural rivers. Equations were
presented giving the apparent shear force acting on the vertical interface between the main
channel and floodplain. The resulted apparent friction factor, fa was shown to increase
rapidly for low relative depth. A method for predicting the discharge of overbank flow of
natural rivers was then presented, by means of a composite friction, fc, which represented the
actual resistance to flow due to the averaged boundary shear force and the apparent shear
force. Equations were also presented giving the composite friction factor from easily
calculated parameters for overbank flow of natural rivers. The discharge obtained using the
proposed methods showed a significant improvement over traditional methods, with an
averaged error of 2.7%.
2.3. OVERVIEW OF THE RESEARCH WORK ON BOUNDARY SHEAR
Distribution of wall shear stress around the wetted perimeter of a channel is influenced by
many factors notably, the shape of channel cross-section, the longitudinal variation in plan
form geometry, the sediment concentration and the lateral-longitudinal distribution of wall
roughness. It is important to appreciate the general three-dimensional flow structures that
exist in open channels to understand the lateral distribution of boundary shear stress. The
interaction between the primary longitudinal velocity U, and the secondary flow velocities,
V and W are responsible for non-uniform wall shear distribution in an open channel flow.
Researchers and engineers have predicted the distribution of wall shear in open channel flow
for both straight and meandering reach.
Einstein’s (1942) hydraulic radius separation method is still widely used in laboratory
studies and engineering practice. Einstein divided a cross-sectional area into two areas and
assumed that the down-stream component of the fluid weight was balanced by the resistance
of the bed. Likewise, the downstream component of the fluid weight in area Aw was balanced
14
Literature Review
by the resistance of the two side-walls. There was no friction at the interface between the
two areas Ab and Aw. In terms of energy, the potential energy provided by area Ab was
dissipated by the channels bed, and the potential energy provided by area Aw was dissipated
by the two side-walls.
Since 1960s, several experimental studies related to boundary shear and its distribution
in open channel flow has been reported by many eminent researchers. Lundgren and Jonsson
(1964) extended the logarithmic law to a parabolic cross-sectional open-channel and
proposed a method to determine the shear stress.
Ghosh and Roy (1970) presented that the boundary shear distribution in both rough and
smooth open channels of rectangular and trapezoidal sections was obtained by direct
measurement of shear drag on an isolated length of the test channel utilizing the technique of
three point suspension system suggested by Bagnold. Existing shear measurement
techniques were reviewed critically. Comparisons were made of the measured distribution
with other indirect estimates, from isovels, and Preston tube measurements, based on
Keulegan’s resistance laws. The discrepancies between the direct and indirect estimates
were explained and out of the two indirect estimates the surface Pitot tube technique was
found to be more reliable. The influence of secondary flow on the boundary shear
distribution was not accurately defined in the absence of a dependable theory on secondary
flow.
Kartha and Leutheusser (1970) expressed that the designs of alluvial channels by the
tractive force method requires information on the distribution of wall shear stress over the
wetted perimeter of the cross-section. The study was undertaken in order to provide some
details on actually measured shear distributions and, hence, to check the validity of available
design information. The latter was entirely analytical in origin and was based either on the
assumption of laminar flow or on over-simplified models of turbulent channel flow. The
experiments were carried out in a smooth-walled laboratory flume at various aspect ratios of
the rectangular cross-section. Wall shear stress was determined with Preston tubes which
were calibrated by a method exploiting the logarithmic form of the inner law of velocity
distribution. Results were presented which clearly suggested that none of then present
analytical techniques could be counted upon to provide any precise details on tractive force
distributions in turbulent channel flow.
15
Literature Review
Ghosh and Kar (1975) reported the evaluation of interaction effect and the distribution
of boundary shear stress in meander channel with floodplain. Using the relationship
proposed by Toebes and Sooky (1967) they evaluated the interaction effect by a parameter
(W). The interaction loss increased up to a certain floodplain depth and there after it
decreased. They concluded that the channel geometry and roughness distribution did not
have any influence on the interaction loss.
Myers (1978) carried out the experimentation using Preston tube to measure shear stress
distributions around the periphery of a complex channel, consisting of a deep section and
one flood plain. Measurements were made with full cross section flow and with flow
confined to the deep, or channel, section. The results were used to quantify the momentum
transfer due to interaction between the channel flow and that over its flood plain,
demonstrating the danger of neglecting this phenomenon in complex channel analysis.
Lateral momentum transfer throughout the channel and flood plain was compared under
isolated and interacting conditions.
Knight and Macdonald (1979) studied that the resistance of the channel bed was varied
by means of artificial strip roughness elements, and measurements made of the wall and bed
shear stresses. The distribution of velocity and boundary shear stress in a rectangular flume
was examined experimentally, and the influence of varying the bed roughness and aspect
ratio were accessed. Dimensionless plots of both shear stress and shear force parameters
were presented for different bed roughness and aspect ratios, and those illustrated the
complex way in which such parameters varied. The definition of a wide channel was also
examined, and a graph giving the limiting aspect ratio for different roughness conditions was
presented. The boundary shear stress distributions and isovel patterns were used to examine
one of the standard side-wall correction procedures. One of the basic assumptions
underlying the procedure was found to be untenable due to the cross channel transfer of
linear momentum.
Knight (1981) proposed an empirically derived equation that presented the percentage
of the shear force carried by the walls as a function of the breadth/depth ratio and the ratio
between the Nikuradse equivalent roughness sizes for the bed and the walls. A series of
flume experiments were reported in which the walls and bed were differentially roughened,
16
Literature Review
normal depth flow was set, and measurements were made of the boundary shear stress
distribution. The results were compared with other available data for the smooth channel
case and some disagreements noted. The systematic reduction in the shear force carried by
the walls with increasing breadth/depth ratio and bed roughness was illustrated. Further
equations were presented giving the mean wall and bed shear stress variation with aspect
ratio and roughness parameters. Although the experimental data was somewhat limited, the
equations were novel and indicated the general behavior of open channel flows with success.
This idea was further discussed by Noutsopoulos and Hadjipanos (1982).
Knight et al. (1984), Hu (1985) and Knight and his associates collected a great deal of
experimental data about the effect of the side walls at different width–depth ratios. With
these data, they proposed several empirical relations which are very helpful in the studies of
open-channel flow and sediment transport.
Knight and Patel (1985) reported some of the laboratory experiments results
concerning the distribution of boundary shear stresses in smooth closed ducts of a
rectangular cross section for aspect ratios between 1 and 10. The distributions were shown to
be influenced by the number and shape of the secondary flow cells, which, in turn, depended
primarily upon the aspect ratio. For a square cross section with 8 symmetrically disposed
secondary flow cells, a double peak in the distribution of the boundary shear stress along
each wall was shown to displace the maximum shear stress away from the center position
towards each corner. For rectangular cross sections, the number of secondary flow cells
increased from 8 by increments of 4 as the aspect ratio increased, causing alternate
perturbations in the boundary shear stress distributions at positions where there were
adjacent contra-rotating flow cells. Equations were presented for the maximum, centerline,
and mean boundary shear stresses on the duct walls in terms of the aspect ratio.
Chiu and Chiou (1986) extended the logarithmic law to rectangular cross-sections and
provided an analytical method to compute the wall shear stress. A parameter estimation
method was developed for a system of three-dimensional mathematical models of flow in
open channels, which did not require (primary flow) velocity data. It broadened the
applicability and effectiveness of the model in scientific investigations into the complex
interaction among the primary and secondary flows, shear stress distribution, channel
characteristics (roughness, slope, and geometry), and other related variables governing all
transport processes in open channels. The results were answered quantitatively many
17
Literature Review
questions that aroused in open channel hydraulics. The difficulty was that the calculation of
boundary shear stress required the knowledge of velocity profile.
Knight and co-workers (1978—92), has led to an improved understanding of the lateral
distributions of wall shear stress in prismatic channels and ducts. Nezu and Nakagawa
(1993) pointed out that the results calculated by Chiu and Chiou revealed large errors.
Rhodes and Knight (1994) deduced results of laboratory experiments on wide smooth
rectangular ducts which were reported in terms of the relationship between duct aspect ratio
b/h and the shear force on the walls expressed as a proportion of the total boundary shear
force, %SFw. The data set (range 0.02
b/h
Knight and Patel (1985) that had range 0.1
b/h
50) overlapped and extended the work of
10. A new form of empirical model was
proposed for the %SFw—b/h relationship. When compared with the %SFw predicted by
assuming the shear force on an element of the boundary to be simply proportional to its
length, the model results deviated from it, with definite maximum and minimum deviations.
Following Einstein`s idea, Yang and Lim (1997) proposed an analytical method to
delineate the two areas as given and determine the boundary shear stress distribution. An
analytical approach for the partitioning of the flow cross-sectional area in steady and
uniform three-dimensional channels was presented. The mechanism and direction of energy
transport from the main flow were analyzed first. Based on the availability of surplus energy
from the main flow, a concept of energy transportation through a minimum relative distance
toward a unit area on the wetted perimeter was proposed. This led to a novel method to
analytically divide the flow area into various parts corresponding to the channel shape and
roughness composition of its wetted perimeter. The demarcated boundary or “division line”
was evaluated using the presented equation. The existence of the division lines, which were
lines of zero Reynolds shear stress within the flow region, had been proved through
comparison with turbulence characteristics measurements. The method was illustrated in a
study of the shear stress distribution in smooth rectangular open channels. Analytical
solutions, valid for all aspect ratios, were derived for the mean side wall and bed shear
stresses, and they compared well with existing experimental data from various researchers.
However, their method was inconvenient for applications because of its implicit and
segmental form except without considering the effect of secondary currents.
18
Literature Review
Patra and Kar (2000) reported the test results concerning 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 off to its sides. They used
five dimensionless channel parameters to form equations representing the total shear force
percentage carried by floodplains. A set of smooth and rough sections were studied with
aspect ratio varying from 2 to 5. Apparent shear forces on the assumed vertical, diagonal,
and horizontal interface plains were found to be different from zero at low depths of flow
and changed sign with increase in depth over floodplain. They proposed a variable-inclined
interface for which apparent shear force was calculated as zero. They presented empirical
equations predicting proportion of discharge carried by the main channel and floodplain.
Yang and McCorquodale (2004) developed a method for computing three-dimensional
Reynolds shear stresses and boundary shear stress distribution in smooth rectangular
channels by applying an order of magnitude analysis to integrate the Reynolds equations. A
simplified relationship between the lateral and vertical terms was hypothesized for which the
Reynolds equations become solvable. This relationship was in the form of a power law with
an exponent of n = 1, 2, or infinity. The semi-empirical equations for the boundary shear
distribution and the distribution of Reynolds shear stresses were compared with measured
data in open channels. The power-law exponent of 2 gave the best overall results while n =
infinity gave good results near the boundary.
Guo and Julien (2005) determined the average bed and sidewall shear stresses in
smooth rectangular open-channel flows after solving the continuity and momentum
equations. The analysis showed that the shear stresses were functions of three components:
(1) gravitational; (2) secondary flows; and (3) interfacial shear stress. An analytical solution
in terms of series expansion was obtained for the case of constant eddy viscosity without
secondary currents. In comparison with laboratory measurements, it slightly overestimated
the average bed shear stress measurements but underestimated the average sidewall shear
stress by 17% when the width–depth ratio becomes large. A second approximation was
formulated after introducing two empirical correction factors. The second approximation
agreed very well (R2 > 0.99 and average relative error less than 6%) with experimental
measurements over a wide range of width–depth ratios.
Dey and Cheng (2005) and Dey and Lambert (2005) developed the expressions for the
Reynolds stress and bed shear stress, assuming a modified logarithmic law of velocity
19
Literature Review
profile due to upward seepage and with stream wise sloping beds. The computed Reynolds
stress profiles were in agreement with the experimental data.
Khatua (2008) and Khatua et al. (2009) undertook a series of laboratory tests for
smooth and rigid meandering channels and developed numerical equations to evaluate and
predict the boundary shear distribution.
2.4. OBJECTIVES OF THE PRESENT RESEARCH WORK
The present research work is intended to study the variations of stage-discharge relationship
with channel roughness, geometry and sinuosity in straight and meandering channels. The
review shows the effect of the parameters such as the bed slope, aspect ratio, sinuosity,
roughness coefficients, flow depth, etc. on the discharge estimation. Using dimensional
analysis, a model for roughness coefficients is planned on which a method of calculation of
discharge carried by meandering channel can be proposed. Again, the boundary shear
distribution for meandering channels can be improved. This should be useful in the solution
of many practical problems and for better understanding of the mechanism of flow. The
present work is also directed towards understanding the underlying mechanism of flow
resistance and flow distribution in the straight and meander channel rather than simulating a
prototype situation. The experimental channel dimensions are small due to space and other
limitations in the laboratory. In view of the above, the research program is taken up with the
following objectives:
¯ To investigate the loss of energy with flow depth for meandering and straight channels
in terms of variation of Manning’s n, Chezy`s C, Darcy-Weisbach f from the
experimental runs.
¯ To propose equations to estimate the percentage of shear carried by the wall of the
meandering channel and to extend the models to the channels of high width ratio and
sinuosity.
20
Literature Review
¯ To study the flow parameters of meandering channel on the basis of which a simple but
accurate method of calculation of discharge carried by a meandering channel can be
proposed.
¯ To validate the developed models using the available experimental channel data from
other investigators.
21
CHAPTER 3
E X P E R I M E NT A L
WORK
EXPERIMENTAL WORK
3.1. GENERAL
The experiments have been conducted in the Fluid Mechanics and Hydraulics Laboratory of
