BOUNDARY SHEAR STRESS DISTRIBUTION IN MEANDERING CHANNELS

BOUNDARY SHEAR STRESS DISTRIBUTION IN MEANDERING CHANNELS
BOUNDARY SHEAR STRESS DISTRIBUTION IN MEANDERING
CHANNELS
A thesis submitted to
National Institute of Technology, Rourkela
In partial fulfillment for the award of the degree
of
Master of Technology in Civil Engineering
With specialization in
Water Resources Engineering
By
Manaswinee Patnaik (211CE4253)
Under the supervision of
Professor K.C. Patra
Department of Civil Engineering
National Institute of Technology, Rourkela
Odisha-769008
May 2013
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
DECLARATION
I hereby declare that this submission is my own work and that, to the best of my knowledge and
belief, it contains no material previously published or written by another person nor material
which to a substantial extent has been accepted for the award of any other degree or diploma of
the university or other institute of higher learning, except where due acknowledgement has been
made in the text.
MANASWINEE PATNAIK
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA
CERTIFICATE
This is to certify that the thesis entitled “Boundary Shear Stress Distribution in Meandering
Channel” is a bonafide record of authentic work carried out by Manaswinee Patnaik under my
supervision and guidance for the partial fulfillment of the requirement for the award of the
degree of Master of Technology in hydraulic and Water Resources Engineering in the
department of Civil Engineering at the National Institute of Technology, Rourkela.
The results embodied in this thesis have not been submitted to any other University or Institute
for the award of any degree or diploma.
Date:
Prof. K.C. Patra
Department of Civil Engineering
Place: Rourkela
National Institute of Technology, Rourkela
Rourkela-769008
i
ACKNOWLEDGEMENTS
I am deeply indebted to National Institute of Technology, Rourkela for providing me the
opportunity to pursue my Master’s degree with all necessary facilities.
I would like to express my hearty and sincere gratitude to my project supervisor Prof. K.C.
Patra whom sincere and affectionate supervision has helped me to carry out my project work. I
would also like to thank Prof. KK Khatua for the support extended by him and his constant
encouragement for the work has been the source of unparalleled enthusiasm for me.
I would like to record my gratitude to all faculty members of Water Resources Engineering who
have constantly guided and inspired me till day in National Institute of Technology, Rourkela.
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 for his useful
assistance and cooperation during the entire course of the experimentation and helping me in all
possible ways.
I wish to thank all of my fellow classmates for their kind help and co-operation extended during
my course of study.
My parents have been my unfailing source of love and inspirations.
Date:
MANASWINEE PATNAIK
ii
ABSTRACT
Precise estimation of boundary shear force distribution is essential to deal with various hydraulic
problems such as channel design, channel migration and interaction losses. Bed shear forces are
useful for the study of bed load transfer where as wall shear forces presents a general view of
channel migration pattern. Meander formation in rivers is an intricate phenomenon that results
from erosion on outer bank and deposition on the inner side. So the analysis of meandering
channels under different geometric and hydraulic condition are necessary to understand one of
the
flow properties such as distribution of boundary shear which is a better indicator of
secondary flows than velocity, on different parameters like aspect ratio, sinuosity, ratio of
minimum radius of curvature to width and hydraulic parameter such as relative depth.
With the purpose of obtaining shear stress distribution at the walls and on the bed of
compound meandering channel, experimental data collected from laboratory under different
discharge and relative depths maintaining the geometry, slope and sinuosity of the channel
constant, are analyzed and confronted. Preston-tube technique is used to collect velocity heads at
various intervals along the wetted perimeter and within the flow that helps to calculate shear
stress values using calibration curves proposed by Patel (1965). The distributions of boundary
shear stress along the channel wetted perimeter are plotted for both in bank and overbank flow
conditions. Based on experimental results, the effect of aspect ratio and sinuosity on wall (inner
and outer) and bed shear forces are evaluated in meandering wide channels (B/H> 5) and having
a sinuosity of 2.04. Equations are developed to determine the percentage of wall and bed shear
forces in smooth trapezoidal channel for in bank flows only. The proposed equations are
compared with previous studies and the model is extended to wide channels. A quasi1D model
Conveyance Estimation System (CES) were then applied in turn to the same compound
iii
meandering channel to validate with the experimental shear velocity which ultimately relates to
the boundary shear stress. It has been found that the CES results underestimate the shear
velocity.
A3D modelling software ANSYS-CFX 13.0 is employed to derive the contours of
longitudinal, lateral and resultant bed shear stress, for a 60 degree meandering channel using
Large Eddy Scale (LES) model.
Key Words:
Aspect ratio; Boundary shear; Compound channel; Conveyance; In-bank flow; Interaction loss;
Meander; Over-bank flow; Preston-tube; Relative depth; Sinuosity
iv
TABLE OF CONTENTS
CHAPTER
DESCRIPTION
PAGE NO.
Certificate
i
Acknowledgements
ii
Abstract
iii
Table of Contents
v
List of Tables
ix
List of Figures and Photographs
x
List of Symbols
xii
1
INTRODUCTION
1-8
1.1
Rivers & Flooding
1
1.2
Boundary Shear Stress
2
1.3
Numerical Modeling
4
1.4
Objectives of the present study
5
1.5
Organization of Thesis
7
2
LITERATURE SURVEY
9-24
2.1
General
9
2.2
Previous Works on Experimental
9
Research for Boundary Shear
2.2.1
Straight Simple Channels
10
2.2.2
Straight Compound Channels
14
v
2.2.3
Meander Simple Channels
17
2.2.4
Meander Compound Channels
18
Overview of Numerical Modeling for
20
2.3
Open Channel Flow
3
EXPERIMENTAL SETUP & PROCEDURE
25-33
3.1
General
25
3.2
Experimental Arrangement
25
3.2.1
Apparatus & Materials used
25
3.2.2
Measuring Equipments
28
Experimental Procedure
29
3.3.1
Measurement of Bed Slope
30
3.3.2
Calibration of Notch
31
3.3.3
Measurement of Normal Depth & Discharge
32
3.3
4
EXPERIMENTAL RESULTS & ANALYSIS
34-55
4.1
General
34
4.2
Stage-Discharge Relationship
34
4.3
Shear Stress Measurement
36
4.3.1
Preston-Tube Technique
36
4.3.2
Shear Stress Contours
39
4.3.2
(a) Simple Meander Channel
39
4.3.2
(b) Compound Meander Channel
41
vi
4.4
Analysis based on experimental results
43
4.4.1
Mean Boundary Shear Stress
43
4.4.2
Distribution of Boundary Shear Stress
44
4.4.2 (a) Simple Meander Channel
44
4.4.2 (b) Compound Meander Channel
46
4.4.3
Shear Force Analysis
49
4.4.4
Development of Model for Percentage Shear Force
50
5
NUMERICAL MODELING
56-69
5.1
General
56
5.2
Geometry Setup
57
5.3
Discretization of Domain (Meshing)
59
5.4
Turbulence
60
5.5
Numerical Model
61
5.5.1
Large Eddy Simulation (LES)
61
5.5.2
Mathematical Model
63
Boundary Conditions
66
5.6.1
Wall
66
5.6.2
Free Surface
66
5.6.3
Inlet and Outlet
66
Numerical Results
67
5.6
5.7
6
CONCLUSIONS AND FURTHER WORK
6.1
Conclusions
70-72
70
vii
6.2
Recommendations for Future Work
REFERENCES
71
73-79
(Appendix A-I) Published and Accepted Papers from the Work
viii
80
LIST OF TABLES
Table No.
Description
Page No.
Table 1
Geometry Parameters of the Experimental Meandering Channel
26
Table 2
Hydraulic parameter for the experimental runs
35
Table 3
Summary of boundary shear force results for the experimental
50
simple meandering channels observed at the bend apex.
Table 4
Summary of percentage shear force results along the wetted
perimeter for the experimental simple meandering channels
observed at the bend apex.
ix
54
LIST OF FIGURES AND PHOTOGRAPHS
Figure No.
Description
Page No.
Photo 3.1(a)
Upstream view of experimental channel
26
Photo 3.1(b)
Side view of experimental channel
26
Photo 3.2 (a)
Series of Pitot-static tube with point gauge
29
Photo 3.2 (b)
Set of piezometers with spirit level
29
Figure 1.1
Schematic influence of secondary flow cells
3
on boundary shear distribution
Figure 1.2
3 D flow structures in open channel
4
(Shiono and Knight, 1991)
Figure 3.1 (a)
Plan of experimental meandering channel at bed level
27
Figure 3.1 (b)
Plan of experimental meandering channel at full bank level
27
Figure 3.2
Schematic diagram of experimental setup
27
Figure 4.1
Plot of stage versus discharge
35
Figure 4.2 (a)
Definition sketch of point locations used in shear stress
37
measurements for simple meander channel at bend apex
Figure 4.2 (b)
Definition sketch of point locations used in shear stress
38
measurements for compound meander channel at bend apex
Figure 4.3 (a)
Shear stress contours of in bank flow for flow depth 1.7 cm
40
Figure 4.3 (b)
Shear stress contours of in bank flow for flow depth 3.8 cm
40
Figure 4.3 (c)
Shear stress contours of in bank flow for flow depth 4.0 cm
40
Figure 4.3 (d)
Shear stress contours of in bank flow for flow depth 5.0 cm
40
x
Figure 4.4 (a)
Shear stress contours of over bank flow for floodplain depth 3.0 cm
41
Figure 4.4 (b)
Shear stress contours of over bank flow for floodplain depth 3.4 cm
42
Figure 4.4 (c)
Shear stress contours of over bank flow for floodplain depth 3.6 cm
42
Figure 4.4 (d)
Shear stress contours of over bank flow for floodplain depth 4.1 cm
42
Figure 4.5
Dimensionless local wall shear stress versus z/H for depths 5 cm
45
and 1.7 cm at inner and outer wall for in bank flow
Figure 4.6
Dimensionless local bed shear stress versus 2y/B for depths 5 cm
46
and 1.7 cm in inner and outer bed region
Figure 4.7
Lateral distribution of shear velocity along the compound cross
48
section of experimental channel for different relative depths
Figure 4.8
Variation of (%SFw)mod with Aspect Ratio
53
Figure 4.9 (a)
Variation of (%SFw)i with Aspect Ratio
53
Figure 4.9 (b)
Variation of (%SFw)o with Aspect Ratio
53
Figure 4.10
Variation of (%SF bed) with Aspect Ratio
55
Figure 4.11
Observed and modeled values of (%SFw)
55
Figure 5.1
Schematic diagram of structured mesh
59
Figure 5.2
Energy cascade process with length scale
62
Figure 5.3
Contours of bed shear stress along one wavelength
68
reach of the 60 degree meandering channel
xi
LIST OF SYMBOLS
B
width of the channel;
Cr
Courant number;
Cs
Smagorinsky constant;
d
outside diameter of the probe;
g
acceleration due to gravity;
G
Gaussian filters;
H
in bank depth of flow;
H'
over bank depth of flow;
h
main channel bank full depth;
k
turbulent kinetic energy;
l
length scale of unresolved motion;
P
wetted perimeter;
Q
discharge;
R
hydraulic radius;
Sr
Sinuosity;
Sij
Resolved strain rate tensor;
S
bed slope of the channel;
y
lateral distance along the channel bed;
z
vertical distance from the channel bed;
α
aspect ratio;
Fluid density;
xii
θ
angle between channel bed and horizontal;
ν
kinematic viscosity;
ε
turbulent kinetic energy dissipation rate;
ω
specific dissipation;
σij
normal stress component on plane normal to i along j;
τij
shear stress component on plane normal to i along j;
ūi', ūj'
time averaged instantaneous velocity component along i,j directions
τ0
overall boundary shear stress;
µ
coefficient of dynamic viscosity;
x*
logarithmic of the dimensionless pressure difference;
y*
logarithmic of the dimensionless shear stress;
∆P
Preston tube differential pressure;
∆h
difference between dynamic and static head;
(%SF) w
percentage shear force at walls;
(% SF) b
percentage shears force on bed;
(% SFw)mod modeled percentage shear force at walls;
(% SFw)act
observed percentage shear force at walls
xiii
Subscripts
a/act.
actual
b, bed
bed
h
hydraulic
r
relative
t
theoretical
T
total
w
wall
mod.
modelled
i, j, k
x, y, z directions respectively
i, inner
inner bank or wall of meandering channel
o, outer
outer bank or wall of meandering channel
xiv
INTRODUCTION
INTRODUCTION
1.1
RIVER AND FLOODING
River and river valleys have been very crucial in the development of civilization. River has
always been main source of water for agriculture, domestic needs, industries etc. Also river
provide as with energy, recreation and transportation routes. Eventually, it becomes hard to
believe that during flood a gentle river inundate its flood plain thereby causing serious damage to
the lives and shelter of the people residing in low-lying areas. Nowadays debate on flooding is
gaining momentum due to combining consequences of climate change. From recent times, river
engineer’s devise solutions by designing flood defenses so as to ensure minimum damage from
flooding. Generally river engineer’s use hydraulic model to make flood prediction. The hydraulic
model incorporates many flow features such as accurate discharge, average velocity, water level
profile and shear stress forecast. Prior to producing hydraulic models capable of modeling all
these flow features detailed knowledge on open channel hydrodynamics is required. In this
regard, first comes the understanding of geometrical and hydraulic parameters of the river
streams. Even the flow properties in rivers vary with the geometrical shape.
