School of Civil Engineering The University of New South Wales a ? Sydney, Australia
Entrainment of Sediment Particles from a Flat Mobile Bed with the Influence of Near-wall Turbulence
by
Alireza Keshavarzy
B.Eng., Shiraz University, Shiraz, Iran M.Eng. Sc., The University of New South Wales, Sydney
A thesis submitted in partial fulfilment of the requirement for the degree of Doctor of Philosophy
1997 L S W 18 APS 1303 Li UR ARY CERTIFICATE OF ORIGINALITY
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 a university or other institute of higher learning, except where due acknowledgment is made in the text.
(Signed)
CERTIFICATE OF ORIGINALITY
3 hereby declare that this submission is my own work and to the best of my knowledge it contains no materials 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 at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to die research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis.
I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and conception or in style, presentation and linguistic expression is acknowledged.
(Signed) Abstract
This study examines the influence of turbulence on the entrainment of sediment particles from a mobile bed. The structure of the turbulent boundary layer is very complicated and, as yet, not fully understood. Consequently, the influence of the boundary layer and the turbulence within the boundary layer on the entrainment of sediment particles is not fully understood.
Turbulence has been defined through a quadrant analysis of velocity fluctuations, which are known as bursting process. Four types of events have been recognised on the basis of a quadrant analysis; these events are sweep, ejection, outward interaction and inward interaction. Of these events, the sweep event has been recognised as the most important event for particle entrainment from the bed.
In this study, the turbulent characteristics of bursting events in an open channel flow was investigated experimentally and applied to the problem of the initiation of sediment particles. The turbulent velocity fluctuations of the flow were measured in two dimensions and analysed. A statistical analysis of the experimental data was undertaken. In this analysis, a Box-Cox transformation was used to convert shear stresses and the angle of inclination to a normally distributed sample. From the statistical analyses of shear stresses in bursting events it was found that, close to the bed, the average magnitude of the shear stress in a sweep event was 140 percent of the time averaged shear stress, that the frequency of occurrence was 30 percent of the time, and that the average inclination angle for the sweep events was 22°.
A stochastic-deterministic mathematical model was developed to define particle entrainment from the bed under the influence of turbulence. This model combines Bagnold’s energy concepts with Einstein’s probability concepts through a combination of the two approaches as postulated in this study. The use of these two concepts was considered necessary in the model as the forces applied to the sediment particles are temporally variable due to the occurrence of turbulence and the associated bursting processes. The model was solved numerically for two alternative cases; these were an instantaneous shear stress and a time-averaged shear stress. When comparing the predicted sediment motion, it was found, for a particular particle diameter, that initiation of particle motion would commence at lower flow rate with consideration of the instantaneous shear stress than would be the case for a time-averaged shear stress using Shields criteria.
An inherent assumption in the model is that sweep events are the primary processes inducing particles into motion. The validity of this assumption was tested by comparing
i the statistics of particles initiated into motion with statistics of the instantaneous shear stresses during sweep events. In order to find the number of particles initiated into motion at any increment of time, a series of sequential video images of particle motion were analysed. Good agreement was found between the statistics of area entrained and the instantaneous shear stresses during sweep events.
The stochastic-deterministic mathematical model was also verified using image processing techniques. The use of a convolution technique with cross-correlation enabled the determination of particle displacement in a given small time increment and consequently the instantaneous velocity of particles. Using these data, the exceedance probabilities of the particle velocities were able to be determined and compared with that predicted by the stochastic-deterministic model. The measured and predicted particle velocities were similar and within the 95% confidence limits.
Finally, using the stochastic-deterministic force balance model, the area entrained from the image processing techniques and the measured turbulent shear stresses of the flow, a modification to Shields diagram was proposed. This modification indicates the probability of a particle being induced into motion.
ii Acknowledgments
I would like to expresss my appreciation to my supervisor Dr. James Ball, for his support, encouragement, comments, discussion and guidance towards the completion of this dissertation. Particularly, his encouragement to express this idea and to publish some papers in this field of study are acknowledged. The suggestions given by Dr. David
Luketina and review of dissertation draft is also acknowledged.
During the period of this work, the writer has received assistance and suggestions from many people to whom he would like to express appreciation. Special thanks are due to Prof.
H.W. Shen for his encouragement and comments during the Stochastic Hydraulics
Symposium held in Mackay, Queensland in 1996. Also thanks to Prof. William Dunsmir,
Head of the Dept of Statistics at the University of NSW, for his statistical suggestions, consulting and review of statistical analysis, Prof. Trinder J. Head of School of Geomatic
Engineering, UNSW, for his suggestions in image processing and review of the final draft of the parts of this dissertation related to image processing and Dr. Wayne Erskine for his suggestion and review some parts of the draft dissertation.
The writer also is grateful to Assoc. Prof. Ron Cox, Director of the Water Research Lab. for his support in the experimental parts of this study. Also special thanks to WRL staff;
James Carley, John Hart, John Baird, Ross Mathews and Margaret Titterton. Also thanks to Mr. Ken Higgs for his encouragement, introduction to the C language and its application in image processing which proved to be a valuable contribution. During study in UNSW,
I benefited from some very close friends, specially Mahmoud Bina, Saied Saiedi, Hamid
Rahimipour, Jafar Nazemosadat, Saied Eslamian and Paul Hogan.
Finally, I would like to express my thanks to my family; to my wife Mina, for her patience, inspiration and teaching my children during our stay in Australia, as well as to my children,
Reza and Zahra for their tolerance in living with a student father. Also I debt my study to my mother, late father, my sisters, brothers, and brother-in-law Mr Safar Keshavarzy for
iii their supporting and encouragement during my study. Thanks to God for giving me wisdom, success and strength. The writer also gives thanks the Ministry of Culture and
Higher Education, Iranian Government, for the sponsoring my scholarship during study in Australia.
iv Table of Contents
Contents Pages
Abstract i Acknowledgments iii Table of Contents v
List of Figures xi
List of Tables xvi Notation xvii
CHAPTER 1: Introduction 2 1.1 Environmental impact of sediment transport 2 1.2 The problem investigated 4 1.3 Objective of the study 6 1.4 Layout of the dissertation 10
CHAPTER 2: Review of Previous Studies 12 2.1 Extent of review 12 2.2 Some basic concepts of sediment transport and Shields diagram 13 2.3 Particle entrainment and instantaneous turbulent shear stress 23 2.4 The turbulent structure of the flow and bursting phenomena 25 2.4.1 Basic concepts and relevant parameters 25 2.4.2 Quadrant analysis of the bursting process 27 2.4.3 Structure of the turbulent boundary layer 30
2.5 Initiation of sediment motion and force balance model 37 2.6 The entrainment function and intensity of particle entrainment 43 2.7 Summary 47
v CHAPTER 3: Experimental Apparatus and Procedure 49 3.1 Introduction 49 3.2 The flume 50 3.2.1 General description 50 3.2.2 Water supply of the flume 54 3.2.3 Bed roughness characteristics 54
3.2.4 Roughness estimation of the flume and F.C. sheet at the bed 56 3.3 Sediment characteristics 58
3.3.1 Definitions 58 3.3.2 Size distribution of sediment particles 58
3.4 Flow velocity measurement 59 3.4.1 Equipment and procedure 60 3.4.2 Determination of shear velocity 65 3.5 Particle movement measurement 66 3.5.1 Equipment 66 3.5.2 Light illumination 66 3.5.3 Method of capturing video images 70 3.5.4 Analysis of the particle motion images 72 3.6 Experimental tests 75 3.6.1 Boundary layer development 75 3.6.2 Scale of turbulence 76 3.6.3 Test conditions 78 3.7 Summary 81
CHAPTER 4: Statistical Analysis of Turbulent Shear Stress 83 4.1 Introduction 83 4.2 Statistical methods for data interpretation 84
vi 4.2.1 Probability density function 84 4.2.2 Gaussian distribution 85 4.2.3 Box-Cox transformation 86
4.3 Analysis of experimental data 87 4.4 Results and discussion 95 4.4.1 Probability distribution of instantaneous shear stresses
during events 95 4.4.2 Instantaneous shear stress of the event classes 98
4.4.3 Frequency of events 105 4.4.4 Angle of the events 108 4.5 Exceedance probability of shear stress in events 110 4.6 Results and conclusions 112
CHAPTER 5: A Mathematical Model for Initiation of Sediment Motion 115 5.1 Introduction 115 5.2 Sediment in a natural stream 116 5.2.1 Mechanism involved 116
5.2.2 Main parameters 117 5.3 Alternative concepts for particle entrainment 118 5.3.1 Energy concept 119 5.3.2 Probability concept 122 5.3.3 Summary of approaches 125 5.4 A force balance model 126 5.4.1 Applied forces 126 5.4.2 Governing equation 126 5.4.3 Force balance model with time averaged conditions 128 5.5 Development of a model including the influence of turbulence 130
vii 5.5.1 Model concepts 130 5.5.2 Formulation of the model with turbulence parameters 131 5.5.2.1 Instantaneous shear stress 132
5.5.2.2 Impinging angle 134 5.5.2.3 Frequency of occurrence 135 5.5.3 Mathematical description of the model 137
5.5.4 Significance of terms in instantaneous model and comparison with time-averaged models 139
5.5.5 Model input 141 5.5.5.1 Drag coefficient 141 5.5.5.2 Friction coefficient 142 5.5.6 Computer programming and solution method 143 5.6 Results and conclusions 146 5.6.1 Evaluation of unknown parameters in the proposed model 147 5.6.2 Comparison of the results from instantaneous and time-averaged models 148 5.6.3 Verification of the proposed model 149 5.7 Summary 152
CHAPTER 6: Image Analysis of Particle Motion and Model Testing 154 6.1 Aims and overview 154 6.2 Digital image processing 155 6.2.1 Introduction to digital image processing 155 6.2.2 Fundamental elements of digital image processing 156
6.3 Statistical tools for image analysis 158 6.3.1 Basic concepts 158 6.3.2 Cross-correlation 159
viii 6.3.3 Auto-correlation function 160 6.3.4 Spectral analysis of the signal 161 6.4 Analysis of images of particles in motion 162
6.4.1 Techniques used 162 6.4.2 The subtraction technique and particle counting 162
6.4.2.1 Definition 162 6.4.2.2 Procedure 163 6.4.2.3 Viewing the particles in new image 164
6.4.2.4 Particle counting 165 6.4.3 The convolution and cross-correlation of the images and particle image velocimetry 169 6.5 Results and discussion 173 6.5.1 Relation between number of particles in motion and instantaneous shear stresses in a sweep event 173 6.5.2 Probability analysis of number of entrained particles and the instantaneous shear stress 180 6.5.3 Modification to the Shields diagram 184 6.5.4 Displacement of particles and velocity detection 190 6.5.5 Verification of the model using experimental data
for particle velocity 192 6.6 Summary and conclusions 196
CHAPTER 7: Results and Conclusions 199
REFERENCES 204
APPENDIX A: Publications 219
ix APPENDIX B: Electromagnetic Velocity Meter 294
APPENDIX C: Velocity Profiles 300
APPENDIX D: Box-Cox Transformation of Sweep and Ejection Events 304
APPENDIX E: Sample Model Output File 334
APPENDIX F: A Sequence of Images with Differences 340
APPENDIX G: Cross-correlation of Sweep Shear Stress and Entrained
Particles 351 List of Figures
Number Page
Figure 2.1 Entrainment-deposition criterion for uniform particles 15
Figure 2.2 Shields diagram for critical shear stress 17 Figure 2.3 Critical shear stress for quartz sediment in water as a function of
grain size 21 Figure 2.4 Extended Shield diagram for cohesiveless granular
sediment in a flat bed 22 Figure 2.5 Four classes of bursting events and the associated quadrants 29 Figure 2.6 Structure of turbulent boundary layer 32 Figure 2.7 Turbulent structure near boundary layer 33 Figure 2.8 The sweep and ejection events 34 Figure 2.9 Eddy structure in wall region and interaction with bed particles 35 Figure 2.10 Action of a group of sweep event over a mobile bed 36 Figure 2.11 Applied forces on sediment particle resting on non-horizontal bed slope 39 Figure 2.12 Comparison of analytical and experimental results 41 Figure 3.1a Photograph of the experimental flume and attachments 51 Figure 3.1b Schematic diagram of open channel flume 53 Figure 3.2 A definition sketch of velocity components in two dimensions 54 Figure 3.3a A photograph of a fixed artificially roughened 56
Figure 3.3b Artificial rough bed with levellers underneath 56 Figure 3.4 Cumulative sediment size distribution curve used in laboratory tests 60
Figure 3.5 Arrangement and setup of the data acquisition equipment 62 Figure 3.6a A simple schematic diagram of the trolley mounting 63 Figure 3.6b A photograph of the trolley mounting 63
xi Figure 3.7 Example of the time series of the velocity measurements from experiment (Test Q-10) 64 Figure 3.8 Colour particles used in mobile bed for image processing 67
Figure 3.9a CCD camera and accessories used in image capturing technique 68 Figure 3.9b Side view of mobile area of the flume used in image
processing test 68 Figure 3.10 Plan view of area of investigation 69 Figure 3.11 Schematic of side view observation area 69 Figure 3.12 Image processing: Capturing process and systems 71 Figure 3.13 Photograph of image processing and digitising equipment 73 Figure 3.14 Two images of the bed and their difference indicating
particle motion 74 Figure 3.15 Auto-correlation of velocity fluctuations 77 Figure 3.16 Non-dimensional turbulence scale of velocity components 78 Figure 4.1 Probability density function of instantaneous shear stress 85 Figure 4.2 Probability density of a normal or Gaussian distribution 85 Figure 4.3 Transformation process of the measured data 86 Figure 4.4 Example of the time series of the velocity measurements 88 Figure 4.5 Normalised velocity profile for some experimental runs 89 Figure 4.6 Computer programming algorithm for analysis of velocity fluctuation 92 Figure 4.7 Variation of turbulence intensity in horizontal direction of flow 94 Figure 4.8 Variation of turbulence intensity in vertical direction of flow 94 Figure 4.9 Normalised Reynolds shear stress with depth 95
Figure 4.10 Frequency distribution of shear stress in sweep event 97 Figure 4.11 Frequency distribution of shear stress in ejection event 97 Figure 4.12 Frequency distribution of shear stress in sweep event 100
xii Figure 4.13 Normal probability plot of shear stress in sweep event 100 Figure 4.14 Frequency distribution of shear stress in ejection event 102 Figure 4.15 Normal probability plot of shear stress in ejection event 102 Figure 4.16 Variation of shear stress in sweep event after transformation 103 Figure 4.17 Variation of shear stress in ejection event after transformation 103
Figure 4.18 Frequency of occurrence of the sweep and outward interaction 107 Figure 4.19 Frequency of occurrence of the ejection and inward interaction 107 Figure 4.20 Variation of inclination angle in sweep event 109
Figure 4.21 Variation of inclination angle in ejection event 109 Figure 4.22 Exceedance probability of shear stress in sweep event 111 Figure 4.23 Exceedance probability of shear stress in ejection event 111 Figure 5.1 Displacement of particles along the bed 120 Figure 5.2 Applied forces on sediment particles at the bed 128 Figure 5.3 Sweep event in a bursting process 131 Figure 5.4 Inclination angle of the sweep event 135 Figure 5.5 Computer programming algorithm for solution of force balance model 145 Figure 5.6 Results of instantaneous particle velocity from proposed model 147 Figure 5.7 Comparison of critical shear velocity for initiation of particle motion resulted from instantaneous and time averaged shear stress models 150
Figure 5.8 Differences in critical shear velocity for two models 150 Figure 5.9 Normalised differences in shear velocity for two models 151 Figure 5.10 Comparison of particle velocity in two different models 151 Figure 6.1 A physical image and a corresponding digital image 157 Figure 6.2 Digitizing an image 158 Figure 6.3 A schematic illustration of image subtraction 166
xiii Figure 6.4 Computer programming algorithm for image subtraction 167 Figure 6.5 Two images of the bed and their difference indicating
particle motion 168 Figure 6.6 An example of application of spatial correlation technique 171 Figure 6.7 Flow-chart of numerical processing in frequency and time domain
for displacement and velocity detection using cross-correlation 172
Figure 6.8 An example of horizontal and vertical velocity fluctuations with respective to their shear stresses and the number of entrained particles 174 Figure 6.9a Number of particles in motion in experimental tests 176 Figure 6.9b Number of particles in motion in experimental tests 177 Figure 6.10a Cross-correlation of instantaneous shear stress in sweep event and number of particles entrained from the bed (test R5-R4) 178 Figure 6.10b Cross-correlation of instantaneous shear stress in sweep event and number of particles entrained from the bed (test R7-R5) 179 Figure 6.11 Inclination of flow velocity of the sweep event 179 Figure 6.12 Frequency distribution of entrained area of a mobile bed for a sequence of images 182 Figure 6.13 Normal probability plot of area entrained for a sequence of images 182 Figure 6.14 Cumulative probability plot of entrained particles from the bed for a sequence of images 183 Figure 6.15 Cumulative probability of instantaneous shear stress in sweep event 183
Figure 6.16 Relation of area entrained with instantaneous shear stress in sweep event 184 Figure 6.17 Modified Shields diagram 187
Figure 6.18 Comparison of present data with Van Rijn’s definition for particle movement 189
xiv Figure 6.19 Comparison of present data with Shields curve for particle movement 189 Figure 6.20 Using FFT and cross-correlation for peak detection technique 191 Figure 6.21 Using FFT and cross-correlation for peak detection technique 191 Figure 6.22 Verification of proposed model and comparison
with previous model 194
Figure 6.23 Comparison between observed and predicted particle velocity 195 Figure 6.24 Residual value of predicted velocity of particles 195 List of Tables
Number Page
Table 2.1: Summary of some previous studies using the force balance model 40
Table 3.1: Properties of the 2 mm sediment used in the laboratory test 59
Table 3.2: Accuracy of particle counting over a specified area 70
Table 3.3: The flow conditions in the rough fixed bed tests 79 Table 3.4: The flow conditions in the rough movable bed tests 80 Table 5.1: Published pick-up relationships 124 Table 5.2: Comparison of variables in time-averaged and instantaneous models 140
Table 6.1: Strategy for the description of subtracted image 164
xvi Notation
Ap Cross-sectional area of the grain exposed to drag and lift
B Width of the channel
B(C) Box-Cox transformation of normalised shear stress
C Normalised shear stress
Co drag coefficient
Ci lift coefficient dg particle diameter
D* dimensionless particle parameter
E pickup rate in mass per unit area and time
E& available energy of the flow
Fd drag force
Fg gravity force
Fi lift force
Fr resistance force g acceleration due to gravity
H total depth of the flow h depth of the flow relative to the bed at any point within the flow k constant
Lx macro-scale of turbulence (eddy size scale)
xvii u macro-scale of turbulence (time scale)
Afd number of particles deposited per unit area and time
Ne number of particles entrained per unit area and time
P pickup probability function
R hydraulic radius
R(t) auto-correlation coefficient
S ratio of density of sediment particle to the density of water at 25° C
So channel slope
Sf energy gradient
T dimensionless bed-shear stress parameter u flow velocity
ü temporal mean velocity in horizontal direction of the flow u* shear velocity of the flow
Up particle velocity
Vf fluid velocity
“p average particle velocity
“i instantaneous velocity fluctuation in horizontal direction of the flow u' velocity fluctuation about mean in horizontal direction of the flow
W* instantaneous shear velocity
A instantaneous particle velocity
xviii turbulent shear velocity turbulent shear velocity in sweep event turbulent shear velocity root mean square of velocity fluctuations in horizontal direction root mean square of velocity fluctuations in vertical direction velocity fluctuation about mean in vertical direction of the flow temporal mean velocity in vertical direction of the flow instantaneous velocity fluctuation in vertical direction of the flow relative velocity of flow coefficient time lag friction coefficient specific weight of the fluid specific weight of sediment grains power of transformation
Prandtl-Von Karman constant (0.40)
Prandtl mixing length kinematic viscosity of the flow instantaneous turbulent shear stress of event turbulent shear stress density of the flow density of sediment grains shear stress at the bed time averaged critical shear stress porosity of the sand bed instantaneous sweep shear stress angle of repose angle of the events angle of sweep event dimensionless shear stress (Shields parameter) instantaneous dimensionless shear stress (Shields parameter) entrainment coefficient Chapter 1 Chapter 1
Chapter 1
Introduction
1.1 ENVIRONMENTAL IMPACTS OF SEDIMENT TRANSPORT
Sediment transport is a significant component of many environmental degradation
problems. The process of sediment transport, however is very complicated due to the
interaction of many parameters. Consequently, after more than five decades of
investigations into sediment transport, there are still many unresolved problems. While
it takes more than two hundred years to form a very thin layer of soil for agricultural
use, it takes substantially less time for this soil to be eroded and transported to a river
channel, where it is sequentially eroded and deposited until reaching the river mouth.
The importance of sediment transport is increasing, as rivers are being used to satisfy multiple purposes. For example, a single river may be used for water supply, flood
2 Chapter 1
control, power generation, irrigation and navigation. The sediment that is scoured and transported from the beds and banks of rivers creates major problems in natural and artificial channels. The dredging and cleaning of navigation canals from deposited sediment is costly and time-consuming, while the entrainment and deposition of sediment in natural rivers causes major problems in the aquatic ecosystem. Sediment transport also causes damage to natural water systems. For example, in areas of erosion, river banks may collapse and threaten property or trees, while in areas of deposition, sediment may accumulate and reduce the navigational capacity of the system.
Various approaches in the investigation of sediment motion problems in natural rivers and natural watercourses have been used. These approaches can be classified arbitrarily as: sediment load estimation, reservoir siltation, degradation and aggregation of channels, local scour around bridge piers, stable channel design, land erosion and soil conservation, flood plain management and flood control, sand and silt extraction, navigation, beach erosion and water quality management They have been adopted for the assessment and analysis of particular problems associated with sediment transport.
While they are sufficient for many problems, the increasing community awareness of the aquatic environment and anthropogenic impacts on the aquatic environment have resulted in a need for the development of new approaches for the assessment and analysis of newly-identified problems.
3 Chapter 1
1.2 THE PROBLEM INVESTIGATED
One aspect of sediment motion which has been recognised to have environmental implications is the incipient motion of particles. This is the point at which particle motion begins; a phenomenon that is not fully understood at present. Incipient motion is dependent on the particle size, particle shape, specific weight of particle and flow conditions. Once motion begins, depending on the sediment size and flow characteristics, sediment particles may slide or roll along the bed (bed load), or be carried along in suspension by the moving water (suspended load).
The motion of sediment particles and the initiation of motion is an important component of many engineering studies. The bed-load transport is also very important in hydraulic works and river training activities. While several researchers, beginning with Hjulstrom
(1935) and Shields (1936), have investigated this problem, there remain some difficulties in describing the entrainment of sediment particles and defining the initiation of particle motion on the bed of open channels.
A large number of phenomena have been suggested as influencing particle entrainment in an open channel flows. However, it is generally accepted that hydrodynamic forces
(e.g. drag force and lift force) are the most important factors in the entrainment of sands and gravels. Einstein (1942; 1950), Shen and Cheong (1973) and Shen and Tabios
(1996) have suggested various stochastic models to define the entrainment of particles from the bed. Studies by Grass (1970; 1982) suggest that these hydrodynamic forces are developed from the bursting process in turbulent flows. When the hydrodynamic forces
4 Chapter 1
overcome the resistance forces, dislodgement occurs. In this situation, particles commence moving by rolling and sliding or by saltation. These particles move spasmodically along the bed until the hydrodynamic forces are sufficient to maintain them in motion. These applied hydrodynamic forces vary with time depending on the turbulent characteristics of the flow and bed form. The investigation of flow turbulence and its impact on the initiation of particle motion from the bed is one of the primary aims of the study reported herein.
The initiation of sediment motion is an important aspect of the sediment transport problem. Most existing critical shear stress models used in sediment transport models are based on the time-averaged shear stress at the bed of a channel, which is defined in terms of the depth, the density of the fluid and the energy gradient, whereas the particles on the bed sustain an instantaneous shear stress which differs from the temporal average shear stress. These differences are derived from the turbulent nature of the flow. Since particle entrainment into the flow occurs as a result of the applied shear stress at the bed, it is necessary to consider the impact of instantaneous shear stress in the development of a sediment entrainment model. In this study, an effort was made to define the entrainment of sediment particles from the bed of an open channel while considering the influence of flow turbulence. In particular, this study was focused on defining the entrainment of non-cohesive sediment particles from a flat bed.
5 Chapter 1
1.3 OBJECTIVE OF THE STUDY
The present study was focused on the investigation of the characteristics of the bursting phenomenon in an open channel flow and, in particular, over a rough bed. The turbulence characteristics of the flow were measured in a flume with a rough bed under differing flow conditions. The magnitude of the shear stress, the frequency and the angle of action for each of the events, were determined from the experimental data. Also determined were the variation of shear stress, the frequency of occurrence and the angle for an individual event at different points through the flow depth. The results obtained from this study were used to assess the impact of flow turbulence on the initiation of motion of sediment particles from the bed of an open channel.