the Civil Engineering Department at the National Institute of Technology, Rourkela, India.
In the laboratory, four types of flumes are fabricated in which channels of varying sinuosity;
geometry and slope are cast inside. Experiments have been conducted in the channels by
changing the geometrical parameters. This chapter discusses in detail the experimental
procedure along with the apparatus required during the present work.
3.2. EXPERIMENTAL SETUP
The present research uses both meandering and straight experimental channels fabricated
inside the tilting flumes separately in the Fluid Mechanics and Hydraulics Engineering
Laboratory of the Civil Engineering Department, at the National Institute of Technology,
Rourkela. The present work is mainly based on the observation taken inside two straight
channels, one of rectangular cross section (Type-I) and another of trapezoidal cross section
(Type-II). The other two channels are Type-III and Type-IV respectively, consisting of
meandering channel. The tilting flumes are made out of metal frame with glass walls at the
test reach. The flumes are made tilting by hydraulic jack arrangement. Inside each flume,
separate meandering/straight channels are cast using 5mm thick Perspex sheets. To facilitate
fabrication, the whole channel length has been made in blocks of 1.20 m length each. The
photo graphs of Type-II, III and IV experimental channels with measuring equipments taken
from the down stream side are shown in Photos 3.1, 3.2, and 3.3 respectively. The detailed
geometrical features of the experimental channels are given in Table 3.1.
22
Experimental work
Photo 3.1
Photo 3.2
Photo 3.3
23
Experimental work
Table 3.1 Details of Geometrical Parameters of the Experimental Channels
Sl
Item description
no.
1 Channel Type
Wave length in down valley
2
direction
3 Amplitude
Geometry of Main channel
4
section
5
Nature of surface of bed
6
Channel width (b)
7
8
9
Bank full depth of channel
Bed Slope of the channel
Meander belt width
Minimum radius of
10 curvature of channel
centerline at bend apex
11 Sinuosity
12 Cross over angle in degree
13 Flume size
Type-I
Type-II
Type-III
Type-IV
Straight
Straight
Meandering
Meandering
-
-
400 mm
2185 mm
-
162
120 mm
0.0019
-
Trapezoidal
(side slope 1:1)
Smooth and rigid
bed
120 mm at
bottom and 280
mm at top
80 mm
0.003
-
120 mm
0.0031
443 mm
685 mm
Trapezoidal
(side slope 1:1)
Smooth and
rigid bed
120 mm at
bottom and 280
mm at top
80 mm
Varying
1650 mm
-
-
140 mm
420 mm
1.00
0.45mx0.4m
x12m long
1.00
2.0m×0.6m×12m
long
1.44
104
0.6mx0.6mx
12m long
1.91
102
2.0m×0.6m×12
m long
Rectangular
Smooth and
rigid bed
120 mm
Rectangular
Smooth and
rigid bed
120 mm
The fabricated channels have the following details.
1.
Type-I channel: The straight channel section has the dimension of 120 mm×120 mm.
The channel is cast inside a tilting flume of 12 m long, 450 mm wide, and 400 mm
deep. The bed slope of the channel is kept at 0.0019. The plan view and details of the
geometrical parameters of the Type-I channel are shown in Fig. 3.1 (a) and (b),
respectively.
Fig. 3.1(a) - Plan View of Type-I Channel inside the Experimental Flume
24
Experimental work
Fig. 3.1(b) Details of Geometrical Parameter of Type- I Channel
2.
Type-II channel: The straight channel section is trapezoidal in cross section. The
channel is of 120 mm wide at bottom, 280 mm wide at top having depth of 80 mm, and
side slopes of 1:1 (Photo 3.1). The channel is cast inside a tilting flume of 12m long,
200 mm wide, and 600 mm deep. The bed slope of the channel is kept at 0.0033. The
plan view and details of the geometrical parameters of the Type-II channel are shown in
Fig. 3.2 (a) and (b), respectively.
Fig. 3.2(a) Plan View of Type-II Channel inside the Experimental Flume
Fig. 3.2(b) Details of Geometrical Parameter of Type- II Channel
25
Experimental work
3.
Type-III channel: The rectangular meandering channel has dimensions of 120
mm×120 mm in cross section. It has wavelength L = 400 mm, double amplitude 2A =
323 mm giving rise to sinuosity of 1.44 (Photo 3.2). This mildly meandering channel is
placed inside a tilting flume of 12 m long, 600 mm wide, and 600 mm deep. The plan
view and details of the geometrical parameters of the Type-III channel are shown in
Fig. 3.3 (a) and (b), respectively.
Fig. 3.3(a) Plan View of Type-III Channel inside the Experimental Flume
Fig. 3.3(b) Details of One Wavelength of Type- III Channel
4.
Type-IV channel: This meandering channel is trapezoidal in cross section. The channel
is 120 mm wide at bottom, 280 mm at top having full depth of 80 mm, and side slopes
of 1:1. The channel has wavelength L = 2185 mm and double amplitude 2A = 1370
mm. Sinuosity for this channel is scaled as 1.91 (Photo 3.3). The plan view and details
of the geometrical parameters of the Type-IV channel are shown in Fig. 3.4 (a) and (b),
respectively.
26
Experimental work
Fig. 3.4(a) Plan View of Type-IV Channel inside the Experimental Flume
Fig. 3.4(b) Details of One Wavelength of Type- IV Channel
3.3. EXPERIMENTAL PROCEDURE
3.3.1. Apparatus and Methodology
Water is supplied to the experimental setup by a recirculating system of water supply (Photo
3.4). Two parallel pumps are used to pump water from an underground sump to the overhead
tank (Photo 3.4a). The overhead tank (Photo 3.4b) has an over flow arrangement to spill
27
Experimental work
excess water to the sump and thus maintain a constant head. From the over head tank, water
is led to a stilling tank (Photo 3.4 c) located at the upstream of the channel. A series of baffle
walls between the stilling tank and channels are kept to reduce turbulence of the incoming
water. At the end of the experimental channel, water is allowed to flow through a tailgate
and is collected in a masonry volumetric tank (Photo 3.4f) from where it is allowed to flow
back to the underground sump. From the sump, water is pumped back to the overhead tank,
thus setting a complete re-circulating system of water supply for the experimental channel.
The tailgate (Photo 3.4e) helps to establish uniform flow in the channel. It should be noted
that the establishment of a flow that has its water surface parallel to the valley slope (where
the energy losses are equal to potential energy input) may become a standard whereby the
conveyance capacity of a meandering channel configuration is assessed.
Water surface slope measurement is carried out using a pointer gauge (Photo 3.5) fitted
to the travelling bridge operated manually having least count of 0.1 mm. Point velocities are
measured using a 16-Mhz Micro ADV (Acoustic Doppler Velocity-meter) at a number of
locations across the predefined channel section. Guide rails are provided at the top of the
experimental flume on which a travelling bridge is moved in the longitudinal direction of the
entire experimental channel. The point gauge and the micro-ADV attached to the travelling
bridge can also move in both longitudinal and the transverse direction of the experimental
channel at the bridge position (Photo 3.6). The micro-ADV readings are recorded in a
computer placed besides the bridge (Photo 3.7 a, b). As the ADV is unable to read the data
of upper most layer (up to 5cm from free surface), a micro -Pitot tube (Photo 3.8, Photo 3.9
a, b, c) of 4 mm external diameter in conjunction with suitable inclined manometer are used
to measure velocity and its direction of flow at the pre defined points of the flow-grid. A
flow direction finder is also used to get the direction of maximum velocity with respect to
the longitudinal flow direction. The Pitot tube is physically rotated normal to the main
stream direction till it gives maximum deflection of manometer reading. The angle of limb
of Pitot tube with longitudinal direction of the channel is noted by the circular scale and
pointer arrangements attached to the flow direction meter.
Discharge in the channel is measured by the time rise method. The water flowing out at
the down stream end of the experimental channel is led to a rectangular measuring tank of
1690 mm long x 1030 mm wide for Type-I channel, 1985 mm long xl900 mm wide for the
Type-III channel and 2112 mm long x 3939 mm wide tank for Type-II and Type-IV channel.
28
Experimental work
The change in the depth of water with time is measured by stopwatch in a glass tube
indicator with a scale having least count of 0.01mm.
A hand-operated tailgate weir is constructed at the downstream end of the channel to
regulate and maintain the desired depth of flow in the flume. The bed slope is set by
adjusting the whole structure, tilting it upwards or downwards with the help of a leaver,
which is termed as slope changing leaver (Photo 3.12). Readings are taken for the different
slopes, sinuosity and aspect ratio.
3.3.2. Determination of Channel Slope
At the tail end, the impounded water in the channel is allowed to remain standstill by
blocking the tail end. For Type-I and Type-II channel, the mean slope is obtained by
dividing the level difference between the two end points of the test reach of 1 m along the
centerline. The level difference between channel bed and water surface are recorded at a
distance of one wavelength along its centerline for Type-III and Type-IV channels. For
meandering channels, the mean slopes for Type-III and Type-IV channels is obtained by
dividing the level difference between these two points by the length of meander wave
along the centerline.
3.3.3. Measurement of Discharge and Water Surface Elevation
A point gauge with least count of 0.1 mm is used to measure the water surface elevation
above the bed of the channel. A measuring tank located at the end of each test channel
receives water flowing through the channels. Depending on the flow rate, the time of
collection of water in the measuring tanks vary between 50 to 262 seconds; lower one for
higher rate of discharge. The time is recorded using a stopwatch. Change in the mean
water level in the tank over the time interval is recorded. From the knowledge of the
volume of water collected in the measuring tank and the corresponding time of collection,
the discharge flowing in the experimental channel for each run of each channel is obtained.
29
Experimental work
Start
Photo 3.4(a)
Photo 3.4(b)
Photo 3.4(f)
Photo 3.4(c)
Photo 3.4(e)
Photo 3.4(d)
Photo 3.4 Recirculating Water System
30
Experimental work
Photo 3.5
Photo 3.6
Photo 3.7(a)
Photo 3.7(b)
31
Experimental work
Photo 3.8
Photo 3.9(a)
Photo 3.9(b)
Photo 3.10
Photo 3.9 (c)
Photo 3.11
Photo 3.12
32
Experimental work
3.3.4. Measurement of Velocity and Its Direction
A 16-MHz Micro ADV (Acoustic Doppler Velocity-meter) manufactured by M/s SonTek,
San Diego, Canada is used for 3-axis (3D) velocity measurement at each grid point of the
channel sections (Photo 3.7a). The higher acoustical frequency of 16 MHz makes the
Micro-ADV the optimal instrument for laboratory study. The Micro ADV with the software
package is used for taking high-quality three dimensional velocity data at different points.
The data is received at the ADV-processor. A computer attached with the processor (Photo
3.7b) shows the 3-dimensional velocity data after compiling with the software package. At
every point, the instrument records a number of velocity data per minute. With the statistical
analysis using the installed software, mean values
of 3D point velocities are recorded for each flow
depth. The Micro-ADV uses the Doppler shift
principle to measure the velocity of small particles,
assuming to be moving at velocities similar to the
fluid (Fig. 3.5 a). Velocities is resolved into three
orthogonal components (tangential, radial, and
vertical) and are measured at 5 cm below the sensor
head, minimizing interference of the flow field, and
allowing measurements to be made close to the bed.
The Micro ADV has excellent features (Fig. 3.5 b)
Fig. 3.5 (a) - Standard Features of a
MicroADV
such as
¯ Three-axis velocity measurement
¯ High sampling rates - up to 50 Hz
¯ Small sampling volume - less than 0.1 cm3
¯ Distance to Sampling Volume-5 cm
¯ Small optimal scatterer - excellent for low flows
¯ High accuracy upto1% of measured range
¯ Large velocity ranges between 1 mm/s to 2.5 m/s
¯ Excellent low-flow performance
¯ No recalibration needed
Fig. 3.5 (b) - Details of Probe of
a MicroADV
¯ Comprehensive software
33
Experimental work
Boasting a sampling volume of less than 0.09 cc and sampling rates up to 50 Hz, the
16-MHz MicroADV is an ideal laboratory instrument for low flow and turbulence studies.
Though the ADV has proven its versatility and reliability in a wide variety of applications,
yet it is unable to read the upper layer velocity, that is, up to 50 mm from free surface. To
overcome the short, a standard Prandtl type micro-Pitot tube in conjunction with a manometer
of accuracy 0.12 mm is used for the measurement of point velocity readings at the specified
location for the upper 50 mm region from free surface across the channel. The results from
the observations have been discussed in the next chapter.
A flow direction meter is used to find the direction of the velocity readings taken by
micro Pitot tube in the experimental channel. It essentially consists of a copper tube at lower
end of which a metal pointer rod tied with a thread is attached, while another pointer at the
upper end of the tube moves over a circular metallic protractor. The copper tube is fixed
with the protractor through a ball bearing arrangement. The thread tied to the pointer rod is
lowered into water that moves along the resultant direction of flow. The copper tube with the
metal pointer rod at upper end is made to rotate till the pointer and thread are in a vertical
straight line parallel to each other. The angle of rotation by upper pointer with the metallic
protractor gives the direction of flow. The deviation of the angle of flow shown by the
pointer with respect to the zero position of the metallic protractor (tangential direction) is
taken as the local direction of the total velocity vector in the channel.
When flow is confined to meander channel, velocity measurements are taken at the
bend apex so as to get a broad picture of flow parameters covering half the meander wave
length. For straight channels, the measurement is taken at the section in the central test
reach. While taking velocity readings using Pitot tube, the tube is placed facing the direction
of flow and then is rotated along a plane parallel to the bed and till it registers relatively a
maximum head difference in the attached manometer. The deviation angle between the
reference axis and the total velocity vector is considered positive, when the velocity vector
is directed away from the outer bank. The total head h reading by the Pitot tube at the
location in the channel is used to give the magnitude of the total velocity vector as U
=2gh, where g is the acceleration due to gravity. Resolving U into the tangential and
radial directions, the local velocity components are determined. While doing so, the tube
coefficient is taken as unit and the error due to turbulence in the computation of U is
considered negligible. Using the data of velocities by Pitot tube and micro-ADV close to the
34
Experimental work
surface of the channels, the boundary shear at various points on the channel bed at the
predefined channel sections are evaluated from the logarithmic velocity distribution
relationship which is described in the next chapter.
All measurements are carried out under quasi-uniform flow condition by maintaining
the out flow through downstream tailgate. Using the downstream tailgate, uniform flow is
maintained for each experimental run and for each channel by maintaining the water surface
slope parallel to the valley bed slope. Experimental results are accessed concerning stagedischarge-resistance relationships for meandering channels with rigid and smooth
boundaries. All observations are recorded in the central test reach for straight channel of
type-I and type-II and at the bend apex of Type-III and type-IV (Photo 3.10 and Photo 3.11)
meandering channels. Details of flow parameter of the experimental channels and the
associated runs are given in Table-3.2.
Table- 3.2 Hydraulic Details of the Experimental Runs
Sl
no
1
2
3
Item description
Type-I
Type-II
Type-III
Type-IV
Channel Type
Number of runs for stagedischarge data
Straight
Straight
Meandering
11
5
15
Meandering
55 (Slopes - 0.003, 0.0042, 0.0053, 0.008,
0.013, 0.015, 0.021)
Slope-0.003 (1479, 2605, 3101, 3717, 4097,
4563, 5994)
Slope-0.0042 (4798, 5329, 5899, 7210, 7809,
8636)
Slope-0.0053 (294, 484, 987, 1742, 2048,
2757, 3224, 3338, 3698, 4191,
4656, 5515, 6396, 7545)
Slope-0.008 (904, 1416, 2165, 4863, 5492,
6768, 8473)
Slope-0.013 (2357, 4877, 5410, 5991, 7181,
8157, 9480)
Slope-0.015 (1319, 2742, 4497, 4884, 6209,
8112, 8397)
Slope-0.021 (1286, 2150, 3346, 4070, 5451,
6123, 9223)
Slope-0.003 (3.1, 4.5, 5.0, 5.6, 5.9, 6.3, 7.4)
Slope-0.0042 (5.7, 6.1, 6.5, 7.2, 7.5, 7.8)
Slope-0.0053 (1.1, 1.4, 2.2, 3.1, 3.4, 4.1, 4.6,
4.7, 4.9, 5.3, 5.6, 6.2, 6.7, 7.3)
Slope-0.008 (1.9, 2.5, 3.3, 5.1, 5.6, 6.2, 7.0)
Slope-0.013 (3.1, 4.8, 5.1, 5.4, 6.0, 6.4, 7.0)
Slope-0.015 (2.1, 3.3, 4.4, 4.6, 5.3, 6.2, 6.3)
Slope-0.021 (1.9, 2.6, 3.4, 3.9, 4.6, 4.9, 6.3)
3
Discharge (cm /s)
Depth of flow (cm)
corresponding to flow
discharge of runs
5
Nature of surface of bed
6
No. of runs for detailed
measurement of 3
dimensional point using
Micro ADV