Broadly streams are classified as straight, braided and meandering. Almost all natural
rivers meander. Natural rivers are seldom straight except for short distances. 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”. Otherwise stated, meandering channels are the channel that winds its way
across the floodplain. The flow path in a meandering channel continuously changes along its
course. Due to this the energy dissipation is not uniform over the meander length. The motion in
Page | 1
INTRODUCTION
meandering channels comprised of two components, the longitudinal component in stream wise
direction which is nearly uniform and gradually varied and transverse component varies
significantly over a meander wavelength. In general, a meander is a bend in a sinuous water
course which is formed when flowing water in a stream erodes the outer bank and widens its
valley. Theoretically a sine generated curve well represents a meander channel. The sinuosity or
meander index which quantifies how much a river course deviates from the shortest possible path
is one of the criterions which control the velocity and shear distribution in meandering channel.
1.2
BOUNDARY SHEAR DISTRIBUTION
When water flows in a channel the force developed in the flow direction is resisted by reaction
from channel bed and side walls. This resistive force is manifested in the form of boundary shear
force. Otherwise stated, tractive
force,
or
boundary
shear
stress,
is
component of the hydrodynamic forces acting along the channel bed.
the
tangential
Distribution of
boundary shear force along the wetted perimeter directly affects the flow structure in an open
channel. Knowledge on boundary shear stress distribution is necessary to define velocity profile
and fluid field. Also computation of bed form resistance, sediment transport, side wall
correction, cavitations, channel migration, conveyance estimation, and dispersion are among the
hydraulic problems which can be solved by bearing the idea of boundary shear stress
distribution.
From theoretical considerations, in steady uniform flow the tractive force is related to
channel bed slope, hydraulic radius and unit weight of fluid. However it is established that such
forces even in straight prismatic channel with simple cross-sectional geometry are not uniform.
Moreover the tractive force is a turbulent quantity composed of a fluctuating component
Page | 2
INTRODUCTION
superimposed on the mean value. The non-uniformity in shear stress is mostly due to this
fluctuating component which is interpreted as secondary currents and is generated by the
anisotropy between the vertical and transverse turbulent intensities, this is given by Gessner,
1973. Although secondary velocity comprises only 2-3% of primary mean velocity it convects
momentum, vorticity and energy towards the corners and subsequently transports them away
along the boundary walls. Tominaga et al. (1989) and Knight and Demetriou (1983) showed that
boundary shear stress increases where secondary currents flow towards the wall and shear stress
decreases as they flow away from the wall. The presence of secondary flow cells in main flow
influences the distribution of shear stress along the channel wetted perimeter which is illustrated
in Fig. 1.1. Other factors that affect the distribution of shear stress in straight channel are shape
of the cross-section, number and structure of secondary flow cells, depth of flow, sediment
concentration and the lateral-longitudinal distribution of wall roughness. In meandering channels
the factors increases by many folds due to accretion in 3-Dimensional nature of flow. Sinuosity
of the meandering channel is considered to be a critical parameter for calculating the percentage
of shear force at channel walls and bed.
Fig.1.1
Page | 3
Schematic influence of secondary flow cells on boundary shear distribution
INTRODUCTION
Compound channel consists of a deep main channel flanked by relatively shallow
floodplains on one or both sides of the main channel. During flood when rivers are at high stage,
the flow from the main channel spills and spreads to the adjacent floodplain. The reduced
hydraulic radius and higher roughness of floodplain result in lower velocities in floodplain as
compared to the main channel. The interaction between the faster moving fluid in main channel
and slower fluid in floodplain result in a bank of vortices as shown by Knight and Hamed (1984),
referred to as “turbulence phenomenon”. Consequently there is a lateral transfer of momentum
that results in an apparent shear stress at the interface of main channel and floodplain which
significantly distort flow and boundary shear stress patterns. The intricate mechanism of
momentum transfer in a straight two stage channel is demonstrated in Fig.1.2.
Fig.1.2
1.3
3 D Flow Structures in Open Channel (Shiono and Knight, 1991)
NUMERICAL MODELLING
Despite of precise results and clear understanding on flow phenomena; experimental approach
has some serious drawbacks such as tedious data collection and data can be collected for limited
Page | 4
INTRODUCTION
number of points due to instrument operation constraints; the model is usually not at full scale
and the three dimensional flow behavior or some complicated turbulent structure which is the
instinct of any open channel flow cannot be effectively captured through experiments. So in
these circumstances, computational approach can be adopted to overcome some of these issues
and thus provide a complementary tool. In comparison to experimental studies; computational
approach is repeatable, can simulate at full scale; can generate the flow taking all the data points
into consideration & moreover can take greatest technical challenge i.e.; prediction of
turbulence. The complex turbulent structures like secondary flow cells, vortices, Reynolds
stresses can be effectively and distinctly identified by numerical modeling which are quite
essential for energy expenditure studies in open channel flows. Many researchers in the recent
years have numerically modeled open channel flows and has successfully validated with the
experimental results. Computational Fluid Dynamics (CFD) is a mathematical tool which is used
to model open channel ranging from in-bank to over-bank flows. Different models are used to
solve Navier-Stokes equations which are the governing equation for any fluid flow. Finite
volume method is applied to discretize the governing equations. The accuracy of computational
results mainly depends on the mesh quality and the model used to simulate the flow.
1.4
OBJECTIVE OF PRESENT STUDY
The present work is aimed to study the distribution of boundary shear stress in simple and compound
meandering channels. The distribution of shear stress along the bed and wall of a meandering channel
depends on width-depth ratio, relative depth, lateral-longitudinal roughness distribution and sinuosity. Out
of these parameters width-depth ratio or aspect ratio and sinuosity plays a major role in estimation of
boundary shear stress distribution in meandering channels. Despite immense interest of investigators in
Page | 5
INTRODUCTION
boundary shear stress, no systematic information about the percentage of shear force carried by walls and
bed of meandering trapezoidal channel is available. Again one of the features of trapezoidal channels is
that there exists an unequal shear drag at the two banks of the channel and this effect becomes more
pronounced when the channel is a meandering one. So here it becomes necessary to analyze the inner and
outer banks of the meander channel separately, which is yet to gain enough concern from the researchers.
Thus an empirical model can be developed for meandering channel to calculate stream wise bed and wall
shear stress. And also models can be developed that describes the percentage shear force at inner wall,
outer wall and bed separately for trapezoidal meandering channels.
Even for compound meandering channels computational works are reported more than experimental
studies. Therefore experimental analysis on distribution of boundary shear stress along the compound
meandering channel can be studied more extensively which can further applied to natural rivers during
flood conditions. These studies should be useful in determining the actual discharge through a meander
channel to solve many hydraulic problems and also it provides a better understanding of flow structure in
open meander channels. The objectives of the present work are summarized as:
Determination of boundary shear stress distribution along the wetted perimeter in simple
meandering channels.
To carry out an investigation concerning the distribution of local shear stress in the main channel
and flood plain of meandering compound channel.
To conduct experiment and analyze experimental data for the investigation of longitudinal wall
and bed shear stress for different flow depths for simple and compound meandering channels.
Page | 6
INTRODUCTION
Development of new mathematical models for evaluation of percentage shear force at wall and
bed incorporating meandering effects and to extend the models to the channels of high aspect
ratio and sinuosity for meandering channels.
To validate local shear stress data in terms of shear velocity with 1D model conveyance
estimation system (CES) for compound meandering channels.
To simulate a 60° simple meandering channel for analyzing the flow phenomena such as
bed shear stress of a meandering channel by Large Eddy Simulation (LES) model using a
CFD tool.
1.5
ORGANISATION OF THESIS
The thesis consists of six chapters. General introduction is given in Chapter 1, literature
survey is presented in Chapter 2, experimental work is described in Chapter 3, experimental
results are outlined and analysis of results are done in Chapter 4, Chapter 5 comprises numerical
modeling and finally the conclusions and references are presented in Chapter 6.
General view on rivers and flooding is provided at a glance in the first chapter. Also the
chapter introduces the concept of boundary shear distribution in meandering channels. It gives an
overview of numerical modeling in open channel flows.
The detailed literature survey by many eminent researchers that relates to the present
work from the beginning till date is reported in chapter 2. The chapter emphasizes on the
research carried out in straight and meandering channels for both in bank and overbank flow
conditions based on boundary shear distribution.
Chapter three describes the experimental programme as a whole. This section explains
the experimental arrangements and procedure adopted to obtain observation at different points in
Page | 7
INTRODUCTION
the channel. Also the detailed information about the instrument used for taking observation is
given.
The experimental results regarding stage-discharge relationship, boundary shear stress for
in bank and overbank flow conditions and mean boundary shear stress are outlined in chapter
four. Also this chapter discusses the technique adopted for measuring boundary shear stress.
Analyses of the experimental results are done. The analysis of shear force for in bank flow
conditions is presented in this chapter
Chapter five presents significant contribution to numerical simulation of in bank
channels. The numerical model and the software used within this research are also discussed.
Finally, chapter six summarizes the conclusion reached by the present research and
recommendation for the further work is listed out.
References that have been made in subsequent chapters are provided at the end of the
thesis.
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LITERATURE SURVEY
LITERATURE SURVEY
2.1
GENERAL
Distribution of boundary shear stress along 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, size and distribution of secondary flow cells and the
lateral-longitudinal distribution of wall roughness. It is quite necessary to take into account 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 boundary shear
distribution in an open channel flow. In earlier times due to 1 Dimensional modeling of flow
emphasis was given to local shear stresses and many empirical models were developed regarding
the distribution of stream wise component of shear stress. But with time many researchers
remarkably noted the presence of secondary velocity in open channel flows due to which
complex mixing occurs giving rise to numerous turbulent structures which affects the velocity
and shear stress distribution and ultimately the conveyance of the channels. So the present
review of literature includes works on experimental research of boundary shear stress for four
channel types followed by numerical studies on open channel flow.