An objective of this study was the analysis of bursting event characteristics and, particularly, sweep events in the region close to the channel bed. This focus resulted from a need to consider the influence of the magnitude of the instantaneous shear stress on the entrainment of sediment particles into the flow. The instantaneous turbulent shear stresses were applied in a simple force balance model to define an entrainment function for the prediction of incipient sediment motion.
There were two distinct components to this study; these were:
A) An experimental investigation for collection of the relevant data.
In the experimental investigation, several approaches were used to investigate the entrainment of sediment particles. These approaches were:
6 Chapter 1
• Measurement of the turbulence characteristics of the flow in a flume over a rough
bed.
In this part, the investigation included measurement of flow characteristics and,
in particular, the velocity fluctuations at different depths of flow from the bed to
the water surface. This measurement was carried out in a laboratory flume over
a rough bed. The turbulence intensity of the flow in the horizontal and vertical
directions, and the Reynolds shear stress and the velocity distribution, at different
depths of flow, were measured.
• Data processing and data preparation:
As part of the experimental work and in order to prepare the experimental data for
later processing, a computer program ( referred to as ANAL ) was written to
calculate the Reynolds shear stress, and time-averaged velocity for two
components, the turbulence intensities of the flow in the horizontal and vertical
directions, and the coefficient of variation of the velocity fluctuations at a
particular depth of the flow.
• Application of image processing techniques to monitor the motion of particles
entrained from a flat movable bed:
In order to monitor particle entrainment at the bed, image capturing and
processing facilities were used. A number of sequences of images are captured
by a CCD camera and recorded on a super VHS® VCR and a Video 8® tape
7 Chapter 1
recorder. These images were digitised for later analysis using IMAGER®
Software.
B) An analysis of the data collected in the experimental component
In this analytical part of the investigation, the approaches used were:
• A theoretical analysis of the data, its subsequent application and validation
The validity and accuracy of collected data were tested by comparing
experimental data with previously collected data from literature.
• An analysis of the characteristics of the bursting process, with particular emphasis
on sweep and ejection events;
In order to analyse the experimental data a Box-Cox transformation was used to
transform the normalised data into a normal probability distribution. The average
instantaneous shear stress for the events in each quadrant were calculated as a
function of the normalised depth within the flow. The same technique was used
to analyse the angle of inclination with the bed for sweep and ejection events. This
approach enabled the mean instantaneous shear stress and angle of applied force
to be determined for each event close to the bed. The frequency of occurrence for
each event was calculated by counting the instantaneous number of occurrences
for events in each quadrant during the test. A computer program ( referred to as
SORT ) was developed to undertake this.
8 Chapter 1
• Incorporation of the turbulence characteristics in a force balance model for
prediction of the initiation of particle motion;
The turbulence characteristics of the flow and, in particular, the characteristics of
sweep events (instantaneous shear stress, frequency and angle of inclination) were
applied in the model. Using the statistics of the flow turbulence and particle
entrainment, the proposed model was used to estimate the probability of the
entrainment of sediment particles from the bed.
• Testing of the proposed model for particle entrainment using image processing
techniques
To obtain the differences between sequential images a computer program
(referred to as DIPT) was written. This computer program read a sequence of
images and produced the difference between each pair of images as a new digital
image. In the new images, the differences between images could be viewed using
image processing software. The second computer program (referred to as
CONVOL) was written to apply convolution technique to the derived images.
Using this computer program enabled the determination of the displacement of
an individual particle for an increment of time. From the second process the
particle velocity could be obtained in a small increment of time.
9 Chapter 1
1.4 LAYOUT OF THE DISSERTATION
This dissertation describes the investigations undertaken to analyse the problems presented above. In Chapter 1 the importance of the study and application is presented.
Also introduced is the problem, with a brief overview of the theory, experimental approach and analytical approach for the investigation. Presented in Chapter 2 are the results of previous studies and basic concepts concerning the entrainment of sediment particles in open channel flows. Chapter 3 outlines a detailed demonstration of the experimental investigation and equipment used in the study. Included in Chapter 3 is a description of the techniques used for the measurement and collection of relevant data.
The statistical analysis of the experimental data, is presented in Chapter 4. The application of the results of the statistical analysis in a force balance model is presented in Chapter 5. Results of the image processing and analysis are presented in Chapter 6 and are related to the force balance model discussed previously in Chapter 5. Finally, conclusions developed during the study are summarised in Chapter 7.
10 Chapter 2
11 Chapter 2
Chapter 2
Review of Previous Studies and Theoretical Concepts
2.1 EXTENT OF REVIEW
Environmental problems involve many interrelated areas of expertise and, as a consequence, there is extensive literature relating to the various aspects of such problems. Since the focus of the study presented in this dissertation is the initiation of sediment particle motion, the published literature reviewed in this chapter is concentrated upon the sediment-related aspects of the topics discussed. For any one topic, therefore, literature which has not been included in the following review exists.
For example, only those papers which discuss turbulence and its relation to sediment motion are reviewed rather than the extensive literature on turbulence in flowing water.
12 Chapter 2
2.2 SOME BASIC CONCEPTS OF SEDIMENT TRANSPORT AND THE SHIELDS DIAGRAM
In particular situations, interaction between non-cohesive grains of sediment resting on
the channel bed and the flow in the channel results in particle motion. These sediment
particles are transported by the flow as a suspended load, bed load or a combination of
bed load and suspended load. The suspended load is transported by the body of the flow,
whereas, the bed-load sediment particles can be moved via several modes (rolling,
sliding or saltation) depending on the magnitude of the shear stress applied by the flow.
For a flow situation which is close to the incipient motion or threshold of motion, sand
and gravel particles will move by rolling or sliding at the bed. Of concern to the present
study is the initiation of that particle motion. In this context, the study deals with those particles which remain in close proximity to the bed and which move by rolling, sliding and occasionally saltating over it.
The initiation of sediment motion is one of the most important and interesting issues in the field of sediment transport. In order to understand the transportation of sediment in a channel or in a river, it is necessary to be able to define the critical shear stress, which is defined as the smallest shear stress necessary to move a sediment particle. The threshold of particle motion is defined as the condition for which the applied forces just exceed the resistance force; at critical shear stress, the condition is defined as incipient motion, or the point of which sediment particles are ready to move but are not yet moving.
13 Chapter 2
In a definition of the threshold of particle motion, Kramer (1935) defined three stages for the threshold of critical motion, which are based on the pick-up intensity or the number of particles which move on a specific area of the bed. These are as follows:
-Weak movement: in this flow condition, several of the smallest particles are in motion at isolated points with a sufficient number for counting.
-Medium movement: in this flow condition, a large number of particles of mean diameter are in motion and can be counted. It does not affect the bed configuration and does not result in an appreciable sediment discharge.
-General movement: indicates the condition in which the largest particles are in motion and movement is occurring at all locations of the bed at all times.
This definition, however, is related to sediment motion in general and not to defining the flow condition when and where the first particle is induced into motion.
Additionally, his definition is based on qualitative criteria of sediment transport stages.
The critical force required to initiate the motion of sediment particles has been associated mostly with two general theories. The first theory which was presented by
Hjulstrom (1935) as a criterion for entrainment-deposition, is based on the cross-sectional mean velocity of the flow (Um) required for the transport of a certain particle size. Figure 2.1 presents this basic entrainment-deposition criterion for a uniform particle size. The diagram also indicates that loose fine sand is the easiest to erode. The resistance to erosion of smaller particles also depends on the cohesive forces
(Hjulstrom 1935).
14 Chapter 2
Erosion
Transportolion
Sedimentation
d odd d (mm)
Figure 2.1 Entrainment-deposition criterion for uniform particles (After Hjulstrom, 1935)
The second theory is based on the time averaged critical shear stress (fc). Du Buat (Graf
1984) used this approach during the late 18th century, but it did not become popular until the beginning of this century when Schoklisch published his results (Graf 1984). Since then, other investigators have used this approach too. As a result there is an extensive range of literature dealing with this aspect of the forces applied to sediment particles resting on the channel bed. In the following discussion only those aspects of the literature related to the Shields diagram are discussed; a more detailed discussion of the forces applied to sediment particles is reviewed in the section 2.5 of this dissertation.
Shields (1936) used the shear velocity (if*) as a measure of the shear stress near the bottom boundary. The shear stress at the bed was introduced by the expression
15 Chapter 2
u* = (To/p){/2 (2.1).
Shields used this term in order to describe his well-known entrainment function. Figure
2.2 is a graphical representation prepared by Vanoni (1975) of the threshold of movement of sediment particles using the principles developed by Shields. This Figure was developed from a dimensional analysis for longitudinal flow using mean values in a turbulent flow. Consequently, it is based on a time-averaged shear stress and not the instantaneous shear stress acting on the sediment particles.
Shields considered that the force acting is a shear force and assumed that the resistance of the particle to motion should depend only on the form of the bed and the submerged weight of the particles. He studied these forces for different flow conditions and showed that the threshold of movement of sediment particles could be represented by a non-dimensional relationship for a uniform sediment in unidirectional, uniform flow.
This relationship can be expressed algebraically as
(Yr-y)ds /fei) (2.2) where Ys and Y are the specific weights of the sediment and fluid, respectively, rc is the time averaged critical shear stress at the bed, <4 is the particle diameter, w*c is the critical shear velocity and v is the kinematic viscosity of fluid. The left part of this relationship is the dimensionless critical shear stress (0C) and the right part of the relationship is a function of the Reynolds Number (/?#)•
16 Chapter 2
As shown in Figure 2.2, Shields (1936) plotted the time-averaged bed shear stress at the
threshold of sediment motion (fc) in a dimensionless form ( 6C ) versus the grain
Reynolds Number (/?//)
u*ds 0 c, (y, - Y)ds N v
He found that 0C has values ranging from 0.035 for R^= 10 to a value of 0.06 for a
Reynolds Number equal to 2.0. For Reynolds Numbers larger than 400, the value of the
Shields parameter at threshold conditions (0C) stays constant at approximately 0.06.
Y,. Em per cu cm f o Amber 1.06 • Lignite 1.27 (Shields) 9 Granite 2.7 c Barite 4.25 Fully developed turbulent velocity profile1 * Sand (Casey) 2.65 + Sand (Kramer) 2.65 1.0 0.8 X Sand (U.S. WES) 2.65 Z l a Sand (Gilbert) 2.65 Z 0a C 0.506 ■ Sand (White) 2.61 - 0.4 s Turbulent boundary layer □ Sand in air (White) 2.10 “ 0.3 ▲ Steel shot (White) 7.9 “
Value of Vo.l(y —1 )gd, I lilioo 0.1 ’nnvtVi 0.08 0.06 0.05 0.04 0.03' 0.02 0.2 0.4 0.6 60 100 1000
Reynolds Number (RN) Figure 2.2 Shields diagram for critical shear stress (After Vanoni, 1975)
Shields determined the critical shear stress of particles by plotting the observed sediment flux versus shear stress, and extrapolated to a zero sediment flux. Essentially, equation (2.2) was obtained from a dimensional analysis and then verified by experimental data, while Figure (2.2) eventually became the well known Shields
17 Chapter 2
diagram. The basis, however, of the Shields diagram was the use of a time averaged
shear stress.
When the Shields diagram has been tested with experimental data, (see, for example
Grass 1970), a considerable scatter has been found in the results. This is due to the fact
that much of the data employed by Shields were collected by other investigators and because of an arbitrary definition of the critical flow condition. This diagram is essentially derived from a best fit to the scattered experimental data and is then extrapolated for other flow conditions. Grass (1970) stated that the scattering of data in the Shields diagram is due to the influence of turbulence and that turbulence is an important factor influencing sediment entrainment at the initiation of motion.
Vanoni (1975) pointed out that the weak movement condition of Kramer (1935) agrees with the Shields critical condition. However, the critical shear stress determined by
Shields for zero sediment discharge does not depend on a qualitative criterion, but rather, as Vanoni (1975) noted, that it is the magnitude of shear stress for zero sediment discharge and this was obtained by extrapolating the relationship between sediment discharge and shear stress determined from experimental data.
The Shields criterion is, therefore, unsuitable for use to define the initiation of particle motion and, hence, it is necessary to investigate the influence of turbulence on the initiation of particle motion.
After Shields, some investigators (e.g. White, 1940; Egiazaroff, 1957; Lane, 1955) derived some relationships for the initiation of motion. White (1940) postulated that
18 Chapter 2
turbulent fluctuations have an important influence on sediment entrainment and noted that, in turbulent flows velocity fluctuations cause a shear stress fluctuation at the bed and, hence, fluctuating forces are applied to the sediment particles. He derived the following relationship by considering the interaction between the drag force and the resistance force on a horizontal bed
0.18 tan0 (2.3) (y*~Y)ds where (p is the angle of repose of the immersed sediment grain in a fluid, fc is the critical shear stress and dg is the sediment particle diameter. He obtained the constant in equation (2.3) from experiments with laminar flow.
Egiazaroff (1957, 1965) proposed a relationship describing the initiation of particle motion in a mobile bed assuming that at the threshold of motion, for a particle, the approach velocity (u) is equal to its free-fall velocity, and using the following equations:
\cD ApP u2 = (Ys-y)f d\ and U = Jj (2.4) in which Q> is the drag coefficient, Ap is the cross sectional area of the grain exposed to drag, u is the velocity that the particle on the bed experiences, d$ is grain size diameter, ys and y are the specific weights of the grain and fluid, respectively, U is cross-sectional mean velocity and/is Darcy-Weisbach friction coefficient. After some manipulation, he derived the relationship:
3 CD(ar + 5.75 log 0.63)2 (2.5)
19 Chapter 2
where 6C is the dimensionless critical shear stress and ar is a constant which, for a hydrodynamically rough bed is equal to 8.5. The value of Q> for quartz of 3-12 mm is about 0.4. Egiazaroff (1965) showed that for Reynolds numbers less than 2, the above relationship has a negative slope and follows the Shields curve. However, the cross-sectional mean velocity does not sufficiently encapsulate the local flow conditions at the bed.
Lane (1955) suggested a value of the critical shear stress (rc) for initiation of motion in clear water flows and non-cohesive material on the bed. He used the d.75 as the representative diameter of the sediment particles and suggested a relationship for the coarse material as
fc = 0.0164d75 ^2 6)
where d.75 is in millimetres (the ^75 means 75 percent of particles are finer), and fc is in lb/ft2. Using Lane’s approach, the recommended value of fc for canals with sand beds is approximately 1.5 to 10 times those given by Shields as shown in Figure 2.3.
20 Chapter 2
T» 10 8
- 6.0
8 8 £y . 4.0 /
/ * 8 // CN
// = 2.0 £ Shields (mean sediment size) — >
(lbs/ft2) Laneclear vsater M
2 1.0 w
2 / / *n I cfl y_R,-4(X) C/5 Line, cle*r wa er y - 06 £
___R. “ 40C stress /
2 / — X 1 7 04 ia O -• 1 -C shear s C/3 ys %
i & -R, - 100 7^ o ■a ------— // == 0.1 ;c 1 Critical u • ; 80* F-^VV32 • F -- 0 06
Temper*ture. 7V^ ------fiO — —r —2-32 f~, 70 “7/40 I f,9£\ //✓50 1
X "\ ' ' 0 t f 1 = = 0.02 i l l
FV•4
R.- 2 = = 0.01 1 —
i
0.1 0.2 0.4 0.6 0.81.0 2 4 6 8 10 20 40 60 100
Sediment size, dg (mm) Figure 2.3 Critical shear stress for quartz sediments in water as a function of grain size (Shields, 1936 and Lane, 1955) (after, Graf, 1984)
Vanoni (1975) pointed out that there are two possibilities for this: Firstly, Lane’s data were obtained in a discharging sediment situation in which the shear stress values were t above the critical condition. Secondly, dunes existed in the canals and Vanoni (1975) suggested these act to increase the critical shear stress.
Gessler (1965, 1970) investigated the probability of critical shear stresses for sediment entrainment from a mobile bed. From an experimental investigation, he found that, for a 50% probability of particle motion the critical shear stress is 20% less than that estimated from the Shields diagram for large Reynolds numbers and that this
21 Chapter 2
discrepancy changes to 5% at a Reynolds number equal to 20. Gessler found that for a
95% probability of no motion, the value of Shields parameter for large Reynolds number flows was 0.024, compared to that derived from Shields diagram (0.06). This discrepancy may be related to the turbulence characteristics of flows. He also assumed that from the Shields curve the probability of motion is 50%, but, acceptance of this assumption seems doubtful.
Mantz (1977) presented an extension of the Shields diagram for the case of fine cohesionless grains (i.e. particle diameters in the range of 0.01 mm to 0.15 mm). The dashed lines in Figure 2.4 indicates the regression line proposed by Mantz based on his data and some from the literature. The extended Shields diagram of Mantz (1977) is suggested for the case of small size particles.
1 1 o While, S.J. (water) • White, S.J. (oil) V Mantz . «5 + Shields 5 ▼ U.S.W.E.S. X Casey * Gilbert J 1______*------1 —------+
°-01 0-1 1 10 100 1000 Reynolds Number (RN)
Figure 2.4 Extended Shields diagram for cohesionless granular flat sediment beds (after Mantz, 1977)
22 Chapter 2
2.3 PARTICLE ENTRAINMENT AND INSTANTANEOUS TURBULENT SHEAR STRESSES
Previous studies (e.g. Raudkivi, 1963; Blinco & Simon, 1974) of sediment movement in turbulent flows have indicated that sediment motion under critical conditions and a low sediment transport rate is intermittent in nature. By experimental investigation,
Einstein and Li (1958) postulated that not only a top layer but also a sediment sublayer is developed and is disrupted in time due to the intermittent nature of the flow. In a fluctuating flow such as a turbulent flow, when the instantaneous shear stress exceeds the critical shear stress and is sufficient to overcome the resisting forces, particles will commence movement along the bed by rolling, sliding or saltating, depending on the magnitude of the available excess shear stresses. The continuing motion of these particles requires the application of forces which exceed the force necessary to maintain particles in motion. However, due to the intermittent nature of turbulent flows, the applied forces fluctuate over time. The forces, therefore, are sometimes lower and sometimes higher than needed to maintain motion. This, results in the intermittent nature of the motion of sediment particles.
Raudkivi (1963, 1990) conducted some experiments over a mobile bed with ripples and concluded that at a place where the shear stress is close to zero, sediment particles are agitated by the generated eddies in a turbulent flow. In a similar manner, Graf (1984) pointed out that the incipient motion of similar-sized particles under a given flow condition was statistical in nature due to the turbulence of the flow. The studies of Cheng
23 Chapter 2
and Clyde (1972), Christensen (1972), Blinco and Simon (1974), and Grass (1971,
1982) found that fluctuations of the instantaneous shear stress about the temporal mean
are the result of the flow turbulence. Sutherland (1967) concluded that the turbulence
of the flow was the main mechanism resulting in the entrainment of particles from the
bed. Furthermore, Grass (1982) and Thome et al. (1989) noted that the mode and rate
of sediment transport changes as a function of turbulence.
Due to the temporal variability of the velocity fluctuations, there is a difficulty in defining the initiation of motion for a particle resting on the channel bed in terms of the
time-averaged shear stress. This arises from the need to consider the instantaneous
turbulent shear stress, which is sometimes lower and sometimes higher than the critical
shear stress for a particle. Schober (1989) indicated that even for a plane bed there are difficulties in defining incipient motion because of the random interactions between the sediment particles and flow. Similar conclusions have been noted by, for example,
Chiew and Parker (1994), who pointed out the inherent difficulties in defining the threshold of sediment transport in open channel flows.
A number of studies have been undertaken to investigate the influence of turbulence and
bursting processes on the entrainment of sediment grains from a bed. Studies of
turbulent fluctuations of shear stress have indicated that the particles are moved most
often by the bursting effect of the turbulent flow. For example, Vanoni (1964, 1975) pointed out that near critical conditions, particle entrainment occurs due to the effect of bursting processes in a turbulent flow. Vanoni (1964) recognised the importance of the
24 Chapter 2
bursting process on sediment entrainment and concluded that the burst frequency is strongly correlated with the number of entrained grains. He adopted the burst frequency as a measure of the rate of sediment movement. From the studies of Thome et al. (1989) and Williams et al. (1990) on sediment motion, it has also been found that the initiation of particle motion is related to the bursting process.
There is a need, therefore, to quantify the threshold conditions for the entrainment of sediment particles under the influence of turbulent structures and, in particular, bursting events.
2.4 THE TURBULENT STRUCTURE OF FLOWS AND BURSTING PHENOMENA
2.4.1 Basic Concepts and Relevant Parameters
There is long history of study in water science and, in particular, the hydraulics and hydrodynamics of flow. One of the more interesting facets of the history of hydraulics and hydrodynamics is Leonardo da Vinci’s (1452-1519) works. These works cover a wide range of subjects including: the motion of water, waves, eddies, falling water, floating bodies, flows in pipes and open channels and hydraulic machines. He was the first to discover the stream-lined shape which presents only a small resistance to a flow.
He also made many discoveries and observations in the field of hydraulics and discovered various laws about the drag of bodies. Additionally, he suggested a technique for observing internal flows by using floating particles in the water; a technique which currently is used in flow visualisation.
25 Chapter 2
Reynolds (1842-1912) observed that when the flow velocity reaches a certain value, a
coloured stream line in the flow is disturbed and mixed with the surrounding fluid. This
method of study, which visualises the inside of a flow, is now called the flow
visualisation technique and has many applications in fluid mechanics research.
Reynolds undertook many experiments using this technique and introduced a
non-dimensional number known as the Reynolds Number, which can be expressed
algebraically as
where RV is the Reynolds Number, u is velocity of the flow, d is a characteristic length
and v is the kinematic viscosity of the flow.
The concept of a boundary layer was proposed first by Ludwig Prandtl (1875-1953).
Prandtl expressed the magnitude of turbulent shear stress in terms of the time-averaged
velocity gradient in depth as
T, = pl2(^)2 (2.8)
where l is the mixing length with a length dimension and tt is turbulent shear stress.
According to Prandtl, / is the average distance travelled by fluid lumps involved in the
turbulent mixing process.
I = K.y (2.9)
where K is a constant (~0.4) and is a proportional factor between / and y, known as
the Prandtl-von Karman constant.
26 Chapter 2
2.4.2 Quadrant Analysis of the Bursting Process
The turbulent structure of flow in a boundary layer over rough and smooth surfaces has been investigated previously using high-speed photography and the hydrogen-bubble technique to visualise the turbulent boundary layer (see for example Kline et al. 1967,
Grass 1971). Kline et al. (1967) conducted a series of experiments over a flat plate to understand the mechanisms of the turbulent boundary layer using hydrogen bubbles generated by the electrolysis of water. They postulated that the slower flow near the wall of the plate was found to be discharged intermittently to the outside layer, while the high-speed fluid away from the wall was discharged continuously to the wall. This distinctive fluid motion is referred to now as the bursting process.
The concept of the bursting process was first introduced by Kline et al. (1967) as a process by which momentum is transferred into the boundary layer.. The bursting process consists of four categories of events; these categories are defined by the quadrant of the event as shown in Figure 2.5 and are sweep («' > 0, v' < 0), ejection
(u' < 0, v' > 0), outward interaction (u' > 0, v' > 0) and inward interaction
0u' <0, v' < 0). The velocity fluctuations ( u' and v'j are defined as variations from the temporal mean velocities in the longitudinal and vertical directions, respectively.
Algebraically, they are defined by
u' = u — u (2.10) and
27 Chapter 2
(2.11) v' — v — V where u and v are the instantaneous velocities in the longitudinal and vertical directions, respectively, and u and v are the temporal mean velocities in the longitudinal and vertical directions. The temporal mean velocities are given by
U (2.12) and
N (2.13) i=\ where N is the number of instantaneous velocity samples.
The bursting process is a process or a mechanism by which momentum is transferred into the boundary layer and has been a focus of many previous studies. The contributions of coherent structures, such as the sweep (quadrant IV) and ejection
(quadrant II) events, to momentum transfer have been extensively studied through quadrant analyses and probability analyses based on two-dimensional velocity information. Studies by Nakagawa and Nezu (1978), and Grass (1971, 1982) have indicated that just above the channel bed, the sweep event is more responsible than the ejection event for the transfer of momentum into and out of the boundary layer. In addition, Nakagawa and Nezu (1978) and Thome et al. (1989) have pointed out that the sweep and ejection events occur more frequently than the outward interaction (quadrant
I) and inward interaction (quadrant III) events.