316, 426,
1061, 1280,
1347
2148, 2307,
1669, 2200,
4766, 6961,
2902, 3249,
2357, 2619,
8147, 8961,
4117, 4548,
2757, 2946,
9739
5058, 5947,
3338, 3698,
6312
4191, 4656,
5596, 5680
1.29, 1.57,
3.44, 4.05,
4.98, 5.31,
5.78, 6.08,
6.41, 7.11,
7.7, 8.55,
9.34,
10.9,11.01
smooth and smooth and smooth and
rigid bed
rigid bed
rigid bed
3.02, 3.44,
4.98, 5.24,
6.21, 6.80, 5.2, 6.4, 7.0,
7.4, 7.8
8.15, 8.82,
9.55, 10.92,
11.48
6
5
6
35
smooth and rigid bed
12
CHAPTER 4
R E SULTS A ND
D I SC US S I O NS
RESULTS AND DISCUSSIONS
4.1. GENERAL
The results of experiments concerning the distribution of velocity, flow, and boundary shear
stress of meandering channels are presented in this chapter. Analysis of results is done for
roughness coefficients and distribution of boundary shear in a meandering channel. The
overall summary of experimental runs for straight channel (Type-I and Type-II) are given in
Table 4.1 (a) and for meandering channels of Type-III and Type-IV it is given in Table 4.1
(b) and Fig. 4.1 (c), respectively. The detailed skeleton of the probe limits are shown in
Fig. 4.1.
Fig. 4.1 Cross Section of Channel for Experimentation
Despite the use of three types of probes (up probe, down probe and side probe) there is
a fringe zone for which the data could not be recorded due to the inherent limitations of the
ADV as discussed in the previous chapter. The division of the channel section is done for
the convenience to take the readings at maximum possible points as shown in Fig. 4.1.
Therefore, the Pitot tube has been used to record the velocity of the fringe zone.
36
Results and Discussions
Table 4.1(a) Experimental Runs for Type-I and Type-II Straight Channels
(2)
SR1
SR2
SR3
SR4
SR5
SR6
SR7
SR8
SR9
SR10
SR11
ST1
ST2
ST3
(3)
0.00106
0.00128
0.00215
0.00231
0.00290
0.00325
0.00412
0.00455
0.00506
0.00595
0.00631
0.00477
0.00696
0.00815
(4)
0.030
0.034
0.050
0.052
0.062
0.068
0.082
0.088
0.096
0.109
0.115
0.052
0.064
0.070
(5)
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
Cross
Wetted Hydraulic
Section
Perimeter Radius
Area
P(m)
R= (A/P)
2
A(m )
(6)
(7)
(8)
0.0036
0.18
0.020
0.0041
0.19
0.022
0.0060
0.22
0.027
0.0063
0.22
0.028
0.0075
0.24
0.031
0.0082
0.26
0.032
0.0098
0.28
0.035
0.0106
0.30
0.036
0.0115
0.31
0.037
0.0131
0.34
0.039
0.0138
0.35
0.039
0.0089
0.27
0.033
0.0118
0.30
0.039
0.0133
0.32
0.042
ST4
0.00896
0.074
0.12
0.0144
0.33
0.044
0.624
1.622
ST5
0.00974
0.078
0.12
0.0154
0.34
0.045
0.634
1.544
Discharge Flow
Depth
Q
(m3/sec) h (m)
Run
Channel Type No
(1)
Type-I
Straight
Rectangular
Channel
Sinuosity=1
Slope=0.0019
Type-II
Straight
Trapezoidal
Channel
Sinuosity=1
Slope=0.0033
Channel
Width
b (m)
Average
Velocity Aspect
(Q/A) Ratio(α)
(m/sec)
(9)
(10)
0.293
3.974
0.310
3.488
0.359
2.410
0.367
2.290
0.389
1.932
0.398
1.765
0.421
1.472
0.430
1.361
0.441
1.257
0.454
1.099
0.458
1.045
0.533
2.308
0.591
1.875
0.613
1.714
Table 4.1(b) Experimental Runs for Type-III Meandering Channel
Channel Type
(1)
Run
No
(2)
MR1
MR 2
MR 3
MR 4
MR 5
Type-III
MR 6
Meandering
MR 7
Rectangular
MR 8
Channel
MR 9
Sinuosity=1.44
MR
10
Slope= 0.0031
MR 11
MR 12
MR 13
MR 14
MR 15
Discharge
Q
3
(m /sec)
Flow
Depth
h (m)
Channel
Width
b (m)
(3)
0.00032
0.00043
0.00135
0.00167
0.00220
0.00236
0.00262
0.00276
0.00295
0.00334
0.00370
0.00419
0.00466
0.00560
0.00568
(4)
0.013
0.016
0.034
0.041
0.050
0.053
0.058
0.061
0.064
0.071
0.077
0.086
0.093
0.109
0.110
(5)
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
37
Cross
Section
Area
2
A(m )
(6)
0.0015
0.0019
0.0041
0.0049
0.0060
0.0064
0.0069
0.0073
0.0077
0.0085
0.0092
0.0103
0.0112
0.0130
0.0132
Average
Wetted Hydraulic
Velocity
Perimeter Radius
(Q/A)
P(m)
R= (A/P)
(m/sec)
(7)
(8)
(9)
0.15
0.011
0.207
0.15
0.012
0.229
0.19
0.022
0.326
0.20
0.024
0.343
0.22
0.027
0.368
0.23
0.028
0.370
0.24
0.029
0.378
0.24
0.030
0.378
0.25
0.031
0.383
0.26
0.033
0.391
0.27
0.034
0.400
0.29
0.035
0.408
0.31
0.037
0.415
0.34
0.039
0.429
0.34
0.039
0.430
Aspect
Ratio(α)
(10)
9.302
7.643
3.488
2.963
2.410
2.260
2.076
1.974
1.872
1.688
1.558
1.404
1.285
1.105
1.090
Results and Discussions
Table 4.1(c) Experimental Runs for Type-I V Meandering Channels
Channel Type Run No
(1)
Meandering
Trapezoidal
Channel
Sinuosity=1.91
Slope=0.003
Meandering
Trapezoidal
Channel
Sinuosity=1.91
Slope=0.0042
Meandering
Trapezoidal
Channel
Sinuosity=1.91
Slope=0.0053
Meandering
Trapezoidal
Channel
Sinuosity=1.91
Slope=0.008
Meandering
Trapezoidal
Channel
Sinuosity=1.91
Slope=0.013
Meandering
Trapezoidal
Channel
Sinuosity=1.91
Slope=0.015
Meandering
Trapezoidal
Channel
Sinuosity=1.91
Slope=0.021
(2)
MT1
MT2
MT3
MT4
MT5
MT6
MT7
MT8
MT9
MT10
MT11
MT12
MT13
MT14
MT15
MT16
MT17
MT18
MT19
MT20
MT21
MT22
MT23
MT24
MT25
MT26
MT27
MT28
MT29
MT30
MT31
MT32
MT33
MT34
MT35
MT36
MT37
MT38
MT39
MT40
MT41
MT42
MT43
MT44
MT45
MT46
MT47
MT48
MT49
MT50
MT51
MT52
MT53
MT54
MT55
Discharge
Q
3
(m /sec)
(3)
0.00148
0.00261
0.00310
0.00372
0.00410
0.00456
0.00599
0.00480
0.00533
0.00590
0.00721
0.00781
0.00864
0.00029
0.00048
0.00099
0.00174
0.00205
0.00276
0.00322
0.00334
0.00370
0.00419
0.00466
0.00552
0.00640
0.00755
0.00090
0.00142
0.00217
0.00486
0.00549
0.00677
0.00847
0.00236
0.00488
0.00541
0.00599
0.00718
0.00816
0.00948
0.00132
0.00274
0.00450
0.00488
0.00621
0.00811
0.00840
0.00129
0.00215
0.00335
0.00407
0.00545
0.00612
0.00922
Flow
Depth
h (m)
(4)
0.031
0.045
0.050
0.056
0.059
0.063
0.074
0.057
0.061
0.065
0.072
0.075
0.078
0.011
0.014
0.022
0.031
0.034
0.041
0.046
0.047
0.049
0.053
0.056
0.062
0.067
0.073
0.019
0.025
0.033
0.051
0.056
0.062
0.070
0.031
0.048
0.051
0.054
0.060
0.064
0.070
0.021
0.033
0.044
0.046
0.053
0.062
0.063
0.019
0.026
0.034
0.039
0.046
0.049
0.063
Channel
Cross
Wetted
Width Section Area Perimeter
2
b (m)
A(m )
P(m)
(5)
(6)
(7)
0.12
0.0047
0.21
0.12
0.0074
0.25
0.12
0.0085
0.26
0.12
0.0099
0.28
0.12
0.0106
0.29
0.12
0.0115
0.30
0.12
0.0144
0.33
0.12
0.0101
0.28
0.12
0.0110
0.29
0.12
0.0120
0.30
0.12
0.0138
0.32
0.12
0.0146
0.33
0.12
0.0154
0.34
0.12
0.0014
0.15
0.12
0.0019
0.16
0.12
0.0032
0.18
0.12
0.0047
0.21
0.12
0.0053
0.22
0.12
0.0066
0.24
0.12
0.0075
0.25
0.12
0.0077
0.25
0.12
0.0083
0.26
0.12
0.0092
0.27
0.12
0.0099
0.28
0.12
0.0112
0.29
0.12
0.0126
0.31
0.12
0.0142
0.33
0.12
0.0026
0.17
0.12
0.0036
0.19
0.12
0.0050
0.21
0.12
0.0087
0.26
0.12
0.0097
0.28
0.12
0.0113
0.30
0.12
0.0133
0.32
0.12
0.0047
0.21
0.12
0.0080
0.25
0.12
0.0087
0.26
0.12
0.0094
0.27
0.12
0.0108
0.29
0.12
0.0118
0.30
0.12
0.0133
0.32
0.12
0.0030
0.18
0.12
0.0050
0.21
0.12
0.0072
0.24
0.12
0.0076
0.25
0.12
0.0092
0.27
0.12
0.0112
0.29
0.12
0.0115
0.30
0.12
0.0026
0.17
0.12
0.0038
0.19
0.12
0.0052
0.22
0.12
0.0062
0.23
0.12
0.0076
0.25
0.12
0.0083
0.26
0.12
0.0114
0.30
38
Hydraulic
Radius
R= (A/P)
(8)
0.023
0.030
0.033
0.035
0.037
0.039
0.044
0.036
0.038
0.040
0.043
0.044
0.045
0.009
0.012
0.017
0.023
0.024
0.028
0.030
0.031
0.032
0.034
0.036
0.038
0.041
0.043
0.015
0.019
0.023
0.033
0.035
0.038
0.042
0.023
0.031
0.033
0.034
0.037
0.039
0.042
0.017
0.024
0.030
0.031
0.034
0.038
0.039
0.015
0.020
0.024
0.027
0.031
0.032
0.038
Average
Velocity
(Q/A) (m/sec)
(9)
0.316
0.351
0.364
0.377
0.388
0.396
0.418
0.476
0.483
0.491
0.522
0.534
0.559
0.215
0.250
0.313
0.368
0.386
0.418
0.428
0.431
0.443
0.457
0.470
0.491
0.509
0.533
0.342
0.391
0.437
0.558
0.564
0.600
0.637
0.504
0.613
0.620
0.638
0.665
0.693
0.713
0.445
0.543
0.623
0.640
0.677
0.727
0.728
0.487
0.566
0.639
0.656
0.714
0.739
0.809
Aspect
Ratio(α)
(10)
3.871
2.667
2.398
2.143
2.034
1.905
1.622
2.105
1.967
1.846
1.667
1.600
1.538
11.429
8.333
5.405
3.834
3.488
2.927
2.637
2.581
2.434
2.264
2.135
1.942
1.788
1.637
6.316
4.800
3.692
2.353
2.162
1.935
1.714
3.871
2.526
2.353
2.222
2.000
1.875
1.714
5.714
3.636
2.727
2.609
2.264
1.951
1.905
3.558
4.558
3.082
2.975
2.553
2.553
3.553
Results and Discussions
4.2. STAGE-DISCHARGE VARIATION FOR MEANDERING CHANNELS
In the present investigation involving the flow in straight and meandering channels,
achieving steady and uniform flow has been difficult due to the effect of curvature and the
influence of a number of geometrical parameters. However, for the purpose of present work,
an overall uniform flow is tried to be achieved in the channels. Flow depths in the
experimental channel runs are so maintained that the water surface slope becomes parallel to
the valley slope. At this stage, the energy losses are taken as equal to potential energy input.
This has become a standard whereby the conveyance capacity of a meandering channel
configuration can be assessed (Shiono, Al- Romaih, and Knight, 1999). Under such
conditions, the depths of flow at the channel centerline separated by one wavelength
distance must be the same. In all the experimental runs this simplified approach has been
tried to achieve. This stage of flow is taken as normal depth, which can carry a particular
flow only under steady and uniform conditions. The stage discharge curves plotted for
different bed slopes for particular sinuosities of channel are shown in Fig. 4.2. The discharge
increases with an increase in bed slope for the same stage.
Fig. 4.2 Stage Discharge Variations in Open Channel Flow
4.3. VARIATION OF REACH AVERAGED LONGITUDINAL VELOCITY
U WITH DEPTH OF FLOW
Plots between the reach averaged velocity U and non-dimensional flow depth for all the
experimental channels for the four types of channels are shown in Fig. 4.3. From the figure it
39
Results and Discussions
can be seen that for all the channels, the increase in reach averaged longitudinal velocity of
flow is nearly in accordance with the increase in depth of flow.
The drop in the reach averaged velocity is higher for Type-IV channel. This may be due
to spreading of water to a wider section for Type-IV channel than Type-I and Type-III
channel sections, (Sinuosity Sr= 1 for Type-I and Type-II, Sr= 1.44 for Type-III, and Sr=
1.91 for Type-IV), resulting additional resistance to flow. For all flow conditions, the rate of
increase of velocity with flow depth is higher for the sinus channel of Type-IV and lower for
straight channel of Type-I. This is mainly due to their large differences in the longitudinal
bed slopes. The slope of the channel is an important parameter influencing the desired
driving force. As for higher slopes (Type-IV, slope=0.021, 0.015, 0.013) in a particular
channel there is increase in the rate of velocity with flow depth.
Fig. 4.3 Variations of Velocity with Flow Depth in Open Channel Flow
4.4. DISTRIBUTION OF TANGENTIAL (LONGITUDINAL) VELOCITY
Measurement of velocity for the experimental channels is mostly recorded by a micro-ADV.
The instrument uses the sign convention for 3-dimensional velocity as positive for ENU
(east, north and upward) and negative for WSD (west, south and down ward) directions
respectively for the longitudinal (Vx), radial (Vy) and vertical components (Vz). For the
experimental channel position, east refers to the direction of longitudinal velocity. The east
probe of ADV is kept in the longitudinal flow direction. Accordingly the other two flow
directions are referred. In the experiments for meandering channels, the readings are taken at
the bend apex with tangential velocity direction taken as east. For radial velocity, positive
40
Results and Discussions
stands for outward and negative stands for inward radial velocity direction. Similarly, for
vertical component of velocity when the ADV readings shows positive, then the velocity
component is upward and if negative, it is in the down-ward direction. The detailed velocity
and boundary shear distribution are carried out for three channels of Type-II, Type-III and
Type-IV of sinuosity 1.00, 1.44 and 1.91, respectively. As the sinuosity of Type-I is similar
as Type-II so, the detailed velocity distribution and evaluation of boundary shear has not
been done. The measurements are taken at any cross section of the channel reach for straight
channels of Type-II and at the bend apex for meandering channels of Type-III (Fig. 4.4a)
and Type-IV (Fig. 4.4b).
Fig. 4.4 (a) Location of Bend Apex (A-A) of
Type-III Meandering Channel
Fig. 4.4 (b) Location of Bend Apex (A-A) of
Type-IV Meandering Channel
For the straight channels of Type-II the radial distribution of tangential velocity with the
boundary shear distribution are shown in Figs.4.5.1-4.5.4. For meander channels, the radial
distribution of tangential velocity in contour form for the runs of meandering channels of
Type-III at bend apex are shown in Fig. 4.6.1 through Fig. 4.6.6. Similarly for Type-IV
channels the radial distributions of tangential velocity at bend apex are shown in Fig. 4.7.14.7.6. The Fig. 4.5.1 - 4.5.4, Fig. 4.6.1- 4.6.6 and Fig.4.7.1-4.7.6 also show the boundary
shear distribution for straight channel of Type-II and meandering channels of Type-III and
Type-IV respectively.
41
Results and Discussions
Fig. 4.5.1 Flow depth h = 5.2 cm (Type-II channel)
Fig. 4.5.2 Flow depth h = 6.4 cm (Type-II channel)
Fig. 4.5.3 Flow depth h = 7.0 cm (Type-II channel)
42
Results and Discussions
Fig. 4.5.4 Flow depth h = 7.4 cm (Type-II channel)
Figs. 4.5.1- 4.5.4 Contours Showing the Distribution of Tangential Velocity and
Boundary Shear for Type-II Straight Channel
Fig. 4.6.1 Flow depth ` = 5.31cm (Type-III channel)
Fig. 4.6.2 Flow depth h = 6.08cm (Type-III channel)
43
Results and Discussions
Fig. 4.6.3 Flow depth h = 7.11 cm (Type-III channel)
Fig. 4.6.4 Flow depth h = 8.55 cm (Type-III channel)
Fig. 4.6.5 Flow depth h = 9.34 cm (Type-III channel)
44
Results and Discussions
Fig. 4.6.6 Flow depth h = 11.01 cm (Type-III channel)
Figs. 4.6.1- 4.6.6 Contours Showing the Distribution of Tangential Velocity and
Boundary Shear for Type-III Meandering Channel at Bend Apex (Section A-A)
Fig. 4.7.1 Flow depth h = 5.3 cm (Type-IV channel)
Fig. 4.7.2 Flow depth h = 5.62 cm (Type-IV channel)
45
Results and Discussions
Fig. 4.7.3 Flow depth h = 5.93 cm (Type-IV)
Fig. 4.7.4 Flow depth h = 6.18 cm (Type-IV channel)
Fig. 4.7.5 Flow depth h = 6.71 cm (Type-IV channel)
Fig. 4.7.6 Flow depth h = 7.33 cm (Type-IV channel)
Figs. 4.7.1 - 4.7.6 Contours Showing the Distribution of Tangential Velocity and
Boundary Shear for Type-IV Meandering Channel at Bend Apex (Section AA)
46
Results and Discussions
4.4.1.Straight Channel
The Type-I straight channel and meandering channel Type-III are classified as deep main
channel (b/h<5) where wall effects are felt throughout the section when compared to shallow
channels. The distribution of tangential velocity in straight channels (e.g. Figs. 4.5.1 to
4.5.4) in contour form at any cross section is almost uniform in nature.
4.4.2.Meandering Channel
From the distribution of tangential velocity in meander channel sections in contour form
(Figs. 4.6.1 to 4.6.6 and Figs. 4.7.1 to 4.7.6) at the locations AA the following features can
be noted.
¯ The contours of tangential velocity distribution indicate that the velocity patterns are
skewed with curvature. Higher velocity contours are found to concentrate gradually at
the inner bank at the bend apex.
¯ Another feature is the location of the thread of maximum velocity is next to inner bank
along the meander channel of experimental runs. For shallow meandering channels the
thread of maximum velocity is located near the outer bank at the bend apex. It indicates
that the effect of secondary circulation is predominant in shallow channels and is less
effective in deep channels.
¯ From the contours of tangential velocity at these sections, it can be observed that the
distribution of tangential velocity does not follow the power law or the logarithmic law.
Under ideal conditions these theoretical velocity distribution laws gives the maximum
velocity at the free water surface, where as the flow in any type of natural or laboratory
channels do not show such a distribution.
¯ Sinuosity of the meander channel is found to affect the distribution of tangential velocity
considerably. The results of channel Type-III and Type-IV (with sinuosity = 1.44 and
1.91 respectively) show irregular tangential velocity distribution. The magnitude and the
concentration of velocity distribution are affected by the curvature of the meander
channel. Similar reports are also seen for deep channels of Kar (1977), Das (1984), and
Khatua (2008) the distribution of tangential velocity as erratic.
47
Results and Discussions
4.5. MEASUREMENTS OF BOUNDARY SHEAR STRESS
Wall shear stress is of great importance in fluid mechanics research, as it represents the local
tangential force by the fluid on a surface in contact with it. There are several methods used
to evaluate wall shear stress in an open channel. The velocity profile method is popularly
used for experimental channels which are described below.
4.5.1. Velocity Profile Method
It is well established that for a regular prismatic channel under uniform flow conditions the
sum of retarding boundary shear forces acting on the wetted perimeter must be equal to the
resolved weight force along the direction of flow. Assuming the wall shear stress τ0 to be
constant over the entire boundary of the channel we can express τ0 as
=ρgRS
(4.1)
0
where, g = gravitational acceleration, = density of flowing fluid, S = slope of the energy
line, R = hydraulic radius of the channel cross section (A/P), A = area of channel cross
section, and P = wetted perimeter of the channel section. From the mixing length theory, the
shear stress for the turbulent flow is given as
0
=
k
2