2.2
PREVIOUS WORKS ON EXPERIMENTAL RESEARCH FOR BOUNDARY SHEAR
The literature review contains a large body of research on the subject of boundary shear stress in
open channel flow. This review intends to present some of the selected significant contribution to
the study of boundary shear stress in open channel flow. Distribution of shear stress in open
channels has been in interest of many investigators from earlier times. Research are done
Page | 9
LITERATURE SURVEY
covering several aspects such as using different channel cross-sections like rectangular and
trapezoidal; different channel geometry such as straight, meandering channel, as well as simple
and compound channel with different channel surface types like smooth and rough channels to
study the factors influencing the boundary shear stress.
2.2.1
STRAIGHT SIMPLE CHANNELS
Earlier works on open channel hydraulics involves experiment in simple straight channel having
rectangular cross section.
Seven decades ago, Leighly (1932) proposed the idea of using conformal mapping to
study the boundary shear stress distribution in open-channel flow. He pointed out that, in the
absence of secondary currents, the boundary shear stress acting on the bed must be balanced by
the downstream component of the weight of water contained within the bounding orthogonals.
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 Ab and Aw
and assumed that the down-stream component of the fluid weight in area Ab is balanced by the
resistance of the bed. Likewise, the downstream component of the fluid weight in area Aw was
balanced 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. However he did not propose any method of determining the exact location of
division line.
Ghosh and Roy (1970) presented the boundary shear distribution in both rough and
smooth open channels of rectangular and trapezoidal sections obtained by direct measurement of
Page | 10
LITERATURE SURVEY
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. 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 experiments were carried out in a smooth-walled laboratory
flume at various aspect ratios of the rectangular cross-section. Wall shear stress measured with
Preston tubes 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 the present
analytical techniques could be counted upon to provide any precise details on tractive force
distribution in turbulent channel flow.
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
Page | 11
LITERATURE SURVEY
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. 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 behaviour of
open channel flows with success. This idea was further discussed by Noutsopoulos and
Hadjipanos (1982).
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 centre position towards each corner. For rectangular cross
sections, the number of secondary flow cells increased from 8 by increments of 4 as the aspect
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LITERATURE SURVEY
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, centreline and mean boundary shear stresses on the duct walls in terms of the aspect
ratio.
Knight and Sterling(2000) observed the distribution of boundary shear stress in circular
conduits flowing partially full with and without a smooth flat bed for a data ranging from
0.375<F<1.96 and 6.5*104<R<3.42*105, using Preston-tube technique. The distribution of
boundary shear stress is shown to depend on geometry and Froude no. The results have been
analysed in terms of variation of local shear stress with perimetric distance and the percentage of
total shear force acting on wall or bed of the conduit. The %SFW results have been shown to
agree well with Knight’s (1981) empirical formula for prismatic channels. The interdependency
of secondary flow and boundary shear stress has been established and its implications for
sediment transport have also been examined.
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.
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LITERATURE SURVEY
Guo and Julien (2005) proposed a method to determine 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.
Lashkar and Fathi (2010) conducted experiments to determine the contribution of wall
shear force on total boundary shear force. A nonlinear regression-based technique was carried
out to analyze the results and develop equations to determine the percentage of wall and bed
shear force on the wetted perimeter of the rectangular channels.
2.2.2
STRAIGHT COMPOUND CHANNEL
Zheleznyakov (1965) was probably the first to investigate the interaction between the
main channel and the adjoining floodplain. He demonstrated under laboratory conditions the
effect of momentum transfer mechanism, which was responsible for decreasing the overall rate
of discharge for floodplain depths just above the bank full level. As the floodplain depth
increased, the importance of the phenomena diminished.
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LITERATURE SURVEY
Ghosh and Jena (1973) and Ghosh and Mehata (1974) reported studies on boundary shear
distribution in straight two stage channels for both smooth and rough boundaries. They found the
distribution of shear is non-uniform and the location of maximum bed and side shear to be some
distance from the centreline and free surface. They related the sharing of the total drag force by
different segments of the channel section to the depth of flow and roughness concentration.
Myers and Elswy (1975) studied the effect of interaction mechanism and shear stress
distribution in channels of complex sections. In comparison to the values under isolated
condition, the results showed a decrease up to 22 percent in channel shear and increase up to 260
percent in floodplain shear. This indicated the possible regions of erosion and scour of the
channel and flow distribution in alluvial compound sections.
Rajaratnam and Ahmadi (1979) studied the flow interaction between straight main
channel and symmetrical floodplain with smooth boundaries. The results demonstrated the
transport of longitudinal momentum from main channel to flood plain. Due to flow interaction,
the bed shear in floodplain near the junction with main channel increased considerably and that
in the main channel decreased. The effect of interaction reduced as the flow depth in the
floodplain increased.
Wormleaton, Alen, and Hadjipanos (1982) undertook a series of laboratory tests in
straight channels with symmetrical floodplains and used "divide channel" method for the
assessment of discharge. From the measurement of boundary shear, apparent shear stress at the
vertical, horizontal, and diagonal interface plains originating from the main channel-floodplain
junction could be evaluated. An apparent shear stress ratio was proposed which was found to be
a useful yardstick in selecting the best method of dividing the channel for calculating discharge.
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LITERATURE SURVEY
It was found that under general circumstances, the horizontal and diagonal interface method of
channel separation gave better discharge results than the vertical interface plain of division at
low depths of flow in the floodplains.
Knight and Demetriou (1983) conducted experiments in straight symmetrical compound
channels to understand the discharge characteristics, boundary shear stress and boundary shear
force distributions in the section. They presented equations for calculating the percentage of
shear force carried by floodplain and also the proportions of total flow in various sub-areas of
compound section in terms of two dimensionless channel parameters. For vertical interface
between main channel and floodplain the apparent shear force was found to be more at low
depths of flow and also for high floodplain widths. On account of interaction of flow between
floodplain and main channel, it was found that the division of flow between the sub-areas of the
compound channel did not follow the simple linear proportion to their respective areas.
Knight and Hamed (1984) extended the work of Knight and Demetriou (1983) to rough
floodplains. The floodplains were roughened progressively in six steps to study the influence of
different roughness between floodplain and main channel to the process of lateral momentum
transfer. Using four dimensionless channel parameters, they presented equations for the shear
force percentages carried by floodplains and the apparent shear force in vertical, horizontal,
diagonal, and bisector interface plains. The apparent shear force results and discharge data
provided the strength and weakness of these four commonly adopted design methods used to
predict the discharge capacity of the compound channel.
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LITERATURE SURVEY
2.2.3
MEANDER SIMPLE CHANNELS
In bank flows in meandering channel are highly three dimensional and exhibit complex turbulent
structures like secondary motions. The phenomenon of secondary motion was first given by
Boussinesq (1868) and Thomson (1876). They studied the influence of secondary motion on
primary velocity distribution. Later on Jia et.al., 2001 showed that secondary motion occurs due
to the imbalance between the driving centrifugal force and the transverse pressure gradient.
Knight, Yuan and Fares (1992) reported the experimental data of SERC-FCF concerning
boundary shear stress distributions in meandering channels throughout the path of one complete
wave length. They also reported the experimental data on surface topography, velocity vectors,
and turbulence for the two types of meandering channels of sinuosity 1.374 and 2.043
respectively. They examined the effects of secondary currents, channel sinuosity, and cross
section geometry on the value of boundary shear in meandering channels and presented a
momentum-force balance for the flow.
Shiono, Muto, Knight and Hyde (1999) presented the experimental data of secondary
flow and turbulence using two components Laser- Doppler Anemometer for both straight and
meandering channels to understand the flow mechanism in meandering channels. They
developed turbulence models and studied the behaviour of secondary flow and centrifugal forces
for both in-bank and over-bank flow conditions. They investigated the energy loss due to
boundary friction, secondary flow, turbulence, expansion and contraction in meandering
channels.
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LITERATURE SURVEY
2.2.4
MEANDER COMPOUND CHANNEL
Flow in compound channel often inundate the adjacent floodplains due to lateral momentum
transfer takes place at the main channel and floodplain interface, which generates more
complicated flow structures than in simple meander channel. Compared to the extensive
literature for straight compound channel, much less work has been reported for compound
meandering channel flows.
Ghosh and Kar (1975) studied 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.
Ervine, Alan, Koopaei, and Sellin (2000) presented a practical method to predict depthaveraged velocity and shear stress for straight and meandering over bank flows. They also
presented an analytical solution to the depth-integrated turbulent form of the Navier-Stokes
equation that includes lateral shear and secondary flows in addition to bed friction. They applied
this analytical solution to a number of channels, at model, and field scales, and compared with
other available methods such as that of Shiono and Knight and the lateral distribution method
(LDM).
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
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LITERATURE SURVEY
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 the sign changes 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 for predicting proportion
of discharge carried by the main channel and floodplain.
Patra and Kar (2004) reported the test results concerning the flow and velocity
distribution in meandering compound river sections. Using power law they presented equations
concerning the three-dimensional variation of longitudinal, transverse, and vertical velocity in
the main channel and floodplain of meandering compound sections in terms of channel
parameters. The results of formulations compared well with their respective experimental
channel data obtained from a series of symmetrical and unsymmetrical test channels with smooth
and rough surfaces. They also verified the formulations against the natural river and other
meandering
compound
channel
data.
Khatua (2008) extended the work of Patra and Kar (2000) to meandering compound
channels. Using five parameters (sinuosity Sr, amplitude, relative depth, width ratio and aspect
ratio), general equations representing the total shear force percentage carried by floodplain was
presented. The proposed equations are simple, quite reliable and gave good results with the
observed data for straight compound channel of Knight and Demetriou (1983) as well as for the
meandering compound channel.
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LITERATURE SURVEY
Khatua (2010) reported the distribution of boundary shear force for highly meandering
channels having distinctly different sinuosity and geometry. Based on the experimental results,
the interrelationship between the boundary shear, sinuosity and geometry parameters has been
shown. The models are also validated using the well published data of other investigators.
2.3 OVERVIEW OF NUMERICAL MODELLING ON OPEN CHANNEL FLOW
For the past three decades, flow in simple and compound meandering channels has been
extensively studied both experimentally and numerically. Various numerical models such as
standard k-ε model, non-linear k-ε model, k-ω model, algebraic Reynolds stress model (ASM),
Reynolds stress model (RSM) and large eddy simulation (LES) have been developed to simulate
the complex secondary structure in compound meandering channel. The standard k- ε model is
an isotropic turbulence closure but fails to reproduce the secondary flows. Although nonlinear kε model can simulate secondary currents successfully in a compound channel, it cannot
accurately capture some of the turbulence structures. ASM is economical because it uses adhoc
expressions to solve Reynolds stress transport equations. But the simulated results by ASM
found to be unreliable. Reynolds stress model (RSM) computes Reynolds stresses by directly
solving Reynolds stress transport equation but its application to open channel is still limited due
to the complexity of the model. Large eddy simulation (LES) solves spatially-averaged NavierStokes equation. Large eddies are directly resolved, but eddies smaller than mesh are modelled.
Though LES is computationally expensive to be used for industrial application but can
efficiently model nearly all eddy sizes.
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LITERATURE SURVEY
Cokljat and Younis and Basara and Cokljat (1995) proposed the RSM for numerical
simulations of free surface flows in a rectangular channel and in a compound channel and found
good agreement between predicted and measured data.
Thomas and Williams (1995) describes a Large Eddy Simulation of steady uniform flow
in a symmetric compound channel of trapezoidal cross-section with flood plains at a Reynolds
number of 430,000. The simulation captures the complex interaction between the main channel
and the flood plains and predicts the bed stress distribution, velocity distribution, and the
secondary circulation across the floodplain. The results are compared with experimental data
from the SERC Flood Channel Facility at Hydraulics Research Ltd, Wallingford, England
Salvetti et al. (1997) has conducted LES simulation at a relatively large Reynolds number
for producing results of bed shear, secondary motion and vorticity well comparable to
experimental results.
