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ABSTRACT
Analysis of fluvial flows are strongly influenced by geometry complexity and large overall
uncertainty on every single measurable property, such as velocity distribution on different
sectional parameters like width ratio, aspect ratio and hydraulic parameter such as relative
depth. The geometry selected for this study is that of a smooth sine generated trapezoidal
main channel flanked on both sides by wide flood plains. The parameters which were
changed in this research work include the overbank flow depth, main channel flow depth,
incoming discharge of the main channel and floodplains.
This paper presents a practical method to predict lateral depth-averaged velocity
distribution in trapezoidal meandering channels. Flow structure in meandering channels is
more complex than straight channels due to 3-Dimentional nature of flow. Continuous
variation of channel geometry along the flow path associated with secondary currents makes
the depth averaged velocity computation difficult. Design methods based on straight-wide
channels incorporate large errors while estimating discharge in meandering channel. Hence
based on the present experimental results, a nonlinear form of equation involving 3
parameters for estimating lateral depth-averaged velocity is formulated. The present
experimental meandering channel is wide (aspect ratio = b/h > 5) and with high sinuosity of
2.04. A quasi1D model Conveyance Estimation System (CES) was then applied in turn to the
same compound meandering channel to validate with the experimental depth averaged
velocity. The study serves for a better understanding of the flow and velocity patterns in
trapezoidal meandering channel.
A commercial code, ANSYS-CFX 13.0 is used to simulate a 60 meander channel
using Large Scale Eddy (LES) model. Contours regarding the velocities in three directions are derived.
INTRODUCTION
Overview of Meandering Channels
River is the author of its geometry. The river network is an open system always tending
towards equilibrium and the hydraulically interdependent factors such as velocity, depth,
channel width and slope mutually interact and self adjust to accommodate the changes in
river plan form geometry and discharge contributed by drainage basin. Straightness of rivers
is a temporary state, the more usual and stable state whereas a meander is characterized by
lower variances of hydraulic factors. Therefore not a single natural river possesses a straight
geometry. Almost all natural rivers meander. In fact Straight River reaches longer than 10 to
12 times the channel widths are non existence in nature.
A meander, in general, is a bend in a sinuous watercourse or river. A meander is
formed when the moving water in a stream erodes the outer banks and widens its valley. As
the inner part of the river has less energy it deposits what it is carrying. As a result a snaking
pattern is formed, the stream back and forth across its down axis valley. Also second reason
of formation of meander as suggested by Leopold (1996) is the secondary circulation in the
cross sectional plane. As Leliavsky (1955) describes the second phenomenon of river
meandering in his book which quotes “The centrifugal effect, which causes the super
elevation may possibly be visualized as the fundamental principles of the meandering theory,
for it represents the main cause of the helicoidal crosscurrents which removes the soil from
the concave banks, transport the eroded material across the channel, and deposit it on the
convex banks, thus intensifying the tendency towards meandering. It follows therefore that
the slightest accidental irregularity in channel formation, turning as it does, the streamlines from their straight course may, under certain circumstances constitute the focal point for the
erosion process which leads ultimately to meander”.
The two critical parameters that govern the flow in open meandering channel are
sinuosity and least centreline radius (rc) to channel width (b) ratio. Sinuosity defines how
much a river course deviates from shortest possible path (how much it meanders). the
sinuosity index is 1 for straight channels whereas for meandering channels it is greater than 1
and can increase to infinity for a closed loop (where the shortest path length is zero) or for an
infinitely-long actual path. Meandering channels are also classified as shallow or deep
depending on the ratio of the average channel width (b) to its depth (h). In shallow channels
(b/h’>5, Rozovskii, 1961) the wall effects are limited to a small zone near the wall which
may be called as "wall zone". The central portion called "core zone" is free from the wall
effects. Whereas in deep channels (b/h’<5) the influence of walls are felt throughout the
channel width. However meandering channels are still a subject of research which involves
numerous flow parameters that are intricately related giving rise to complex three
dimensional motions in the flow. Due to this the 1 D and 2D modeling of open channel flows
fail to estimate the discharge precisely. So a paradigm shift is towards the study of three
dimensional modelling of open channel flows that can capture and take into account the
complicated unseen phenomena called "turbulence".
1.2 Velocity Distribution in Open Channels
The knowledge of velocity distribution helps to know the velocity magnitude at each point
across the flow cross-section. It is also essential in many hydraulic engineering studies
involving bank protection, sediment transport, conveyance, water intakes and
geomorphologic investigation. Despite several researches on various aspects of velocity
distributions in curved meander rivers, no systematic effort has yet been made to establish the relationship between the dominant meander wavelength, discharge and the velocity
distributions. In straight channel velocity distribution varies with different width-depth ratio,
whereas in meandering channel velocity distribution varies with aspect ratio, sinuosity,
meandering making the flow more complex to analyze. Compound channels are all the way
different and velocity distribution is a combination of flood plain and main channel (straight
or meandering). In laminar flow max stream wise velocity occurs at water level; for turbulent
flows, it occurs at about 5-25% of water depth below the water surface (Chow, 1959).