28 Chapter 2
Of the alternative quadrants for bursting events, the sweep event has been recognised in previous studies as being the most important event for the entrainment of sediment particles from the bed. This importance arises from the angle of the velocity vector which is angled towards the bed as implied in Figure 2.5. Of particular concern is the magnitude of the horizontal component of the velocity vector and the angle at which the velocity vector impacts on the bed particles.
/ (n) (I) Ejection Outward Interaction
(u' < 0, v' > 0) Cu’ > 0, v' > 0)
\ /
Inward Interaction Sweep
Cu' <0, v' < 0) («’ > 0, v' < 0)
(HI) (IV)
Figure 2.5 Four classes of bursting events and their associated quadrants
The four types (quadrants) of bursting events identified have different influences on the rate, and mechanisms, of sediment entrainment in a turbulent flow. In studies of sediment transport by, for example, Williams (1990) and Thome et al. (1989), it has been shown that sediment entrainment occurs from the bed most frequently during sweep events and only occasionally during outward-interaction events, whereas the transport of suspended sediments depends primarily on ejection events.
29 Chapter 2
These bursting events impose a rapid and significant fluctuation of forces on the bed.
It is these fluctuations that are considered to have a significant influence on the
entrainment of sediment particles from the bed, with a temporally-variable sediment
transport rate being the result. These instantaneous forces lower the local pressure near
the bed and hence particles may be ejected from the bed (Raudkivi (1990). He postulated that even sheltered particles can be entrained by this mechanism. The entrainment of sediment particles, therefore, is substantially influenced by bursting events and needs to be investigated.
2.4.3 Structure of the Türbulent Boundary Layer
Sutherland (1967), Corino and Brodkey (1969), Grass (1971) and Kim et al. (1971)
investigated the structures of the turbulent boundary layer and showed that in turbulent flow, high-speed eddies from the main body of flow penetrate into the boundary. This process imposes a high momentum pulse to the boundary layer and as a result a low momentum eddy is ejected to the outer region. They also showed that motion in the viscous sublayer is rarely laminar and that it is a misnomer to call it the laminar sublayer.
When a layer of flow with high velocity passes over another layer of flow with a low velocity, eddies form between the two regions of flow. These high-speed eddies from the outlayer impinge upon the boundary layer and to the sublayer resulting in a disturbance of the viscous flow. As the eddies penetrate through the boundary layer, they transfer momentum into and out of the boundary layer. This process retards the flow velocity in the outer region and reduces the energy of the flow. The initial high
30 Chapter 2
momentum eddy which penetrates the boundary is known as a sweep event and the low momentum eddy which ejects from the boundary layer is known as an ejection event.
The major generation and dissipation of turbulent energy occurs through these processes.
The effect of boundary roughness on the bursting process was investigated by Grass
(1971). He conducted a series of experiments over smooth and rough beds and postulated that the sweep and ejection events such as are depicted in Figure 2.6, were present irrespective of the bed roughness. Grass et al. (1991) also pointed out that the bursting phenomena is a common feature of the turbulent boundary layer irrespective of the wall roughness condition and that its characteristics remain constant for geometrically similar 3-12 mm roughness elements.
Grass (1971) concluded that beyond a certain distance from the boundary, the turbulence intensity becomes independent of boundary roughness and depends entirely on the shear velocity (m*) and/or the Reynolds Number (R)y). He also pointed out that the effect of the sweep event appears to be mainly in a region close to the boundary, while the effects of the ejection event occur far from the boundary. Additionally, he suggested that the sweep event is mostly responsible for a high momentum transfer toward the boundary layer, while ejection represents a low momentum transfer outwards from the boundary layer.
31 Chapter 2
Figure 2.6 Structure of the turbulent boundary layer (after Grass, 1971)
The structure of the turbulent boundary layer consists of a series of hairpin (horseshoe) vortices (Smith & Walker, 1990; Smith et al., 1991). It has been shown that sweep and ejection events have the form of a hairpin vortex. Smith and Walker (1990) postulated that once an initial hairpin vortex is generated within the boundary layer, it generates subsidiary hairpin vortices on the flanks of the original vortex and secondary hairpin vortices upstream at its rear (see Figure 2.7).
32 Chapter 2
Primary Hairpin
Subsidiary Hairpin
Low Speed Streaks
Secondary Hairpins Subsidiary Hairpin
Figure 2.7 Turbulence structure near a boundary layer (after Smith & Walker, 1990)
Although the flow structure in a boundary layer is controlled by the viscosity of the fluid, it consists of an arranged complex of low and high speed streaks (Raudkivi & Callander,
1975). In describing the structure of the turbulent flow in the boundary layer, Townsend t (1956, 1976) suggested an organised structure.
Yalin (1992) pointed out that sweep and ejection events are the result of the formation of eddies ( denoted by e\ e"... in Figure 2.8) between the two regions of the flow with high and low-speed velocities. Shown in Figure 2.8 are the actions of sweep and ejection events in the flow. These processes are common features of both smooth and rough walls for sweep and ejection events.
33 Chapter 2
:|pr E)j ection
VZZZZZZZW/ Sweep
Figure 2.8 Sweep and ejection events (after Yalin, 1992)
Recent studies of sediment entrainment in turbulent flow have found that the coherent structures of turbulent flow and, in particular, the bursting phenomenon are the main
t mechanisms of sediment entrainment. These investigations have mostly been related to the turbulent structure of flow over rough boundary layers. There has been a major focus on the relationship between bursting phenomena and the transport of sands and gravels in marine and river environments; see, for example, Rao et al. (1971), Offen and
Kline (1975), Jackson (1976) and Sumer and Oguz (1978). Most of these studies of the bursting process and flow turbulence were undertaken in a laboratory flume. They used a quadrant analysis of turbulent fluctuations of the instantaneous velocities in the
34 Chapter 2
horizontal and vertical flow directions and agreed that flow turbulence is a process of net momentum transfer into the boundary layer of the flow. Sutherland (1966, 1967) pointed out that turbulence is the main mechanism for the entrainment of sediment particles from the bed. He also suggested that entrainment results from an interaction between eddies and sediment grains (see Figure 2.9).
DIRECTION OF MEAN FLOW
IMPINGING EDDY DEPARTING EDDY
_EDGE OF WALL LAYER
Figure 2.9 Eddy structure in the wall region and interaction with bed particles (after Sutherland, 1966)
Studies by Drake et al. (1988), Thome et al. (1989), Williams et al. (1989), Williams
(1990), Nelson et aL (1995) and Bennett and Best (1995) indicate that close to the bed, most of the sediment entrainment of sand and coarse materials occurs during sweep events. However, the characteristics of the bursting event and, in particular, the sweep event (shear stress magnitude, frequency of occurrence and angle of inclination towards the bed) over a rough boundary wall remain to be investigated.
35 Chapter 2
Additionally, the effects of sweep events have been investigated in some studies (for example Williams & Kemp, 1971; 1972) over large transverse scales for a group of sweep events. It is recognised that a group of sweep events produces a group of transverse lineation which spreads out downstream and consequently forms a transverse ridge on the mobile bed (Best ,1992). Shown in Figure 2.10 is a group of sweep events acting together to form a ridge over a mobile bed. These lineation at right angle to the direction of flow may result in a group of low speed streaks over the ridge with alternating high speed streaks between the ridges (Williams & Kemp, 1971; 1972).
A group of sweep events
Direction of the flow
Sediment particles
Velocity vector in sweep event
Figure 2.10 Action of a group of sweep events over a mobile bed.
In spite of the importance of bursting events and, in particular, sweep events on the entrainment function of particles from the bed, little work has been undertaken in order to define an entrainment function associated with this process. It has been emphasised recently by Nezu and Nakagawa (1993) that the bursting process and coherent structures
36 Chapter 2
in turbulent flows, particularly over a rough bed in an open channel, should be studied in greater detail, as these mechanisms remain unclear.
2.5 INITIATION OF SEDIMENT MOTION AND THE FORCE BALANCE MODEL
An entrainment function for sediment particles generally is based on a consideration of the forces acting on a particle resting on the bed. These forces include: a) resistance force or submerged weight of particles, and b) agitation forces (predominantly drag and lift forces).
At incipient motion, the agitating forces and resistance forces are equal. The review of previous work indicates that it is unrealistic to ignore the temporal variation in the hydrodynamic force components.
Yalin (1972) stated that in a steady, uniform flow of water and for an equilibrium transport of sediment particles, seven basic parameters are needed to define the flow and sediment motion conditions; these parameters are the:
-density of water (p),
-density of sediment (ps),
-kinematic viscosity (v),
-particle diameter (4),
-flow depth (H),
37 Chapter 2
-shear velocity (w*)> and
-acceleration due to gravity (g).
These seven basic parameters can be reduced to a set of four dimensionless parameters which are:
-a mobility parameter, ul/(S-l)gdg,
-a particle Reynolds Number, R^=u*ds/v
-a depth-particle size ratio, H/dg, and
-a relative density parameter, S=py /p.
From theoretical considerations, the forces influencing entrainment of sand and gravel into the flow from the bed of an open channel are the drag force and the resistance force.
These applied forces are shown in Figure 2.11. The magnitudes of the agitating force, in the form of drag and lift forces are given by
Fd = \cDpApU2r (2.14)
FL = \cLp A„ U2r (2.15) while the resistance force is given by
Fr = Fs X ß (2.16) where Fg is submerged weight which is
Fg = £jz d3s(ps- p)g (2.17)
38 Chapter 2
and Fd is the drag force, Fi is the lift force, Fr is the resistance force, Fg is the submerged weight of the sediment particle, ß is the friction coefficient, Co and Q, are the drag and lift coefficients for the sediment particle, Ap is the cross-sectional area of the grain exposed to drag, which is assumed to be 0.25nds2, p is the density of water, ps is the density of the sediment particle, 4 is the particle diameter, g is the acceleration due to gravity and Ur is the relative velocity which is given by
Ur = (uf- up) (2.18) where u^is the flow velocity and Up is the particle velocity.
*7777777777777777777 77777777777777777777
Figure 2.11 Applied forces on sediment particles resting on a non-horizontal bed slope
The horizontal forces applied to a sediment particle on the bed of a horizontal plane are the resistance force and a horizontal drag force (Fd). This drag force induces an agitation force upon the bed sediments, which thus causes the particles to move by rolling and sliding. White (1940), Egiazaroff (1965), and Graf (1984) derived their theoretical
39 Chapter 2
results for sediment motion by balancing the fluid drag forces and resistance forces (see
Table 2.1).
Table 2.1 A summary of previous studies on the initiation of motion
Concept Authors Year Definition
rC _ r,U*C ds. Shields, A. 1936 (Yf-Y)ds n v ’ The Drag White, C.M. 1940 -—T<\ , = 0.18 tan0 Force (y*-y)ds Concept Lane, E.M. 1955 fc = 0.0164 Vanoni 1966 rc = Atyf-y)^ cos Ikeda (1982) conducted a series of experimental tests in a wind tunnel with a mobile bed on both flat and laterally sloping beds. This experimentally obtained data was used to validate an analysis of the incipient motion of sediment particles. The following relationship was used to calculate the dimensionless critical shear stress in the analysis. tan0 Oc (2.19) FJi 1 + K. tan0) A where (j> is the angle of repose of particles or the friction angle, Fd is the dimensionless drag force, FL is the dimensionless lift force and K = FjF d. From an investigation of the dimensionless critical shear stress on a flat bed, as shown in Figure 2.11, it was concluded that for K=0, the curve accurately predicted the experimental data for a wide range of u*dAi. Therefore, the lift force can be neglected in estimating the critical shear 40 Chapter 2 stress but Ikeda (1982) recommended its inclusion for safety and for providing a lower boundary for experimental value on incipient motion. Consequently, only the forces denoted as Fj and Fg in Figure 2.11 need to be considered ■ I I 11 o Casey • Tlson a Shields a Iwagaki □ Present study u*d/v Figure 2.12 A comparison of analytical and experimental results (after Ikeda (1982)) Bagnold (1954,1956) proposed a theory for bed particle entrainment based on the work done by the fluid to transport a sediment particle. He introduced the concept of dispersive grain pressure on the bed surface and assumed that at low bed load motion, the critical bed shear stress is equal to the turbulent bed shear stress at the threshold of motion. Using the same concept, Engelund and Fredsoe (1976) introduced a mathematical model for sediment transport containing time-averaged quantities rather than an instantaneous description of the forces. This model is a force balance model for the estimation of bed load and particle velocity on the bed of an open channel. The mean transport velocity of the bed load materials is up , when they are moving. 41 Chapter 2 Both Engelund and Fredsoe (1976) and Bagnold (1956) introduced flow velocity (Uf) at a distance of about one or two grain diameter from the fixed bed, as a function of shear velocity (w*) as (uj- = a w*), where a is a coefficient. Additionally, in their model, the agitating force is represented in the form of a drag force and is given by Fd = 2C° p Ap (au*~ up)2 (2.20) The model equation then expresses the equilibrium of agitating and resisting forces Fd = Fr (2.21). At incipient motion up is equal to zero and hence from equations (2.16), (2.17) and (2.21) it can be shown that - Jc„ <“> where 6C is the Shields parameter, and m* is the time-average shear velocity, Up is the average particle velocity, ß is a friction coefficient, Q> is the drag coefficient and a is a coefficient which was suggested by Engelund and Fredsoe (1975) to be in the range of 6-10. James (1990) presented a model for predicting the entrainment conditions for non-uniform, non-cohesive sediments by balancing the forces applied to the particles. He used a force balance model and analysed the moments of the forces acting on the particles. He concluded that, due to the need to explain variability in the entrainment of sediment particles, the particle geometry is the most important factor in defining entrainment conditions. However, in his study, the influence of turbulence and its 42 Chapter 2 intermittent nature was not considered. As turbulence will also introduce variability in the entrainment of sediment particles, the conclusions developed by James (1990) may be modified through a more complete analysis. A number of studies have investigated entrainment from a bed by considering particle saltation; see, for example, Francis (1973), Abbott and Francis (1977) and Fenton and Abbott (1977). Their studies mostly considered the general motion of particles jumping into the flow from the bed under time-averaged shear stresses and also the situation in which these shear stresses exceeded the critical shear stress. Consequently, the results and conclusions of these studies are not appropriate for defining the initiation of motion. Recently, Ling (1995) proposed a criterion for the incipient motion of sediment particles based on time-averaged forces. A deterministic approach was used with consideration of drag and lift forces in the definition of incipient motion. In this criterion, a probabilistic parameter was not considered for inclusion of the variability of flow and sediment characteristics. Additional studies were suggested, however, for the inclusion of the influence of turbulence on incipient motion. Thus, it remains necessary to develop an entrainment function which includes the effects of the flow turbulence. 43 Chapter 2 2.6 THE ENTRAINMENT FUNCTION AND THE INTENSITY OF PARTICLE ENTRAINMENT An understanding of sediment entrainment and local scour depends on knowledge of the pick-up and deposition of sediment grains from the bed surface. Even the simple case of equilibrium sediment transport for a uniform flow over a flat bed involves entrainment and deposition. In an equilibrium situation, the entrainment and deposition rates are equal. In local scour, the condition is not steady and in equilibrium because the entrainment rate is greater than the deposition rate. Various researchers have investigated the entrainment rate of grain particles based on their definitions of flow and sediment conditions. In general, the conditions are based on average shear stresses, for example Yalin (1977), Van Rijn (1984), or the critical velocity of the flow such as LeFeuve et al. (1970), or an equilibrium situation such that of Einstein (1950). However, the turbulent structure of flow and, in particular, bursting processes, have been recognised as being important mechanisms which influence the sediment entrainment from the bed and must be considered. Usually, the entrainment rate of particles from the bed is defined as the probability or number of particles picked up from the bed in a specified area and time. The probability of grain entrainment is a function of the excess effective shear stress at the bed. A definition of the entrainment intensity or probability on the basis of a time-averaged shear stress and/or velocity of the flow is not sufficient for estimation of an entrainment function. In some entrainment function models such as; the Gessler (1970), Engelund 44 Chapter 2 and Fredsoe (1976), Van Rijn (1984), and Bridge and Bennett (1992) models, it is assumed that the effective shear stress is normally distributed. However, as is discussed in later sections, the distribution of effective shear stress is not normally distributed close to the bed in turbulent, open-channel flows. Einstein (1950) expressed an entrainment function based on: the probability of particles being entrained from the bed in a unit area and time, and the particle volume and the average distance travelled from the moment they are eroded until the moment they are deposited on the bed. He assumed that, under uniform flow conditions and equilibrium sediment transport, the number of particles eroded from the bed is equal to the number of particles deposited on the bed surface per unit area and time. However, the sediment transport at the bed is not always at equilibrium conditions, particularly in scouring conditions. He also related the average travel distance to the probability of a saltating particle being deposited when striking the bed surface. However, particles are not only transported by saltation, particularly at low sediment transport rates and at the initiation of motion. Despite these difficulties, Einstein (1950) proposed that the total average travel distance of particles must increase with increasing bed shear stress and as a consequence, he proposed a relation for the entrainment function as follows: E = £ [(yr-yKl0 5 P (2.23) where E = entrainment rate in mass per unit area and time, P = probability that the lift force exceeds the resistance force, 45 Chapter 2 § = a coefficient, dg = particle diameter and ys and y = specific gravities of the sediment and fluid, respectively. Fernandez Luque and van Beek (1976) conducted a series of experiments in a closed conduit rectangular flume and counted the deposited particles on the bed at different bed slopes and particle sizes. They found that the probability of a particle being deposited when striking the bed surface is independent of the time-averaged bed shear stress and they concluded that it contradicts Einstein’s theory for bed load transport. Yalin (1977) assumed that a sediment particle is entrained when it leaves the bed surface to perform a jump and assumed that the jump duration is proportional to the ratio of the particle diameter to shear velocity at the bed. The analysis of Yalin (1977) rested on non-fluctuating or time-averaged shear velocity and forces. According to Van Rijn (1993), Yalin’s definition of the entrainment rate differs considerably when compared to Einstein’s definition. The instantaneous turbulent shear stresses have been recognised as significantly influencing the motion of sediment particles. However, the inclusion of these instantaneous shear stresses into a model describing how sediment particles are induced into motion has not been attempted previously. In this study, an effort was made to apply instantaneous shear stresses and, in particular, those during sweep events within a force balance model. 46 Chapter 2 2.7 SUMMARY The significance of turbulence on sediment entrainment from the bed of a channel has been pointed out by many previous researchers. In studies of turbulence, the bursting process has been recognised as the main mechanism by which momentum is transferred into the boundary layer and has a significant influence on the entrainment and movement of sediment particles. In fact, Grass (1971) and Kline et al. (1967) postulated that the eddy impinging upon the boundary layer during the bursting process is the main mechanism of sediment entrainment. Further studies by Thome et al. (1989), Williams (1990), Nelson et al. (1995) and Best (1992) have indicated that, close to the bed, most of the sediment entrainment occurs during the sweep event. Thus, the magnitude and probability of the instantaneous shear stress in the sweep event is an important factor for defining particle entrainment from a bed. However, most of the previous investigations of the bursting process have been carried out in smooth walled, man-made channels. Hence, the mechanisms of the bursting processes over the rough boundaries in natural rivers remains poorly understood. A definition of the incipient motion based on the time-averaged shear stress at the bed is not appropriate. Therefore, a model which includes the effects of turbulence is required for a definition of entrainment of particles from the bed. Such a model is the focus of this study. 47 Chapter 3 Chapter 3 Chapter 3 Experimental Apparatus and Procedure 3.1 INTRODUCTION AND AIMS Modelling the entrainment of sediment particles from a bed under the influence of turbulence is of great importance due to the many environmental problems associated with sediment motion. As discussed in the previous chapter, the structure of flow turbulence over a rough bed in open channels and rivers is very complicated and is still not completely understood. Further investigation, therefore, is needed. In this experimental study, attention was placed on the entrainment of sediment particles from a rough, flat, movable bed and the influence of turbulence on this was examined. To investigate the entrainment of sediment particles in turbulent flows within open channels, an experimental investigation was carried out at the Water Research 49 Chapter 3 Laboratory (WRL), University of New South Wales. The experiments were conducted over a rough bed in a flume with different flow conditions for different test series. The experimental program for this study consisted of two fundamental parts : • an investigation of the turbulent structure of flow in an open channel over a rough bed, and • image processing to monitor the entrainment of sediment particles at the bed. This chapter provides an overview of the experimental equipment, experimental setup, measuring techniques and procedures for collecting data. 3.2 THE FLUME 3.2.1 General Description The experiments undertaken during this study were carried out in a non-recirculating, tilting rectangular flume of 0.61 m width, 0.60 m height and 35 m length. The flume was supported and strengthened by steel framework. It was equipped with facilities for changing the bed slope for different experiments. A schematic of the assembly of the flume with some of the facilities used in this study is shown as Figures 3.1a and 3.1b. 50 Chapter 3 Figure 3.1a Photograph of the experimental flume and attachments The flume was also equipped with additional facilities which were not needed for this research; this equipment included a sediment feeder and a bed leveller as detailed by Saeidi (1993). The side walls of the flume were made of glass, making it possible to observe and record the flow characteristics. The glass side walls also made flow visualisation possible during the experimental tests. The bed of the flume was constructed with movable fibre cement sheets (F.C. sheets). These sheets were covered by sand particles of 2 mm nominal diameter. Construction of the flume in this manner enabled experimentation with different bed roughnesses. This facility was used to measure flow velocities over a fixed rough bed under different flow conditions. Minor modifications to the flume were necessary in order to measure the instantaneous 51 Chapter 3 velocity components and, furthermore, to observe the sediment entrainment process on a fine scale at the bed by using an image processing technique. The flume was constructed of six parts and each part was mounted on two steel beams which rested on four spiral jacks in order to adjust each part of the flume independently for different slope adjustments. Each part of the flume was joined to the next part by hinges in order to adjust the bed slope of the flume. In addition, each part of the flume was joined to the bed by a flexible, water-tight material, allowing for ease of adjustment and prevention against leaks. Two components of the instantaneous velocity fluctuations (horizontal and vertical directions) were measured using a small Marsh-McBimey Electromagnetic velocity meter. Shown in Figure 3.2 is a diagrammatic definition of the velocity components in the horizontal and vertical directions, where u is the instantaneous velocity component in the horizontal direction and v is the instantaneous velocity component in the vertical direction. 52 Chapter 3 53 Chapter 3 Flow direction Figure 3.2 A definition sketch of velocity components in two directions 3.2.2 Water Supply to the Flume The WRL laboratory is located downstream of Manly Dam and close to the wall of the dam. Water supply for the flume is provided from Manly Dam without pumps for discharges up to 125 litres per second. Extra discharge was available through the use of pumps. The turbulence of the flow at the inlet to the flume was dissipated by a head tank. The flow rate in the flume can be measured by a Bray Series 30 Water Pattern 250 mm butterfly valve. The flow rate was controlled using an EL-o-Matic model EL350 electric actuator, connected to a computer, located in the supply pipe to the flume head tank. 3.2.3 Rough Bed Characteristics On the bed of the flume a series of fibre cement sheets (F. C.) with a thickness of 20 mm were installed at the same level and slope over the bed. The F.C. sheets were covered 54 Chapter 3 by sand particles of nominal sieve size D50 = 2 mm for this study. As a result, it was possible to perform experiments with different bed roughnesses simply by substituting F.C. sheets of different surface roughnesses over the bed of the flume. The temporary false bed of the flume consisted of 900 mm length segments that were 605 mm wide and 20 mm thick. Each segment was designed as a module that rested on the flume bottom as a bed and was supported on four pieces of metal (levellers) attached to the flume bottom. The height of the metal pieces was consistent in order to ensure that the sheets had the same slope as the base of the flume. The F.C. sheet surface was initially given two or three coats of a clear polyurethane varnish (ESTAPOL®), in order to fill all of the small pore holes on the F.C. sheet. A single layer of sediment particles was bonded to the F.C. sheet by screening over the wet polyurethane uniformly. In order to retain particle roughness, an effort was made to prevent any glue drying on top of the sediment grains. After the curing of the polyurethane, the particles were sufficiently well bonded to the sheet surface so as to not be transported by the water flow. Shown in Figure 3.3a is a photograph of the sand particles glued to the surface of the F.C. sheet. Also shown in Figure 3.3b is the installation of the F.C. sheet and the use of the bed supports and levellers. 