Rameshwaran P, Naden PS.(2003) analyzed three dimensional nature of flow in
compound channels.
Sugiyama H, Hitomi D, Saito T.(2006) used turbulence model consists of transport
equations for turbulent energy and dissipation, in conjunction with an algebraic stress model
based on the Reynolds stress transport equations. They have shown that the fluctuating vertical
velocity approaches zero near the free surface. In addition, the compound meandering open
channel is clarified somewhat based on the calculated results. As a result of the analysis, the
present algebraic Reynolds stress model is shown to be able to reasonably predict the turbulent
flow in a compound meandering open channel.
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LITERATURE SURVEY
Kang H, Choi SU. (2006) used a Reynolds stress model for the numerical simulation of
uniform 3D turbulent open-channel flows. The developed model is applied to a flow at a
Reynolds number of 77000 in a rectangular channel with a width to depth ratio of 2. The
simulated mean flow and turbulence structures are compared with measured and computed data
from the literature. It is found that both production terms by anisotropy of Reynolds normal
stress and by Reynolds shear stress contribute to the generation of secondary currents.
Jing, Guo and Zhang (2008) simulated a three-dimensional (3D) Reynolds stress model
(RSM) for compound meandering channel flows. The velocity fields, wall shear stresses, and
Reynolds stresses are calculated for a range of input conditions. Good agreement between the
simulated results and measurements indicates that RSM can successfully predict the complicated
flow phenomenon.
Cater and Williams (2008) reported a detailed Large Eddy Simulation of turbulent flow
in a long compound open channel with one floodplain. The Reynolds number is approximately
42,000 and the free surface was treated as fully deformable. The results are in agreement with
experimental measurements and support the use of high spatial resolution and a large box length
in contrast with a previous simulation of the same geometry. A secondary flow is identified at
the internal corner that persists and increases the bed stress on the floodplain.
Kim et al. (2008) analyses three-dimensional flow and transport characteristics in
two representative multi-chamber ozone contactor models with different chamber width
using LES.
Wang et.al., (2008) used different turbulence closure schemes i.e., the mixing-length
model and the k-ε model with different pressure solution techniques i.e., hydrostatic assumptions
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LITERATURE SURVEY
and dynamic pressure treatments are applied to study the helical secondary flows in an
experiment curved channel. The agreements of vertically-averaged velocities between the
simulated results obtained by using different turbulence models with different pressure solution
techniques and the measured data are satisfactory. Their discrepancies with respect to surface
elevations, super elevations and secondary flow patterns are discussed.
Balen et.al., (2010) performed LES for a curved open-channel flow over topography. It
was found that, notwithstanding the coarse method of representing the dune forms, the
qualitative agreement of the experimental results and the LES results is rather good. Moreover, it
is found that in the bend the structure of the Reynolds stress tensor shows a tendency toward
isotropy which enhances the performance of isotropic eddy viscosity closure models of
turbulence.
Beaman (2010) studied the conveyance estimation using LES method.
Esteve et.al., (2010) simulated the turbulent flow structures in a compound meandering
channel by Large Eddy Simulations (LES) using the experimental configuration of Muto and
Shiono (1998). The Large Eddy Simulation is performed with the in-house code LESOCC2. The
predicted stream wise velocities and secondary current vectors as well as turbulent intensity are
in good agreement with the LDA measurements.
Ansari et.al., (2011) presented the use of (CFD) to determine the distribution of the bed
and side wall shear stresses in trapezoidal channels. The impact of the variation of the slant
angles of the side walls, aspect ratio and composite roughness on the shear stress distribution is
analyzed. These equations derived compute the shear stress as a function of three components.
The results show a significant contribution from the secondary currents and internal shear
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LITERATURE SURVEY
stresses on the overall shear stress at the boundaries. This work also extends previous work of
the authors on rectangular channels.
Larocque, Imran, Chaudhry (2013) presented 3D numerical simulation of a dam-break
flow using LES and k- ε turbulence model with tracking of free surface by volume-of-fluid
model. Results are compared with published experimental data on dam-break flow through a
partial breach as well as with results obtained by others using a shallow water model. The results
show that both the LES and the k –ε modeling satisfactorily reproduce the temporal variation of
the measured bottom pressure. However, the LES model captures better the free surface and
velocity variation with time.
From literature survey, it is found that very limited work on boundary shear stress has
been reported for meandering channels. Although adequate literature is available on numerical
studies that make use of different turbulence models for modeling compound meandering
channels but the literature lacks substantial experimental works for compound meandering
channels.
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EXPERIMENTAL SETUP AND PROCEDURE
EXPERIMENTAL SETUP AND PROCEDURE
3.1. GENERAL
Estimation of boundary shear stress in a meandering channel is typical in the sense that many
unseen flow parameters comes into play due to which the three-dimensional nature of flow
increases. Also it is established by many researchers that the secondary current affects the
distribution of boundary shear stress in open channel flow. Owing to this the evaluation of
discharge capacity in a meandering channel is a complicated process and is dependent on precise
prediction of shear force carried by different boundary elements of a channel. The present
research work utilises the flume facility available in the Fluid Mechanics and Hydraulic
Engineering Laboratory of the Civil Engineering Department at the National Institute of
Technology, Rourkela, India. The basic objective behind these experiments is to conceive better
understanding on the variation of distribution of boundary shear stress due to variation of flow
and sinuosity under uniform flow conditions. The following section provides a brief overview of
details of hydraulic and geometric parameters of the present meandering channel, experimental
arrangements, measuring equipments and procedure used in the process of data collection.
3.2 EXPERIMENTAL ARRANGEMENTS
3.2.1 Apparatus and Materials Used
The experiments are conducted in channel built in a long tilting flume made up metal frame with
glass walls of size 15m long; 4m wide and 0.5m deep. The flume can be tilted with the help of
hydraulic jack arrangement for different bed slope arrangements. The channel is cast using 6mm
thick Perspex sheet, having Manning’s n value 0.01. The experimental meandering channel is
Page | 25
EXPERIMENTAL SETUP AND PROCEDURE
trapezoidal at cross-section and all measurements were taken at central bend apex. The detailed
information on geometric parameters of meandering channel is provided in the table below.
Table.1 Geometry Parameters of the Experimental Meandering Channel
Sl. No.
Highly Meander channel
Item Description
1.
2.
3.
Wave length in down valley direction
Amplitude
Geometry of main channel section
4.
5.
6.
7.
8.
9.
10.
11.
12.
Main channel width (B)
Bank full depth of main channel
Top width of compound channel (B')
Slope of the channel
Meander belt width (BW)
Nature of surface bed
Sinuosity(Sr)
Cross over angle in degree
Flume size
4054mm
2027mm
Trapezoidal
(side slope 1:1)
330mm at bottom
65mm
460 mm
0.0055
2357mm
Smooth and rigid bed
2.04
90
15m*4m*0.5m
Photographs of the experimental meandering channel from two different views with measuring
equipments are shown in Photos.3.1 (a, b) whereas Figs.3.1 (a, b) show the plan view of half
meander wavelength with dimensional details at bed and at bank full level (i.e., at 6.5 cm)
respectively.
Photo.3.1 (a) View of Experimental Channel
Page | 26
Photo.3.1 (b) Side View of Experimental Channel
EXPERIMENTAL SETUP AND PROCEDURE
Fig.3.1(a) At Bed Level
Fig.3.1(b) At 6.5cm above Bed
For the sake of experiment, two tanks namely overhead tank made up of reinforced cement
concrete (RCC) and masonry volumetric tank at the downstream of the channel are constructed.
Various arrangements are done within the flume to convey water to the channel. Those are
stilling chamber, baffle walls, head gate, rectangular notch, square wire mesh, travelling bridge
and tail gate. The experimental arrangements also consists of an underground sump, water
supply devices, two parallel pumps etc. The plan view of full length experimental channel with
other arrangements is shown in Fig.3.2.
Fig.3.2
*After Mohanty et.al. (2012)
Page | 27
*Schematic Diagram of Experimental Setup
EXPERIMENTAL SETUP AND PROCEDURE
3.2.2 Measuring Equipments
A pointer gauge, located on a mobile instrument carriage, is used to measure the water
level at different locations along the flume to an accuracy of 0.1 mm. Five unequally spaced
micro-Pitot tube each of them having 4.6 mm external diameter is used in conjunction with five
manometers placed inside a transparent fibre block fixed to a wooden board. A spirit level is
positioned at the top of the wooden board to maintain the verticality of manometers. On the
experimental flume, main guide rails are provided on which a travelling bridge is moved in the
longitudinal direction of the entire experimental channel. The point gauge and a micro-Pitot tube
are attached to the travelling bridge with secondary guide rails allowing the equipments to move
in both longitudinal and the transverse direction of the experimental channel. A rectangular notch
arrangement made at the upstream of the channel is calibrated to establish stage-discharge
relationship and to estimate the theoretical discharge whereas a piezometer fitted to the tail tank
for actual discharge through the channel. The measuring equipments and the devices are
arranged and calibrated properly to carry out experiments in the channel. The following
photographs show the measuring devices used for data collection. The photographs of series of
pitot static tube fitted to stand with point gauge and set of piezometers fitted to wooden board to
record pressure with spirit level are shown in Photos. 3.2(a) and 3.4(b) respectively.
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EXPERIMENTAL SETUP AND PROCEDURE
Photo 3.2(a)
Series of Pitot static tube
Photo .3.2(b)
Set of piezometers with spirit level
3.3 EXPERIMENTAL PROCEDURE
Two parallel pumps are used to pump water at the rate up to 200 lit/sec from an underground
sump to the overhead tank. The water in overhead tank is maintained at constant head so that the
excess water returns to the sump again. The water is conveyed to the flume by two different
supply pipelines. To reduce large disturbances in the outgoing flow from the pumps, water is
first conducted into a stilling tank from where it is led to an adjustable vertical gate along with
series of baffle walls in upstream section sufficiently ahead of rectangular notch to reduce
turbulence and velocity of the incoming water. From the rectangular notch water is made to fall
on a wire mesh provided just below the notch. Water is then directed to the channel to flow
under gravity through a smooth bell mouth transition section to improve the inflow conditions
from the inlet tank to a specific channel. Finally the water at the downstream end is allowed to
flow through another adjustable tailgate and is collected in a masonry volumetric tank from
where it is again flow back to the underground sump. From the sump, water is then pumped back
to the overhead tank, thus a complete re-circulating system of water supply for the experimental
channel is established. The adjustable tailgates were used to achieve uniform flow for a specific
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EXPERIMENTAL SETUP AND PROCEDURE
flow depth. Since in uniform flow conditions, the energy slope (Se), the water surface slope (Sw)
and the bed slope (S0) are all equal, i.e; Se = Sw = S0. It is only under this condition that the depth
and velocity can be assumed to be constant at all cross sections, before any measurement could
be taken in the channel, uniform flow conditions had to be achieved. The adjustable tailgates at
the downstream end of the flume were used for this purpose. All the measurements are taken at
the bend apex of the third wave reach of the experimental channel from the upstream end to
achieve a fully developed flow. Observations are recorded for different flow depths, only under
steady and uniform conditions.
3.3.1 Measurement of Bed Slope
The water in the channel is kept still by blocking the adjustable tail gate provided at the
downstream end of experimental channel. With the help of a pointer gauge the bed and water
surface level are recorded at a certain point (say A) in standstill condition of water. Towards
downstream, at another point (say B) the bed and water surface level are again noted. The
elevation between these two points is given by (∆A - ∆B). This is repeated for number of points
along the channel centerline for distance of one wavelength. The mean slope for the meandering
channels may be obtained from,
Slope = Σ(∆A - ∆B) / L
Where,
∆A = Level difference of channel bed and water surface at point A
∆B = Level difference of channel bed and water surface at point B
L = length of meander wave along the centerline.