 du
h `2 
 dh
2
(4.2)
where, u = the velocity at location h` from the wall, k = Von Karman’s constant which has a
value of approximately 0.40 for most of the flows. The shear stress
can be assumed to be equal to that at the boundary
true by measurements. Substituting,

0 



u =
 *

0),
0
close to the boundary
as is indeed shown to be reasonably
in equation (4.2), integrating and taking
u = 0 at h` = h o where, ho is the distance from the channel bottom at which logarithmic law
indicates zero velocity, the following equation results
 h` 
 h` 
u 1  h `  2 .3
= ln  =
log  =5 .75 log  
u* k  h 0  k
 h0 
 h0 
Or
 h` 

u = 5.75 u log 
 h0 
(4.3)
Equation (4.3) is known as Prandtl-Karman law of velocity distribution and is generally
found applicable over the entire depth of flow. In a curved channel, there is variation of the
retarding shear force in the longitudinal and transverse direction and the variation is
dependent on the location at the bend and the radius of curvature.
48
Results and Discussions
Nikuradse’s experiment has enabled the evaluation h` for different types of boundaries.
For smooth boundary h`= d`/107, where as for rough boundary h`=Ks/30. Combining, with
equation (4.3) we get,
 hu 
u
 + 5.5 8
= 5.75log 
u*


For smooth boundary
u
= 5.75log
u*
and for rough boundary
 30 h 
 h

 = 5.75log 
 ks 
 ks
(4.4a)