55 Chapter 3 Figure 3.3a A photograph of a fixed artificially roughened bed Sand Particles "jj jjjjjjjjj jjj jjjjjjjjjj. Leveller Flume bed Figure 3.3b Artificial rough bed installation with levellers underneath 3.2.4 Roughness Estimation of the Flume and F/C Sheets at Bed In order to calculate the roughness of the flume, a series of tests with different discharges were carried out to estimate the total roughness of the flume. The water surface and 56 Chapter 3 depth of the flow were measured at different stations along the flume. The tests were carried out under steady flow conditions. The “Standard Step Method” (Chow, 1959; Henderson, 1966) was used to calculate flow depths along the flume. The calculated and measured depths were compared using error analysis. By trial and error, the roughness which gave the closest agreement between the calculated and observed water surface profiles was adopted as being appropriate for the equivalent roughness of the flume. The equivalent Manning’s roughness coefficient for the flume computed from this method was determined to be about 0.0136. In conjunction with the above experimental tests, the method of Ven Te Chow (1959, p. 136) for estimation of the equivalent roughness of an open channel with composite roughness was used to determine the bed roughness of the flume (roughness of F/C sheets at the bed). The wall roughness coefficient of the flume was estimated from Chow (1959, Table 5.6) to be about 0.009 for a glass surface wall. Using the determined composite roughness of 0.0136 and the estimated wall roughness of 0.009, the average value of the bed roughness for the F/C sheets was calculated, using equation (3.1), as being 0.0175 (/^ =0.0175). 57 Chapter 3 ^2/3 where n^=equivalent roughness of the flume, f,vv=wetted perimeter of both walls, P\j-wetted perimeter of the bed, /i^roughness of the wall, n^,=roughness of the bed and P is the total perimeter. 3.3 SEDIMENT CHARACTERISTICS 3.3.1 Definitions Sediment characteristics are determined by the size, shape, fall velocity and bulk density of sediment particles. Of all the properties of sediment particles, the size of the sediment is most commonly used in sediment transport research. The sediment particles in a river bed consist of different shapes from round to flat and needle-like. Due to the extreme irregularity in shape and size, particle size is usually defined by the volume, fall velocity and size of sieve mesh through which the particle will pass. Garde and Ranga Raju (1977) and Shen and Julien (1993) stated the following definitions for sieve particle size diameter: Sieve diameter: the sieve diameter of a sediment particle is defined as the square size opening in a sieve through which a given sediment particle will just pass. Shen and Julien (1993) stated that in most studies it has been shown that the sieve diameter is 90 percent of the nominal diameter. In most cases a series of sieves is used for the 58 Chapter 3 separation of sediments into various size grades provided that the particles are larger than 0.0625 mm in diameter. Sieve openings are usually square in shape. 3.3.2 Size Distribution of Sediment Particles Since both flow resistance and sediment movement are directly related to the sediment size distribution, it is important to obtain an accurate measurement of sediment size distribution in a stream (Vanoni, 1975). The most common and reliable method used to measure the sediment size distribution is a laboratory sieve analysis. In this method a series of sieves are fitted over one another with the mesh size decreasing in a downward direction. A sediment sample is placed on the top sieve and the sieve column is usually shaken with an automatic shaking machine. Some precautions must be taken prior to a sieve analysis. These include the size of test sample, the type of sieving (dry or wet), the sieve sizes, the shaking method and also the shaking duration. In this study a sieve analysis method was used in order to find the size distribution of the sediment particles. The results of the sieve analysis, and some other properties of the sediment particles, are listed in Table 3.1, while the sediment size distribution is shown in Figure 3.4. Table 3.1: Properties of the 2 mm sediment particles used in the laboratory test Mesh Opening 2.80 2.56 2.36 2.00 1.40 1 0.075 (nun) % Finer 100 86 61 50 5 0.2 0.05 by mass 59 Chapter 3 Grain Size (mm) Figure 3.4 Cumulative sediment size distribution curve used in laboratory tests 3.4 FLOW VELOCITY MEASUREMENT 3.4.1 Equipment and Procedure The longitudinal and vertical components of the instantaneous velocity were measured using a small electromagnetic velocity meter (henceforth referred to as EMC). A schematic representation of the velocity measurement devices with the experimental set-up is shown as Figures 3.5, 3.6a and 3.6b. The use of a trolley mounting device enabled the EMC to be moved to any location in the vertical and transverse flow directions for measurement of the velocity fluctuations. Details of the trolley are shown in Figure 3.6a with a photograph shown in Figure 3.6b. During an experimental run, measurements of turbulent velocity fluctuations were performed at a variety of depths at the centre-line of the flume, 7.00 metres downstream from the inlet of the flow. 60 Chapter 3 Electromagnetic velocity meters have been used to measure turbulent flow velocities for many years. The first reported use was by Bowden and Fairbum (1956) who measured the velocity components of the flow in two directions at depths up to 22 metres. Recently, reasonably small EMCs have been manufactured for use in laboratory flumes. The advantage of the EMC is the absence of moving parts, which can be affected by dirty water. In this study, the EMC was preferable to other instruments due to the unclean water from Manly Dam. The EMC probe used in this experimental study was a Marsh-McBimey Electromagnetic Current Meter (Model-523) with two pairs of electrodes, enabling the measurement of two velocity components of the flow. The probe was installed in the flume and connected to a computer using a DAS 16 analogue-to-digital converter. Computer software (Lotus Measure®) was used for the data acquisition. In order to avoid noise in the system, batteries (DC power) were preferred for the operation, rather than 240 V AC current. In addition to this, prior to any test being undertaken, the EMC was tested in a bucket of still water. For sensitivity and calibration of the EMC a series of tests was undertaken initially in order to get reasonable and accurate data. In order to minimise the temporal variation of velocity fluctuations the velocities were collected for 2-3 minutes during the experimental test at each point within the flow. The gain accuracy of the EMC was ±2% over the full range (0-300 cm/sec) and the long term zero drift was < 2.1 cm/sec. In this experimental study the maximum velocity 61 Chapter 3 of the flow was less than 100 cm/sec; hence, the gain accuracy of the EMC was reasonable. More specifications of the EMC are presented in Appendix B. Discrete signals are usually recorded by sampling from continuous signals. There is a sampling rate, generally referred to as the Nyquist rate, that guarantees the recoverability of a signal (Oppenheim & Schafer, 1975). Nyquist’s sampling theorem states that under some conditions it is possible to completely represent a continuous signal by means of a set of numbers. These numbers correspond to the level of the signal at regular sampling intervals. For any signal, the Nyquist rate is twice the highest frequency content of that signal (f>2fmax). Here this condition was considered for collection of the sampling rate. Application of an auto-correlation technique resulted in the highest frequency in the data collected being 15 Hz and hence a sampling and frequency rate of 30 Hz was adopted. /y Trolley } (Interface) ( Electromagnetic ( Hume and Locatlon ( Computer) Velocimeter) 0 measurement) Figure 3.5 Arrangement and setup of the data acquisition equipment 62 Chapter 3 Vertical Movement Horizontal Movement Figure 3.6a Simple schematic diagram of trolley mounting Figure 3.6b A photograph of the trolley mounting 63 Chapter 3 An example of the time record of both horizontal and vertical velocity fluctuations about the mean obtained in the experiments is given in Figure 3.7. Figure 3.7 Example of the time series of the velocity measurement from Test Q-10 The flow velocity components were obtained as described above and stored in a computer file for subsequent analysis. These velocities were analysed by means of a specially-written computer program which was developed to calculate the time averaged velocity in the horizontal and vertical directions, the overall mean shear stress, the velocity fluctuations, the mean shear stress in each bursting event and to count the number of events in each quadrant. In addition, the computer program was able to calculate the magnitude of the velocity vector and its angle from the horizontal in a clockwise direction. 64 Chapter 3 3.4.2 Determination of Shear Velocity The accurate determination of the shear velocity at the bed of an open channel was an important part of the experimental study. Without an accurate determination of the shear velocity, it was not possible to determine other parameters of interest. The shear velocity in this experimental study was determined from a linear extrapolation of the Reynolds shear stress [r = -pu'v'] to the bed, in which x is the Reynolds shear stress at any point in the flow depth. This results in the use of (3.2) where x0 is shear stress at the bed, u* is shear velocity at the bed and p is the density of the fluid. These Reynolds stresses were determined from the simultaneous measurements of two instantaneous velocity components in the horizontal and vertical directions. As reviewed and discussed by Nezu and Nakagawa (1993), this method is much more accurate than alternative methods. This approach, therefore, was adopted for the determination of the shear velocity at the bed. In addition to determining the Reynolds shear stress at the channel bed, the Reynolds shear stresses at a variety of depths within the flow were calculated from the velocity fluctuations and variations of the determined shear stresses with respect to the flow depth. The relative depths at which measurements were made for each experimental run are given in Appendix C. o 65 Chapter 3 3.5 PARTICLE MOVEMENT MEASUREMENT 3.5.1 Equipment The second aspect of the experimental component of this study was a determination of particle motion under varying flow conditions. The movement of sand particles (Figure 3.8) in the test section (7.00 m downstream from the inlet) was observed and recorded with two cameras which recorded 25 frames per second. These cameras were • A super VHS Panasonic CCD camera (Figure 3.9a) with a resolution of 400 lines per frame; and, • a Sony 8mm video camera with 200 lines per frame resolution. These images were recorded on video tape and transferred to a computer to produce a sequence of digital images for later analysis. In order to get a clear picture, a slide projector was used to illuminate the specific area. A schematic representation of the image capturing equipment set-up is shown in Figure 3.9b. The situation and position of the desired observation area on the flume bed is shown in Figure 3.10. 3.5.2 Light Illumination Illumination is a very important part of an image-capturing system as the quality of the captured image or picture depends upon the object illumination. The area of ideal illumination is based on the light and the images are best captured with the camera directed perpendicularly to that plane of light. With a normal light source (e.g. lamp) illumination is very difficult and therefore it is necessary to design a restricted illumination system. In order to produce a narrow beam of light, a slide projector was 66 Chapter 3 used to illuminate the observation area. The slide projector was equipped with an illumination lens at the front. It was installed at one side of the flume and focused on the observation area. The projector lens was adjustable in order to focus light on the observation area. A schematic of the illumination arrangement is shown as Figure 3.11. In order to avoid distortion of the images by waves on the water surface and to improve the image quality, a flexible plastic sheet mounted on a wooden frame was adjusted just at the water surface. This plastic sheet had enough flexibility to result in no disturbance to the water surface along the observation area. The plastic sheet spanned an area of 30 by 20 cm. Careful attention was made to avoid air bubbles in the flow. Thus, a polystyrene foam sheet was used at the inlet of the flume to clear dust, other floatable material and air bubbles from the water. These precautions were taken to ensure that a clear image was captured. Figure 3.8 Colour particles used as mobile bed for image processing 67 Chapter 3 Figure 3.9a CCD camera and accessories used in the image-capturing technique CCD Camera ssaa Fixed Bed Area \ 1 » \ 1 1 Observed Mobile Area \ 1 1 \ 1 1 ...... v... .v • ...... 1 ____\______JL___ i___ i__Z__ V { / ------\3------IIJ \ ____ J " ~——- Adjustable part Figure 3.9b Side view of the mobile area of the flume used in the image processing test 68 Chapter 3 SO mm £ ijjjjjjjjjjjjitjjjjjj^jjjjjjjjjjjjj iJJJJJJJJJJJJJJJJJJJJJjJJJJJJJJJJJ iJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ Q sjjjjjjjjjjjjjjjjjjjj ijjjjjjjjjjjjjjjjjjjj Observation Area 1 Row i JJJJJJJJJJJJJJJJJJJJTTTTTTTTi s * J i J J J J J J J J J J J J J J J J J J JJ JJ J J JJ^f J J J J J J i J J J J J J J J J J J J J J J J J J J J J J J J^Tj J J J J J J J -JJJJJJJJJJJJJ JJyrTj jj j j j j j j j JJJJJJJJJJJJJ IJJJJJJ VJJJJJJJJJJJ JJJJJJJJJJJJJ IJJJJJJ JJJJJJJJJJJJJ JJJJJJJJJJJJJ IJJJJJJiJJJJJJJJJJJJJ J J J JJJJJJJJJJ IJJJJJJ J J J JJ J JJ J J J J J yy JJJJJJJJJJJJJ IJJJJJJ JJJJJJJJJJJJJ JJJJJJJJJJJJJ iJJJJJJJJJJJJJJJJJJJ .JJJJJJJJJJJJJ JJJJJJJJJJJJJ JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ tJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ iJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ iJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ iJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ j j j i j i i i ) ) j I i j j i j i j j i j ) i < i ( I )*)-*< 610 mm * Figure 3.10 Plan view of area of investigation Flexible Plastic Sheet Light Projector Sediment Particles Figure 3.11 Schematic of sideview observation area 69 Chapter 3 3.5.3 Method of Capturing Video Images In this study an attempt was made to observe sediment particle motion in the centre of a flume over a mobile bed. Consequently, a movable bed was prepared using sand particles of 2 mm diameter with a thickness of 65 mm on the bed and 7.00 m downstream of the flume inlet. The bed was screeded in order to produce a very flat mobile bed. Sediment particles were painted using a special dye in four colours (see Figure 3.8). The dye was diluted with methanol and mixed with the sand particles until the methanol evaporated. There was no effect on the sand characteristics as a result of this painting. These coloured particles were used for some preliminary tests, but, in the actual experimental tests, where video images were captured, only white particles were used. A CCD camera was installed at the top of the flume to observe any movement of the sand particles in the observation area. The camera was set up normal to the bed of the flume and focused over the observation area of the movable bed. A schematic representation of the image capturing system is shown in Figure 3.12. A steady flow condition was created in the flume in order to produce an incipient sediment motion condition. The experimental tests were carried out in different flow conditions and with different levels of sediment motion. For each experimental test and level of sediment motion, a sequence of images was recorded for later analysis using the super VHS Video and the Video8 recorders while being observed simultaneously on a TV screen. The CCD camera was able to capture images at a rate of 25 frames per second. At the same time, the velocity fluctuations were recorded for subsequent correlation with the entrainment of the sediment particles. 70 Chapter 3 During the experimental runs, the flow velocities were measured at the centreline of the flume. The velocities were measured in two dimensions at different points within the flow from the bed to the water surface. Water surface profiles were also measured during the experimental tests with a measurement accuracy of it 0.5 mm. The captured images and recorded velocity fluctuations were used later to find a probability distribution of instantaneous shear stress and particle entrainment. CCD Camera Flume Sediment particles SVHS time lapse Video Recorder 8 mm Video Recorder Figure 3.12 Image capturing: Capturing process and systems 71 Chapter 3 3.5.4 Analysis of the Particle Motion Images The captured images were digitised using Image Maker® Software and a 486, 50 MHz computer. The images were played on a video player connected to the computer and were passed through a hardware card and displayed on the monitor. This software was able to capture up to 25 frames per second depending on the captured frame size. For this analysis, a range of 5-25 frames/second were captured and digitised. The recorded video images were digitised into an array of 384 x 288 pixels, each pixel being quantised in 8 bits with light intensities ranging from 0 (for black) to 255 (for white). The Grey format was selected due to its low requirements for storage and transfer to the computer. The images were digitised as a sequence of frames in VSF format (Video Sequence Format) saved as a single, large computer file. The digitised images which were saved as a video sequence could then be split into a series of still pictures. These images were converted to a bitmap format (BMP) with a grey scale to minimise processing time in the computer. A photograph with a schematic representation of the image digitising process and systems is shown in Figures 3.13. Using the commercial UNIX software (XV® Software), these BMP images were converted to Portable Grey Map (PGM) files in a raw binary format. In order to analyse the images a computer program was required to read the binary files and to process them for any analysis. In this study the aim of the image analysis was to derive the difference between two sequential images in time to ascertain the number of particles entrained in an instant of time. Thus, a specifically written computer program in the Turbo C (and or C++) language was used to determine the differences 72 Chapter 3 between a sequence of images. The C language was chosen as it is a powerful computer language, suitable for image processing. Figure 3.13 Photograph of image processing and digitising equipment The program opens two images in their binary format and calculates the difference of intensity between them at each pixel with this difference being stored in a file with the same image format. By viewing the new image with image processing software (in this case, the software used was XV®), the entrained and deposited particles can be seen. Examples of the images and their differences are shown as Figure 3.14. As can be seen from Figure 3.14(c), the entrained particles in the new image are black in colour while the deposited particles are white in colour. In this chapter, the focus has been on an introduction to the experimental facilities and the procedures used for image capturing and digitisation. A full description of the image processing techniques employed to analyse the images are presented later in chapter 6. 73 Chapter 3 Figure 3.14 Two sequence of images with difference; a) Image I, b) Image II, a-b) Difference II-I 74 Chapter 3 3.6 EXPERIMENTAL TESTS 3.6.1 Boundary Layer Development Most previous studies have focused on the turbulent characteristics of fluid flow in a closed conduit and over a smooth bed. The characteristics of the flow in an open channel and over a rough bed as in a river flow are still not understood (Nezu and Nakagawa, 1993). To investigate this problem, the turbulent characteristics of the flow in an open channel and over a rough bed were tested. In these experimental tests an approach similar to Nikuradse’s experiment was used. As mentioned earlier the rough fixed bed was made up of a dense layer of 2mm uniform sand glued to F.C. sheets with a thickness of 1 or 2 grain diameters. The height of the roughness was estimated to be =4.0ri=0.4 mm using an average of 15 individual random points. This was close to the A*=2d5o which was suggested by Einstein (1949). To create a rough mobile bed, uniform 2 mm sand was screed over the bed with a thickness of 65 mm. The test section selected was 7.0 m downstream of the entrance in order to ensure a fully-developed flow in the test section of the flume. Bauer (1954) studied boundary layer development over a smooth bed and suggested that the required length can be evaluated using: „ _ 0.048^4/5 5I/15RI/12 o e where H is the flow depth, SQ is slope of the bed, ^ is Reynolds number and x is the required length to achieve a fully-developed flow. The value of x was calculated using 75 Chapter 3 this method. The length required to achieve a fully-developed flow was verified as occurring at x ^ 5.00 m, for a smooth bed (for U=0.8 m/sec, So=0.003 and H=0.1 m). As the roughness is very important to the production of a fully-developed flow, the development of the boundary layer occurred earlier than predicted by this method. Additionally, in order to get a fully-developed flow in the flume earlier than evaluated by the above equation, some guiding, straightening and stabilising devices were used in the entrance of flume as shown in Figure 3.1. In addition to the above criterion, a symmetrical velocity distribution about the centre line, and also a maximum velocity just below the water surface (see Figure 4.5), confirmed the developed boundary layer. 3.6.2 Scale of Turbulence In boundary layer shear flow, there exists fluid volume, with an average dimension equal to the eddy size (Gordon, 1974). The eddy-like structures can be defined by their length and time-scales (Taylor, 1935). These structures are similar to the macro-scales in a turbulent shear flow. According to Taylor (1935) the Eulerian definition of the integral time-scale (£,) at a point of flow is defined as 00 (3.3). o In equation (3.3), R(At) is the auto-correlation function of the velocity fluctuation u(t) and is given as u{t).u{t + At) dt R{At) = (3.4) o 76 Chapter 3 In equation (3.4), At is the delay time or lag in the auto-correlation function. The time-scale and eddy size scales at the macro-scale can be computed using auto-correlation. The auto-correlation function between values of velocity fluctuations is required as an input. A typical graphical representation of the auto-correlation function applied to the velocity fluctuations (1/25 sec) is shown in Figure 3.15. This was obtained using the discrete form of equations (3.3) and (3.4). At (sec) Figure 3.15 Auto-correlation of velocity fluctuations Figure 3.15 indicates that an anti-phase condition in auto-correlation occurred at 0.5 seconds, 1.5 seconds, etc., and the period of the auto-correlation is 1.0 seconds. This indicates that the flow condition was reversed at these time lags. Further, the results indicate that the high correlation approaches zero at the limit of the large eddies. Using Kolmogorov’s definition of macro-scale (Tennekes and Lumley, 1972, Townsend, 1976), the corresponding length scales or eddy sizes were determined by first 77 Chapter 3 calculating the area under each curve, to obtain the time scale (Lt), and then multiplying by the mean velocity U, which is given by Therefore, the eddy length scale was obtained by Lx = Lt X Ü (3.6) The macro-scale of turbulence was calculated from the longitudinal component of the velocity fluctuations for one of the experimental tests (Test R5-R4) and the results in a non-dimensional form are presented in Figure 3.16. Figure 3.16 Non-dimensional turbulence scale of velocity components Figure 3.16 indicates that Lx(u)/H increases with increasing flow depth in the region where d/H<0.50 and remains almost unchanged where d/H>0.50. Additionally, the values of Lx(U)/H were in the range 0.15 78 Chapter 3 3.6.3 Test Conditions At the section tested, the velocity of the flow in two dimensions (horizontal and vertical directions) was measured and the turbulence intensities and Reynolds shear stress were calculated. In order to calculate the shear velocity at the bed, the water surface was measured at some point upstream and downstream of the test section. The experimental tests were carried out in steady, uniform and steady, nonuniform flow conditions. In order to stabilise the flow conditions in the flume, a tail gate installed at the end of the flume was used. A summary of the experimental tests is given in Tables 3.2 and 3.3. Table 3.2 The flow conditions in the rough, fixed-bed experiments 1 2 3 4 5 6 7 8 9 10 11 12 Test E F G H J K L M N O P Q Q(l/s) 63.7 76.3 52 58 73.6 40.2 61 30.3 22 48.5 79.8 97.5 H(mm) 355 154 276 120 145 120 212 154 70 166 230 265 T(oC) 15 15 15 15.5 14 13.5 13.5 13 12.2 13 13 12.6 sexier2) 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 U(m/s) 0.29 0.81 0.31 0.79 0.83 0.55 0.47 0.32 0.51 0.48 0.57 0.60 Fr 0.16 0.66 0.19 0.73 0.70 0.51 0.33 0.26 0.62 0.37 0.38 0.37 Flow S& S & S & S& S & S & S& S& S& S& S& S& Condition N U N U U NNN N N N N S = Steady flow condition N = Non-uniform flow U = Uniform flow 79 Chapter 3 Table 3.3 The flow conditions in the rough, movable-bed experiments H Q U T Fr Flow Condition Tests So (mm) (I/s) X (m/s) (oC) 10“2 • R1-R1 95 31.6 0.5 0.54 18 0.56 steady & Non-uni form R1-R2 85 31.6 0.5 0.61 18 0.67 steady & Uniform R5-R1 125 37.8 0.5 0.49 16.5 0.44 steady & Non-uni form R5-R2 110 37.8 0.5 0.56 16.5 0.54 steady & Non-uni form R5-R3 107 37.8 0.5 0.58 16.5 0.57 steady & Non-uni form R5-R4 105 37.8 0.5 0.59 16.5 0.58 steady & Non-uni form R5-R6 103 37.8 0.5 0.60 16.5 0.60 steady & Uniform R7-R1 152 48.1 0.5 0.52 15.5 0.43 steady & Non-uni form R7-R2 140 48.1 0.5 0.56 15.5 0.48 steady & Non-uni form R7-R3 134 48.1 0.5 0.59 15.5 0.51 steady & Non-uni form R7-R4 130 48.1 0.5 0.61 15.5 0.54 steady & Non-uni form R7-R5 127 48.1 0.5 0.62 15.5 0.56 steady & Uniform R9-R1 140 43.0 0.5 0.50 17.3 0.43 steady & Non-uni form R9-R2 128 43.0 0.5 0.55 17.3 0.49 steady & Non-uni form R9-R3 121 43.0 0.5 0.58 17.3 0.53 steady & Non-uni form R9-R4 116 43.0 0.5 0.61 17.3 0.57 steady & Non-uni form R9-R5 114 43.0 0.5 0.62 17.3 0.59 steady & Uniform 80 Chapter 3 3.7 SUMMARY An experimental investigation was carried out in the Water Research Laboratory at the University of New South Wales to investigate the turbulence structure of the flow. In this experimental investigation the velocity fluctuations of the flow in two directions were measured over a rough bed in a flume. Different flow conditions were used in these experimental tests. The velocities of the flow were measured at various depths from the bed to the water surface. Additionally, in order to monitor the initiation of particle motion at the bed, an image processing facility was used in the experimental tests. The motion of sediment particles was analysed using image processing techniques. An analysis of the captured images was carried out with some specifically-written computer programs and computer software packages. One of these was a specifically written computer program in the C++ language that compared a sequence of two images and gave the difference between these two images. 81 Chapter 4 82 Chapter 4 Chapter 4 Statistical Analysis of Turbulent Shear Stress 4.1 INTRODUCTION In this study, the characteristics of the bursting processes and, in particular, the sweep event were investigated in a flume with a rough bed. As previously described, the instantaneous velocity fluctuations of the flow were measured in two dimensions using a small electromagnetic velocity meter and the turbulent shear stresses were determined from these velocity fluctuations. A statistical analysis of the experimental data was undertaken and this analysis forms the basis of this Chapter. Determined from this statistical analysis were the mean instantaneous shear stress in bursting events, the angle of action and standard error of estimate for the events. Presented in this chapter is a discussion of the statistical analysis and the results obtained from the analysis. One important aspect of the analysis, however was the need to use a Box-Cox transformation 83 Chapter 4 to produce a normally distributed sample. More details of the transformation are presented in the following text. This analysis determined the temporal mean velocity in the longitudinal and vertical directions, the temporal mean shear stress, the velocity fluctuations, the mean shear stress for events occurring in each quadrant, and the number of events in each quadrant during the sampling period. The analysis also determined the mean angle from the horizontal of events in quadrants II (ejection events) and IV (sweep events). 4.2 STATISTICAL METHODS FOR DATA INTERPRETATION 4.2.1 Probability Distribution Function The probability distribution of instantaneous shear stress depends on the intensity of the turbulence within the flow. It can be used to estimate the probability of shear stress applied to the sediment particles at the bed. The skewness of a random variable indicates the asymmetry in a probability density function. If the distribution is skewed to the left side, then the skewness of the distribution is positive and if the distribution is skewed to the right side, the skewness is negative. The probability of a variable x lying between r j and r2 is defined as; (4.1) where/f t) is the probability density function. In Figure 4.1 a typical probability density function is shown. 