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(1)
EXPERIMENTAL SETUP AND PROCEDURE
3.3.2 Calibration of Notch
The accurate estimation of discharge can be obtained from the rectangular notch fitted at the
upstream of the channel. For which the notch is needed to be calibrated first before measuring
the discharge. The actual discharge passing through channel is recorded from a volumetric tank
of 208666 cm2 provided at outfall of the channel. A piezometer connected to this tank gives rise
in height of water in volumetric tank over a time interval. Variation of time depends on the rate
of flow from the channel e.g., the time of collection of water in the measuring tanks vary
between 60 to 300 seconds; lower one for higher rate of discharge. Time is recorded using a
stopwatch with respect to the collection of water in the tank and the rise in height of water is
obtained from measuring scale attached to the piezometer. Finally change in the mean water
level in the tank over the time interval is recorded. The volume of water collected in the tank is
given by
V = Ah
(2)
From the knowledge of the volume of water collected in the measuring tank and the
corresponding time of collection, the actual discharge in the experimental channel for each run
is obtained by,
Qa = V/t
(3)
For theoretical discharge, the height of water above notch is measured by a point gauge
arrangement made at the notch in a wooden platform. Theoretical discharge is given by,
Qa = Cd
Page | 31
2
3/ 2
L 2g H n
3
(4)
EXPERIMENTAL SETUP AND PROCEDURE
The coefficient of discharge for each run is calculated as per equation given below,
(5)
Q a = C d Qt
Where A = Area of volumetric tank i.e., 208666 cm2, h = Height of water in the volumetric tank,
Q a = Actual discharge, V = volume of water in tank,
t = time interval, Qt = Theoretical
discharge, L = Length of the notch, H n = Height of water above the notch, g = Acceleration due
to gravity, C d = Coefficient of discharge calculated from notch calibration
Hence the rectangular notch is calibrated for the purpose of present experiment and the coefficient of
discharge Cd is found to be 0.71.
3.3.3
Measurement of Normal Depth and Discharge
Once the notch is calibrated and the coefficient of discharge is made fixed, the discharge ‘ Q a ’for
each run is calculated as for the equation given below.
Qa = C d
2
3/ 2
L 2g H n
3
where
Q a = Actual discharge
C d = Coefficient of discharge calculated from notch calibration
L = Length of the notch
H n = Height of water above the notch
g = Acceleration due to gravity.
Experimental results concerning stage-discharge relationships for meandering channels with rigid
Page | 32
(6)
EXPERIMENTAL SETUP AND PROCEDURE
boundaries are accessed. A pointer gauge located on the travelling bridge was used to measure
the flow depth at bend apex in the channel for a given discharge. For meandering channel, the
level difference of bed and water surface at outer bank is recorded by pointer gauge which
gives depth at the outer bank. Likewise the procedure is repeated at inner bank to get normal
depth. The mean depth of these two depths is taken as normal depth for a particular discharge.
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EXPERIMENTAL RESULTS AND ANALYSIS
EXPERIMENTAL RESULTS AND ANALYSIS
4.1 GENERAL
The experimental results concerning the distribution of boundary shear along the wetted
perimeter and flow has been presented in this section. The stage-discharge relationship from inbank to over-bank flow situation for meandering compound channel is shown in Fig.4.1.
Analysis of results is done for distribution of boundary shear stress in meandering channels and
shear force results are derived accordingly. Empirical models are developed for percentage of
shear force carried by bed and inner and outer banks to better understand the underlying flow
mechanism in meandering channels.
4.2 STAGE- DISCHARGE RELATIONSHIP
Making a flood prediction while using hydraulic model which incorporates various flow features,
is not an easy task. Researches have shown that the structures of the flow are even more difficult
to analyze for compound meandering channel, due to an increase in 3-Dimensional nature of
flow (Shiono, Al-Romaih, and Knight 1999). In the present experimentation involving flow in
simple meandering channel, steady and uniform flow has been tried to achieve. Flow depths in
the experimental channel runs are so maintained that the water surface slope becomes parallel to
the valley slope to minimize the energy losses. Under such conditions, the depths of flow at the
channel centerline along one wave reach must be the same. This depth of flow is considered as
normal depth, which can carry a particular flow only steady and uniform condition. The stage
discharge curve plotted for the present meandering channel is shown in Fig.4.1 The figure show
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EXPERIMENTAL RESULTS AND ANALYSIS
discharge against the stage from in-bank to over-bank flow situations. Also the hydraulic
parameters of each experimental runs for in bank and overbank flow are listed in Table 2.
Table 2. Hydraulic parameter for the experimental runs
Runs
INBANK FLOW
1
2
3
4
5
6
OVERBANK
FLOW
1
2
3
4
5
Discharge Q
(in lit/s)
Flow depth
(in cm)
1.165
1.872
3.803
4.153
6.055
7.5
1.7
2.5
3.8
4.0
5.0
5.8
18.564
33.227
42.513
46.635
52.533
8.9
9.5
9.9
10.1
10.6
Relative
depth
β
Froude No.
(Fr)
Reynolds No.
(R)
(H/h)
0.2615
0.3846
0.5846
0.6154
0.7692
0.8923
(H'-h /H')
0.495
0.441
0.468
0.471
0.484
0.473
3847.90
5832.49
10851.62
11701.64
16035.97
18953.12
0.2697
0.3158
0.3434
0.3564
0.3868
0.314314
0.397014
0.464403
0.495639
0.521894
28389.52
39091.89
46843.45
53654.54
62962.90
Fig.4.1 Plot of Stage versus Discharge
Page | 35
EXPERIMENTAL RESULTS AND ANALYSIS
4.3 SHEAR STRESS MEASUREMENTS
Shear studies in open channel flow has many implications such as bed load transport, channel
migration, momentum transfer etc. Bed shear forces are useful for the study of bed load transfer
where as wall shear forces presents a general view of channel migration pattern. There are
several methods used to evaluate bed and wall shear stress in an open channel. The Preston-tube
method is an indirect estimate for shear stress measurements and is widely used for experimental
channel which is described below. In the following section, results regarding the distribution of
boundary shear stress along with the contours of local shear stress is shown and discussed. Also
the mean boundary shear stress results are discussed in details.
4.3.1 Preston-tube Technique
Using Preston’s technique (1954) together with calibration curves of Patel’s (1965) local
boundary shear stress measurements were made around wetted perimeter of the present
meandering channel. Preston developed a simple shear stress measurement technique for smooth
boundaries in a fully developed turbulent flow using a Pitot tube. Based on the law of the wall
assumption (Bradshaw and Huang, 1995), i.e. the velocity distribution near the wall can be
empirically related to the differential pressure between the dynamic and static pressures,
Preston presented a non-dimensional relationship between the differential pressures, ∆P and the
boundary shear stress, το:
(7)
Page | 36
EXPERIMENTAL RESULTS AND ANALYSIS
Where, d is the outside diameter of the tube, ρ is the density of the flow, ν is the kinematic
viscosity of the fluid and F is an empirical function. Following this work, Patel (1965) presented
definitive calibration curves for the Preston tube defined in terms of two non-dimensional
parameters which are used to convert pressure readings to boundary shear stress:
,
(8)
The calibration of x*and y* for different regions of the velocity distribution (i.e. viscous sub
layer, buffer layer and logarithmic layer) is expressed by three different formulae:
for 0 < y* < 1.5
y* = 0.8287 – 0.1381x* + 0.1437x*2 - 0.006x*3
for 1.5 < y* < 3.5
for 3.5 <y* < 5.3
INNER
@0.8 H
@0.4 H
@0.6 H
OUTER
H
Fig 4.2 (a)
Definition sketch of point locations used in shear stress measurements for
simple meander channel at bend apex (All location spacing’s are in centimeters)
Page | 37
EXPERIMENTAL RESULTS AND ANALYSIS
@ 0.4(H'-h)
INNER
@0.6(H'-h)
OUTER
H'
h
0.24
3.25
Fig 4.2 (b)
0.46
Definition sketch of point locations used in shear stress measurements for
compound meander channel at bend apex
(All location spacing’s are in meters)
In the present case, all shear stress measurements are taken at the bend apex due to minimum
curvature effect. The pressure readings were taken using Pitot tube. These are placed at the
predefined points of the flow-grid in the channel, facing the flow which is demonstrated in Fig.
4.2. (a) and 4.2 (b) for in bank and overbank flow conditions respectively. The manometers
attached to the respective Pitot tubes are used to measure head difference. The differential
pressure was then calculated from the readings on the vertical manometer:
∆P = ρg∆h
Where ∆h is the difference between the two readings from the dynamic and static, g is the
acceleration due to gravity and ρ is the density of water. Here the tube coefficient is taken as unit
and the error due to turbulence considered negligible while measuring velocity. Each
experimental runs of the channel are carried out by maintaining the water surface slope parallel
to the valley slope to achieve the steady and uniform flow conditions.
Page | 38
EXPERIMENTAL RESULTS AND ANALYSIS
4.3.2
Shear Stress Contours
Most of hydraulic formulae assume that the boundary shear stress distribution is uniform over
the wetted perimeter. However it is established that such forces even in straight prismatic
channel with simple cross-sectional geometry are not uniform. For meandering channel the
distribution in shear stress widely varies from point to point along the wetted perimeter.
Therefore, a study has been carried out to investigate the distribution of local shear stress along
the channel boundary. Boundary shear stress measurements at the bend apex of a meander path
covering a number of points in the wetted perimeter have been obtained from Patel’s formulae
for only in bank flows.
4.3.2 (a) Simple meander channel
Boundary shear force is a turbulent quantity composed of a fluctuating component
superimposed on the mean value. The non-uniformity in shear stress is mostly due to this
fluctuating component which is interpreted as secondary currents and is generated due to the
anisotropy between the vertical and transverse turbulent intensities (Gessner, 1973). Even in
straight prismatic channel with simple cross-sectional geometry the distribution of shear stress is
not uniform. The interdependency of boundary shear stress and secondary flow has been
recognized since long; as such it becomes difficult to understand either phenomenon without
recourse to the other. Also the local shear stress largely depends on the velocity gradient near to
the boundary and consequently on the pattern of isovels. Boundary shear stress distributions
Page | 39
EXPERIMENTAL RESULTS AND ANALYSIS
along with local shear stress contours are obtained for four different depths of flow at bend apex
section of the meander path and are shown in Fig.4.3 (a, b, c and d).
Fig. 4.3(a)
Flow Depth (h) = 1.7 cm
Fig. 4.3(c) Flow depth (h) = 4 cm
Fig. 4.3(b) Flow depth (h) = 3.8 cm
Fig. 4.3(d) Flow depth (h) = 5cm
Fig.4.3 (a) - Fig.4.3 (d) Contours showing the local shear stresses for in bank flow at the
bend apex and graph showing distribution of boundary shear stress.
(Longitudinal shear stress contours are in N/m2)
Page | 40
EXPERIMENTAL RESULTS AND ANALYSIS
From the above shear stress contours the following inferences can be made:
i.
The location of thread of maximum local shear stress is found to occur near the inner wall
than outer wall.
ii.
The distribution of boundary shear stress along the wetted perimeter in meandering
channels follows sinusoidal (non-uniform) pattern. The distributons of boundary shear
stress are used to highlight secondary flow currents, as the boundary stress is usually a
better indicator of their presence than isovel patterns (Knight and Patel, 1985).
iii.
A remarkable feature of all the distribution is that a high value of shear stress persists
near the air-water interface. This is likely to happen as in these regions the surface
velocities are quite large.
4.3.2 (b) Compound meander channel
Fig.4.2(b) shows the grid points where the local shear stress measurements are made and the
shear stress contours for four relative depths are derived which are depicted in Fig.4.4 (a)-4.4 (b).