 + 8.5

(4.4b)
An indirect method to calculate the boundary shear stress is the graphical plotting of
velocity distribution based on the work of Karman and Prandtl. Let u 1 and u2 are the time
averaged velocities measured at h`1 and h`2 heights respectively from the boundary. From
the closely spaced velocity distribution observed at the vicinity of the channel bed and the
wall we can take a difference of u and h` between two points 1 and 2 close to each other.
Substituting u1, h`1, u2, and h`2 in equation (4.4b), taking a finite difference and by
substituting u * =
0
u* =
we can write equation (4.4b) as
u 2 − u1
M
1
=
5.75 log 10 (h' 2 / h' 1 ) 5.75
0
Again by substituting u * =
0
where, M =
=
(4.4c)
in equation (4.4b) we can rewrite it as
 M 
 5.75 


2
(4.4d)
u 2 − u1
u 2 − u1
=
= the slope of the semi-log plot of
log 10 (h' 2 / h' 1 ) log 10 (h' 2 ) − log 10 (h' 1 )
velocity distributions near the channel bed and the wall.
Using the Micro-ADV, velocities readings close to boundary for each flow is recorded
for boundary shear evaluation. Knowing the value of M (slope of the semi-logarithmic plot
between distances from boundary
against the corresponding velocity values) and using
equation (4.4d), the local shear stress very close to the boundary are estimated.
49
Results and Discussions
4.6. DISTRIBUTION OF BOUNDARY SHEAR STRESS
Most of hydraulic formulae assume that the boundary shear stress distribution is uniform
over the wetted perimeter. However, for meander channel geometry, there is a wide
variation in the local shear stress distribution from point to point in the wetted perimeter.
Therefore, there is a need to evaluate the shear stress carried by the channel boundary at
various locations of meander path. Boundary shear stress measurements at one section of
straight channel and at the bend apex of a meander path covering a number of points in the
wetted perimeter have been obtained from the semi-log relationship of velocity distribution.
For each run of the experiment, shear stress distributions are found. The distribution of
boundary shear along the channel perimeter for Type-II straight channel is shown in Figs.
4.5.1-4.5.4. Again, for meander channels of Type-III and Type-IV, the distribution of
boundary shear along the channel perimeter at bend apex of the meander path is shown in
Figs. 4.6.1-4.6.1 and Figs. 4.7.1-4.7.6. The total perimeter is marked and calculation of shear
stress is carried out for both the walls and bed of the channel. The slope of the semi log-plot
(M) close to the boundary at different points along the wetted perimeter of channels is
calculated using equation (4.4d). Comparisons of mean shear obtained by the velocity
distribution approach and energy gradient methods for straight and meander channels are
presented in Table 4.2.
Table 4.2 Comparison of the Boundary Shear Results for Straight Channel and
Meandering Channels at Bend Apex-AA
Overall shear
Overall Shear Force
Wetted
Flow
Discharge Cross Section
Force by Energy
by Velocity
Channel Type
Run No
Perimeter
Depth(m)
(m3/s)
Area (m2)
Gradient
Distribution
(m)
Approach (N/m)
Approach (N/m)
(1)
(2)
(3)
(4)
(5)
(6)
(9)
(10)
ST1
0.052
0.00477
0.00894
0.267
0.290
0.310
Straight
ST2
0.064
0.00696
0.01178
0.301
0.381
0.396
Trapezoidal
Channel
ST3
0.070
0.00815
0.01330
0.318
0.431
0.459
(Type-II,
ST4
0.074
0.00896
0.01436
0.329
0.465
0.467
Sinuosity =1)
ST5
0.078
0.00974
0.01536
0.340
0.497
0.502
MR6
0.0531
0.00236
0.00637
0.2262
0.194
0.203
Mildly Meandering MR8
0.0608
0.00276
0.00730
0.2416
0.222
0.234
Rectangular
MR10
0.0711
0.00334
0.00853
0.2622
0.260
0.273
Channel
MR12
0.0855
0.00419
0.01026
0.2910
0.312
0.310
(Type-III,
MR13
0.0934
0.00466
0.01121
0.3068
0.362
0.371
Sinuosity =1.44)
MR15
0.1101
0.00568
0.01321
0.3402
0.402
0.430
MT23
0.0530
0.00419
0.00917
0.2699
0.477
0.483
Highly Meandering MT24
0.0562
0.00486
0.00990
0.2789
0.515
0.544
Trapezoidal
MT25
0.0593
0.00512
0.01063
0.2877
0.553
0.586
Channel
MT26
0.0618
0.00552
0.01124
0.2948
0.584
0.610
(Type-IV,
MT27
0.0671
0.00640
0.01255
0.3098
0.653
0.707
Sinuosity =1.91)
MT28
0.0733
0.00755
0.01417
0.3273
0.737
0.794
50
Results and Discussions
For the experimental channels, the mean shear found from the velocity distribution
approach agrees well with the mean value computed from energy gradient approach. The
energy gradient states that the weight of water in the direction of flow in the channel is
equal to the resistance offered by the channel. The following features can be noted from
the figures of boundary shear distribution.
4.6.1. Straight Channel
¯ Symmetrical and uniform nature of boundary shear stress distribution is found for
straight channel of Type-II when compared to the meandering channels of Type-III and
Type-IV.
¯ The boundary shear at the channel junctions are generally found to be more than that
compared to other points of the wetted perimeter.
¯ The overall mean value of boundary shear in the channel increases proportionately with
the flow depth.
¯ Total shear carried by the wetted perimeter of the channel compares well with the
energy gradient approach.
4.6.2. Meandering Channels
¯ On comparison of the results with straight uniform channel, it can be seen that there is
asymmetrical nature of shear distribution especially where there is predominant
curvature effect.
¯ The over all mean value of boundary shear stress obtained through the velocity
distribution approach compares well with that obtained from energy gradient approach.
¯ Maximum value of wall shear occurs significantly below the free surface and is located
at the inner walls.
51
Results and Discussions
4.7. ROUGHNESS COEFFICIENTS IN OPEN CHANNEL FLOW
Distribution of energy in a channel section is an important aspect that needs to be addressed
properly. While using Manning’s equation, selection of a suitable value of n is the single
most important parameter for the proper estimation of velocity in an open channel. Major
factors affecting Manning’s roughness coefficient are the (i) surface roughness, (ii)
vegetation, (iii) channel irregularity, (iv) channel alignment, (v) silting and scouring, (vi)
shape and size of a channel, and (vii) stage-discharge relationship. Assuming the flow to be
uniform and neglecting all non-friction losses, the energy gradient slope can be considered
equal to the average longitudinal bed slope S of a channel. Under steady and uniform flow
conditions, we use the equations proposed by Chezy, Darcy-Weisbach (1857), or Manning
(1891) to compute the section mean velocity carried by a channel section proposed as
Manning’s
1 2
U= R 3 S
n
(4. 5)
Chezy’s equation
U = C RS
(4. 6)
Darcy-Weisbach’s equation
U=
8gRS
f
(4. 7)
where, A = the channel cross-sectional area, g = gravitational acceleration, R = the hydraulic
mean radius of the channel section, f = the friction factor used, n = Manning’s resistive
coefficient, and C = Chezy’s channel coefficient.
Sinuosity and slope have significant influences for the evaluation of channel discharge.
The variation of resistance coefficients for the present experimental meandering channels are
found to vary with depth, aspect ratio, slope and sinuosity and are all linked to the stagedischarge relationships.
52
Results and Discussions
Table 4.3 (a) Experimental Results for Straight Channels of Type-I and Type-II
Showing n, C and f (Sinuosity=1)
Channel Type
Average
Discharge Flow Channel Cross
Wetted
Velocity
Run
Q
Depth Width Section Perimeter
(Q/A)
No
3
2
(m /sec) h (m) b (m) Area (m )
(m)
(m/sec)
(1)
(2)
SR1
SR2
SR3
SR4
Type-I Straight
SR5
Rectangular
SR6
Channel
SR7
Slope=0.0019
SR8
SR9
SR10
SR11
ST1
Type-II Straight
ST2
Trapezoidal
ST3
Channel
ST4
Slope=0.0033
ST5
(3)
0.00106
0.00128
0.00215
0.00231
0.00290
0.00325
0.00412
0.00455
0.00506
0.00595
0.00631
0.00477
0.00696
0.00815
0.00896
0.00974
(4)
0.030
0.034
0.050
0.052
0.062
0.068
0.082
0.088
0.096
0.109
0.115
0.052
0.064
0.070
0.074
0.078
(5)
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
(6)
0.0036
0.0041
0.0060
0.0063
0.0075
0.0082
0.0098
0.0106
0.0115
0.0131
0.0138
0.0089
0.0118
0.0133
0.0144
0.0154
(7)
0.18
0.19
0.22
0.22
0.24
0.26
0.28
0.30
0.31
0.34
0.35
0.27
0.30
0.32
0.33
0.34
(8)
0.293
0.310
0.359
0.367
0.389
0.398
0.421
0.430
0.441
0.454
0.458
0.533
0.591
0.613
0.624
0.634
(All Measurements in S.I. unit)
Darcy
Manning`s
Chezy`s Weisbach
S n Roughness
Friction
C
n
Factor f
(11)
3.96
3.97
3.97
3.98
3.99
3.96
3.97
3.96
3.99
3.97
3.96
5.13
5.13
5.08
5.04
5.00
(10)
0.0110
0.0110
0.0110
0.0109
0.0109
0.0110
0.0110
0.0110
0.0109
0.0110
0.0110
0.0110
0.0102
0.0112
0.0101
0.0111
(12)
47.389
48.109
49.987
50.327
51.143
51.163
51.950
52.168
52.748
52.909
52.953
50.692
52.023
52.140
52.041
51.903
(13)
0.0242
0.0235
0.0218
0.0215
0.0208
0.0208
0.0202
0.0200
0.0196
0.0194
0.0194
0.0212
0.0201
0.0200
0.0201
0.0202
Table 4.3 (b) Experimental Results of Meandering Channel of Type-III Showing n, C
and f (Sinuosity=1.44)
Channel Type
(1)
Run
No
Discharge Flow
Q
Depth
(m3/sec) h (m)
Cross
Average
Channel
Wetted
Section
Velocity
Width
Perimeter
Area
(Q/A)
b (m)
(m)
2
(m )
(m/sec)
(All Measurements in S.I. unit)
Darcy
Manning`s
Chezy`s Weisbach
S n Roughness
Friction
C
n
Factor f
(2)
MR1
(3)
0.00032
(4)
0.013
(5)
0.12
(6)
0.0015
(7)
0.15
(8)
0.207
(11)
5.14
(10)
0.0108
(12)
43.279
(13)
0.0419
MR2
MR3
MR4
Type-III
Meandering MR5
Rectangular MR6
Channel
MR7
Slope=0.0031 MR8
MR9
MR10
MR11
MR12
0.00043
0.00135
0.00167
0.00220
0.00236
0.00262
0.00276
0.00295
0.00334
0.00370
0.00419
0.016
0.034
0.041
0.050
0.053
0.058
0.061
0.064
0.071
0.077
0.086
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.0019
0.0041
0.0049
0.0060
0.0064
0.0069
0.0073
0.0077
0.0085
0.0092
0.0103
0.15
0.19
0.20
0.22
0.23
0.24
0.24
0.25
0.26
0.27
0.29
0.229
0.326
0.343
0.368
0.370
0.378
0.378
0.383
0.391
0.400
0.408
5.11
5.01
4.93
4.88
4.79
4.75
4.68
4.66
4.61
4.60
4.56
0.0109
0.0111
0.0113
0.0114
0.0116
0.0117
0.0119
0.0120
0.0121
0.0121
0.0122
44.203
47.561
47.594
48.099
47.504
47.433
46.865
46.897
46.739
46.966
46.881
0.0401
0.0347
0.0346
0.0339
0.0347
0.0348
0.0357
0.0356
0.0359
0.0355
0.0357
53
Results and Discussions
Table 4.3 (c) Experimental Runs for Meandering Channel of Type-IV Showing n, C and
f (Sinuosity=1.91)
Channel Type Run No
(2)
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.003
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.0042
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.0053
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.008
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.013
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.015
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.021
(1)
MT1
MT2
MT3
MT4
MT5
MT6
MT7
MT8
MT9
MT10
MT11
MT12
MT13
MT14
MT15
MT16
MT17
MT18
MT19
MT20
MT21
MT22
MT23
MT24
MT28
MT29
MT30
MT31
MT32
MT33
MT34
MT35
MT36
MT37
MT38
MT39
MT40
MT41
MT42
MT43
MT44
MT45
MT46
MT47
MT48
MT49
MT50
MT51
MT52
MT53
MT54
Discharge
Q
(m3/sec)
(3)
0.00148
0.00261
0.00310
0.00372
0.00410
0.00456
0.00599
0.00480
0.00533
0.00590
0.00721
0.00781
0.00864
0.00029
0.00048
0.00099
0.00174
0.00205
0.00276
0.00322
0.00334
0.00370
0.00419
0.00466
0.00090
0.00142
0.00217
0.00486
0.00549
0.00677
0.00847
0.00236
0.00488
0.00541
0.00599
0.00718
0.00816
0.00948
0.00132
0.00274
0.00450
0.00488
0.00621
0.00811
0.00840
0.00129
0.00215
0.00335
0.00407
0.00545
0.00612
Average
Wetted
Flow Channel Cross
Velocity
Depth Width Section Perimeter
(Q/A)
2
h (m) b (m) Area (m )
(m)
(m/sec)
(4)
(5)
(6)
(7)
(8)
0.031 0.12
0.0047
0.21
0.316
0.045 0.12
0.0074
0.25
0.351
0.050 0.12
0.0085
0.26
0.364
0.056 0.12
0.0099
0.28
0.377
0.059 0.12
0.0106
0.29
0.388
0.063 0.12
0.0115
0.30
0.396
0.074 0.12
0.0144
0.33
0.418
0.057 0.12
0.0101
0.28
0.476
0.061 0.12
0.0110
0.29
0.483
0.065 0.12
0.0120
0.30
0.491
0.072 0.12
0.0138
0.32
0.522
0.075 0.12
0.0146
0.33
0.534
0.078 0.12
0.0154
0.34
0.559
0.011 0.12
0.0014
0.15
0.215
0.014 0.12
0.0019
0.16
0.250
0.022 0.12
0.0032
0.18
0.313
0.031 0.12
0.0047
0.21
0.368
0.034 0.12
0.0053
0.22
0.386
0.041 0.12
0.0066
0.24
0.418
0.046 0.12
0.0075
0.25
0.428
0.047 0.12
0.0077
0.25
0.431
0.049 0.12
0.0083
0.26
0.443
0.053 0.12
0.0092
0.27
0.457
0.056 0.12
0.0099
0.28
0.470
0.019 0.12
0.0026
0.17
0.342
0.025 0.12
0.0036
0.19
0.391
0.033 0.12
0.0050
0.21
0.437
0.051 0.12
0.0087
0.26
0.558
0.056 0.12
0.0097
0.28
0.564
0.062 0.12
0.0113
0.30
0.600
0.070 0.12
0.0133
0.32
0.637
0.031 0.12
0.0047
0.21
0.504
0.048 0.12
0.0080
0.25
0.613
0.051 0.12
0.0087
0.26
0.620
0.054 0.12
0.0094
0.27
0.638
0.060 0.12
0.0108
0.29
0.665
0.064 0.12
0.0118
0.30
0.693
0.070 0.12
0.0133
0.32
0.713
0.021 0.12
0.0030
0.18
0.445
0.033 0.12
0.0050
0.21
0.543
0.044 0.12
0.0072
0.24
0.623
0.046 0.12
0.0076
0.25
0.640
0.053 0.12
0.0092
0.27
0.677
0.062 0.12
0.0112
0.29
0.727
0.063 0.12
0.0115
0.30
0.728
0.019 0.12
0.0026
0.17
0.487
0.026 0.12
0.0038
0.19
0.566
0.034 0.12
0.0052
0.22
0.639
0.039 0.12
0.0062
0.23
0.656
0.046 0.12
0.0076
0.25
0.714
0.049 0.12
0.0083
0.26
0.739
54
(All Measurements in S.I. unit)
Darcy
Manning`s
S n Roughness Chezy`s C Weisbach
Friction
n
Factor f
(9)
(10)
(11)
(12)
5.47
0.0100
53.10
0.0278
5.02
0.0109
51.09
0.0300
4.94
0.0111
50.96
0.0302
4.83
0.0113
50.57
0.0307
4.84
0.0113
51.01
0.0301
4.78
0.0115
50.78
0.0304
4.66
0.0118
50.45
0.0308
6.04
0.0107
53.54
0.0273
5.93
0.0109
52.98
0.0279
5.84
0.0111
52.59
0.0284
5.90
0.0110
53.81
0.0271
5.92
0.0110
54.26
0.0266
6.08
0.0107
56.00
0.0250
6.78
0.0107
42.57
0.0433
6.58
0.0111
43.26
0.0419
6.47
0.0113
45.18
0.0384
6.34
0.0115
46.34
0.0365
6.33
0.0115
46.81
0.0358
6.26
0.0116
47.40
0.0349
6.09
0.0120
46.70
0.0360
6.07
0.0120
46.64
0.0360
6.05
0.0120
46.89
0.0357
6.02
0.0121
47.07
0.0354
6.02
0.0121
47.37
0.0349
7.71
0.0116
42.90
0.0426
7.58
0.0118
43.78
0.0409
7.38
0.0121
44.14
0.0402
7.49
0.0119
47.43
0.0349
7.26
0.0123
46.45
0.0363
7.31
0.0122
47.41
0.0349
7.31
0.0122
48.13
0.0338
8.72
0.0131
40.65
0.0474
8.53
0.0134
42.01
0.0444
8.33
0.0137
41.39
0.0458
8.32
0.0137
41.64
0.0452
8.23
0.0138
41.74
0.0450
8.31
0.0137
42.45
0.0435
8.18
0.0139
42.24
0.0439
9.50
0.0129
39.12
0.0512
9.10
0.0135
39.83
0.0494
9.02
0.0136
40.92
0.0468
9.05
0.0135
41.30
0.0460
8.92
0.0137
41.46
0.0456
8.89
0.0138
42.08
0.0443
8.80
0.0139
41.79
0.0449
10.97
0.0132
37.67
0.0553
10.76
0.0135
38.57
0.0527
10.55
0.0137
39.15
0.0511
10.10
0.0144
38.15
0.0539
10.10
0.0143
38.96
0.0517
10.13
0.0143
39.40
0.0505
Results and Discussions
4.7.1.Variation of Manning’s n with Depth of Flow
The experimental results for Manning’s n with depth of flow for channels investigated
are plotted in Fig. 4.8. 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 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 slopes. For narrow
channels the decrease in the value of n with depth is mainly due to the decrease of the
resistance to flow and for wider channels the values of n increases with sinuosity and
channel slope. For steeper slope Manning’s n is less and for milder it is more. Again, for
highly sinuous channels the values of n become large indicating that the energy loss is more
for such channels.
Fig. 4.8 Variation of Manning`s n with Flow Depth
4.7.2.Variation of Chezy`s C with Depth of Flow in Open Channel
The variation of Chezy’s C with depth of flow for the channels investigated for different
slopes is shown in Fig. 4.9. It can be seen from the figure that the meandering channels,
exhibits a steady increase in the value of C with depth of flow. Chezy`s C is found to
decrease with increase of aspect ratio indicating that the meander channel consumes more
energy as the depth of flow increases. For a channel with increase in slope, the Chezy`s C is
less for a particular stage.
55
Results and Discussions
Fig. 4.9 Variation of Chezy`s C with Flow Depth
4.7.3.Variation of Darcy-Weisbach f with Depth of Flow in Open Channel
The variation of friction factor f with depth of flow for the straight and meandering channels
are shown in Fig. 4.10. The behavioral trend of friction factor f is decreasing with flow
depth.
Fig. 4.10 Variation of Darcy - Weisbach f with Flow Depth
From the Figs. 4.8 and 4.10, it is seen that the roughness coefficients n and f are not
behaving in similar manner for the experimental channels. The cause of this behavior may
be due to the large variation of hydraulic radius (R) with depth of flow. By comparing
equations 4.6 and 4.8, the relationship between the coefficients with hydraulic radius (R)
56
Results and Discussions
can be expressed as f =
8g
R
1
3
n2 . For the present study, dimensional analysis is done to
evaluate the variation of roughness coefficients with depth of flow for meandering
channels which is discussed in the next section. Some of the graphs have been omitted in
the Figs. 4.8, 4.9 and 4.10 to maintain clarity of the graphs.
4.8. DIMENSIONAL ANALYSIS
Dimensional analysis offers a method for reducing complex physical problems to the
simplest (i.e., most economical) form prior to obtaining a quantitative answer. Bridgman
(1969) explains it thus: "The principal use of dimensional analysis is to deduce from a study
of the dimensions of the variables in any physical system certain limitations on the form of
any possible relationship between those variables. The method is of great generality and
mathematical simplicity".
Accurate discharge assessment for meandering channel requires dimensional analysis.
In quantitative analysis of physical events one seeks mathematical relationships between the
numerical values of the physical quantities that describe the event.
A physical equation must be dimensionally homogeneous. Every correct physical
equation—that is, every equation that expresses a physically significant relationship between
numerical values of physical quantities—must be dimensionally homogeneous. Dimensional
analysis reduces the number of variables that must be specified to describe an event.
4.8.1. Evaluation Of Roughness Coefficients
The important variables affecting the stage-discharge relationship are considered to be
velocity U, hydraulic radius R, viscosity ν, gravitational acceleration g, bed slope S,
sinuosity S r, aspect ratio α, and it can be expressed functionally as,
φ {U , S , S r , α , g , R ,ν }
(4. 8)
The dimensionless group used because of its similarity to the traditional roughness
coefficient equations for one dimensional flow in a channel is expressed as
57
Results and Discussions
U =
1
R
n
2
3
S
1
2
10
1  R 3S
UR
= 
ν
n ν 2