84 Chapter 4 X'* *P(x) f( X) X Xi X2 Figure 4.1 Probability density function of instantaneous shear stress (x) 4.2.2 Gaussian Distribution A distribution must be fitted to a random variable before any statistical analysis of a variable. One common distribution is the Gaussian or normal distribution. The Gaussian distribution is bell-shaped and is symmetrical about the mean. The distribution is expressed as; (4.2) where ox is the standard deviation of x and x is the mean shear stress. A typical Gaussian distribution is shown in Figure 4.2. /(X) x X Figure 4.2 Probability density of a normal or Gaussian distribution 85 Chapter 4 4.2.3 Box-Cox Transformation The original data for a variable in a real situation is not always normally distributed. The general purpose of the transformation is to provide a model to convert the original time series data into a normal distribution. The transformation can be applied to convert the original data to a transformed domain for analysis and subsequent inverse transformation to predict the analysed variables in the original data domain. The power Box-Cox transformation (Box and Cox, 1964) is a useful model to transform the original data into normally distributed data. The overall procedure for making a prediction using a Box-Cox transformation model is displayed in Figure 4.3. Its application to the instantaneous shear stresses of bursting events is explained in Section 4.5.2. Original data Box-Cox transformation Predict variable in transformed domain Inverse Box-Cox transformation Predict variable in original domain Figure 4.3 Transformation process of the measured data. 86 Chapter 4 4.3 ANALYSIS OF EXPERIMENTAL DATA As discussed in Chapter 3, velocity components in both longitudinal and vertical directions were measured and stored in a digital format for later analysis. A sample of the two-dimensional velocity fluctuations with the quadrant of the bursting events indicated is shown in Figure 4.4. It was expected that, with these experiments, the effects of the Reynolds number on the flow conditions would be considered indirectly; the validity of this assumption was assessed by comparing the non-dimensional velocity profiles determined during the experiments with the data of Vanoni (1964) and Kironoto and Graf (1994). Shown in Figure 4.5 are the measured velocity profiles normalised by the cross-sectionally averaged longitudinal velocity and the height above the bed (d) normalised by the flow depth (H). Not all the experimental profiles are shown in this figure, but those shown are representative of all profiles obtained. Velocity profiles not shown in this figure are presented in Appendix C. 87 Chapter 4 Outward Interaction (I) Ejection (II) 5 100- r~ 1 100‘ 1 5»- s' n ! tu 3 U- vv y ! yywr^1^1 AAnorAifS v -50- 'Vyv V: \j^ -iV -100-J -100- • 1 i1 i ^ 50- 1 “1 1 “o' \ Hka i AA Aa o- A*vlv\fl\jv vu wvyuvi v ywyi/vyv -25- V YPmi« Vw VHWVvV V -25-^ 50- jU - III' Time (Sec) Time (Sec) Inward Interaction (in) Sweep (IV) r t 100- ? 100- 1 50- I 50- 0- U Aj V\ -aaA. j, / I nA .J:A h IllAtflvh *3 p*w*\ \yvp \f m -50- 'v^vniWW -50-1 -100- -100-1 , i 1? 50- 1 ! 1 ni\ki^\ N/-V A,yA*A.— > °- nl# IriWy'--' r'rU/mVw Vvri y -25- -25-' 50------1------1------r————i— Time (Sec) Time (Sec) Figure 4.4 Example of the time series of the velocity measurement from the experiment 88 Chapter 4 - E □ F + G ♦ J X K o L A M A O X p O Q ■ Vanoni(1964) • Kironoto & Graf (1994) Figure 4.5 Normalised velocity profile for some experimental runs Data obtained from the experiments was analysed to calculate the time-averaged velocity, velocity fluctuations, root mean square of the velocity fluctuations, and the Reynolds shear stresses. t The velocity fluctuations ( u', v') are defined as the variations from the temporal mean velocities in the longitudinal and vertical directions. As presented earlier, they are defined by u! = u — U (4.3) and v' = v - v (4.4) 89 Chapter 4 where u and v are the instantaneous velocities in the longitudinal and vertical directions respectively, and u and v are the temporal mean velocities in the longitudinal and vertical directions. These temporal mean velocities are given by (4.5) /=i and '-hi* (4.6) i=i where N is the number of instantaneous velocity samples. The root mean squares of the velocities (turbulence intensity) in the longitudinal and vertical directions were determined by Urms = yfu^ = ^~ (4'7^ Vrms = M ^^4'8^ respectively, while the Reynolds shear stresses were calculated from the velocity fluctuations using r' — — pu'v' (4.9) (4.10) /= l r = — pu'v' (4.11) 90 Chapter 4 where u^ and v^ are the turbulence intensities in the longitudinal and vertical directions respectively, and f is the time averaged Reynolds shear stress and r' is the instantaneous Reynolds shear stress. The shear velocity was determined using w* = fj~P (4.12) where u* is shear velocity, R is the hydraulic radius, x0 is the shear stress at the bed, g is the gravitational acceleration and Sf is the friction gradient which, for steady flow conditions, is equal to the energy gradient. A computer program was written in order to calculate the mean, standard deviation, coefficient of variation, velocity fluctuations in two directions and also the Reynolds shear stress. The computer program also enables the counting of the number of events in each quadrant, the calculation of the angle of each event at any instant in time, and the calculation of the magnitude of the shear stress for each event. The instantaneous shear stresses were normalised by the Reynolds shear stress of the flow. The algorithm for the computer program is shown as Figure 4.6. 91 Chapter 4 Read u* and v' Call Subroutine Standard Compute Mean, STD, Coefficient of Variation Velocity Fluctuations and Time-averaged Shear Stress Call Subroutine Sort Compute the Number of Events, Reynolds Shear Stress, Angle, and Vector magnitude for each event in Bursting Process Quadrant I Quadrant II Quadrant III Quadrant IV u' v' 0, Ql.out 32.out Q3.out Figure 4.6 Computer Programming Algorithm for Analysis of Velocity Fluctuations 92 Chapter 4 Shown in Figures 4.7 and 4.8 are the turbulence intensities normalised by the shear velocity as a function of the measurement height above the channel bed (d) normalised by the flow depth (H). These normalised turbulence intensities were compared with previously published data. As shown in Figures 4.7 and 4.8, substantial agreement between the published data sets was obtained except the turbulence intensity in horizontal direction particularly at d./H=0.10-0.40. This may be due to the fact that the previous data used in Figures 4.7 and 4.8 was collected partly in a wind tunnel (i.e. Läufer, 1050), with smooth bed condition which differs from water flow in open channel with rough bed. Additionally, there is more spread in the turbulence intensity measured in the present study, particularly, that taken close to the water surface (d/H^ 1.0). This is due to the wave action and the shear between the water surface and air above it. Similar to the previously published data, it was found that the turbulence intensity in the longitudinal direction decreased from the bed to the water surface while the turbulence intensity in the vertical direction was not found to change significantly over the flow depth. In addition, a comparison of normalised Reynolds shear stress with previously published data was undertaken. Shown in Figure 4.9 are the non-dimensional Reynolds shear stress as a function of the non-dimensional vertical location; the data obtained during this study, as indicated in this figure, compares favourably with that previously published. As explained earlier, the shear velocity at the bed was determined using a linear fit to the Reynolds shear stress and extrapolation to the bed. As discussed by Nezu and Nakagawa (1993) this is the most accurate method for the determination of flow shear velocity at the bed. Also shown in Figure 4.9 is the Reynolds stress variation with depth and extrapolated to the bed. 93 Chapter 4 A Test-K Horizontal Turbulence Intensity 0 Test-L + Test-M □ Test-N 3 T o Test-0 X Test-P X Test-Q ▲ Nakagawa& °>^ <& 0 o Nezu (1975) XA ■ Grass (1971) A O • Laufer(1950) O I X X (P ♦ McQuivey& 2 - o+ + □ o A X Richardson(1969) * : o o 0.6 0.8 d/H Figure 4.7 Variation of normalised horizontal turbulence intensity as a function of normalised flow depth A Test-K Vertical Turbulence Intensity + Test-M □ Test-N O Test-O y X Test-P X Test-Q ^ Nakagawa 0.6 0.8 dm Figure 4.8 Variation of normalised vertical turbulence intensity as a function of normalised flow depth 94 Chapter 4 O Test-O ^ Test-P A Test-Q X Test-k O Test-M + Test-N ♦ Nakagawa&Nezu A Grass(1971) 0 Läufer (1950) (1975) McQuivey & • Lyn(C-1)0987) ▲ Lyn(C-2)(1987) ❖ Lyn(C-3)(19871 Richardson (1969) 0.8 V 0.6 0.4 0.2 Figure 4.9 Normalised Reynolds shear stress as a function of normalised depth 4.4 RESULTS AND DISCUSSION An analysis of the experimental data indicated that the characteristics of the event classes (sweep, ejection, outward and inward interactions) differed over the flow depth. The following discussion outlines important aspects obtained from the analysis of the t experimental data. In order to analyse the experimental data, it was necessary to apply the statistical techniques discussed earlier. 4.4.1 Probability Distribution of Instantaneous Shear Stresses in Events The probability distribution of instantaneous shear stress in each quadrant was calculated and normalised with the time-averaged shear stress. Shown in Figure 4.10 95 Chapter 4 and 4.11 are the frequency distribution of instantaneous shear stresses in sweep and ejection events. As apparent in these figures, the frequency distribution of instantaneous shear stresses in each quadrant was not normally distributed. To facilitate the processing of the measured data and producing an easily understood model it is required to transform the data to a normal distribution. In order to obtain a normal distribution, a statistical tool was necessary to transform the original data to a Gaussian distribution. The Box-Cox transformation was used for this purpose. 96 Chapter 4 Frequency distribution of shear stresses Sweep events 20 - r'/r Figure 4.10 Frequency distribution of shear stress in an sweep event Frequency distribution of shear stresses Ejection events 20 - r'/r Figure 4.11 Frequency distribution of shear stress in an ejection event 97 Chapter 4 4.4.2 Instantaneous Shear Stress of the Event Classes The magnitude of the mean instantaneous shear stress for each quadrant was found to be different, and to differ from the overall temporal mean shear stress for the flow. The instantaneous shear stress for each event was normalised by the total mean shear stress at that point within the flow. Expressed algebraically, the non-dimensional instantaneous shear stress was determined by C = — (4.13) u f where r' is the instantaneous shear stress and r is the mean shear stress at that point in the flow. In order to transform the normalised data into a normally distributed parameter, a Box-Cox power transformation (see Box and Cox, 1964) was used to transform the original data denoted by the symbol C to the transformed data denoted by B(C). A Box-Cox power transformation is defined, for non-zero values of A, by B(C) = if X * 0 (414) and, for zero values of A, by B(C) = ln(C + k) if X = 0 (4'15) where k is a constant and A is the transformation power. The K and A were determined by trial and error in order to produce a distribution which was close to normal. If all of the values in the time series are greater than zero, then the constant k is usually set to zero (Box and Cox, 1964). Application of the transformation to the data set comprising more than 50 sets of quadrant IV events (sweep events) and 50 sets of quadrant II events 98 Chapter 4 (ejection events) resulted in a average value of A, equal to 0.280 and 0.272, respectively. Hence, the first case of the transformation equation (4.14) is relevant here. The inverse transformation which is the transformation B“1 of B(C), for this situation, is given by C = [A x B{C) + l]* -K (4.16). The frequency distribution of the instantaneous shear stresses for events in quadrant IV (sweep) after application of this transformation is shown in Figure 4.12 for one experimental run (Test-K). More examples of these frequency distributions are shown in Appendix D. The test for normality used in this study is the probability plot correlation coefficient test discussed by Looney and Gulledge (1985). In this test, the more normal a data set is, the closer it plots to a straight line on a normal probability plot. To test for normality, this linearity was tested by computing the linear correlation coefficient between the data and their normal quantiles or normal scores ( the linear scale on a probability plot). Samples from a normal distribution will have a correlation coefficient very close to 1.0. As data depart from normality, their correlation coefficient will decrease below 1.0. As shown in Figure 4.12, the transformed data are almost normally distributed. A good normal distribution can be concluded from Figure 4.13 and consideration of the correlation coefficients of 0.997 obtained for quadrant IV events; similar correlation coefficients were obtained for quadrant IV events for the other experimental runs detailed in Appendix D. application of the shear stress for As discussed by Looney and Gulledge (1985) and Helsel and Hirsch (1992), the critical correlation coefficients for acceptance of the normality hypothesis with a 0.05 level of significance is 0.993 for events in quadrants IV; this critical value is less than that obtained suggesting that the normality hypothesis can be accepted with a high level of confidence. 99 Chapter 4 Frequency Distribution of Shear Stress Sweep Event B(C) Figure 4.12 Frequency distribution of shear stress in a sweep event Normal Probability Plot (Sweep Event) .999 "1 j 7 { .05 “ 4 I------\ B(C) Average: 0.364769 Nof data: 314 Correlation^.9 97 Figure 4.13 Normal probability plot of shear stress in a sweep event 100 Chapter 4 The frequency distribution of shear stresses for ejection events (quadrant II) was also calculated and is shown in Figure 4.14. In a manner similar to that used with Figure 4.13, consideration of Figure 4.15 indicates that the instantaneous shear stresses after transformation are normally distributed. In addition, the correlation coefficient of 0.993 between a theoretical normal distribution and the obtained data indicates a good normal distribution; similar correlation coefficients were obtained for quadrant II events for the other experimental runs detailed in Appendix D. In this case, the critical correlation coefficient is 0.991 for a level of significance of 0.05; this value is lower than that calculated for the data. Therefore, the hypothesis of a normal distribution for this data can be accepted as well. Mean values of the transformed data B(C) for each quadrant (J) were calculated using n (4.17) J=\.A The inverse Box-Cox transformation was applied to these mean values to enable determination of the mean instantaneous shear stress ratio for events in a particular quadrant There is a probability density function associated with the instantaneous shear stresses at any point within the flow for each event The probability density function represented in Figures 4.16 and 4.17, is for a 50% exceedance probability. For events in quadrant IV (sweep events), it was found that the magnitude of the shear stress close to the bed of the channel was approximately 148% of the overall mean shear stress while for events in quadrant II, it was found that the magnitude of the shear stress was approximately 135% of the temporal mean shear stress. Furthermore, as shown in Figures 4.16 and 4.17, it was found that the mean shear stress for events in quadrants II and IV increased with elevation above the bed to the water surface. 101 Chapter 4 Frequency Distribution of Shear Stress Ejection Event B(C) Figure 4.14 Frequency distribution of shear stress in an ejection event Normal Probability Plot (Ejection Event) ---- L _ J___ .001 ~ f------i Average: 0.626845 B(C) Nof data:294 correlation=0.993 Figure 4.15 Normal probability plot of shear stress in an ejection event 102 Chapter 4 Variation of shear stress in sweep event - -+2SEE X X XX X - -2SEE X - X X ' SEE= Standard Error of Estimation Figure 4.16 Variation of shear stress in a sweep event after transformation Variation of shear stress in ejection event +2SEE -2SEE - X ” - X ■ SEE= Standard Error of Estimation Figure 4.17 Variation of shear stress in an ejection event after transformation 103 Chapter 4 Regression equations were developed for the ratio of the mean instantaneous shear stress in quadrant II and IV to the temporal mean flow shear stress; these relationships are plotted with 95% confidence limits (i.e. +/- 2SEE) in Figures 4.16 and 4.17. Algebraically, the relationship for quadrant IV events was Q = 1.48 + 1.13^ (4.18) and for quadrant II events was CT = 1.35 + 1.83t7 (4.19) Correlation coefficients and the standard error of estimate for these relationships were 0.73 and 0.36 respectively for quadrant IV events, and 0.81 and 0.38 respectively for quadrant II events. Regression equations for the other quadrants were not determined as this study focussed on the events in quadrants II and IV. A particular focus of this study was the analysis of bursting event characteristics and particularly sweep events in the region close to the channel bed. This focus resulted from the need to consider the influence of the instantaneous shear stress magnitude on the entrainment of sediment particles into the flow. From the analysis described above, the magnitude of the instantaneous shear stress during quadrant IV events on average is approximately 1.48 times the temporal mean shear stress in the region near the channel bed. It is expected, therefore, that particles which would not move at the temporal mean shear stress are able to move during sweep, or quadrant IV, events due to the higher induced shear force. This does not mean, however, that sediment particles are induced into continuous motion but rather are dislodged from their current location. 104 Chapter 4 4.4.3 Frequency of Events The instantaneous shear stress data was divided into different classes based on the quadrant or phase of the bursting process. The frequency of each quadrant was determined by (4.20) and N=fjnJ (4.21) i = 1 where P is the frequency of each event class (quadrant), is the number of occurrences of each event class, N is the total number of events, and the subscript k represents the individual quadrants (J =1...4). The frequency of events in each of the four quadrants was determined from the experimental data and plotted in relation to the normalised flow depth. Shown in Figures 4.18 and 4.19 are the variations in the determined frequency of occurrence with respect to the normalised depth for each of the four quadrants. It can be seen that the frequency of an individual event varies with the quadrant of the event and the normalised depth. From these figures, it can also be seen that the frequency of events in quadrants IV and II (sweep and ejection events), particularly those close to the bed of an open channel, is higher than the frequency of events in quadrants I and III (outward and inward interaction events). Close to the bed, quadrant IV and quadrant II events have a frequency of approximately 30%, whereas that of events in quadrants I and III is 105 Chapter 4 approximately 20%. As a result, it can be concluded that outward and inward interaction events should occur approximately 30 percent less frequently than sweep and ejection events (= x 100). Additionally, the frequency of the sweep and the ejection events decreases with the depth from the bed to the water surface, whereas those of the outward and inward interactions increase with the depth. As shown in Figures 4.18 and 4.19, the frequency of all events in all quadrants approaches a value of 25 % near the free water surface which suggests that all events are equally probable. The relationship between the frequency of an event in a quadrant and the normalised depth was investigated with regression relationships developed for estimating the probability of events at a defined normalised depth in quadrants IV and II. The regression relationship for events in quadrant IV (sweep events) was 31.9 - [7.2 jj] (4.22) 100 and that for events in quadrant II (ejection events) was 31.3 - [8.9$ (4.23) P 2 IÖÖ Correlation coefficients for these relationships were 0.77 and 0.83 respectively. 106 Chapter 4 30 -- 25 -- o Oo a i5 -- fa 10 -- Isotropic turbulence A sweep O O.I. Figure 4.18 Frequency of occurence of sweep and outward interaction a is - Isotropic turbulence O ejection o I.I. Figure 4.19 Frequency of occurence of ejection and inward interaction 107 Chapter 4 4.4.4 Angle of the Events The force applied to sediment particles on the channel bed during events in quadrant IV depends upon the inclination angle of the force to the bed. The angle of the applied force for individual events was determined from the ratio of the turbulent velocity fluctuations; expressed algebraically, the angle of an event was determined from 6j = arctan| -^7 J (4.24) where is the angle of the event measured from the horizontal. Similar to the analysis of the shear stress magnitude, it was necessary to apply a Box-Cox transformation to the calculated angles. The mean angle of events in a particular quadrant after application of the transformation was determined then as a function of the normalised depth. The relationships developed for the mean event angle, for events in quadrants IV and II, were e4 = 21.6 + 22.3^ (4.25) and 0, = 18.8 + 21.2^ (4.26) respectively. Where 04 and 02 && w degrees. Correlation coefficients for these relationships were 0.67 and 0.71 respectively. Shown in Figures 4.20 and 4.21 are the mean angles of the applied force due to events in quadrants IV (sweep) and II (ejection). It can be seen from Figure 4.20 that the mean angle of sweep events increases with the depth from the bed to the free water surface. Close to the bed, the mean angle of sweep events is approximately 20°-22° while, at d/H=0.5, the mean angle increases to nearly 35°. In a similar manner, the mean angle of events in quadrant II is shown in Figure 4.21 as a function of the normalised depth. Once again, the mean angle of the event increases 108 Chapter 4 with the normalised depth; this increase is from approximately 18° to nearly 32° at d/H=0.5. Variation of sweep angle in depth 50 -- • 40 -- O O a O Ggs 20 -- -C 10 -- Figure 4.20 Variation of inclination angle in a sweep event Variation of ejection angle in depth 30 V 20 B 15 i- 10 Figure 4.21 Variation of inclination angle in an ejection event 109 Chapter 4 4.5 EXCEEDANCE PROBABILITY OF SHEAR STRESS IN EVENTS In order to get a normal distribution, the quantiles of the data were plotted against quantiles of the standardised theoretical normal distribution. As discussed previously in this Chapter, the probability plot correlation coefficient test was used. After the construction of a normal probability plot, the probability distribution function or exceedance probability of data was plotted. In this study, the exceedance probability of the instantaneous shear stress for each event was calculated for analysis. Shown in Figures 4.22 and 4.23 are two examples of the cumulative probability for the instantaneous shear stress in sweep and ejection events. From Figure 4.22, the magnitude of instantaneous shear stress for a desired exceedance probability can be estimated using equation (4.16). In this study the magnitude of the instantaneous shear stress for an exceedance probability of 50% was calculated for all quadrants. Additionally, exceedance probability distributions were determined for the instantaneous shear stresses in quadrants II and IV. 110 Chapter 4 (Sweep Event) 100-1 “ ” I------T ■%” I------1" ~ T ------1 0--! B(C) Figure 4.22 Cumulative probability of shear stress in a sweep event (Ejection Event) I ♦ I 50 — 1 0“i B(Q Figure 4.23 Cumulative probability of shear stress in an ejection event 111 Chapter 4 4.6 RESULTS AND CONCLUSIONS The motion of sediment particles and, particularly, the initiation of that motion is an important component of many water quality problems. As the first stage of a study to investigate the entrainment of sediment particles into the flow, the influence of flow turbulence and the associated coherent structures on the shear stress applied to sediment particles was investigated and reported herein. The investigation was based on the analysis of the turbulence characteristics of flows in a laboratory flume with a mobile bed. From this analysis of the experimental data, it was concluded that • The measured data was consistent with previously published data. • The magnitude of the mean shear stress in each quadrant (sweep, ejection, outward and inward interactions) was different from the overall temporal mean shear stress at a defined depth within the flow. For quadrant IV events, the mean shear stress was approximately 148% of the overall mean shear stress in the region near the bed with 95% confidence limits and increased to approximately 190% at d/H=0.50. In quadrant II events, the shear stress ratio is about 135% near the bed with 95% confidence limits and increases to approximately 200% at the d/H=0.55. The shear stress ratios of the quadrant I and HI events are low compared to quadrant IV and II events. • The frequency of each event was determined also. It was found that, near the bed, the frequency of quadrant IV and II events was approximately 30%, whereas that of quadrant I and El events was only 20%. Consequently, the events in quadrants I and III occur 30% less often than events in quadrants II and IV. 112 Chapter 4 • The average angle of action for events in quadrants II and IV were found to vary with the depth from the bed of the channel to the free water surface. Close to the bed, the mean angle of sweep events is approximately 22° while, the mean angle increases to nearly 35° at d/H=0.5. The mean angle of events in quadrant II is approximately 18° close to the bed increasing to nearly 32° at d/H=0.5. The implication of these results is that the instantaneous forces applied to a sediment particle are higher than the time averaged bed shear stress used in Shields diagram. Consequently, it is suggested that some sediment particles will be induced into motion at lower flow rates than those obtained from the Shields diagram. This motion, however, need not be continuous which was the criteria used by Shields in defining critical shear stress. In order to include the effect of turbulence characteristics to the initiation of particle motion over a mobile bed, it is necessary to incorporate them in a force balance model. This will be presented and discussed in the next chapter. 113 Chapter 5 114 Chapter 5 Chapter 5 A Mathematical Model for the Initiation of Sediment Motion 5.1 INTRODUCTION The initiation of sediment motion is an important component in the study of sediment transport. Most existing critical shear stress models used in sediment transport models are based on the time-averaged channel bed shear stress which is defined in terms of the flow depth, the density of the flowing fluid and the friction slope, whereas particles on the bed sustain instantaneous shear stresses which differ from the temporal average shear stress. These differences derive from the turbulent nature of the flow in most natural channels. Since entrainment of particles into the flow occurs as a result of the applied shear stress at the bed, consideration of the the temporal variation in the shear stress is necessary in the development of a physically based sediment entrainment model. 