INNER
OUTER
Fig.4.4 (a)
Page | 41
Depth of flow, (H' -h) = 3.0 cm; H = 9.5cm
EXPERIMENTAL RESULTS AND ANALYSIS
INNER
OUTER
Fig.4.4 (b)
Depth of flow, (H' -h) = 3.4 cm; H = 9.9cm
OUTER
INNER
Fig.4.4(c)
Depth of flow, (H' -h) = 3.6 cm; H = 10.1cm
INNER
OUTER
Fig.4.4 (d) Depth of flow = (H' - h) = 4.1 cm; H = 10.6cm
Fig.4.4 (a) - Fig.4.4 (d) Contours showing the local shear stresses for over bank flow at the
bend apex. (Longitudinal shear stress contours are in N/m2)
Page | 42
EXPERIMENTAL RESULTS AND ANALYSIS
From the above local shear stress contours of compound meander channel for different flow
depths, the following remarkable features can be noted:
i.
The pattern of overbank shear stress contours is somewhat similar to that of in bank shear
contours.
ii.
The lowest shear stress contour lines prevail at outer main channel bottom corner and its
concentration increases with the increase in flow depth over the flood plain.
iii.
Another feature which is discernible from the contour diagram is that the maximum value
of local shear stress occurs at inner floodplain wall and its concentration increases as the
flow depth over the floodplain increases.
iv.
Interestingly the local average shear stress appears more prominent in the left floodplain
region.
4.4 ANALYSIS
4.4.1 MEAN BOUNDARY SHEAR STRESS
River scientists have found that water in reality does not move as an ideal frictionless fluid.
Water during its motion is resisted by various forces and in doing so it consumes energy.
Understanding this variation in resisting forces is of primary importance in shear studies. Water
is impelled downstream by force of gravity. This force so developed in flow direction is resisted
by reaction from channel boundary. If the flow is uniform, velocity does not change downstream
and one may conclude from Newton’s first law of motion that the driving and resisting forces
must be in balance.
The relevant forces are,
Page | 43
EXPERIMENTAL RESULTS AND ANALYSIS
Driving force (Ws) = downward component of the total weight of water,
Ws = W Sinθ = ρgAL Sinθ
(9)
Resisting Force (F0) = boundary shear stress* bed area,
F0 = τ0PL
Now Ws = F0,
(10)
ρgAL Sinθ = τ0PL
τ0 = ρg(A/P) Sinθ
(11)
(12)
The ratio (A/P) is known as hydraulic radius, Rh (m). Substituting (A/P) as Rh and Sinθ as slope S
in equation (12) we get,
τ0 = ρg RhS = γRhS
(13)
τ0 is referred as overall mean boundary shear stress or ‘depth-slope product’ because hydraulic
radius normally is approximated by the mean depth (h) of the channel. Equation (13) denotes the
variation of shear stress within the section. Furthermore this equation is strictly valid only for
uniform flow. Also τ0 to some extent applies reasonably well to gradually varied flow as in
experimental channel in which, at least for short reaches of channel, the flow approximates
uniform conditions.
4.4.2 DISTRIBUTION OF BOUNDARY SHEAR STRESS
4.4.2 (a) Simple meander channel
The wall and bed shear stresses are normalized with their mean values. The variation of
normalized inner and outer wall with z/H and the variation of normalized inner and outer bed
shear stress with 2y/B for two types of aspect ratio are shown in Fig.4.3 and Fig.4.4 respectively.
The influence of secondary currents on the boundary shear stress distribution in meandering
Page | 44
EXPERIMENTAL RESULTS AND ANALYSIS
channel may be established from Figs.4.5 and 4.6. With reference to Fig.4.6, for lower aspect
ratio (B/H=6.6) the maximum bed shear stress is displaced from centerline line towards a corner;
whereas at higher aspect ratio (B/H=19.4) the maximum bed shear stress returns to the bed
centerline position for inner bed region. For outer bed region the maximum bed shear stress
occurs at the centerline position irrespective of the aspect ratio. Similarly the maximum value of
inner and outer wall shear stress does not occur at the free surface (z/H=1) but at some
intermediate position between the free surface and the bed (Fig.4.5).The wall and bed shear
distributions exhibit certain perturbations which may be explained in terms of the number and
distribution of secondary flow cells within the cross section (Knight et.al., 1984). As seen from
Fig.4.5, the maximum wall shear stress is positioned next to the free surface (i.e., at 0.8 depth)
for low depth flow whereas for high depth flow the maximum wall shear stress occurs towards
bed (i.e., at 0.08-0.4depth).
Fig.4.5 Dimensionless local wall shear stress versus z/H for flow depths 5cm and 1.7 cm at
inner and outer wall
[
Page | 45
EXPERIMENTAL RESULTS AND ANALYSIS
Fig.4.6 Dimensionless local bed shear stress versus 2y/B for flow depths 5cm and 1.7 cm in
inner and outer bed region
4.4.2 (b) Compound meander channel
Overbank flow cases are complicated processes to model due to the presence of turbulent
structures ranging from large to small scales which transfers momentum from faster moving
fluids in main channel to relatively slower fluid in flood plain while increasing and decreasing
the conveyance respectively. Therefore a lot of experimental and numerical approaches has
been recently adopted to quantify this lateral momentum transfer at the main channel and flood
plain interface. This accretion in technical knowledge compel operating authorities should use
recent improved knowledge on conveyance to reduce uncertainty in flood predictions. Taking
this into account a team of experts led by HR Wallingford introduced a new Conveyance
Estimation System (CES) which is being adopted in England, Wales, Scotland and
Northern Ireland. CES has been recommended for use in natural rivers, artificial straight and
meandering channels for estimating conveyance, computing stage discharge relationship and
also a number of flow parameters like depth averaged velocity, boundary shear, shear velocity,
Page | 46
EXPERIMENTAL RESULTS AND ANALYSIS
energy coefficients etc. The software includes a roughness advisor, conveyance generator, and
an uncertainty estimator. The conveyance generator is based on 1D Shiono and Knight
conveyance estimation method (SKM). Previously conveyance estimation methods incorporate
only roughness parameter such as Manning’s n, Chezy’s C and Darcy Weisbach f, but SKM
consists three calibration constants: Weisbach f, dimensionless eddy viscosity λ, transverse
gradient of secondary flow term Г. Therefore it precisely models the flow to reduce
uncertainties in the estimation of river flood levels, discharge and velocities.
Hr = 0.316
Hr = 0.343
Page | 47
EXPERIMENTAL RESULTS AND ANALYSIS
Hr = 0.356
Hr = 0.387
Fig.4.7 Lateral distribution of shear velocity along the compound cross section of
experimental channel at bend apex for different relative depths
The boundary shear stress values are expressed in terms of shear velocity for different relative
depths and are validated with one-dimensional modeling software conveyance estimation system
(CES) which is shown in Fig. 4.7. Series 1shows the experimental values and series 2 shows the
values yielded through CES. Some important inferences can be drawn from these diagrams
which are summarized below:
Page | 48
EXPERIMENTAL RESULTS AND ANALYSIS
i.
In case of the present compound meandering channel, the maximum value of
boundary shear stress occurred at the junction between inner main channel bank and
the flood plain. However the boundary shear stress decreases in the main channel but
again rises at the junction between outer main channel bank and the flood plain. But
this time the increment in boundary shear stress is less as compared to the increase at
the inner main channel bank and the flood plain junction.
ii.
The minimum boundary shear stress occurs towards the outer bank region in the main
channel.
iii.
This trend of distribution of boundary shear stress gives enough indication of presence
of secondary flow at main channel corner and main channel-flood plain interaction
regions which is substantially affected by the large amount of momentum
transportation between the main channel and flood plain (Jing et. al., 2009).
iv.
The quasi-1D model conveyance estimation system (CES) underestimates boundary
shear stress values whereas it successfully reproduces velocity values.
4.4.3
SHEAR FORCE ANALYSIS
The measured local shear stresses are integrated over the respective wetted perimeter to obtain
boundary shear force per unit length of the channel. The energy gradient states that the weight of
water in the direction of flow in the channel is equal to the resistance offered from the channel
boundary.
Shear Force per unit length of wetted perimeter, (By Energy Gradient) F = ρgAS
Page | 49
EXPERIMENTAL RESULTS AND ANALYSIS
For the experimental channels, the mean shear found from the Preston- tube agrees well
with the mean value computed from energy gradient approach. The shear forces calculated from
measured shear stresses along the channel boundary are tabulated below in Table 3.
Table 3. Summary of boundary shear force results for the experimental simple meandering
channels observed at bend apex.
Expt.
Runs
Discharge
(cm3/s)
Average
velocity
(cm/s)
Flow
Depth
(cm)
Aspect
Ratio
SFbed
(SFw) i
(SFw) o
(N)
(N)
(N)
SFT
(Actual)
SFT
(Theoretical)
(ρgAS)
% of
error
for
SFT
M1
1165.29
19.75
1.7
19.4
0.300152
0.013155
0.006727
0.320034
0.318281
0.55
M2
1872
21.09
2.5
13.2
0.477705
0.027889
0.019468
0.525061
0.478851
8.80
M3
3803
27.19
3.8
8.68
0.665384
0.05687
0.04318
0.765433
0.754507
1.43
M4
4153.37
28.06
4
8.25
0.667682
0.062045
0.038288
0.768014
0.798534
-3.97
M5
6055.41
31.87
5
6.6
0.779612
0.142777
0.044873
0.967262
1.025145
-5.98
M6
7500
33.33
5.8
5.69
0.897967
0.167863
0.065055
1.130886
1.214203
-7.37
4.4.4
DEVELOPMENT OF MODEL
Flow and velocity distribution are more complex in meandering channels than the straight
channels due to influence of continuous changing curvatures and the flow path. The flow
structure becomes 3-dimensional influencing greatly to the asymmetrical nature of boundary
shear distribution between inner and outer sides. This necessitates separate analysis of two banks
(inner and outer) of meandering channel. Many investigators have developed empirical equations
for shear force by fitting equations to the data. Knight and Hamed (1984), Knight and Patel
(1985), Patra (1999) and Khatua (2008) have used the experimental data’s to show the variation
Page | 50
EXPERIMENTAL RESULTS AND ANALYSIS
of percentage of measured wall shear force to the total boundary shear force for different aspect
ratios. It is ascertained that percentage of total shear force carried by walls decreases with
increase in aspect ratio.
Knight et.al (1984) showed that for straight channel the percentage of shear in wall SFw
varied exponentially with the aspect ratio [i.e., α = B/H], that can be expressed as,
(14)
By plotting on logarithmic scale and using linear regression of (% SFw) against α (= b/h), we
can write
b

log 10 (% SF w ) = 1 .4026 log 10  + 3  + 3 + 2 .67
h


(15)
where, b is the base width and h the depth of flow of water. Comparing (1) and (2) we get
b

m = 2.30259[ A1 Log 10  + A2  + A3 ]
h

(16)
that gives
m = −3.23Log10 (α + 3) + 6.146
(17)
It proves that m is a function of the aspect ratio [i.e. m = f (b/h) = f (α)].
Finally,
Page | 51
(18)
EXPERIMENTAL RESULTS AND ANALYSIS
Khatua and
Patra (2010) used two types of meandering channels with smooth and rough
surfaces having sinuosity 1.21 and 1.22 for a range of aspect ratio 1.01 < α < 2.45 to further
modify (5) to include meandering effect, given as
%SFW = e m + 2.15Sr
−1.06
Ln(30.692×α )
(19)
where m = −3.23Log10 (α + 3) + 6.146
Incorporating experimental results of present study, (6) is now improved for higher ranges of
aspect ratio 5.7 < α < 19.4 and the modified equation as follows,
(20)
where (% SFw )mod .= modeled percentage shear force wall of both the walls, Sr = the sinuosity of
meandering channel and α = the aspect ratio.