So,
1
U =C R S
Again, since
2
1
2
UR C 3 2 12
= R S
ν
ν
and
UR  8
= 
ν
 f
1
2
1
2
(4.9a)
and
=>








 8g
U = 
RS 

 f
1
2
(4.9b)
 gR 3 S

2
 ν



1
2
(4.9c)
where, n = Manning’s roughness co-efficient, C = Chezy`s roughness co-efficient and f =
Darcy-Weisbach friction factor.
4.8.1.1.
Derivation of Roughness Coefficients using Dimensional Analysis
Sinuosity (Sr) and is inversely related to the velocity therefore, it is in denominator. Eq. (4.8)
may be rewritten in the following form as
 g . R 10 3
 1
UR
= φ  , S ,α , 
 2 13
ν
Sr
ν m
The dimensional group
 g .R10 3 ν 2 m 13 






(4.10)
is used because of its similarity to the traditional
Manning`s equation for one dimensional flow in a prismatic channel. As manning`s
1
roughness coefficient is dimensionally non-homogenous, a length factor m 3 is considered to
it to make it homogenous. After dimensional analysis, we obtain a relation between the
parameters.
Now,
 gR103 S 12 
 2 1 
 ν m 3 
vs.
URS r 


 να 
is plotted using the new collected experimental data
together with data from previously collected for meandering channel for different slopes,
sinuosity, geometry etc in an attempt to find a simple relationship between the dimensionless
groups for meandering channel shapes, under different hydraulic conditions.
58
Results and Discussions
Fig. 4.11 Calibration Equation between  gR
10
3
S

1
2
 ν m 3
1
2



and
URS r  for


 να 
Manning`s n
After plotting between dimensionless parameters, the best possible combination is found to
be given by the power equation having a regression correlation of 0.98 as,
 gR10 3 S 12 
URS r 
 = 7k  2 1 

 να 
 ν m 3 
0. 86
(4.11)
The newly developed relationship for Manning`s n in a meandering channel is expressed as
Manning`s n
n =
S rν 0 .72 S 0 .07 m 0 . 29
7 k α g 0 . 86 R 1 . 2
(4.12)
where, k is a constant of value 0.001found from the experiments.
Similarly, eq. (4.8) can also be rewritten for Chezy`s C and Darcy Weisbach f using
dimensional analysis. Darcy Weisbach f is a dimensionally homogenous but Chezy`s C is
dimensionally non-homogenous. So, the dimensional group (g .R 3 ν 2 ) is used because of its
similarity to the traditional Chezy`s equation for one dimensional flow in a prismatic
channel. Both give similar equation which is expressed as
 1
UR
, S ,α
=φ
ν
Sr
 gR 3
, 
2
 ν
59

 

(4.13)
Results and Discussions
Then,
 gR 3 S 1 2 


2
 ν

vs.
 URS

 να
r



is plotted using the new collected experimental data together
with data from previously collected for meandering channel for different slopes etc, in an
attempt to find a simple relationship between the dimensionless groups for meandering
channel shapes, under different hydraulic conditions.
Fig. 4.12 Calibration Equation for between
 gR3S 12 
 2 
 ν

and
URS r  for


 να 
Derivation of
Chezy`s C and Darcy Weisbach f
After plotting between dimensionless parameters, the best possible condition is found to be
given by the power equation having a regression correlation of 0.97 is,
 gR 3 S 12 
URS r 
=k ×