115 Chapter 5 Presented in this Chapter is the theoretical basis and results of the application of instantaneous shear stresses in a simple mathematical force balance model. Utilisation of this force balance enables the definition of an entrainment function for sediment particles on the bed of an open channel. As the instantaneous shear stresses are probabilistic in nature, the resultant entrainment function has a probabilistic nature also. 5.2 SEDIMENTS IN NATURAL CHANNELS 5.2.1 Mechanisms Involved As mentioned earlier, the entrainment of a sediment particle from the bed requires the application of forces which exceed the critical force for that particle. The critical force is defined as the force required to induce motion of the particle. Estimation of the critical force is obtained from the resistance force or the immersed weight of the sediment particle at the initiation of motion. When the applied force exceeds the resistance force, the particle commences movement along the bed; this movement may be by rolling, sliding or saltation depending on the magnitude of the available shear stress. For situations near the threshold of movement, i.e. the initiation of motion, rolling and sliding are the predominant modes of bed load transport for sand and gravel particles. In a study of the influence of turbulent agitation of particles, Graf (1984) pointed out that the incipient motion of similar sized particles under a given flow condition was statistical in nature due to the turbulence of the flow. Furthermore, as previously discussed in Chapter 2, Grass (1982) and Thome et al. (1989) noted that the mode and rate of sediment transport changes as a function of turbulence. In more detailed studies 116 Chapter 5 of these processes, Williams (1990), Williams et al. (1989), Drake et al. (1988), Bennet and Best (1995) and Nelson et al. (1995) all concluded that, of the bursting processes, the sweep event was the most important process for particle entrainment from the bed. Due to the temporally variable nature of turbulence, there is a difficulty in defining a deterministic criterion for the initiation of motion for a sediment particle resting on a channel bed. Schober (1989) stated that even for a flat bed there are difficulties in defining incipient motion because of variabilities in the flow behaviour and sediment characteristics. Similar conclusions have been noted by, for example, Chiew and Parker (1994) who pointed out the difficulty of sediment entrainment in open channel flow, and Ball and Keshavarzy (1995) who discussed the difficulty of defining incipient motion of sediment particles. This difficulty in the definition of incipient motion arises from the instantaneous turbulent shear stress which has a statistical nature. As explained above, not only the magnitude of the effective shear stress but also the statistical distribution of the instantaneous effective shear stress is important for the entrainment of sediment particles. Therefore, consideration of this variability in the instantaneous effective shear stress in a force balance model is desirable and is one of the aims of this study. 5.2.2 Main Parameters As mentioned earlier, in a steady uniform flow of water and sediment particles, seven basic parameters are needed to define the sediment flow conditions. These parameters are: the density of water (p), the density of sediment grains (ps), the kinematic viscosity 117 Chapter 5 (v), the particle size (J5), the flow depth (H), the shear velocity («*), and the gravitational acceleration (g). From a combination of the above parameters, three dimensionless parameters can be obtained for an open channel flow. These three parameters, namely the Shields parameter, the particle Reynolds number and the specific density parameter, are the most important in defining a sediment entrainment function. These dimensionless parameters are used for the definition of sediment particle motion through the application of force balance models. As discussed in previous sections, however, the shear velocity (u*) is not constant but rather varies with the flow turbulence. Consequently, there has been a need to introduce empirical coefficients to account for the temporal variability of the induced shear stresses. Physically, a stochastic-deterministic approach is reasonable because of its ability to incorporate the turbulent shear stresses into the model. However, in order to achieve this, it is necessary to ascertain the variability of the instantaneous shear stress through a quadrant analysis of the flow turbulence as discussed in previous chapters. The focus of this chapter, therefore, is to incorporate the variability of the instantaneous shear stress into a force balance model. 5.3 ALTERNATIVE CONCEPTS FOR PARTICLE ENTRAINMENT In order to study particle entrainment from the bed, two kinds of approaches have been used in the past The first approach uses an energy concept to model the particle 118 Chapter 5 entrainment with the available energy of the flow. The second approach uses a probability function to predict the probability of particle motion in a mobile bed. 5.3.1 Energy Concept Bagnold (1966, 1979) proposed an approach based on the physical concept of the available energy and power in the flow. The power concept was previously suggested and applied to sediment transport by Rubey (1933), Knapp (1938), and later Bagnold (1956). With this concept the flowing fluid is regarded as a transporting mechanism for particle movement. Bagnold (1956, 1966, 1973) considered that the energy of the flow is responsible for the transporting of the sediment material from the bed. He assumed that the energy of the flow is equal to the work done to the sediment particles. Based on Bagnold’s theory the useful work done by the flow on the sediment can be defined as work rate = available stream power - unutilised power (5.1) or, in an alternative form, it can be written as work rate = available stream power X efficiency (5.2) The mean available power supply to a column of the fluid per unit bed area (w) can be written as to = PZQS?— (5.3) flow width where; Q is the discharge of the stream, and SQ is the bed slope. 119 Chapter 5 With this theory it is assumed that the granular material is transported in the vicinity of the bed. If the submerged weight of a single spherical particle with a diameter (<4) moving over the unit area of the bed surface is Fg, where Fg = f ( Ys-Y)d3s (5-4) and Up is its velocity then, during a time interval equal to unity, the weight Fg is displaced from position 1 to position 2 by up (Figure 5.1). 1 2 Up _> ^ 1 ©; Fr Fr Fg Fg Figure 5.1 Displacement of particles along the bed Suppose Fr-Fg.taruj), then the friction force opposing the motion is Fg.tan§, where <|) is the angle of repose of the sediment particles. The work done on the sediment particles (Wb) by the flow per unit time, therefore, is: Wb = Fg .tan0. Up (5.5) 120 Chapter 5 This work according to Bagnold is the “useful work” done by the flow. The product of Fg.Up in Equation (5.5) is the weight of the granular material passing through a unit width per unit time and thus is the transport rate qs. Hence, Wb = qs X tan0 (5.6) where qs is the bed load, since it is assumed that the weight of sediment Fg is moving in the vicinity of the bed. For determination of the available energy for the production of useful work Wb, Bagnold gave the expression Fa = [(r-rc)] uf (5.7) where up is the flow velocity in the vicinity of the bed. It can easily be verified that Ea has the same units as Wb, namely those of work (or energy) per unit time per unit area of the bed surface. Consequently, the efficiency et, can be written as: = useful work _ Wb €b available energy Ea where Wb is the useful work and Ea is the available energy. Although this concept has been widely used for the estimation of sediment transport, its application to the definition of incipient motion of sediment particles is limited due to its neglect of the variability of the applied forces and the consequent variability of sediment entrainment. Furthermore, the energy concept proposed by Bagnold (1956) is based on the particles in motion rather than the forces at the initiation of motion, which is the focus of this investigation. 121 Chapter 5 5.3.2 Probability Concept In an equilibrium condition of sediment transport the entrainment and deposition of sediment particles can be defined as (5.9) where Ne is the number of particles eroded per unit time per unit area of the bed, and Nj is the number of particles deposited per unit time per unit area of the bed. This expression of the equilibrium condition of sediment entrainment is a fundamental assumption derived from the continuity of mass and was used in Einstein’s original bed load transport equation (Einstein, 1950). The number of particles eroded per unit time per unit area (Ne) is a function of the probability that a particle is eroded (P), the bed area associated with one particle (ds2) and the time required to bring the particle into motion (t); this function can be expressed as Ne = f(P, ds, t) (5.10) Einstein (1950) stated that the time to detach the particles from the bed is proportional to the time required for a grain to fall a distance equal to its own diameter (ds) at a velocity equivalent to the steady settling velocity of the grain ( Vs ) and is expressed as: t = £ ( dt/V, ) (5.11) where § is a dimensionless coefficient. Based on these assumptions (i.e. equilibrium transport conditions and a normal distribution of lift force), Einstein (1950) introduced his stochastic approach for particle entrainment from the bed as follows: 122 Chapter 5 E = s [(Ys-y ) ds]°-5.P (5-12) where E = entrainment rate in mass per unit area and time, P = probability that instantaneous lift force exceeds the resistance forces and that a sediment particle is picked up by the flow, ^ = a coefficient, ys = specific weight of particle Y = specific weight of the fluid and ds = median particle diameter. Other researchers such as Fernandez Luque and Van Beek (1976), Yalin (1977), Nakagawa and Tsujimoto (1976, 1980), De Ruiter (1982, 1983) and Van Rijn (1984, 1986) introduced relationships for the pickup of particles from the bed. These relationships are given in Table 5.1. In this Table, D* is a dimensionless particle parameter which is given by D* = ds[{S-\)g/v2]1//3, Tis a dimensionless shear stress parameter which is given by T = (z' — rc)/rc , ö is the standard deviation of shear stress, B is the width of channel and W is the porosity of the sand bed. All other parameters have been defined previously. 123 Chapter 5 Table 5.1 Published pick-up relationships Source Equation * E _ tPs(6-0c)'5 -Fernandez Luque and Van Beek (1976) us-i)gdsr'/2 3^ * f i 0.035 -Nakagawa and Tsujimoto (1976, 1980) E ~ ^ 6 £ pA(S-l)gds]~'V -Yalin (1977) E = £ psu* P * E- tPsP -De Ruiter (1982, 1983) L r 1-1/2 [(S-i;{£}(Mtan())] * -Einstein (1950) E = f [( Ys-y ) d, ]0S.P ps[(s~i)gds]'/2D03pn -Van Rijn (1984) h 3030 -Shen and Cheong (1980) E = Ys( l-V )B J «7p(y)dy F in unit r»f wpipht pp.r unit timp * as presented by Van Rijn (1993) Most entrainment function models, for example those proposed by Shen and Cheong (1973,1980), Nakagawa and Tsujimoto (1976,1980) and Van Rijn (1984), are based on probability distributions of length and jump height of sediment particles. Van Rijn (1993) compared the entrainment rates using alternative relationships with collected experimental data by fitting the § coefficients. He found that the entrainment functions of Einstein and Yalin were not very dependent on the sediment size. However, those of Nakagawa and Tsujimoto (1976,1980) and Fernandez Luque and Van Beek (1976) were 124 Chapter 5 strongly dependent on the sediment size while the entrainment function of De Ruiter (1982) was weakly dependent on the sediment size. An important feature of the pick-up functions investigated by Van Rijn was the inclusion of a parameter directly related to the probability of a particle being entrained. It was through this parameter that Van Rijn calibrated the relationships for his comparison. However, as discussed in previous sections, the statistical variability of the entrained sediment particles should not only be related to a calibration factor or fitting parameter, but also to the influence of the turbulence of the flow. 5.3.3 Summary of Approaches As discussed above, both the energy and the probability approaches have been applied to the analysis of sediment motion. However, there are limitations to these approaches due to their inherent assumptions. These limitations can be mitigated through a combination of the two approaches postulated in this study. The probability aspects of particle motion are introduced through consideration of the turbulence characteristics of the flow while the energy aspects are considered through an instantaneous force balance model. In this manner, a stochastic-deterministic model for particle motion is developed. 125 Chapter 5 5.4 A FORCE BALANCE MODEL 5.4.1 Applied Forces As mentioned in Chapter 2, the total forces applied to a sediment particle on the bed of a horizontal plane are divided into two major groups, which are: • agitating forces; and • resistance forces. The agitating forces are produced primarily as a result of a pressure differential between the upstream and downstream sides of the particle. Conversely, the resistance forces, such as frictional forces, arise from the particle weight, which tend to prevent movement of the sediment particles. When the agitating forces are equal to the resistance forces, the particles will be in on the threshold of motion and dislodgment may occur. Under this condition the magnitude of bed shear stress is considered to be the critical shear stress value for that particle. The critical shear stress of a particle is basically a function of the particle density, size, shape, roughness and arrangement of the individual particles in the bed surface. With the exception of the arrangement of the individual particles on the bed surface, these parameters are those identified earlier as being important for the definition of sediment motion. 5.4.2 Governing Equations From theoretical considerations, the forces influencing entrainment of sand and gravel from the bed of an open channel into the flow are the drag force and the resistance force. 126 Chapter 5 These applied forces are shown in Figure 5.2. The magnitude of the agitating force in the form of drag is given by Fd = \cDp ApU2r (5-13) As discussed in Chapter 2, Ikeda (1982) pointed out that the lift force is not important for entrainment and, hence, can be neglected in models describing particle entrainment from the bed. The resistance force, or Fr is given by Fr = Fg x ß (5.14) where Fg = ±7id](p;rp)g (5-15) where F Ur = (au*-Up) (5.16) 127 Chapter 5 in which u* is the shear velocity and up is the particle velocity and a is a coefficient. At the bed the time-averaged shear velocity is given by w* = (t0/ p)05 = (g R Sf)05 ^5'17^ where u* is the shear velocity at the bed, z0 is the shear stress at the bed, p is the flow density, g is the gravitational acceleration, R is the hydraulic radius and Sf is the energy gradient of the flow. d 4------ Figure 5.2 Applied Forces on Sediment Particles at the Bed 5.4.3 Force Balance Model with Time-Averaged Conditions A number of alternative sediment transport models have been proposed. The basic concept of these models is similar; therefore, only the model proposed by Engelund and Fredsoe (1976) is discussed in detail herein. An important aspect of these models is the 128 Chapter 5 consideration of the average shear stress at the bed rather than the instantaneous variable shear stress at the bed. Engelund and Fredsoe (1976) introduced a sediment transport model based on analysis and experimentation. Their approach was based on the idea of Bagnold (1954, 1956, 1966) for describing bed load transport. They used a force balance model for estimation of the bed load and particle velocity at the bed of an open channel. The theoretical formulation of their force balance model is based on the time-averaged shear stress at the bed and was introduced as \cD p^(a«,-«p)2 = Id)( ps-p)g.ß (5'18) where u* is the average shear velocity, up is the average particle velocity, ß is friction coefficient (ß=tan Engelund and Fredsoe (1976) suggested that the value of a is between 6 and 10. To derive the probability of particle entrainment, Engelund and Fredsoe (1976) assumed that the probability (P) of entrainment of a particle at the bed is a certain fraction of the particles in a single layer. They introduced a probability relationship which was a function of the Shields parameter. Consequently, Engelund and Fredsoe (1976) suggested that the probability of entrainment of a particle can be obtained by knowing the friction coefficient (ß), Shields parameter (0) and the critical dimensionless shear stress (6C). They defined this probability as 129 Chapter 5 (5.19). P 1 + This model has been developed through consideration of the time-averaged shear stress. Therefore, in this model the inclusion of turbulence was not considered. As the entrainment of particles at the bed is intermittent in nature, the incorporation of turbulence is necessary in the entrainment function. This is the focus of this study. It is this approach which was used in the development of the mathematical model discussed in the following section. 5.5 DEVELOPMENT OF A MODEL INCLUDING THE INFLUENCE OF TURBULENCE 5.5.1 Model Concepts Essentially the entrainment of particles in an open channel is based on the applied effective bed shear stress on the particles, which is induced by the flowing water. The flowing water is the only energy source causing particles to move and the energy is produced mostly by the turbulence of the flow. The inclusion of turbulence as a parameter in an entrainment function also is very difficult but necessary for defining the initiation of motion. Ling (1995) pointed out that turbulence must be incorporated into a definition of particle motion but, as yet, no generally applicable definition has been published. The initiation of motion for a sediment particle resting on the bed is controlled by the applied instantaneous shear stress at the bed. As previously discussed, these shear 130 Chapter 5 stresses are generated by the bursting processes, with the most important event for the entrainment of sediment particles being the sweep event. This importance arises from the sweep event applying a force towards the bed in the flow direction. The applied shear stress resulting from a sweep event depends on the magnitude of the turbulent shear stress and its probability distribution. Shown in Figure 5.3 is a schematic representation of how the sweep event applies force towards the bed and induces the motion of sediment particles. Flow direction Figure 5.3 Sweep event in a bursting process (after Yalin, 1992) 5.5.2 Formulation of the Model with Turbulence Parameters To include the influence of turbulence, it is necessary to incorporate turbulence parameters and, in particular, those relevant to sweep events in the new model. In order to formulate the model with the inclusion of turbulence, the parameters derived in Chapter 4 are incorporated into the model in the following sections. 131 Chapter 5 5.5.2.1 Instantaneous shear stress in the model Using the Reynolds shear stress relationship for time-averaged and instantaneous shear stresses at any point of the flow, i.e. f = -pwV (5.20) the value of time-averaged shear stress and instantaneous shear stress can be obtained. As discussed in Chapter 4, the magnitude of the instantaneous shear stress differs from the time-averaged shear stress and can be obtained from a quadrant analysis. Therefore, consideration of the instantaneous turbulent shear stress derived from the quadrant analysis must be considered in a model for estimating the entrainment of sediment particles. The following development incorporates the instantaneous shear stresses in quadrant IV (sweep) events. If the instantaneous shear stress is xISS and the time averaged shear stress at the same point in the flow is x, then the ratio of instantaneous to time-averaged shear stress at that point of the flow for a sweep event is: Tiss (5.21) a or tiss a XT f where a is a dimensionless variable representing the variability of shear stress during sweep events. Now, a second relationship between the turbulent shear stress and the time-averaged shear stress is (5.22). r = r + r' 132 Chapter 5 where x' is turbulent shear stress, r is the total shear stress and r is the time-averaged shear stress. Therefore, the value of turbulent shear stress can calculated as follows: (5.23) r' = r — r From a conventional Reynolds decomposition u* — u* + u\ (5.24) and (5.25) r = p(M* + w'*)2 = p[ui + u'l'2 + 2 w*w'*] (5.26) r = p\ui + h'2] (5.27) Substituting equations (5.26) and (5.27) into equation (5.23), results in t' = t — t = p[2u*u'* 4- M*'2 — w'2] (5.28) The instantaneous shear stress applied to the particles at the bed during sweep events can be determined from equations (5.27) and (5.28) as [ 2w*m'* + u*'1 + u'l ] p^ul + u'l (5.29) [ w2 + k'2 ] p^ul + u'l [a - l} The variable a in equation (5.21) is a dimensionless parameter which represents the shear stress variability during sweep events. As discussed in Chapter 4, the variable a was found from the statistical analysis of experimental data, and has a mean value of approximately 1.48 for sweep events close to the bed; this is the value reported by 133 Chapter 5 Keshavarzy and Ball (1996a, 1997). Consequently, there is a probability density function associated with the variable a . The turbulent shear stress t' therefore has a probability distribution associated with it. In order to compare the results from the proposed model with those of the time-averaged model, the value of a for an exceedance probability of 50% was selected. The model can be tested, however, for any exceedance probability from 0 to 100%. 5.5.2.2 Impinging angle As discussed in Chapter 4, in addition to determining the magnitude of the shear stress during sweep events, the impinging angles were determined and statistically analysed. The results of this analysis were presented in detail in Chapter 4 and were reported by Keshavarzy and Ball (1996a, 1997). The effectiveness of the applied shear stress to mobilise the sediment also depends on the impinging angle towards the bed. If the flow velocity of a sweep event is more tangential to the bed, then the applied force on particle movement is more effective in sediment entrainment. If the velocity in the sweep event tends to be normal to the bed, the applied force for sediment entrainment has a minimum effect on entrainment. Thus the angle of velocity in the sweep event has an important influence on sediment entrainment. Shown in Figure 5.4 is how the angle of the sweep event influences the applied force on a sediment particle. The horizontal component of the velocity vector in the sweep event can be determined using; 134 Chapter 5 Uf = us' X cos {Os) (5 30) where us' and 6S are as illustrated in Figure 5.4. The impinging angle of the sweep event (quadrant IV) was shown in Chapter 4 with respect to the relative depth of the flow. There is a probability distribution function associated with the angle at any point. The mean impinging angle of the sweep event was reported in Chapter 4 to be about 22° near to the bed. Flow direction JJJ JJJ JJJJliJJJ \JJJJ JJJ JJJ Figure 5.4 Inclination angle of the sweep event 5.5.2.3 Frequency of occurrence The entrainment of sediment particles from the bed not only depends on the shear stress magnitude but also on the frequency distribution of the sweep events. In addition to the instantaneous shear stress magnitude, the frequency of occurrence of a sweep event was determined from velocity fluctuations recorded during experimental tests. The 135 Chapter 5 frequency of occurrence of the sweep events is reported in Chapter 4 and was reported by Keshavarzy and Ball (1996a, 1997). Close to the water surface the flow has an isotropic turbulence while close to the bed the turbulence is not isotropic. From the statistical analysis, the probability of sweep events was found to be about 30 percent close to a rough bed. The shear velocity incorporated in the proposed model is defined as a function of the shear velocity in a sweep event, the impinging angle and the probability of occurrence of the sweep event. The instantaneous turbulent shear velocity in the new model w* is defined as a function of the turbulent shear velocity in the sweep event u\s and is written as; M* = H* + U\s (5.31). The turbulent shear velocity in a sweep event u\s can be defined as a function of shear stress magnitude, angle of inclination and frequency of occurrence, and can be expressed as; (5.32) where a is a dimensionless parameter which represents the shear stress variability in sweep events, 6S is the impinging angle of sweep and P is the probability of occurrence of the sweep event in a bursting process. Therefore, the shear velocity is incorporated in the proposed model through a combination of these new variables based on a quadrant analysis of the bursting events. 136 Chapter 5 It is this shear velocity applied in the instantaneous model, instead of the time-averaged shear velocity in the previous models. 5.5.3 Mathematical Description of the Model The development of this model is based on the incorporation of the instantaneous shear velocity derived from a quadrant analysis of the bursting process instead of the time-averaged shear velocity used in the previously developed model (e.g. equation (5.18)). In order to define the initiation of motion of sediment particles with the influence of turbulence included, the turbulence parameters presented earlier were applied in a force balance model. This model is based on the following relationships, which include the effects of the instantaneous shear stress. The formulation of the force balance model is expressed as: \cD Ps£.(au,-up)2 = |rf?( Pri>)gß (5.33) where w* is the instantaneous shear velocity and uP is the instantaneous particle velocity along the bed. The difference between equations (5.33) and (5.18) relates to the incorporation of the instantaneous shear velocity in the left hand side rather than the conventional time-averaged shear velocity. In Equation (5.33) the right hand side of the expression represents the resistance force, while the left-hand side represents driving forces. If a variable kj is substituted as a coefficient for the left hand side, 137 Chapter 5 (5.34) 2 CdP^p and a variable k2 is introduced on the right hand side of Equation (5.33): (5.35). ^2 - Ps~P)g-ß Equation (5.33) can be written as a 2nd order equation; kx{au* — Up)2 = k2 (5.36) Therefore equation (5.33) can be written as (aw* — Up)2 = k2/kx (5.37) , A A s2 f Ps~P)gß (au*-Up) =“------(5.38). lr 2 P 4 Then, equation (5.38) can be expressed as 2 - 4(^-1)^ _ kl (au* - up) (5.39) 3Cr> kl and expanding the left hand side using equation (5.31) gives: m* + u\s ]2.a2 + u2p ~[2a(u* + w'*5 )] mp} = ——— (5.40) where Cd is the drag coefficient, ß is a friction coefficient and ds is the particle size diameter. The dimensionless specific gravity of sediment particles within a fluid of density p can be defined as S =(U (5.41). 138 Chapter 5 Therefore, Equation (5.39) can be written in an equivalent form as 4« iff k2/kx (5.42) 3Ö.C,, in which, the instantaneous Shields parameter is presented here as a function of the instantaneous turbulent shear velocity (w*) and is given by a2 6 u* (5.43) (S-l)gds A where 0 is the instantaneous dimensionless shear stress (instantaneous Shields parameter in terms of instantaneous shear velocity). This dissertation has introduced a number of instantaneous parameters which are analogous to the time-averaged parameters in classical models. A comparison of these parameters is presented in the next section together with a discussion of their importance. 5.5.4 Significance of Terms in Instantaneous Model and Comparison with Time-Averaged Models As previously discussed, flow turbulence is important for the initiation of particle motion from the bed of an open channel. The importance of this is due to the fact that turbulence can agitate particles from the bed and entrain them into the flow. Of particular interest in this study is the importance of the bursting process in the entrainment of sediment particles from the bed. Consequently, instantaneous shear stresses applied to the sediment particles have been considered through incorporation 139 Chapter 5 of bursting processes in the proposed model rather than using the time-averaged stress. The importance of the variables and comparison in both models are shown in Table 5.2. Table 5.2 Comparison of variables in time-averaged and instantaneous models Time-averaged models Instantaneous model Parameter (1) (2) Shear Ü* U* = M* + U\s velocity A Particle Up Up velocity Force \CDPnf (au. uP)2- i CDP^f(au.-uP)2 = Balance ps-p)gß ^d3s( ps-p)g.ß Shields A *2 Parameter 0 - “* 0 - u* (S-\)gds (S-l)gds Drag coefficient cD = mN) = f(u.) CD = f(RN) = /(«*) li* Particle os II A u*ds Reynolds “ v Number As is shown in Table 5.2, instead of the time-averaged shear velocity presented in column 1 row 1, the instantaneous shear velocity presented in column 2 row 1 is incorporated in the new model. The above development in the modelling of the entrainment function resulted in an estimation of the instantaneous particle velocity as presented in column 2, row 2. This instantaneous particle velocity was generated from the instantaneous shear forces applied to the sediment particles due to turbulence. 140 Chapter 5 Therefore, this instantaneous shear velocity resulted in an instantaneous Shields parameter as presented in column 2 row 4, with a probability distribution function associated with it. As the Reynolds number and drag coefficient are functions of shear velocity, the inclusion of the instantaneous shear velocity resulted in an instantaneous Reynolds number and consequently a drag coefficient. The major benefits (advantages) of the proposed model are: • Inclusion of the effects of turbulence through consideration of the instantaneous applied forces in a force balance model; • that a probability density function is associated with instantaneous shear stresses and can be used to consider the temporal variability of the flow due to turbulence. An additional advantage of the proposed model is the estimation of the instantaneous particle velocity at the bed by incorporation of the instantaneous shear velocity. 