In addition, a non linear best fit relation between the percentage of wall shear force (% SFw)
and the channel aspect ratio (α) in Fig.4.8 takes the following form,
(21)
where x = aspect ratio
Page | 52
and
y = (% SFw)mod.
EXPERIMENTAL RESULTS AND ANALYSIS
Fig.4.8
Variation of (%SFw)mod with Aspect Ratio
Since one of the objective of present study is to analyze inner and outer wall shear separately,
the computed inner wall shear force (SFw)inner and outer wall shear force (SFw)outer is nondimensionalised with total shear force to get (%SFw)inner and (%SFw)outer respectively. Variation
of (%SFw)inner and (%SFw)outer with aspect ratios are shown in Fig.4.9(a) and Fig.4.9(b)
respectively.
Fig.4.9 (a) Variation of (%SFw)i with Aspect Ratio Fig.4.9 (b) Variation of (%SFw)o with Aspect Ratio
Page | 53
EXPERIMENTAL RESULTS AND ANALYSIS
Comparing Figs.4.9 (a) and 4.9 (b), it is clear that the percentage of shear force at the inner wall
is more than that of the outer wall. Usually it is found that the average shear force at inner wall is
1.3 times the average shear force at the outer wall at low in bank flows whereas the average
shear force at inner walls increases by 3 folds than outer wall for higher flow depths. Also, the
variation of (% SFbed) with aspect ratio for the observed values is shown in Fig.4.10. It is
surprisingly found that, for low in bank flows average shear force at bed is 15 times higher than
that of average shear force at wall whereas at higher depths the variation of shear force decreases
to 4. The experimental measurements of bed shear force reveals the dominating influence of bed
than wall in meandering channels at higher aspect ratio. The values of percentage shear force at
the bed, at inner and outer wall has been provided in Table 4.
Table 4. Summary of percentage shear force results along the wetted perimeter for the
experimental simple meandering channels observed at bend apex.
Modeled
Percentage
shear force on
walls
(%SFw)mod.
6.43401
Percentage
shear force at
inner wall
(%SFw)inner
Percentage
shear force at
outer wall
(%SFw)outer
Percentage
shear force on
bed
(%SFb)
19.4
Actual
Percentage
shear force on
walls
(%SFw)act.
6.212461
4.110434
2.101953
93.78761
13.2
10.13039
9.589239
6.757115
3.51718
92.55157
8.68
13.07096
14.63293
7.429735
5.641221
86.92904
8.25
13.06392
15.37401
8.078583
4.985335
86.93608
6.6
19.40013
18.8445
14.761
4.639132
80.59987
5.69
19.70742
20.95215
14.69801
5.949956
79.54673
Aspect Ratio
(B/H)
Page | 54
EXPERIMENTAL RESULTS AND ANALYSIS
R2=0.9539
Fig.4.10 Variation of (%SF bed) with Aspect Ratio
Fig.4.11 Observed and modeled values of (%SFw)
The variation of error in the computation of (%SFw) by the proposed (20) against observed
(%SFw) is plotted in Fig.4.11 giving the least error which shows adequacy of developed
equation.
Page | 55
NUMERICAL MODELING
NUMERICAL MOELING
5.1
GENERAL
Computational Fluid Dynamics (CFD), is a branch of fluid mechanics that uses numerical methods and
algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the
calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary
conditions. With high-speed supercomputers, better solutions can be achieved. Ongoing research yields
software that improves the accuracy and speed of complex simulation scenarios such as transonic or
turbulent flows. It has started around 1960 and with the process of improvement in computer processor
speed, CFD simulation is now showing astounding accuracy. The CFD based simulation relies on
combined numerical accuracy, modeling precision and computational cost.
The fundamental basis of almost all CFD problems is the Navier–Stokes equations, which define
any single-phase fluid flow. These equations can be simplified by removing terms describing
viscosity to yield the Euler equations. Further simplification, by removing terms describing
vorticity yields the full potential equations. Finally, for small perturbations in subsonic and
supersonic flows these equations can be linearized to yield the linearized potential equations.
There is no direct solution of the equation for flow. The N-S in vector form for single phase
incompressible fluid flow can be expressed as:
(22)
Where σij and τij are normal and shear stress component on any assumed plane normal to i
along j direction. ūi', ūj' are time averaged instantaneous velocity component along i,j
Page | 56
NUMERICAL MODELING
directions. p = pressure, µ = co-efficient of viscosity, ρ = density. The process of the numerical
simulation of fluid flow using the above equation generally involves four steps and the details
are:
(a)Problem identification
1. Defining the modeling goals
2. Identifying the domain to model
(b) Pre-Processing
1. Creating a solid model to represent the domain (Geometry Setup)
2. Design and create the mesh (grid)
3. Set up the physics
•
Defining the condition of flow (e.g turbulent, laminar etc.)
•
Specification of appropriate boundary condition and temporal
condition.
(c) Solver
1. Using different numerical schemes to discretize the governing equations.
2. Controlling the convergence by iterating the equation till accuracy is achieved
(d) Post processing
1. Visualising and examining the results
2. Considering revisions to the model
5.2
GEOMETRY SETUP
The fluid flow governing equations (momentum equation, continuity equation) are solved based
on the discretization of domain using the Cartesian co-ordinate system. This procedure involves
Page | 57
NUMERICAL MODELING
dividing the continuum into finite number of nodes. The CFD computations need a spatial
discretization scheme and time marching scheme. Mainly the domain discretization is
based on Finite element, Finite Volume and Finite Difference Method. Finite Element
method is based on dividing the domain into elements. The numerical solution can be
obtained in this method by integrating the shape function and weighted factor in an
appropriate domain. This method is suitable with respect to both structured and unstructured
mesh. The application of Finite Volume method needs dividing the domain into finite number of
volumes. Here the specified variables are calculated by solving the discreitized equation in the
centre of the cell. This method is developed by taking conservation law in to account.
Finite Volume method is suitable for applying in unstructured domain. Finite Difference method
is based on Taylor's series approximation. This method is more suitable for regular domain.
The meandering open channel corresponding to one wavelength of five wavelengths is
continuously constructed in the experimental flume. In the experimental flume, five meander
waves are fabricated for the case of Sr = 1.29, where the sinuosity, Sr, is defined as the ratio of
the meandering channel length to the meander wavelength. One meander wavelength has
trapezoidal cross-section with following dimensions: bed width, 330mm; top width, 460mm;
main channel depth, 65mm; meander wavelength, 4170 mm and central angle of the bend; 120◦.
Each section has a 60° circular bend with the centreline radius of 1.1 m followed by a 0.68m
straight reach which approximates a sine generated meandering curve. For the present case of
numerical solution, the flow is simulated at 50 mm depth of water in the meander channel.
Page | 58
NUMERICAL MODELING
5.3
DISCRETIZATON OF DOMAIN (MESHING)
The discretization of complex computational domain is critical. These kinds of domain don‘t
coincide with the co-ordinate lines with that of a structured grid, which leads to approximation of
the geometry. The only procedure to represent complex computational domain is to use a
stepwise approximation. But such an approximation is also arduous and quite time
consuming. Further, the stepwise approximation introduces truncation error and that can
be overcome by providing very fine Cartesian mesh. In whichever way the domain is
discretised, based on mesh methods such as any of those mentioned above, care has to be taken
in order to produce a good mesh. A mesh with too few nodes could lead to a quick solution, yet
not a very accurate one. However a very dense mesh of nodes will potentially waste
computational time and memory. Usually more nodes are required within areas of interest, such
as near wall and wake regions, in order to capture the large variation of fluid properties expected
in these regions. Thus, structure of grid lines causes further wastage of computer storage due to
un-necessary refinement. In this study, the flow domain is discretized using structured grid and
body-fitted coordinates. The detailed meshing of the flow domain with two views is shown in
Fig.5.1
Fig. 5.1
Page | 59
Schematic diagram of structured mesh
NUMERICAL MODELING
5.4
TURBULENCE
Turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This
includes low momentum diffusion, high momentum convection, and rapid variation of pressure
and velocity in space and time. Turbulence occurs when the inertia forces in the fluid become
significant compared to viscous forces, and is characterized by a high Reynolds Number. In
principle, the Navier-Stokes equations describe both laminar and turbulent flows without the
need for additional information. However, turbulent flows at realistic Reynolds numbers span a
large range of turbulent length and time scales, and would generally involve length scales much
smaller than the smallest finite volume mesh, which can be practically used in a numerical
analysis. The Direct Numerical Simulation (DNS) of these flows would require computing
power which is many orders of magnitude higher than available in the foreseeable future. To
enable the effects of turbulence to be predicted, a large amount of CFD research has concentrated
on methods which make use of turbulence models. Turbulence models have been specifically
developed to account for the effects of turbulence without recourse to a prohibitively fine mesh
and direct numerical simulation. Most turbulence models are statistical turbulence model, as
mentioned below.
Turbulence Models
Algebraic (zero-equation) model
k-ε, RNG k-ε
Shear stress transport
k-ω
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NUMERICAL MODELING
Reynolds stress transport model (second moment closure)
k-ω Reynolds stress
Detached eddy simulation (DES) turbulence model
SST scale adaptive simulation (SAS) turbulence model
Smagorinsky large eddy simulation model (LES)
Scalable wall functions
Automatic near-wall treatment including integration to the wall
User-defined turbulent wall functions and heat transfer
The two exceptions to this in ANSYS CFX are:
• Large Eddy Simulation Theory
• Detached Eddy Simulation Theory
However open-channel flows are characterized by complicated flow structures, even for simple
geometry, such as that of a rectangular channel. This is largely due to wall and free surface
boundaries. The secondary currents are generated in open channel flows because free surface and
walls reduce the turbulence intensity in the direction normal to the surface or the walls and lead
to anisotropy of turbulence. The three dimensional nature of turbulent flow can be decomposed
into mean part and fluctuation part, which is called Reynolds decomposition. Gravity and
channel geometry are mainly responsible for turbulent flow for this particular condition.
5.5
NUMERICAL MODEL
5.5.1
Large Eddy Simulation (LES)
The three major types of turbulence methodologies are Direct Numerical Simulation (DNS),
Large eddy Simulation (LES) and k-epsilon modelling. k-epsilon models the turbulent flows by
Page | 61
NUMERICAL MODELING
time or space averaging. But it is not suitable for transient flows because the averaging process
removes most of the important characteristics of a time-dependent solution. On the other hand,
Direct Numerical Simulation, attempts to solve all time and spatial scales. As a result, the
solution is very accurate. However to resolve almost all ranges of scales, the spatial and temporal
grids would need to be extremely small in the order of Kolmogorov length and time scale,
resulting in a problem which would take an extraordinarily long time to solve, making it
computationally intensive.
Fig. 5.2
Energy cascade process with length scale
One compromise between these two models is Large Eddy Simulation. Large Eddy Simulation
directly solves large spatial scales (DNS), while modeling the smaller scales (k-epsilon). First,
the larger scales that carries the majority of the energy, which accounts for 80% of turbulent flow
energy and hence is more important. Second, the smaller scales have been found to be more
universal, and hence are more easily modeled. The resulting methodology is a hybrid between
these two methods, which involves the filtering of the Navier-Stokes equations to separate those
Page | 62
NUMERICAL MODELING
scales which will be modeled from those which will be solved for directly. Subsequently LES
method simulates large scale turbulent motions directly, while the unresolved small
scale motions are modeled through the use of a Smagorinsky model. This model captures
larger scale motion such as DNS, as well as it covers the effects of small scales of eddies by
using sub-grid scale (SGS) model.