2
 να 
 ν

0.86
(4.14)
Using he relation obtained from equation (4.14) and equation (4.9b), the newly developed
relationship for Chezy`s C in a meandering channel is expressed as
Chezy`s C
C = k ×
g 0 . 95 R 1 . 35 α
S 0 . 02 ν 0 . 9 S r
where k is a constant of value 0.001found from the experiments.
60
(4.15)
Results and Discussions
Again, from equation (4.14) and equation (4.9c), a relationship for Darcy Weisbach f is
developed which is expressed as
8ν 1 . 8 S r S 0 . 04
0 . 01 2 α 2 R 2 . 7 g 0 . 9
2
Darcy-Weisbach f
f =
(4.16)
Thus the newly developed equations of roughness coefficients as obtained from equations
4.12, 4.15 and 4.16 can be summarized as
S rν 0 . 72 S 0 . 07 m 0 . 29
7 k α g 0 .86 R 1 . 2
Manning`s n
n =
Chezy`s C
C = k ×
Darcy Weisbach f
f =
g 0 . 95 R 1 . 35 α
S 0 . 02 ν 0 . 9 S r
8ν 1 . 8 S r S 0 . 04
0 . 01 2 α 2 R 2 . 7 g 0 . 9
2
Here, k is a constant of value 0.001found from the experiments.
4.9. DISCHARGE ESTIMATION USING THE PRESENT APPROACH
For the given geometry of channels, the values of n, c and f are calculated using the
equations (4.12), (4.15) and (4.16).
Then, the corresponding discharge can be easily
calculated. Graphs are plotted using the above mentioned equations, showing the variations
of observed and calculated (modeled) discharge in Figs. 4.13(a), 4.13(b) and 4.13(c) for
Manning`s n, Chezy`s C and Darcy Weisbach f, respectively.
Fig. 4.13 (a) Variation of Observed and Modeled Discharge Using Manning`s n
61
Results and Discussions
Fig. 4.13 (b) Variation of Observed and Modeled Discharge Using Chezy`s C
Fig. 4.13 (c) Variation of Observed and Modeled Discharge Using Darcy Weisbach f
From the figures 4.13 (a, b, c) it is clear that the present formulation gives better
results. The calculated (modeled) discharge using the newly developed equations of
manning`s n, Chezy`s C and Darcy-Weisbach f from equations 4.12, 4.15 and 4.16
respectively, is in good agreement with the observed value. The calculated discharge for the
meandering channels of Type-III and Type-IV are more close to the observed value.
4.10. APPLICATION OF OTHER METHODS TO THE PRESENT CHANNEL
Some published approaches of discharge estimation for meandering flow are discussed and
applied to the experimental data of the present very wide and highly sinuous channels to
know their suitability for such geometry. The standard error of estimation of discharge by
the present equation is compared well with some of the methods of other investigators which
are shown in Fig. 4.14(a), 4.14(b) and 4.14(c) for n, C and f, respectively.
62
Results and Discussions
Fig. 4.14 (a) Standard Error of Estimation of Discharge using Proposed Manning`s
Equation and other established Methods
Fig. 4.14(b) Standard Error of Estimation of Discharge using Proposed Chezy`s
Equation other established Methods
63
Results and Discussions
Fig. 4.14(c) Standard Error of Estimation of Discharge using Proposed Darcy
Weisbach f and other established Methods
From Fig. 4.14 (a) the overall standard error of discharge from the proposed method
using Manning`s n is 14.39 whereas, it is 30.04 for LSCS method, closely followed by the
traditional method with standard error of 31.32 and the Shino Knight method with a value of
46.9. Similarly, from Fig. 4.14 (b) the overall standard error of discharge for the proposed
method using Chezy`s C is 18.08 compared to 28.5 for LSCS method. In this case, the
conventional method gives standard error of 31.32 followed by Shino and Knight method
with standard error of 46.9. Further using Darcy Weisbach f for discharge calculation, the
standard error for the present method is 18.07. This is in comparison to 20.79 for
conventional method, 29.79 and 46.8 for LSCS and the Shino Knight method, respectively
(Fig. 4.14 c). From the Figs. 4.14(a, b, c), it reveals that the proposed approach gives better
discharge results as compared to other approaches for the present experimental channels.
4.11. BOUNDARY SHEAR DISTRIBUTION IN MEANDERING CHANNEL
Information regarding the nature of boundary shear stress distribution is needed to solve
a variety of river hydraulics and engineering problems such as to give a basic understanding
of the flow resistance relationship, to understand the mechanism of sediment transport, to
design stable channels, revetments etc. The tractive force is commonly used to describe the
average shear of the flow exerted on the wetted perimeter.
64
Results and Discussions
The complex three-dimensional flow structures in a channel leads to a complicated
patterns of boundary shear stress distribution and becomes even more complicated when the
channel is meandering. One line of approach to predict the distribution around the wetted
perimeter of a given channel for a certain flow rate is from a theoretical approach,
concerning a detailed knowledge of the distribution of numerous turbulence coefficients.
The other one is to adopt an empirical approach, fitting equations to the data obtained. The
first step of in deducing an empirical approach would be to obtain equations for the mean
boundary shear stress or force on particular boundary elements. Then, it would be possible
to fit equations for the local variations around the mean on a particular boundary element.
This has been attempted many investigators as Knight (1981), Knight and Hamed (1984),
Knight and Patel (1985), Patra (1999), Khatua (2008), Kean et al. (2009) etc.
The percentage of the total shear force carried by the walls (%SFW) is a useful
parameter expressed as %SFW = (100 SFW) / SF. Similarly, the percentage of the total shear
force carried by the bed, given as %SFB = (100 SFB) / SF, where %SFW =Percentage of shear
force which is carried by the walls, %SFB= percentage of shear force carried by the base and
SF= total shear force.
Some of the works of previous investigators are like Ghosh and Roy (1970), Myers
(1978) and Noutsopoulos and Hadjipanos (1982), etc in smooth rectangular channels, Patel
(1984) and in smooth rectangular ducts in smooth trapezoidal channels with 45° side walls
slope, and with uniformly roughened trapezoidal channels. They have used experimental
data either for uniformly roughened or smooth trapezoidal channels by the use of duct and
open channel flow data together. The %SFW values show a systematic reduction with
increase in aspect ratio (α =b/h), typical of an exponential function first proposed by Knight
(1981).
Knight et.al (1981, 1984) showed that for a straight channel the % of shear in wall
%SFW varied exponentially with the aspect ratio [i.e.α= b/h)]. It can be expressed as
% SFw = e m
(4.17)
By plotting separately to log-log scales and assuming a simpler linear relationship
between %SFw and (α=b/h), we have
log 10 (% SF w ) = 1 . 4026 log 10 (
65
b
+ 3 ) + 2 . 67
h
(4.18)
Results and Discussions
Comparing equation 4.17 and 4.18 we can write
b