Typical stream wise velocity contour lines (isovels) for flow in various cross sections are
shown in Fig. 1.1.
Power law
An alternative function for the velocity distribution is the “power law”. The general form of
this law is proposed as (Barenblatt and Prostokishin, 1993; Schlichting, 1979):
u+ = C4(z+
)
m
, where C4 and m are the coefficient and exponent of the power law.
1.2.3 Chiu's velocity distribution
An alternative approach from the stated empirical velocity distribution equations is the
method developed by Chiu (1987, 1989). Based on the probability concept and entropy
maximization principle, Chiu derived a new two-dimensional equation for the velocity field.
This equation is capable of describing the variation of velocity in both vertical and transverse
directions with the maximum velocity occurring on or below the water surface. It can also accurately describe the velocity distribution in regions near the water surface and channel
bed, where most the existing measuring devices face problems.
1.3 Depth-Averaged Velocity
It is quite difficult to model flows in meandering trapezoidal channel as the inner and outer
banks exert unequal shear drag on the fluid flow that ultimately controls the depth- averaged
velocity. Depth-averaged velocity means the average velocity for a depth ‘h’ is assumed to
occur at a height of 0.4h from the bed level.
1.3.1 AIM OF PRESENT RESEARCH
It is concluded from literature review that very less work has been done regarding lateral
distribution of depth-averaged velocity in meandering trapezoidal channel. However lack of
qualitative and quantitative experimental data on the depth averaged velocity in meandering
channels is still a matter of concern. The present study aims at collecting velocity data from
wide meandering channel trapezoidal at cross section. The objective of the present work is
listed as:
To study the distribution of stream wise depth-averaged velocity at channel bend apex
for single flow depth. Also to study its variation at different flow depths for in bank
and overbank flow conditions.
To propose equations for simple meandering channel trapezoidal at cross section to fit
experimental data.
To validate the depth averaged velocity data with quasi one dimensional model
Conveyance Estimation System (CES) for overbank flow conditions.
To simulate a 60 degree simple meandering channel using Large Eddy Simulation
(LES) model and to derive 3 dimensional velocity contours for the same.
ORGANISATION OF THESIS
The thesis consists of seven 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 in Chapter 4 and analysis based on experimental results are done in
Chapter 5, Chapter 6 comprises numerical modelling and finally the conclusions and
references are presented in Chapter 7.
General view on rivers and flooding is provided at a glance in the first chapter. Also
the chapter introduces the concept of velocity distribution in meandering channels. It gives an
overview of numerical modelling 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 velocity 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 the channel. Also the detailed information about the instrument used for taking
observation is given.
The experimental results regarding stage-discharge relationship, velocity for in bank
and overbank flow conditions and depth average velocity are outlined in chapter four. Also
this chapter discusses the technique adopted for measuring depth average velocity.
Analyses of the experimental results are done in Chapter 5. The analysis of depth average
velocity distribution pattern for in bank and overbank flow conditions is presented in this
chapter.
OVERVIEW
Distribution of flow velocity in longitudinal and lateral direction is one of the important
aspects in open channel flows. It directly relates to numerous flow features like water profile
estimation, shear stress distribution, secondary flow, channel conveyance and host to other
flow entities. Various factors that affect the velocity distribution such as channel geometry,
types and patterns of channel, channel roughness and sediment concentration in flow has
been critically studied by many eminent researchers in the past. Prandtl (1932) developed the
general form of velocity distribution, which is generally considered as P-vK law, this law was
derived by assuming the “shear stress is constant”, and can be applied near bed, but has been
applied in outer flow region with modification of von karman constant like Milikan(1939) ,
Vanoni (1941). The P-vK law was derived by taking shear stress as constant whereas the
shear stress is not constant in turbulent layer (outer zone) in open channel flow. Milikan
(1939) suggested that actual velocity distribution consists of logarithmic part and correction
part, where the correction part considers the outer layer into account. So detailed literature
review is done for four different channel types followed by works on numerical modelling for
open channel flows.
2.2 PREVIOUS EXPERIMENTAL RESEARCH ON VELOCITY DISTRIBUTION
2.2.1 STRAIGHT SIMPLE CHANNEL
Coles (1956) proposed a semi-empirical equation of velocity distribution, which can
be applied to outer region and wall region of plate and open channel. He generalized the
logarithmic formula of the wall with tried wake function, w(y/8).This formula is asymptotic
to the logarithmic equation of the wall as the distance y approaches the wall. This is basic
formulation towards outer layer region.