5.5.2
Mathematical Model
Large Eddy Simulation (LES) is about filtering of the equations of movement and decomposition
of the flow variables into a large scale (resolved) and a small scale (unresolved) parts. Any flow
variable f can be written such as:
(23)
Where
the large scale part, is defined through volume averaging as:
Where, G(x – ) is the filter function (called the hat filter or Gaussian filter). After performing
the volume averaging and neglecting density fluctuations, the filtered Navier-Stokes equations
become:
(24)
The non linear transport term in the filtered equation can be developed as:
Page | 63
NUMERICAL MODELING
In time, averaging the terms (2) and (3) vanish, but when using volume averaging, this is no
longer true.
Introducing the sub-grid scale (SGS) stresses, τij, as:
(25)
we can rewrite the filtered Navier-Stokes equations as:
with
(26)
Above equation is the basis of the LES technique. LES explicitly models smaller scale motions
thereby reducing the computational cost to a larger extent than DNS in which effort is given in
modeling smaller scales whereas larger scales predominantly contains energy and anisotropy.
Sub-Grid Scale Models
The non-linear transport of energy generates ever-smaller scales like a cascade process until it
reaches the size of Kolmogorov scales as shown in Fig.5.2. The essence of LES is to account for
this energy cascade from resolved large scales to the unresolved sub-grid scales. This is the role
of the SGS model. The most popular class of SGS models is the eddy-viscosity type, based on
(variants of) the Smagorinsky model (Smagorinsky, 1963).
Smagorinsky Model
Page | 64
NUMERICAL MODELING
The Smagorinsky model can be thought of as combining the Reynolds averaging assumptions
given by Lij + Cij = 0 with a mixing-length based eddy viscosity model for the Reynolds SGS
tensor. It is thereby assumed that the SGS stresses are proportional to the modulus of the strain
rate tensor,
of the filtered large-scale flow:
(27)
To close the equation, a model for the SGS viscosity VSGS is needed. Based on dimensional
analysis, the SGS viscosity can be expressed as:
where l is the length scale of the unresolved motion (usually the grid size ∆=(vol)1/3 ) and qSGS is
the velocity of the unresolved motion. In the Smagorinsky model, based on an analogy to the
Prandtl mixing length model, the velocity scale is related to the gradients of the filtered velocity:
where
(28)
This yields the Smagorinsky model for the SGS viscosity:
(29)
with Cs the Smagorinsky constant. The value of the Smagorinsky constant for isotropic
turbulence with inertial range spectrum:
is
(30)
For practical calculations, the value of Cs is changed depending on the type of flow and mesh
resolution. Its value is found to vary between a value of 0.065 (channel flows) and 0.25. Often a
value of 0.1 is used.
Page | 65
NUMERICAL MODELING
5.6
BOUNDARY CONDITIONS
5.6.1
Wall
A no-slip boundary condition is the most common boundary condition implemented at
the wall and prescribes that the fluid next to the wall assumes the velocity at the wall, which is
zero.
U=V=W=0
5.6.2
Free Surface
Here, Symmetry Boundary condition is used for the free-surface. This condition follows that, no
flow of scalar flux occurs across the boundary. In applying this condition normal velocities are
set to zero and values of all other properties outside the domain are equated to their values at the
nearest node just inside the domain. Here the experimental bulk velocity of the flow is initially
approximated as
W = 0.319 m/s, V = 0, U= 0 and
5.6.3
Inlet and Outlet Boundary Condition
To initialize the flow a mean velocity was specified over the whole inlet plane and is computed
by Uin=Q/A, where Q is the flow discharge of the channel and A is the cross section area o f the
inlet. A pressure gradient was further specified across the domain to drive the flow. In order to
specify the pressure gradient the channel geometries were all created flat and the effects of
gravity and channel slope implemented via a resolved gravity vector. It represents the angle
between the channel slope and the horizontal, the gravity vector is resolved in x, y and z
components as (ρgsinθ 0 -ρgcosθ).
Page | 66
NUMERICAL MODELING
Where θ = angle between bed surface to horizontal axis. Here, the x component denotes the
direction responsible for flow of water along the channel and the z component is
responsible for creating the hydrostatic pressure. From the simulation, z component of the
gravity vector (-ρgcosθ) is found to be responsible for the convergence problem of the solver.
Opening type boundary condition is imposed at the outlet.
5.7
NUMERICAL RESULTS
ANSYS-CFX 13.0 solver manager is used to carry out the simulation process. Here the
advection term is discretized with bounded central difference scheme and transient terms
are discretized with Second order scheme. Courant number (Cr) is controlled between 0 - 0.5
and the transient time step size is taken as 0.001 s. After that, the equation is iterated over and
over till desirable level of accuracy of 10-6 of residual value is achieved.
Page | 67
NUMERICAL MODELING
Fig. 5.3 Contours of bed shear stress (longitudinal, lateral and resultant) along one
wavelength reach of the 60° meandering channel
Page | 68
NUMERICAL MODELING
One of the turbulent flow properties, bed shear stress is simulated for the 60 degree simple
meandering channel. The simulation results show that the flow patterns are similar to
experimental studies. Figure 5.3 shows the contours of bed shear in longitudinal, lateral and
resultant directions.
Tominaga and Nezu (1991) noticed that the flow is considered to be uniform incompressible
fully developed turbulent flow at a test section of 7.5 m. As it can be seen from the above
figures that the flow becomes fully developed at third bend apex of the channel reach. The
numerically modeled resultant bed shear stress at the third bend apex shows similar pattern of
distribution as that of experimental bed shear stress. The contours of experimental shear stress as
shown in Figures 4.3 (a)-4.3(d) depicts the presence of maximum thread of shear stress values
towards inner bed region, as such the modeled bed shear stress reveals almost the same
distribution pattern. Figure 5.3 demonstrates that the maximum resultant shear stress occurs near
the inner side of the main channel as the flow enters the curved part from the straight reach. But
in the straight reach region (at the point of inception) the thread of maximum shear stress
gradually shifts to channel centerline. It is seen that the numerical simulation is in reasonable
agreement with the experimental measurements.
Page | 69
CONCLUSIONS
CONCLUSIONS AND FURTHER WORK
6.1
CONCLUSIONS
Experiments are carried out to investigate the effect of sinuosity and channel aspect ratio on
the boundary shear in a meandering channel. Point to point observations are made at bend
apex of meandering channel for wall shear data at different aspect ratio and for a higher
sinuosity (Sr = 2.04) of 90° bend meander channel. Based on analysis and discussions of the
experimental investigations certain conclusions from the present work are as discussed
below:
For simple meander channel, results show that shear force values are skewed and
more shear force is observed at the inner wall than outer wall contrary to the findings
for the narrow and deep channels.
It is also observed in case of simple meandering channel that, as the aspect ratio in the
main channel decreases the ratio of shear force between the inner and outer walls
increase indicating that the shear force per unit length at inner wall increases faster
with respect to the outer walls. Similarly, it can be seen that though the bed and wall
shear increases with depth of flow in main channel, the rate of increase in wall shear
is nearly four times the rate of increase in bed shear giving rise drastic decrease in the
ratio of bed to wall shear with decrease in aspect ratio.
The proposed (20) in this study can estimate wall shear forces for simple meandering
channels provided channel geometry, width, flow depth and sinuosity are known. The
equation is good for channels with higher aspect ratio.
Sinusoidal distribution of boundary shear stress along the wetted perimeter is
observed which confirms the presence of secondary currents in meandering in bank
flows.
Page | 70
CONCLUSIONS
Functional relationship between percentage of shear force carried by bed, inner and
outer wall of meandering channel have been analysed as a function of depth ratio.
Power expressions with higher correlation degrees have been obtained at the bed and
inner wall whereas exponential relationship holds good at outer wall.
Interestingly it is found that the pattern of overbank shear stress contours is somewhat
similar to that of in bank shear contours. A trend of distribution of boundary shear
stress has been extensively studied in compound meandering channel which gives
enough indication of presence of
secondary flow at main channel corner and main
channel-flood plain interaction regions that is substantially affected by the large
amount of momentum transportation between the main channel and flood plain.
For compound meandering channel, the observed boundary shear stress data in terms
of shear velocity is validated with CES and it is concluded that unlike velocity/depthaveraged velocity, CES underestimates shear velocity.
Numerical analysis has been performed for the fully developed turbulent flow of a 60°
simple meandering channel using Large Scale Eddy model in ANSYS CFX 13.0. The
bed shear stresses results are thus derived in the form of contour which is in
agreement with the experimental results.
6.2
RECOMMENDATIONS FOR FUTURE WORK
The present research is restricted to sole channel geometry, nature of surface and sinuosity of
the meandering channel. The work thereof leaves a broad spectrum for other investigators to
explore many intricate flow phenomena such as secondary currents, turbulent intensities and
vortices that significantly affects the distribution of boundary shear stress in simple and
compound meandering channel. Evaluation of boundary shear stress distribution has been
performed for the simple and compound meandering channels involving limited data. The
Page | 71
CONCLUSIONS
equations developed may be improved by incorporating more data from channels of different
geometries and sinuosity. The future scope of the present work may be summarized as:
1. Distribution of boundary shear stress components in lateral and vertical directions can be
evaluated which has implications for the sediment transport studies.
2. The present work lacks shear force analysis for over bank flow conditions. The
percentage of floodplain and main channel shear can be estimated and models can be
developed incorporating present data.
3. The current data can be used to validate with data of other investigators and natural
rivers.
4. Modeling by conventional methods is not reliable due to its instinct one dimensional
modeling of flow. Though the computational methods effectively capture intricate turbulent
structures in flow but require huge computer resources. Thus analytical methods based on
mechanism of energy transportation, momentum and continuity equations can be solved with
least approximation.
5. The channel here is smooth and rigid. Further investigation for the distribution of
boundary shear stress may also be carried out for mobile beds and by roughening the
channel bed.
6. LES and other turbulence closure models like k-ɛ, k-ω, RSM etc can be used to
simulate various channel geometry with different hydraulic conditions.
Page | 72
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Page | 79
Appendix A-I
Publications from the Work
A: Published
1. Patnaik, M., Patra.K.C., Khatua, K.K., Mohanty, L. (2012) “Boundary Shear Distribution
in Highly Sinuous Meandering Channels” Proceedings of National Conference on
Hydraulic and Water Resources, IIT Bombay, India, HYDRO 2012, 1233-1242.
2. Mohanty, L., Patra.K.C., Khatua, K.K., Patnaik, M. (2012) “Depth-Averaged Velocity
Distribution in Trapezoidal Meandering Channels” Proceedings of National Conference
on Hydraulic and Water Resources, IIT Bombay, India, HYDRO 2012, 625-634.
3. Patnaik, M., Mohanty, L., Patra.K.C. (2013) “Wall and Bed Shear Distribution in
Meandering Channels” Symposium on Sustainable Infrastructure Development, IIT,
Bhubaneswar, Odisha, India, IWMSID 2013, 374-382.
4. Mohanty, L., Patnaik, M., Patra.K.C. (2013) “Lateral Distribution of Depth-Averaged
Velocity in Trapezoidal Meandering Channels” Symposium on Sustainable Infrastructure
Development, IIT, Bhubaneswar, Odisha, India, IWMSID 2013, 383-389.
5. B: Accepted for Publication
1. Patnaik, M., Patra.K.C., Khatua, K.K., Mohanty, L. (2012) “Modeling Boundary Shear
Stress in Highly Sinuous Meandering Channels” accepted for ISH Journal of Hydraulic
Engineering, Taylor & Francis Group, UK.
2. Mohanty, L., Patra.K.C., Khatua, K.K., Patnaik, M. (2012) “Modeling Depth-Averaged
Velocity in Trapezoidal Meandering Channels” accepted for ISH Journal of Hydraulic
Engineering, Taylor & Francis Group, UK.
Page | 80
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