5.5.5 Model Input 5.5.5.1 Drag coefficient Values of the sediment particle drag coefficient suitable for use in the model can be obtained by two methods. The first of these methods was suggested by Engelund and Fredsoe (1976) for sand particles while the second method was suggested by Morsi and Alexander (1972). In this study, the value of the drag coefficient as suggested by Engelund and Fredsoe (1976) for sand particles was used for comparison. As an 141 Chapter 5 alternative for the proposed model, it is possible to calculate the drag coefficient by a method suggested by Morsi and Alexander (1972). Using the method of Morsi and Alexander (1972) the value of the drag coefficient can be obtained as a function of a Reynolds number. CD = mN) (5.44) They introduced a series of relationships between the drag coefficient and Reynolds number. Assuming that the grains are spherical, with size equal to ds. CD — 24/Rtf Rtf < 0.1 CD = 22.13/Rn + 0.0903/R% + 3.69, 0.1 < RN < t.O CD = 29.1661/Rn + 3.8889/R^, + 1.222 , 1.0 < RN < 10 (5.45) CD = 46.5/Rn + 116.61/Rn + 0.6167, 10 < RN < 100 in which p _ u* ds (5.46) nN ~ v where R/y is the particle Reynolds number, m* is the shear velocity and v is the kinematic viscosity of the flow. 5.5.5.2 Friction coefficient The friction angle or pivoting angle (Li & Komar, 1986) or the angle of repose (Miller & Byrne, 1966; Engelund & Fredsoe, 1976) expresses the resistance to removal of the grain by the flow. Engelund and Fredsoe (1976) suggested a value for the angle of 142 Chapter 5 repose to be about 27° for sand particles. Terzaghi et al. (1996) reported a typical value for the angle of repose to be equal to 26°. Using this angle, the friction coefficient can be determined: ß = tan0 (5-47) Here, the angle of repose suggested by Engelund and Fredsoe (1976) was used to estimate the resistance coefficient using Equation (5.47). This angle was used for a comparison between the time-averaged and instantaneous force balance models. To verify the instantaneous force balance model, an angle of repose was measured for sediment particles used in this experimental study, using ISO standard procedure (ISO 4324-1977) to evaluate the angle of repose for 2 mm sand particles. This analysis resulted in an average angle of repose equal to 27.2°, which is very close to the angle reported by Engelund and Fredsoe (1976) and Terzaghi et al. (1996). This angle was used to estimate the friction coefficient in the instantaneous model. 5.5.6 Programming and Solution Method This section describes the computer program developed for the implementation of the mathematical model presented in the Section 5.5. Two interrelated computer programs were developed; one for the instantaneous model and another for a time-averaged model. The software programs were written in the FORTRAN language (Microsoft V5.1) and compiled on a PC. In addition, this was linked to Grapher® Software in order to display the results diagrammatically on a computer screen. A flow chart indicating the algorithm used to find of the numerical solution is given in Figure 5.5. 143 Chapter 5 The proposed model was solved numerically using the computer via an iterative method for two unknown parameters; these were the instantaneous particle velocity and the instantaneous shear velocity. Two sets of equations were used simultaneously to compute the two alternative cases, these were an instantaneous case and a time-averaged case. In each iteration, for the two sets of equations (instantaneous shear stress and time-averaged shear stress), particle velocities and shear velocities were computed numerically. The results of the computer program were stored in a new file. In this file, the Reynolds number, the Shields parameter, the shear velocity and the particle velocity for the two different models were stored for later reference and processing. An example of the output file is presented in Appendix E. A discussion of the results obtained from the model is presented in the next section, including a comparison with the time-averaged model. 144 Chapter 5 ^ Start ^ Read and print input data Set initial condition of the release Shear velocity Compute initial parameters of the model „^"Select drag coefficient [orsi & Alexander \ method / Method (1972) Engelund and Fredsoe Method (1976) Computation of Shields Parameter Compute instantaneous particle velocity with the influ ence of turbulence and instantaneous shear stress ^ ^Compute the time averaged particle velocity with the timi averaged shear stress (Engelund and Fredsoe (1976))^^ 1=1+1 / Output File Plot the Results /-______Results__ C STOP ^ Figure 5.5 Algorithm for the Solution of the Force Balance Model 145 Chapter 5 5.6 RESULTS AND DISCUSSIONS As mentioned earlier, from the experimental study of turbulence it was found that the magnitude of the shear stress in a sweep event is much higher than that of the time-averaged shear stress at a point in the flow and that sweep events occur more frequently than other events. The magnitude, probability of occurrence and angle of instantaneous shear velocity in the sweep event was presented in Chapter 4 and reported by Keshavarzy and Ball (1996a, 1997). The turbulence parameters (sweep shear stress, impinging angle and the probability of occurrence of a sweep event) which were derived from the experimental data in a flume over a rough bed was applied in a force balance model to calculate the instantaneous particle velocity in the mobile bed. The model was developed algebraically and was presented in Section 5.5 in detail. The developed model describes the entrainment of a sediment particle and movement over a flat bed and in steady flow conditions. The instantaneous particle velocity in the model is computed for a particular uniform particle size and under different flow conditions. For a particular flow condition, the instantaneous shear velocity is calculated and consequently gives the instantaneous Shields parameter. The model predicts instantaneous particle velocity for any flow condition. The results of the model are shown in Figure 5.6 for the instantaneous particle velocity versus the dimensionless Shields parameter. Due to the fact that no instantaneous particle velocity data could be found in the literature, the results from the instantaneous model could not be compared with 146 Chapter 5 published data. Recent developments in image processing and computer technology provided a technique for measuring and collecting instantaneous data. In this study, therefore, an effort was made to measure particle velocity using experimental and analytical facilities associated with image processing. Predicted particle velocity at the bed from the model 1 Instantaneous model, present model Cd = 0.6, ds=2mm, a = 10 Figure 5.6 Results of instantaneous particle velocity from the proposed model 5.6.1 Evaluation of Unknown Parameters in the Proposed Model The model which was developed in the previous sections was solved to evaluate the new terms incorporated in the model. The unknown parameters are; • the non-dimensional instantaneous particle velocity up/u* A ___ • the non-dimensional instantaneous Shields parameter 0/0 • the non-dimensional time-averaged particle velocity mp/m*. 147 Chapter 5 In addition to the above terms, the model calculates the Reynolds number and other necessary terms for both the instantaneous and time-averaged models. 5.6.2 Comparison of the Results from the Instantaneous and Time-Averaged Models To compare the predicted instantaneous particle velocity from the model proposed in Section 5.5, the time-averaged model was also solved to evaluate the time-averaged particle velocity. In order to compare the results, the instantaneous velocity of particles was determined for a 50% exceedance probability in the proposed model. The particle velocities were normalised using the shear velocity. The model was solved for different particle sizes to predict the initiation of motion due to the instantaneous and the time-averaged shear stresses. The magnitudes of the critical shear velocities for the instantaneous shear stress and the time-averaged shear stress for different particle sizes were compared as shown in Figure 5.7. For a particular particle diameter, it can be seen that the shear velocity for the initiation of particle motion in the instantaneous model is lower than in the time-averaged model. For different particle diameters (lmm-20mm) the differences between the critical shear velocities are shown in Figure 5.8. The normalised differences in shear stress are shown in Figure 5.9. After solving the equations, the dimensionless instantaneous velocity of particles for two different cases (with and without the influence of turbulence) was calculated. The numerical results obtained from the instantaneous and time-averaged models are shown in Figure 5.10. In this figure, it can be seen that the predicted instantaneous particle velocity is higher than that the time averaged velocity for the time-averaged conditions. 148 Chapter 5 In particular, it was predicted that particles with a diameter equal to 2 mm start to move at ^ /~v r [0/0]ur> =0.84. For the same flow conditions, with the time-averaged model, it was predicted that particles with the same size start to move with a higher shear velocity, which is equivalent to a ratio of instantaneous shear stress to time-averaged shear stress of [6/0]0-5= 1.00. Also, for a value of [6/6]05 = 1.2008, the value of mp/m*=3.3857, while the value of wp/w*=1.6723. As a result, the initiation of particle motion by an instantaneous shear stress will occur at a lower flow rate than that predicted using a time averaged shear stress. This agitation, which results in the initiation of motion, is due to the turbulent shear stress imposed on the particles on the bed. Additionally it can be seen in Figure 5.10 that for any flow condition, the instantaneous particle velocity is much higher than the time-averaged particle velocity. 5.6.3 Verification of the Proposed Model To verify results from the instantaneous model, the instantaneous velocities of the particles must be obtained. A review of the available literature revealed no available instantaneous velocity data which was suitable for verification of the results. In order to determine the instantaneous particle velocity of a particle at the bed, an image processing technique was used. Using this technique enabled the capture of a series of images of the bed and the subsequent determination of the particle velocity. Details of the image processing technique and verification of the model are presented in the next chapter of this dissertation. 149 Chapter 5 0.10 r i Ä...A A 0.09 instant ^ t time averaged model 0.08 a| A • 4 j A 1 s—' 0.07 j # J •4 A > * ♦ T 0.06 ___ LajL.... _4 I O 8 4 i > ! f 1 h-1 0.03 __ iL • ___ 1 -- _ i Sfl i> 1 0.02 A 1 j •■e i *E 0.01 i U 0.00 -Zr: d 6 8 10 12 14 Particle diameter, d (mm) Figure 5.7 Comparison of Critical shear velocity for initiation of particle motion resulting from instantaneous and time-averaged shear stress models 0.02 t 0.02...... - * O ♦ ♦ ❖ ? 0.01 -- c c 0.01 -- 0.00 -I------1------1------1------1------1 —-r -I------1------1------1------1------1------1------1------1------1------1 0 2 4 6 8 10 12 14 16 18 20 Particle diameter, d (mm) w*c = Critical shear velocity ( time averaged model ) = Critical shear velocity ( instantaneous model ) Figure 5.8 Differences in critical shear velocity for the two models 150 Chapter 5 -■ - ^ "O- ^ ^ ^ ^ ^ Particle diameter, d (mm) w*f = Critical shear velocity ( time-averaged model ) m*c = Critical shear velocity ( instantaneous model ) Figure 5.9 Normalised differences for the two models Predicted velocity fluctuation of particle at the bed 9 - ...... t...... i" “'1...... 1...... !' 6 -...... f...... i...... i...... I...... --...... i...... 1...... 1...... f...... .. !..... 1 - ...... 1..... Cd = 0.6, ds=2mm, a = 10 Figure 5.10 Comparison of particle velocity in the two different models 151 Chapter 5 5.7 SUMMARY The development of a mathematical model for describing the entrainment and movement of sediment particles on a flat bed in an open channel has been presented in this chapter. A simple instantaneous force balance model was used to estimate the initiation of motion and also the particle velocity at the channel bed. The influence of turbulence on the entrainment of sediment particles from the bed was considered through the incorporation of the bursting process, and in particular the sweep event, in a force balance model. The effect of the sweep on the entrainment of sand particles has been reported by many researchers, however little attempt has been made to predict the actual sediment transport rate under the influence of turbulence and bursting events. In this study the influence of the sweep event on the initiation of motion resulted in an indication of the variation of sediment motion with the effects of turbulence. It was found that due to the instantaneous turbulent shear stress in the sweep event, particles start to move earlier than predicted by time averaged shear stress models. It was also found that in any flow condition, after incipient motion, the instantaneous particle velocity is much higher than the time averaged particle velocity. In order to define the initiation of particle motion, it is necessary to accurately measure the initiation of motion. In this study, an image processing technique was selected in order to observe particle motion. This technique is able to measure the time of motion and also the travel distance in an of increment time. The image processing technique is explained in detail, with the results obtained, in Chapter 6. 152 Chapter 6 Chapter 6 Chapter 6 Image Analysis of Particle Motion and Model Testing I 6.1 AIMS AND OVERVIEW Despite the importance of the initiation of individual sediment particles, no general definition yet exists. One reason for this is the difficulty of observing sediment particles at the initiation of motion. Recently, attention has been focused on the use of image processing techniques for observing the motion of sediment particles. In several studies, for example those by Nelson et al. (1995), Best (1992) and Drake et al. (1988), image processing techniques were used as a tool to investigate the intermittent nature of particle entrainment over a bed. Nelson et al. (1995) and Best (1992) attempted to define the formation of ripples and dunes with the effect of turbulence and, in particular, 154 Chapter 6 the bursting process. As discussed in previous chapters, there is a difficulty in the definition of incipient particle motion over a flat bed with the influence of turbulence. The application of image processing techniques has the potential to assist in understanding the processes influencing the threshold of particle motion. Motion picture photography is uniquely capable of detailed observation, quantitative tracking, and the measurement of bed particle movement in clear water. This technique can show the entrainment of particles from the bed, the settlement of particles, the speed of particle movement, the transport mode, and the resting periods of a particle on the bed. With the capture and collection of this data, it is possible to develop a statistical description of particle entrainment. In this study, attention has been paid to define the initiation of sediment particle motion over a mobile bed with the influence of bursting processes. In order to understand this process an image processing technique was used to observe the particle movement in an instant of time over a desired area. In this chapter this observed motion of particles is characterised and a statistical description of the initiation of particle motion is presented. 6.2 DIGITAL IMAGE PROCESSING 6.2.1 Introduction to Digital Image Processing The utilisation of image processing has increased significantly in many scientific fields over recent years. Present trends indicate a continuation of the explosive growth of applications of digital image processing in the future. The application of digital image 155 Chapter 6 processing in fluid mechanics helps to understand practical problems, particularly in the study of environmental degradation problems. In the initial development stages of this technique, there were limitations in computer technology; however, with recent advances in computer software and hardware, these are now of less concern. Digital image processing, or the manipulation of images by computer is a relatively recent development in the visualisation techniques applied to fluid mechanics problems. These techniques require a wide range of hardware, software and theoretical supports as well as a broad range of knowledge in fields such as electronics, optics, mathematics, photography and computers. 6.2.2 Fundamental Elements of Digital Image Processing Digital image processing requires a computer with two special items of equipment, these are; • an image digitiser and, • an image display device in order to digitise, process and display one image. The physical image is divided into small regions called picture elements or pixels. Because computers work with numerical values rather than pictorial data, an image must be converted to a numerical form before commencing processing. This is achieved through a digitising process whereby an image is converted to a digital format and stored in a rectangular array of numbers in a file which represents a physical image. The integer number at each array 156 Chapter 6 element represents the brightness of the image at each pixel location and, hence, the brightness of the corresponding point in the actual image. A physical image and its corresponding digital image is represented in Figure 6.1. The conversion process of an image from pictorial to digital is called digitisation. At each pixel location, the brightness of the image is sampled and quantised. This process produces an integer for each pixel which represents the brightness or darkness of the image at that point. After digitisation, the image is represented by a rectangular array of integers with each integer representing a single pixel. Each pixel has an integer location or address which corresponds to the row and column number and an integer value called a grey scale. This array of digital data is representative of the original image. jjjj Physical Image Digital Image Figure 6.1 A physical image and a corresponding digital image 157 Chapter 6 A digital image is an image f(x,y) that has been discretised both in spatial coordinates and brightness. A digital image can be considered a matrix whose row and column indices identify a point in the image and whose corresponding matrix element value identifies the grey level at that point. The elements of such a digital array are called image elements, picture elements or pixels. Pixel is a commonly used abbreviation of "picture element". A representative diagram of image digitisation is shown in Figure 6.2. Figure 6.2 Digitising an image 6.3 STATISTICAL TOOLS FOR IMAGE ANALYSIS 6.3.1 Basic Concepts A digital image is a function of light intensity and the intensity value of the primary colours (red, green and blue) that are arranged to generate different levels of colour. Each small part of a black and white image has a light intensity between level 0 (Black) 158 Chapter 6 to level 255 (White). For a colour image, in addition to the light intensity, three levels of basic colour (primary colour) are required which are arranged in 3 columns for a digital image. In a digital image, the level of primary colours also varies between 0 to 255. The signal in image processing is a physical quantity of the image which varies temporary and spatially. In a digital image the signals are a series of digits or numbers which are organised in an array in a file. A signal may be continuous or discrete. In image processing the signals are in the form of a discrete signal. In order to analyse a series of sequential images spatially and temporally, it is necessary to apply statistical tools to the digital images. This section introduces some basic terms, concepts and analytical methods of digital signal processing in image processing. 6.3.2 Cross-Correlation This function determines the correlation between two signals. It is useful to determine relations between the two signals of a process. This method is used to determine relationships between two series of signals which have been collected from two separate sources. The cross-correlation function of two discrete signals/fm,rc) and g(m,n) is given by the expected value E as; Rjcy(m, n) = E[f(n,m).g(m,n)] (6.1) 159 Chapter 6 + oo +oo X X f(k' + m’1 + ^ k = — oo / = - oo Rxy(m, n) (6.2). + 00 + 00 + 00 + 00 ^ x /(*>*> X X *(fc/) k = — O0 / = —oo £ = —00 / = —00 Whereas the cross-correlation of two continuous signals yfrj and g(7) is given by /^(<*>) = + *)1 (6-3) where 0 to 1. A cross-correlation coefficient of 1 indicates a high correlation between two signals. One important application of cross-correlations is to find the spatial displacement of two sequential images. 6.3.3 Auto-Correlation Function Another useful statistical function for the analysis of a signal in image processing is the auto-correlation function. The auto-correlation function is applied to a series of signals from a source with signals from the same source with a time lag. The auto-correlation function is the relationship between some instances of a signal and the values at other instances of the same source and is given by RX( The auto-correlation function is usually used for analysis of a signal in the time domain. It gives a lag, for which two signals are best correlated. 160 Chapter 6 6.3.4 Spectral Analysis of the Signal The Fourier transform of a general sequence g(t) is given by r + 00 Gif) (6.5) where /is the frequency and the inverse Fourier transform is r + 00 git) (6.6). Thus these relations may be written as git) ~ G(f) (6.7). The spectrum of a signal is the Fourier transform of the signal (Hipel & McLeod, 1994) and the power spectrum of a signal is a function of frequency and indicates the distribution of power. The instantaneous energy of a signal g(t) is given by E(l) = lg(/)l 2 (6.8). Based on Parseval’s theorem, the total energy of a signal E is given by r + 00 E \g(t)\2dt \G(f)\2df (6.9) The spectral analysis of a signal can be performed in the frequency domain using a Fast Fourier Transform function. 161 Chapter 6 6.4 ANALYSIS OF IMAGES OF PARTICLES IN MOTION 6.4.1 Techniques Used In order to analyse the captured images, two different techniques were used in this study. These techniques were • a probability analysis of the entrained particles determined by counting the number of particles in motion at an instant. This approach was useful for obtaining a probability of exceedance of particles in motion in time, with respect to the probability of exceedance of shear stresses of the bursting process at the bed. Experimental aspects of the image processing were explained previously in Chapter 3. In the next section the application of this method will be explained in detail together with the results obtained. • application of some statistical tools to determine the displacement of particles between images. Cross-correlations and Fourier transforms were used for the convolution technique which was explained in Section 6.3. 6.4.2 The Subtraction Technique and Particle Counting 6.4.2.1 Definition The difference between two images f(x,y) and g(x,y), can be expressed as; h(x,y) = f(x,y)-g(x,y) (6.10) where h(x,y) is a new image. Two images can be compared by computing the difference between the light intensities at all pairs of corresponding pixels from image flx,y) and image g(x,y). Here, this technique was used and a sequence of images was compared 162 Chapter 6 to find the number of particles which were entrained and deposited over a specified area and in a given time increment. 6.4.2.2 Procedure As mentioned earlier, a sequence of images was captured using a high resolution CCD camera during the experimental tests and these images were recorded on tape for later processing. A more complete description of the experimental equipment was presented previously in Chapter 3. For analysis of the images, each image was digitised into an array of 384 by 288 pixels. Two different types of format were selected in the digitising process, a BMP format in colour (24 b/p) and a PGM grey scale (8 b/p) subformat. The PGM grey format was selected for its lower storage requirements and the ease of processing and file transfer between different computers. The images were digitised using a 486, 50 MHZ computer using Image Maker software driving a frame grabber in order to convert video film to a series of separate frames and hence to store them as a group of files in the computer. Initially, this software captures a sequence of frames as a large file and then, in another process, converts this file to a series of frames with the desired format. It was this series of images that were analysed to investigate the initiation of sediment motion. In this part of the study the purpose was to derive the difference between two sequential images in order to ascertain the number of particles entrained in an instant of time. To meet this aim, and in order to analyse the images, a specially written computer program 163 Chapter 6 was required to read the binary files and to process them for analysis. This program was developed in the C++ language due to its utility and capability for image processing. The flowchart and algorithm are shown in Figure 6.4. 6.4.2.3 Viewing of particles in new image In order to view the difference between two images, as an image, the subtracted light intensities must be kept between 0 and 255, where the entrained and deposited particles are denoted by black and white spots, respectively. To keep the light intensities in this range, a value of 255 must be added to the light intensities derived from the subtraction of the two images and then the sum divided by two. From this a visual comparison of images can be obtained. Shown in Table 6.1 is an illustration of how the entrained and deposited particles are represented in the produced image. This process is illustrated in Figure 6.3. Table 6.1: Strategy for the description of subtracted image Diff = Imaee II- Imaee I if Diff = positive (+) White if Diff = 0 (No change) Grey if Diff = negative (-) Black Shown in Figure 6.5 are two images with the computed difference between these two images. From this figure, it is clear how many particles were entrained or deposited in a time increment. If the two images of particles on the bed are compared carefully, the difference between the images can be observed in the produced image. In Figure 6.5a the entrained particles are marked as red spots and the deposited particles as yellow spots; good correlation between the corresponding locations of black and red spots in 164 Chapter 6 the images is apparent. For statistical analysis of the entrained particles, a sequence of 81 images was compared in each experimental run, except test Rl-Rl and test R1-R2 (200 images). Further examples are presented in Appendix F. 6.4.2.4 Particle counting A ’’manual” counting of entrained particles was preferred due to the need for interpretation to ensure accurate counting of the particles. This need for interpretation arises from the potential for particles to be agitated but not moved. In this case the movement appears as a curved shadow line in the produced image and, therefore, was not considered to be a moved particle. An example of this effect is shown in Figure 6.5. Using these derived images, the number of the entrained and deposited particles in an instant of time were manually counted. The accuracy of counting for one image was presented for 10 repeats by one person and compared with a group of people. The counting accuracy for one person was higher than that of the group due to the counting of unexperienced people. Also, the judgement and decision-making of one person is easy, and is more consistant for one image and for all repeats. In order to determine the accuracy of the manually counted number of entrained particles, several people were requested to count the number of entrained particles in an image. Statistical analysis of the independent estimates resulted in the estimated accuracy being zh 3.6%. The statistical analysis of these data are shown in Table 6.1. In order to compare the accuracy of the manually counted number of entrained particles, from several people, and repetition by one person, the number of entrained particles was 165 Chapter 6 also counted in the same image by the author. Statistical analysis of the independent estimates resulted in the estimated accuracy being zh 2%. Table 6.1 Accuracy of particles counted over a specified area Person No. of particles counted Repeats No. of particles counted 1 17 1 15 2 16 2 17 3 16 3 16 4 16 4 16 5 18 5 14 6 14 6 17 7 12 7 15 8 15 8 16 9 16 9 16 10 16 10 17 Mean=15.78, Median=16, Mode=16 Mean=15.9, Median=16, Mode=16 Standard Error of Estimation = 0.58 Standard Error of Estimation = 0.31 Standard Deviation = 1.74 Standard Deviation = 0.994 Confidence T/imit (95%) =114 Confidence Limit (95%) = 0.616 diff=+ve white Figure 6.3 A schematic illustration of image differentiation 166 Chapter 6 Figure 6.4 Computer Programming Algorithm for image subtraction 167 Chapter 6 Figure 6.5 a) Two images with their difference; b) Sequence of images showing particle movement 168 Chapter 6 6.4.3 Convolution and Cross-Correlation Technique of Images A two dimensional convolution of two images f(x,y) and g(x,y) is defined as /(*, y) * g{X, y) <*> F{u, v)G(w, v) (6.11) where F(u,v) is the Fourier transform of function f(x,y) and G(u,v) is the Fourier transform of the function g(x,y). The importance of convolution in frequency domain analysis lies in the fact that if f(x) *g(x) and F(u)G{u) constitute a Fourier transform pair then, ßx)*g(x) is the Fourier transform of F(u)G(u) (Gonzalez and Woods, 1992). Equation (6.11) indicates that the convolution in the* andy domain can be obtained also by taking the inverse Fourier transform of the product F(u)G(u). fix, y)gix, y) <=> F(u, v) * G(w, v) (6.12). Using a cross-correlation technique [using Equation (6.1) and (6.2)] between two images, the displacement of the two images can be found. The cross-correlation algorithm in the time domain involves a long time for the computer to process the images. Hence, this technique is not suitable for real-time processing of the images but rather for post-processing of recorded images. In addition, this method is useful only for small image sizes e.g. 32 X 32 or 64 X 64 pixels. In this study, this method was mostly used to estimate the displacement of particles. A frame size of 64x64 was selected from the centre of the flume with a frequency of 25 frames per second. An example of the application of the spatial correlation technique to two artificial images is shown in Figure 6.6. Consider a sediment particle on the bed of an open 169 Chapter 6 channel tagged at time t\ in grid coordinate 10, 10 (x=10, y=10) in Figure 6.6a. The tagged particle at time t2 is in a new position, 20, 20 (x=20, y=20) in Figure 6.6b. The tagged particle is displaced therefore by a distance of dx=10 and dy=10 at At=t2~ti. The spatial correlation coefficientin equation 6.7, between the light intensity fields of image one (Ij) and the light intensity of image two (I2), as a function of pixel displacement was computed using the cross-correlation technique. The location of the peak in defines the displacement vector (Figure 6.6c). In addition to the simple cross-correlation, in order to simplify and enable a quick computational process, a method described by Willert and Gharib (1991) was also used with some modifications. In this method a Fast Fourier Transform is used to convert the time domain signals of images to the frequency domain. Applying cross-correlation to the spectral energy of two images gives a new two-dimensional matrix, but in the frequency domain. Using an Inverse Fast Fourier Transform gives the displacement of the image for an increment of time. By knowing the displacement and time increment between two sequential images, the velocity can be calculated simply. This method is commonly used for the velocity detection of particles (Adrian, 1991). The process is fast and needs little computer time, however careful attention is necessary in the frequency domain. The results from both methods were similar for similar images. Shown in Figure 6.7 is the algorithm for the peak detection and velocity detection techniques. 