REVIEW OF LITERATURE
9 | P a g e
Coleman (1981) proposed that the velocity equation for sediment-laden flow consists
of two parts, as originally discussed by Coles for clear-water flow. Also he has revealed that
the von Karman coefficient is independent of sediment concentration. The elevation of the
maximum velocity and the deviation of velocity from the logarithmic formula at the water
surface are functions of the aspect ratio of the channel. The log-law is developed into an
equation applicable to the whole flow including the region near the water surface for various
boundary conditions. The wake law describes the velocity distribution below the maximum
velocity point.
M. Salih Kirkgoz et al. (1997) measured mean velocities using a Laser Doppler
Anemometer (LDA) in developing and fully developed turbulent subcritical smooth open
channel flows. From the experiments it is found that the boundary layer along the centre line
of the channel develops up to the free surface for a flow aspect ratio . In the turbulent
inner regions of developing and fully developed boundary flows, the measured velocity
profiles agree well with the logarithmic "law of the wall" distribution. The "wake" effect
becomes important in the velocity profiles of the fully developed boundary layers.
Sarma et al. (2000) tried to formulate the velocity distribution law in open channel
flows by taking generalized version of binary version of velocity distribution, which
combines the logarithmic law of the inner region and parabolic law of the outer region. The
law developed by taking velocity-dip in to account.
Wilkerson et al. (2005) using data from three previous studies, developed two models
for predicting depth-averaged velocity distributions in straight trapezoidal channels that are
not wide, where the banks exert form drag on the fluid and thereby control the depthaveraged
velocity distribution. The data they used for developing the model are free from the
effect of secondary current. The 1st model required measured velocity data for calibrating the
model coefficients, where as the 2nd model used prescribed coefficients. The 1st model is recommended when depth-averaged velocity data are available. When the 2nd model is used,
predicted depth-averaged velocities are expected to be within 20% of actual velocities.
Knight et al. (2007) used Shiono and Knight Method (SKM), which is a new
approach to calculating the lateral distributions of depth-averaged velocity and boundary
shear stress for flows in straight prismatic channels, also accounted secondary flow effect. It
accounts for bed shear, lateral shear, and secondary flow effects via3 coefficients- f, λ, and Γ
—thus incorporating some key 3D flow feature into a lateral distribution model for stream
wise motion. This method used to analyze in straight trapezoidal open channel. The number
of secondary current varies with aspect ratio. It is three for aspect ratio less than equal to 2.2
and four for aspect ratio greater than equal 4.
Afzal et al. (2007) analyzed power law velocity profile in fully developed turbulent
pipe and channel flows in terms of the envelope of the friction factor. This model gives good
approximation for low Reynolds number in designed process of actual system compared to
log law.
Yang (2010) investigate depth-average shear stress and velocity in rough channels.
Equations of the depth-averaged shear stress in typical open channels have been derived
based on a theoretical relation between the depth-averaged shear stress and boundary shear
stress. Equation of depth mean velocity in a rough channel is also obtained and the effects of
water surface (or dip phenomenon) and roughness are included. Experimental data available
in the literature have been used for verification that shows that the model reasonably agrees
with the measured data.
Oscar Castro-Orgaz (2010) reanalysed the available data on turbulent velocity profiles
in steep chute flow, to determine general law by taking into account both the laws of the wall
and wake. Once the velocity profile is defined, an equivalent power-law velocity
approximation is proposed, with generalised coefficients determined by the rational approach.
The results obtained for the turbulent velocity profiles were applied to analytically determine
the resistance characteristics for chute flows.
Albayrak et al. (2011) combined the detailed acoustic Doppler velocity profiler
(ADVP ), large-scale particle image velocimetry (LSPIV) and hot film measurements to
analyse secondary current dynamics within the water column and free surface of an open
channel flow over a rough movable (not moving) bed in a wider channel, with a higher bed
roughness and at higher Reynolds number.
Kundu and Ghoshal (2012) re-investigated the velocity distribution in open channel
flows based on flume experimental data. From the analysis, it is proposed that the wake layer
in outer region may be divided into two regions, the relatively weak outer region and the
relatively strong outer region. Combining the log law for inner region and the parabolic law
for relatively strong outer region, an explicit equation for mean velocity distribution of steady
and uniform turbulent flow through straight open channels is proposed and verified with the
experimental data. It is found that the sediment concentration has significant effect on
velocity distribution in the relatively weak outer region.
2.2.2 STRAIGHT COMPOUND CHANNELS
Wormleaton and Hadjipanos (1985) measured the velocity in each subdivision of the
channel, and found that even if the errors in the calculation of the overall discharge were
small, the errors in the calculated discharges in the floodplain and main channel may be very
large when treated separately. They also observed that, typically, underestimating the
discharge on the floodplain was compensated by overestimating it for the main channel. The
failure of most subdivision methods is due to the complicated interaction between the main
channel and floodplain flows.