m = 2 . 30259 [ A1 Log 10  + A 2  + A 3 ]
h


or m = − 3 . 23 Log
10
(α
+ 3 ) + 6 . 146
(4.19)
It proves that m is a function of the aspect ratio [i.e. m = F (b/h) = F (α)].
Table 4.4(a) Summary of Experimental Results Showing Overall Shear Stress in
Straight Channels of Type-I and II (Sinuosity=1)
Channel Type Run No
(1)
Type-I
Straight
Rectangular
Channel
Slope=0.0019
Type-II
Straight
Trapezoidal
Channel
Slope=0.0033
Cross
Wetted Average Overall Mean
Flow
Discharge Q
Channel
Section Perimeter Velocity Shear Stress
Depth
3
(m /s)
Width (m)
2
2
Area (m )
(m)
(m/s)
h (m)
(N/m )
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
SR1
SR 2
SR3
SR4
SR5
SR6
SR7
SR8
SR9
SR10
SR11
ST1
ST2
ST3
ST4
ST5
0.00106
0.00128
0.00215
0.00231
0.00290
0.00325
0.00412
0.00455
0.00506
0.00595
0.00631
0.00477
0.00696
0.00815
0.00896
0.00974
0.030
0.034
0.050
0.052
0.062
0.068
0.082
0.088
0.096
0.109
0.115
0.052
0.064
0.070
0.074
0.078
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.0036
0.0041
0.0060
0.0063
0.0075
0.0082
0.0098
0.0106
0.0115
0.0131
0.0138
0.0089
0.0118
0.0133
0.0144
0.0154
0.18
0.19
0.22
0.22
0.24
0.26
0.28
0.30
0.31
0.34
0.35
0.27
0.30
0.32
0.33
0.34
0.293
0.310
0.359
0.367
0.389
0.398
0.421
0.430
0.441
0.454
0.458
0.533
0.591
0.613
0.624
0.634
0.37
0.41
0.51
0.52
0.57
0.59
0.64
0.67
0.69
0.72
0.73
1.08
1.27
1.35
1.41
1.46
Table 4.4(b) Summary of Experimental Results Showing Overall Shear Stress in
Meandering Channel of Type-III (Sinuosity=1.44)
Channel Type
Run No
(1)
(2)
MR1
MR 2
MR 3
MR 4
MR 5
MR 6
MR 7
MR 8
MR 9
MR 10
MR 11
MR 12
MR 13
MR 14
MR 15
Type-III
Meandering
Rectangular
Channel
Slope= 0.0031
Flow Channel
Cross
Discharge
Depth Width
Section
3
Q (m /s)
2
h (m)
(m)
Area (m )
(3)
(4)
(5)
(6)
0.00032 0.013
0.12
0.0015
0.00043 0.016
0.12
0.0019
0.00135 0.034
0.12
0.0041
0.00167 0.041
0.12
0.0049
0.00220 0.050
0.12
0.0060
0.00236 0.053
0.12
0.0064
0.00262 0.058
0.12
0.0069
0.00276 0.061
0.12
0.0073
0.00295 0.064
0.12
0.0077
0.00334 0.071
0.12
0.0085
0.00370 0.077
0.12
0.0092
0.00419 0.086
0.12
0.0103
0.00466 0.093
0.12
0.0112
0.00560 0.109
0.12
0.0130
0.00568 0.110
0.12
0.0132
66
Wetted
Perimeter
(m)
(7)
0.15
0.15
0.19
0.20
0.22
0.23
0.24
0.24
0.25
0.26
0.27
0.29
0.31
0.34
0.34
Average
Velocity
(m/s)
(8)
0.207
0.229
0.326
0.343
0.368
0.370
0.378
0.378
0.383
0.391
0.400
0.408
0.415
0.429
0.430
Overall Mean
Shear Stress
2
(N/m )
(9)
0.32
0.38
0.66
0.74
0.83
0.86
0.90
0.92
0.94
0.99
1.03
1.07
1.11
1.18
1.18
Results and Discussions
Table 4.4(c) Summary of Experimental Results Showing Overall Shear Stress in
Meandering Channel of Type-IV (Sinuosity=1.91)
Channel Type
(1)
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.003
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.0042
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.0053
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.008
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.013
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.015
Type-IV
Meandering
Trapezoidal
Channel
Slope=0.021
Run No
(2)
MT1
MT2
MT3
MT4
MT5
MT6
MT7
MT8
MT9
MT10
MT11
MT12
MT13
MT15
MT16
MT17
MT18
MT19
MT20
MT21
MT22
MT23
MT24
MT25
MT26
MT27
MT28
MT29
MT30
MT31
MT32
MT33
MT34
MT35
MT36
MT37
MT38
MT39
MT40
MT41
MT42
MT43
MT44
MT45
MT46
MT47
MT48
MT49
MT50
MT51
MT52
MT53
MT54
MT55
Discharge
Q
3
(m /s)
(3)
0.00148
0.00261
0.00310
0.00372
0.00410
0.00456
0.00599
0.00480
0.00533
0.00590
0.00721
0.00781
0.00864
0.00048
0.00099
0.00174
0.00205
0.00276
0.00322
0.00334
0.00370
0.00419
0.00466
0.00552
0.00640
0.00755
0.00090
0.00142
0.00217
0.00486
0.00549
0.00677
0.00847
0.00236
0.00488
0.00541
0.00599
0.00718
0.00816
0.00948
0.00132
0.00274
0.00450
0.00488
0.00621
0.00811
0.00840
0.00129
0.00215
0.00335
0.00407
0.00545
0.00612
0.00922
Flow
Depth
h (m)
(4)
0.031
0.045
0.050
0.056
0.059
0.063
0.074
0.057
0.061
0.065
0.072
0.075
0.078
0.014
0.022
0.031
0.034
0.041
0.046
0.047
0.049
0.053
0.056
0.062
0.067
0.073
0.019
0.025
0.033
0.051
0.056
0.062
0.070
0.031
0.048
0.051
0.054
0.060
0.064
0.070
0.021
0.033
0.044
0.046
0.053
0.062
0.063
0.019
0.026
0.034
0.039
0.046
0.049
0.063
Channel
Width (m)
(5)
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
0.12
67
Cross
Section
2
Area (m )
(6)
0.0047
0.0074
0.0085
0.0099
0.0106
0.0115
0.0144
0.0101
0.0110
0.0120
0.0138
0.0146
0.0154
0.0019
0.0032
0.0047
0.0053
0.0066
0.0075
0.0077
0.0083
0.0092
0.0099
0.0112
0.0126
0.0142
0.0026
0.0036
0.0050
0.0087
0.0097
0.0113
0.0133
0.0047
0.0080
0.0087
0.0094
0.0108
0.0118
0.0133
0.0030
0.0050
0.0072
0.0076
0.0092
0.0112
0.0115
0.0026
0.0038
0.0052
0.0062
0.0076
0.0083
0.0114
Wetted
Perimeter
(m)
(7)
0.21
0.25
0.26
0.28
0.29
0.30
0.33
0.28
0.29
0.30
0.32
0.33
0.34
0.16
0.18
0.21
0.22
0.24
0.25
0.25
0.26
0.27
0.28
0.29
0.31
0.33
0.17
0.19
0.21
0.26
0.28
0.30
0.32
0.21
0.25
0.26
0.27
0.29
0.30
0.32
0.18
0.21
0.24
0.25
0.27
0.29
0.30
0.17
0.19
0.22
0.23
0.25
0.26
0.30
Average
Velocity
(m/s)
(8)
0.316
0.351
0.364
0.377
0.388
0.396
0.418
0.476
0.483
0.491
0.522
0.534
0.559
0.250
0.313
0.368
0.386
0.418
0.428
0.431
0.443
0.457
0.470
0.491
0.509
0.533
0.342
0.391
0.437
0.558
0.564
0.600
0.637
0.504
0.613
0.620
0.638
0.665
0.693
0.713
0.445
0.543
0.623
0.640
0.677
0.727
0.728
0.487
0.566
0.639
0.656
0.714
0.739
0.809
Overall Mean
Shear Stress
2
(N/m )
(9)
0.66
0.88
0.96
1.04
1.08
1.14
1.28
1.48
1.56
1.63
1.76
1.81
1.87
0.63
0.90
1.18
1.27
1.45
1.57
1.60
1.67
1.77
1.85
1.98
2.11
2.25
1.19
1.49
1.84
2.59
2.76
3.00
3.28
2.87
3.99
4.21
4.39
4.75
4.99
5.33
2.43
3.48
4.34
4.49
5.00
5.59
5.69
3.13
4.04
4.99
5.55
6.29
6.60
7.92
Results and Discussions
The overall summary of the experiments runs concerning the distribution of shear
force and boundary shear stress are presented in Table 4.4 (a) for Type-I and Type-II,
Table 4.4 (b) for Type-III and Table 4.4 (c) for Type-IV channels respectively.
4.11.1. Analysis of Boundary Shear Results in Meandering Channels
For the present analysis, the published meandering channel data of Patra (1999) along
with the current meandering channel data are used. Patra (1999) provided shear force
data of two types of meandering channel having smooth and rough surfaces having
sinuosity (S r) 1.21 and 1.22 and aspect ratios varying from α =1.01 to 2.45. Now for
the four types of meandering channels having sinuosity 1.21, 1.22, 1.44 and 1.91, the
variation of wall shear with aspect ratio and sinuosity are shown in Fig. 4.15 (a, b)
respectively.
Fig. 4.15 (a) Variation of Percentage of Wall Shear with Aspect Ratio
In the Fig. 4.15(a), it is observed that the difference factor (DF) curves of the observed three
channels remains almost parallel. The nature of variations of difference factor (DF) with
respect to aspect ratio is same for all channels, irrespective of the difference in their
sinuosity. Again, the plots in Fig 4.15 (b) show that the change in difference factor (DF) for
channels having higher sinuosity is very small, irrespective of the change in geometry
(aspect ratio) where as, there is a wide change in difference factor for channels of low
sinuosity. The best fit relationship for the difference factor with aspect ratio is obtained
from Fig. 4.15 (a) and with sinuosity is obtained from Fig 4.15 (b).
68
Results and Discussions
Fig. 4.15 (b) Variation of Percentage of Wall Shear with Sinuosity
For meandering channel the distribution of shear is complicated as compared to that
of a straight channel. From Fig. 4.15 (a, b) it is clear that wall shear increases with
channel aspect ratio (b/h =α) and decreases with sinuosity (S r), which are the most
significant parameter influencing the flow mechanism in meandering channels.
The best fit functional relationships of the difference factor with the parameters
obtained from the plots (Fig. 4.15 a, b) is given as
Difference factor = F1 (Sr
-0.67
) and
F2 ln (30.692 × α )
(4.20)
Combining all the dependable parameters the difference factor is composited as
Difference factor =
2 . 15 S r
− 0 . 67
(4.21)
Ln (30.692 × α )
Equation (4.17) is further modified to incorporate meandering effect and is written as
%SFW = e m + 2.15S r
where, m = − 3 . 23 Log
10
(α
−0.67
Ln(30.692× α )
(4.22)
+ 3 ) + 6 . 146
The percentage of wall shear is calculated for Type-III and Type-IV channels, and
the two channels of Patra (1999) (S r= 1.21 and 1.22) using (4.22) and the results are
presented in Table 4.5 and Table 4.6, respectively. The calculated values are compared
with the actual value and the percentages of error for SFW for all the channels are given
in Table 4.5 and Table 4.6.
69
Results and Discussions
Table 4.5 Results of Experimental Data for Boundary Shear Distribution in Type-III
and Type-IV Meandering Channels
Discharge
Expt.
Sinuosity
(cm3/sec)
Runs
(1)
MR6
MR8
MR10
MR12
MR13
MT23
MT24
MT25
MT26
MT27
(2)
1.44
1.44
1.44
1.44
1.44
1.91
1.91
1.91
1.91
1.91
(3)
2357
2757
3338
4191
4656
4656
5122
5515
6396
7545
Flow
Depth α= b/h
h (cm)
(4)
5.31
6.08
7.11
8.55
9.34
5.62
5.93
6.18
6.71
7.33
(5)
2.260
1.974
1.688
1.404
1.285
2.135
2.024
1.942
1.788
1.637
Total
shear
stress
(N/m2)
(6)
0.857
0.918
0.990
1.072
1.111
1.257
1.286
1.308
1.352
1.397
SF1
SF2
(7)
0.02
0.03
0.03
0.04
0.08
0.10
0.13
0.14
0.17
0.19
(8)
0.11
0.10
0.11
0.19
0.15
0.24
0.25
0.25
0.30
0.34
% SFw
% of
% SFw (calculated
SF3
error for
(Actual) from eq.
% SFw
4.27)
(9)
(10)
(11)
(12)
0.10 52.647 52.618
-0.057
0.11 58.214 56.103
-3.627
0.13 58.087 60.099
3.464
0.23 58.352 64.691
10.863
0.16 61.735 66.823
8.241
0.20 55.165 52.864
-4.171
0.21 57.264 54.262
-5.243
0.23 59.420 55.335
-6.875
0.24 57.187 57.465
0.486
0.26 57.557 59.731
3.778
Table 4.6 Results of Experimental Data of Kar (1977) and Das (1984) (from Patra, 1999)
Flow Aspect
Expt. Runs/ Discharge
Depth Ratio
3
Sinuosity (cm /sec)
(cm) α= b/h
(1)
1.21(smooth)
1.21(rough)
1.21(smooth)
1.21(rough)
1.21(smooth)
1.21(rough)
1.21(rough)
1.21(smooth)
1.21(smooth)
1.22(rough)
1.22(smooth)
1.22(smooth)
1.22(rough)
(2)
750
750
1088
1088
2350
2350
2960
2960
3300
1400
1400
3000
3000
(3)
4.08
4.08
5.09
5.2
7.59
7.61
8.6
8.65
9.4
7.45
7.52
9.85
9.65
(4)
2.451
2.451
1.965
1.923
1.318
1.314
1.163
1.156
1.064
1.342
1.330
1.015
1.036
Total
shear
stress
2
(N/m )
(5)
0.882
0.882
0.990
1.000
1.183
1.184
1.241
1.243
1.281
1.790
1.797
1.985
1.971
By Integration Method
SF1
SF2
SF3
(6)
4.90
3.69
8.14
4.95
12.14
7.70
9.46
15.57
16.92
14.53
18.80
24.63
19.30
(7)
10.00
8.50
13.00
10.00
20.00
11.00
13.00
21.00
22.00
17.70
13.00
19.00
20.50
(8)
5.71
4.10
10.18
11.00
22.77
13.86
16.34
26.82
31.96
12.67
7.52
18.72
30.88
% SFw
% SFw (calculated
(Actual) from eq.
4.27)
(9)
(10)
51.475
51.434
47.821
51.434
58.498
57.075
61.464
57.619
63.579
66.992
66.216
67.054
66.495
69.906
66.869
70.039
68.962
71.906
60.572
66.509
66.938
66.732
69.522
72.893
70.996
72.446
% of
error
for %
SFw
(11)
-0.081
7.555
-2.434
-6.256
5.367
1.266
5.131
4.740
4.269
9.800
-0.307
4.849
2.043
The variation of computed percentage of wall shear force of wetted perimeter, using
equation (4.22), with the observed value of Type-III, Type-IV and that of Patra is shown
in Fig. 4.16. The graph shows comparisons between the calculated and observed values.
The calculated value is in good agreement with the observed value, irrespective of the
variations in the aspect ratios.
Again, the variation of computed percentage of shear force error at the wall
perimeter with the observed value of all these meandering channels are plotted in Fig.
4.17. The plot gives the percentage of error in predicting the SFw distribution vs.
70
Results and Discussions
different flow depths (aspect ratio). From the figure it can be seen that the error in
predicting the % SFw is very less. The value of average error is mostly in the range ±5.
The figures 4.16 and 4.17 show the adequacy of the developed equation.
Fig. 4.16 Variation of Observed and Modeled Value of Wall Shear in Meandering
Channel
Fig. 4.17 Percentage of Error in Calculating %SFW with Values of Aspect Ratios
71
CHAPTER 5
C O NC LU S I O N S
CONCLUSIONS
5.1. GENERAL
Experiments are carried out to examine the effect of channel sinuosity, and cross section
geometry on the wall shear in a meandering channel. Point to point wall shear data of
meandering channels by different aspect ratio and varying sinuosity is studied. The
study is also extended to a meandering channel of higher sinuosity (S r = 1.91). Based on
analysis and discussions of the experimental investigations certain conclusions can be
drawn.
The conclusions from the present work are as discussed below:
¯
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.
¯
It is an established fact that the influences of all the forces that resist the flow in an
open channel are assumed to have been lumped to a single resistance coefficient in
terms of n, C and f. 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.
¯
Dimensional analysis has been carried out to predict the resistance coefficients in a
meandering channel. These roughness coefficients are found to depend on the four
dimensionless parameters namely, sinuosity, aspect ratio, longitudinal slope of the
channel and Reynolds number of the flow.
¯
Using the proposed equations (4.12), (4.15) and (4.16) the stage discharge
relationship in a meandering channel can be adequately predicted. The equation is
found to give better discharge results as compared to the other established methods.
72
Conclusions
¯
Measurement of boundary shear from point to point along the wetted perimeter of
meandering channels of different sinuosity and geometry are performed. The overall
mean value of wall shear stress obtained through the velocity distribution approach
compares well with that obtained from energy gradient approach.
¯
Comparing the results of wall shear stress distribution of meandering channel with
straight channel, it can be seen that there is asymmetrical nature of shear distribution
especially where there is predominant curvature effect. Maximum value of wall shear
occurs significantly below the free surface and is located at the inner walls.
¯
The wall shear distribution in meandering channel is also found to be function of
dimensionless parameters like aspect ratio and sinuosity. An equation (4.22) has been
developed to predict the wall shear distribution in meandering channel. The proposed
models give less error for the present meandering channel as well as for the
meandering channels of other investigators.
5.2. SCOPE FOR FUTURE WORK
The present work leaves a wide scope for future investigators to explore many other aspects
of meandering channel analysis. Evaluation of flow, energy loss aspects, and boundary shear
stress distribution has been performed for the meandering and straight channels using
limited data. The equations developed may be improved by incorporating more data from
channels of different geometries and sinuosity. Further investigation is required to study the
flow properties and develop models using numerical approaches. The channels here are
rigid. Further investigation for the flow processes may also be carried out for mobile bed.
73
R E FR E NC E S
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78
Dissemina tion of Work
DISSEMINATION OF WORK
A: Published
1.
Nayak, P.P., Khatua, K.K., Patra, K.C. (2009) “Evaluation of roughness coefficients in
a meandering open channel flow”. 7TH International R&D Conference on Development
and Management of Water and Energy Resources 4-6 February 2009, Bhubaneswar
(Orissa), India.
2.
Nayak, P.P., Khatua, K.K., Patra, K.C. (2009) “Variation of Resistance Coefficients in a
Meandering Channel”. Proceedings of Advances in Environmental Engineering (AEE09), 14-Nov-2009 to 15-Nov-2009, National Conference Advances in Environmental
Engineering, Department of Civil Engineering, National Institute of Technology,
Rourkela, India.
3.
Nayak, P.P., Khatua, K.K., Patra, K.C., Pradhan A.K. (2009) “Ground water
prospecting by soil resistivity meter - A case study”. Proceeding of Water Resources
Management in 21st century, NITRAAB 2009.
4.
Khatua, K.K., Patra, K.C., Nayak, P.P., Sahu, M. (2010) “Wall Shear Distribution in
Meandering Channel”. Institute of Engineers India (IEI), February, 2010. (Obtained
gold medal for the best paper)
B: Accepted for Publication
1.
K.K.Khatua, K.C.Patra, N. Sahoo, Nayak, P. P. (2010) Evaluation of boundary shear
Distribution in a Meandering Channel”. The ninth International Conference on HydroScience and Engineering (ICHE 2010), 2-5 August, 2010
2.
Nayak, P.P., Khatua, K.K., Patra, K.C., (2010) “Roughness evaluation for meandering
channels” International conferences in Advances in Fluid Mechanics, AFM2010,Portugal, Lisbon, September,2010.
C: Communicated
1. Sahu, M., Nayak, P.P., Khatua, K.K. (2010) “Meandering effect for Evaluation of
roughness coefficients in open channel flow” Tenth International Conference on
Modeling, Monitoring and Management of Water Pollution, WATER POLLUTION
2010, Wessex Institute of Technology, Bucharest, Romania.
79
Brief Biodata of Author
BRIEF BIODATA OF THE AUTHOR
NAME
: PINAKI PARASANNA NAYAK
FATHER’S NAME
: Mr. AMIYA KUMAR NAYAK
MOTHER’S NAME
: Mrs. JYOTSNAMAYEE NAYAK
DATE OF BIRTH
: 02-03-1983
NATIONALITY
: INDIAN
PRESENT ADDRESS
: M.TECH (RES.), DEPT. OF CIVIL
ENGINEERING, NATIONAL
INSTITUTE OF TECHNOLOGY,
ROURKELA - 769008
: D/O Mr. A. K. NAYAK,
PERMANENT ADDRESS
PLOT
No.
573,
SAHIDNAGAR,
BHUBANESWAR – 751007
: B.TECH, CIVIL ENGINEERING,
EDUCATIONAL QUALIFICATIONS
2006 (WITH Hons.)
+2 SCIENCE, 2001
H. S. C., 1999
80
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