170 Chapter 6 Figure 6.6 An example of the application of the spatial correlation technique 171 Chapter 6 - Sequential images Image I Image II Image HI -Difference between images; Diff H-I Diff III-II image I, II and III as input j\m,n) g(m,n) - Calculating 2D FFT for images I and II Cross-Correlation - Cross-correlation in spatial (frequency domain) frequency domain Cross-Correlation -Cross-correlation in spatial (time domain) time domain - Inverse FFT - Peak detection for displacement) - Velocity calculation Figure 6.7 Flow chart of numerical processing in frequency and time domain for displacement and velocity detection using cross-correlation 172 Chapter 6 6.5 RESULTS AND DISCUSSION 6.5.1 Relationship Between Number of Particles in Motion and Instantaneous Shear Stresses in Sweep Events The entrainment of particles from a mobile bed in an open channel flow has been investigated in several studies, for example by Einstein and Li (1958), where it was pointed out that this process is stochastic in nature due to the effect of turbulence. The number of entrained particles over a specified area will vary with time. The entrained particles at any time depends on the instantaneous turbulent shear stress arising from the velocity fluctuations and the instantaneous shear stresses at the bed. Shown in Figure 6.8 are the velocity fluctuations of the flow with time and the corresponding instantaneous shear stress at the bed. Also shown in Figure 6.8 is the number of particles in motion, at an instant of time, and consequently how the entrained number of particles varies with time. The entrainment process can be defined by considering the instantaneous shear stress in sweep events and also the instantaneous number of particles in motion. In a study of sediment entrainment from the bed, Williams et al. (1989) and Nelson et al. (1995) investigated and found a high correlation between the streamwise velocity component and the sediment flux. Additionally, Nelson et al. (1995) found that the transport rate tends to be higher when the vertical velocity and the Reynolds momentum flux are angled towards the bed. They found the best correlation between the sediment flux and the streamwise velocity component to occur with a lag of 0.1 seconds and, consequently, 173 Chapter 6 they suggested a measuring frequency of 10 Hz to get the best results. A frequency rate of 10 Hz was thus selected for this investigation. -100- -150- 2000- 1000- -1000- -2000- Time (Sec) Figure 6.8 An example of flow velocity fluctuations in the horizontal and vertical directions, with respect to shear stresses and particles in motion (Test R1-R2) In a sweep event the velocity of the flow is directed in a streamwise direction and towards the bed. Therefore, the sweep event applies forces to the particles resting on the bed. As discussed in Chapter 4, the average angle of the velocity vector in a sweep 174 Chapter 6 event is 22° from the horizontal towards the bed. As a result, consideration of only the streamwise flow velocity is not sufficient to define the intermittent nature of the entrainment process. Rather, it is necessary to examine the correlation of instantaneous shear stresses in sweep events and particle entrainment from the bed. In this part of the study, the major intention is to investigate the effect of the instantaneous shear stresses during sweep events on the particle motion. The above relationship was investigated using a cross-correlation analysis between instantaneous shear stresses in sweep events and the number of particles entrained in a time increment. The number of entrained particles were counted in a sequence of produced images derived from the subtraction of sequential recorded images ( see Figures 6.9a and 6.9b ). The dimensionless shear stress in sweep events was computed also from a time series of the velocity fluctuations which was recorded simultaneously with the recording of the images. A cross-correlation analysis was undertaken between the number of entrained particles and the instantaneous shear stress in a sweep event. A good correlation was found between the number of particles in motion and the instantaneous shear stress in a sweep event. Shown in Figures 6.10a and 6.10b are two examples of the results determined from the cross-correlation analysis between the instantaneous shear stress in sweep events and the number of particles induced into motion. Similar results from other tests are shown in Appendix H. A consideration of these results, shows that to the nearest 0.1s, no significant lag exists. As a result, the particle entrainment at the initiation of motion is highly correlated with sweep shear stress. 175 Number of particle in motion in an increment of time(0.1 sec) Figure 6.9a: Entrained particles Time 176 Test Test Test in (sec) different R5-R1 R5-R2 R5-R6 experimental MA tests a AW Chapter 6 Chapter 6 30- Test R7-R1 25- 20 15- 10 5- xy^~V /-\ /\aA~ 0 I I I I I I I I 30 Test R7-R2 25- 20 15 10 5 0 40 Test R7-R3 30 20 10 0 30 Test R7-R4 25 20 15 10 5 0 30 Test R7-R5 25 20 15 10 5 Time (sec) Figure 6.9b: Entrained particles in different experimental tests 177 Chapter 6 e .2 t3 o 1/3Y oC/3 uUi Lag O (sec) Figure 6.10a Cross-correlation of instantaneous shear stress in sweep events and number of particles entrained from the bed (Test R5-R4) In sweep events, the velocity fluctuations are located in quadrant IV (downwards from the horizontal direction of flow), with an average angle of 0=22°, see Figure 6.11, and therefore the contribution of positive streamwise flow velocity and negative vertical flow velocity to the sweep shear stress is high. The results obtained in this study are in agreement with the results obtained by Williams et al. (1989) and Nelson et al. (1995). The relationship obtained in this study between instantaneous shear stress in a sweep and the entrained particle numbers shows no significant lag between the instantaneous shear stress in a sweep event and particle entrainment. 178 Chapter 6 Figure 6.10b Cross-correlation of instantaneous shear stress in a sweep event and number of particle entrained from the bed (Test R7-R5) Flow direction JJJ Figure 6.11 Inclination of flow velocity of the sweep event 179 Chapter 6 6.5.2 Probability Analysis of the Number of Entrained Particles and the Instantaneous Shear Stress The entrainment of a single particle from a mobile bed is very complicated due to the intermittent nature of the motion. The most appropriate way to investigate this is to use the probability density function of entrained particles and the relative instantaneous shear stress in sweep events. In this study, the relationship between the instantaneous shear stresses in sweep events and the number of particles in motion was investigated using a probability distribution analysis. The probability density function was adopted as an appropriate approach for the investigation of the contribution of sweep events to the entrainment process. This was pointed out by Nakagawa and Nezu (1978), who suggested that the consideration of a bursting event must be treated in the form of a probability distribution function. The percentage of the area which was eroded from a defined area of the bed was investigated with respect to the instantaneous turbulent shear stress of bursting events. It was assumed that the observation area occupied by particles of an arbitrary shape is proportional to the mean diameter of the size fraction. In order to compute the fraction of area entrained, the number of particles entrained was counted and compared with the total disturbed area. For this case the D50 of the particles resting on the bed was 2 mm and the observation area was 25 square centimetres. Thus, each particle has an area fraction of (a/A), where a is the area of the particle and is defined as a=jtd2/4, and A=L2 is the sample area of the bed. The frequency distribution of the entrained particles was 180 Chapter 6 calculated for the experimental data and is shown as Figure 6.12 with the normal probability plot given in Figure 6.13. The cumulative probability of the area entrained (percentage of area entrained) from the bed was calculated and is shown in Figure 6.14. The cumulative probability of the instantaneous shear stress was also calculated for the same experimental period and is shown in Figure 6.15. Therefore, for the same cumulative probability, the magnitude of instantaneous shear stress and respective entrained area were derived and correlated. This comparison is shown as Figure 6.16 in which the area entrained is shown to correlate well with the instantaneous shear stress in sweep events. This relationship was also confirmed by cross-correlation as shown in Figures 6.9 and 6.10. Regression equations for these relationships are shown also in Figure 6.16. 181 Chapter 6 Frequency Distribution of Area Entrained (Test R5-R4) 30 - 20 - 10 - a/A(%) Figure 6.12 Frequency distribution of the entrained area of a mobile bed Normal Probability Plot Area Entrained (Test R5-R4) .999 - .99 - .80 - .50 - .05 - a/A(%) Average: 1.8232 Anderson-Darling Normality Test Std Dev: 0.745289 A-Squared: 0.967 N of data: 200 p-value: 0.015 Figure 6.13 Normal probability plot of area entrained 182 Chapter 6 Frequency Distribution of Area Entrained (Test R5-R4) 100 — ■4—» CD O cd Q_ CD > jfl E o■3 o -...... •- 0.0 0.5 1.0 1.5 2.0 2.5 a/A(%) Figure 6.14 Cumulative probability plot of particles entrained from the bed Frequency Distribution of Shear Stress in Sweep Event (Test R5-R4) 100 — Z3 E3 o o —- -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 B-C(t/T) Figure 6.15 Cumulative probability of instantaneous shear stress in a sweep event 183 Chapter 6 Figure 6.16 Relationship between area entrained and the instantaneous shear stress in sweep event 6.5.3 Modification to the Shields Diagram Gessler (1971) pointed out that the Shields diagram is inconvenient to use and suggested a modification to the Shields diagram. He stated that the probabilistic nature of the entrainment process depends only on the statistical nature of turbulence. In Gessler’s approach, attention was paid to deriving the probability that grains of a specific size would remain as an armour coat material. He assumed that fluctuations of the shear stress about the mean are normally distributed, and that if the value of mean shear stress at the bed is equal to the critical shear stress for a specific grain size, the probability of motion is 50%. However, some recent studies (for example, Thome et al., (1989)) concluded that the bursting process, and specifically the sweep event, was the only effective mechanism for producing a high velocity flow towards the bed which results 184 Chapter 6 in erosion. As shown in Chapter 4, the effective shear stress or sweep shear stress is not normally distributed. Therefore, even though Gessler’s work was valuable at the time, recent findings suggest that investigations of the bursting processes present a better approach. Other studies such as those by Grass (1971) and Mantz (1977) suggest some extensions to the Shields diagram at a low Reynolds number. Mantz (1977) used some collected experimental data for fine cohesionless grains (e.g. 15, 30, 45, and 66 pm) and used regression analysis of the collected data. Mantz’s extension of the Shields diagram was for a low particle Reynolds number (Rn<10). In their extensions it was shown that the number of particles entrained was lower than that predicted by the Shields diagram. In this section, an effort was made to make some additional modifications to the Shields diagram through a probability distribution function. Firstly, an investigation was carried out on the turbulent characteristics of the flow and bursting processes to get a probability density function for intermittent instantaneous shear stresses. Also, an investigation was made of intermittent particle entrainment from the bed and a probability distribution was derived. These simultaneous probabilities were used to modify the Shields diagram through the addition of the probability density function of particle motion. The probability distribution of instantaneous shear stresses in sweep events was investigated using a statistical analysis of the data. The procedures used were presented and discussed in detail in Chapter 4. 185 Chapter 6 A probability distribution function is associated with shear stresses applied to the bed during the sweep event. The probability for a level of shear stress can be obtained using an exceedance probability plot of the instantaneous shear stresses. For each experimental run and at a specified time, the instantaneous shear stress for various exceedance probabilities (from 0% to 100%) were obtained. In order to incorporate the probability function of the instantaneous shear stress into the Shields diagram, the shear stress for 50% exceedance probability was selected and computed. A statistical analysis of the experimentally observed particle entrainments from the bed was undertaken and an exceedance probability distribution obtained. As a result, for each exceedance probability between 0% and 100%, the percentage of the area entrained was obtained. In order to apply the instantaneous shear stresses and entrained area (percentage) to the Shields diagram, a 50% exceedance probability was selected and employed. These results were applied to the Shields diagram to add information relevant to the initiation of motion. This modification to the Shields diagram enables the entrainment of particles with a defined probability of occurrence to be obtained from the Shields diagram. As shown in Figure 6.17, at each point for the 2 mm particle size, there is a probability associated with the entrainment process. The associated probability density function at each point varies from 0% to 100%. As an example, the percentage of entrained area at the bed shown in this figure is derived from the 50% exceedance probability of entrained particle within time increment between images. The entrainment of particles 186 Chapter 6 also varies from zero percent to 100% in a line on the Shields diagram which is depicted as a dashed line for 50% exceedance probability in this figure. The above modification to the Shields diagram enables estimation of the initiation of particle motion for a desired probability and particle size. dsa = 2mm [ Shields Curve caption ** percentage of area entrained Figure 6.17 Modified Shields diagram Many previous studies, for example by Gessler (1971), Van Rijn (1993) and Graf and Pazis (1977), presented qualitative and quantitative definitions and evaluations of the Shields diagram based on assumptions or experimental data. Van Rijn (1993) presented some qualitative definitions for the transport of sediment stage and compared these with the Shields diagram. Shown in Figure 6.18 is a representation of Van Rijn’s (1993) 187 Chapter 6 definitions. The experimental results obtained from this study for 2mm particles are also shown in Figure 6.18 for comparison. Van Rijn (1993) defined seven stages for sediment transport as presented in Figure 6.18. In Van Rijn‘s (1993) definitions, the Shields curve represents permanent grain movement at all locations of the bed. Additionally, stage five is defined as ‘frequent particle movement at all locations’. However, the results obtained from this study and also those of Petroff (1993), which are shown in Figures 6.18 and 6.19, indicate that Van Rijn’s definition does not agree with the results. The results obtained from this study for a 2 mm particle size and from Petroff (1993) for a 4.8 mm particle size agree in terms of the expected percentage area in motion. In order to modify the Shields diagram, a series of dashed lines were drawn in Figure 6.19 to predict the percentage of area entrained below the Shields curve. It is apparent from Figure 6.19 that close to the Shields curve the percentage of area entrained is about 10 percent, with an exceedance probability of 50%. This contradicts the assumption made by Gessler (1971) that assumed 50% of particles are in motion on Shields curve. Hence, predictions using Gessler’s definition will be excessively conservative. The numbers shown in Figure 6.19 are the percentages of entrained area with an exceedance probability of 50%. 188 Chapter 6 d50 = 2 mm d50 = 4.8 mm Percentage of movement in area Note: Van Rijn’s definition (1993) Present study ■ Petroff 119931 •- 1-occasional particle movement at some locations ffl 0.3% IS 7.0% 2-frequent particle movement at some locations 3.0% m 9% 3-frequent particle movement at many locations 5.6% IS 12.1% 4- frequent particle movement at nearly all locations m 8.0% 5- frequent particle movement at all locations 6-permanent particle movement at all locations o 14.1% 7-general transport (initiation of ripples) Figure 6.18 Comparison of present data with Van Rijn’s definition for particle movement 4.8mm percentage of area entrained Figure 6.19 Comparison of present data with the Shields curve for particle movement 189 Chapter 6 6.5.4 Displacement of Particles and Velocity Detection In order to find the average displacement of particles in a mobile bed, the convolution technique and a cross-correlation method were used together. A sequence of images was selected for processing. Using the subtraction method explained in Section 6.3.1 and the algorithm shown in Figure 6.3, the differences between the images were computed. Thus a series of secondary images (differences between the sequences of images) was produced. By application of a cross-correlation technique to the captured images, the correlation peak was found to move away from the origin by the average spatial displacement of particles in that region. A high cross-correlation coefficient i.e. close to 1, is observed where many particle images match up with their corresponding spatially shifted partners. Shown in Figures 6.20 and 6.21 is a graphical illustration of the cross-correlation and peak detection technique between two images. The cross-correlation between these images gives a large spike displaced from the origin and gives the displacement or spatial shift of particle images from one region to the next. The highest correlation peak is considered to represent the best match of particle images between two images. As the result of this procedure, the instantaneous velocity fluctuation of particles can be obtained over a time increment. 190 Chapter 6 Velocity Detection of Partcle(im6-im7) PIXELS 0 0 PIXELS Figure 6.20 Using FFT and cross-correlation for peak detection technique o 0.9 ^ PIXELS PIXELS Figure 6.21 Using FFT and cross-correlation for peak detection technique 191 Chapter 6 6.5.5 Verification of the Model Using Experimental Data for the Particle Velocity Most of the proposed models for the entrainment of sediment particles from the bed are based on the time averaged shear stress at the bed. As discussed previously, a consideration of time averaged shear stresses for the entrainment of sediment particles is not sufficient to define the initiation of motion due to the intermittent nature of turbulent flow. In order to define particle entrainment from the bed, a model was proposed in Chapter 5. This model is based on a physical understanding of a bursting event and, in particular, the sweep event. In order to validate this model, a series of experimental tests were conducted to measure the instantaneous velocity fluctuation of sediment particles. These tests were carried out in the flume described in Chapter 3. The intermittent nature of sediment motion was observed with an image processing technique, together with a mathematical analysis. At the same time, the turbulent characteristics of the flow were also measured. An analysis of the turbulent velocity fluctuations was carried out to determine a probability distribution of the shear stresses in the sweep event. Exceedance probabilities were able to be derived for given values of shear stress. A statistical analysis was also applied to the velocity fluctuations of particles, and particle velocities for different exceedance probabilities were derived. 192 Chapter 6 The data from this technique was applied to the model presented in Chapter 4 and the results from the model are shown in Figure 6.22. As seen in Figure 6.22, the experimental data of particle velocities fits well with the results computed from the model. This model is novel in nature because in addition to the inclusion of the bursting process in the model, an associated probability distribution function for instantaneous shear stress and entrained are are also incorporated. A comparison of observed and predicted values of the particle velocities shown in Figure 6.23 also indicates a good agreement between the results of the model and the data from the experimental tests. The correlation coefficient between the observed and predicted value was 0.997. As shown in Figure 6.24, the standard error of estimation for the predicted particle velocity is 95% confidence interval, when compared to the measured particle velocity. 193 Chapter 6 Figure 6.22 Verification of proposed model and comparison with previous model (for 50% exceedance probability) 194 SEE ere* Observed Standard Figure Observed 6.24 Error Residual of Estimation and predicted value Predicted of of predicted 195 values the predicted of value particle particle of velocity particle velocities r= velocity 0.977 Chapter 6 Chapter 6 6.6 SUMMARY AND CONCLUSIONS The influence of instantaneous turbulent shear stresses on the entrainment of sediment particles was investigated using an image processing technique. In addition to the measurement of instantaneous turbulent shear stresses in sweep events, the movement of particles over a mobile bed were observed and analysed using an image processing technique. In order to find the number of particles in motion in an instant of time over a mobile bed, a subtraction technique was used to get the differences between sequences of images. In this method, a series of secondary images were produced, which contained the differences between images. By counting the number of entrained particles, the percentage of area eroded in an instant of time was obtained. A high correlation between the area entrained and the instantaneous shear stress was found. Additionally, applying a convolution technique and a cross-correlation tool to the sequence of images, the displacement of particles over the mobile bed was obtained. To apply these techniques to the images, a numerical procedure was used. In this numerical algorithm, the image data were converted from the time domain to the frequency domain using a two-dimensional Fast Fourier transform. The cross-correlation was carried out upon the data in the frequency domain and finally an inverse Fourier transform was applied to the data to convert it back to the time domain. With this technique, the displacement of particles from the origin was found by obtaining the peak cross-correlation of the motion. The peak cross-correlation was shown as a spike on the image pixels. By counting the number of pixels associated with 196 Chapter 6 the displacement, calibration was possible with the distance of movement. By knowing the time between images, the instantaneous velocities of the particles were obtained. The experimental data obtained for the velocities of the moving particles was used to verify the stochastic-deterministic model presented in Chapter 5. Good agreement between the predicted and observed particle velocities was obtained. Finally, using an exceedance probability for the area entrained and the instantaneous shear stress close to the bed, a modification to the Shields diagram was proposed. This modification to the Shields diagram (see Figure 6.19) enables the estimation of particle motion for a desired probability of shear stress and particle size. This modification was carried out for 2mm and 4.8mm particle sizes. 197 Chapter 7 198 Chapter 7 Chapter 7 Results and Conclusions The motion of sediment particles and, particularly, the initiation of that motion is an important component of many water quality problems. In the study reported herein, the entrainment of sediment particles from a movable bed into the flow was investigated under the influence of flow turbulence. This investigation was based on an analysis of the monitored characteristics of flows turbulence during steady flow in a laboratory flume with a movable bed. Details of the experimental flume were presented in Chapter 3 while the statistical analysis of the flow turbulence was discussed in Chapter 4. As presented in Chapter 4, the statistical analysis of the experimental data resulted in the following conclusions. • The magnitude of the mean shear stress in each quadrant (sweep, ejection, outward and inward interactions) was different from the overall temporal mean shear stress at a defined depth within the flow. 199 Chapter 7 • For quadrant IV or sweep events, the mean shear stress was approximately 148% of the overall temporal mean shear stress in the region near the bed to within 95% confidence limits and increased to approximately 190% at d/H=0.50. • In quadrant II events, the shear stress ratio was about 135% near the bed to within 95% confidence limits and increased to approximately 200% at d/H=0.55. • The shear stress ratios of the quadrant I and III events were low compared to quadrant IV and II events. • The frequency of each event was determined also. It was found that, near the bed, the frequency of quadrant IV and II events was approximately 30%, whereas that of quadrant I and III events was only 20%. Consequently, the events in quadrants I and HI occurred 30% less often than events in quadrants II and IV. • The average angle of action for events in quadrants II and IV were found to vary with depth from the bed of the channel to the free water surface. Close to the bed, the mean angle of sweep events was approximately 22°, while the mean angle increased to nearly 35° atd/H=0.50. The mean angle of events in quadrant II was approximately 18° close to the bed, and to nearly 32° at d/H=0.50. The implication of these results is that the instantaneous forces applied to a sediment particle are higher than those suggested through application of time-averaged shear stress such as those used in the development of the Shields diagram. Consequently, it is suggested that there is a defined probability that sediment particles will be induced 200 Chapter 7 into motion at lower flow rates than those obtained from a Shields diagram. This motion, however, need not be continuous. Sediment particles, as discussed in Chapter 5, are induced into motion when agitating forces exceed the resistance forces. Analysis of the flow turbulence resulted in the de velopment of exceedance probabilities for the magnitude of the instantaneous shear stresses applied to the sediment particles. As a consequence, the force balance model used to define particle motion in this study incorporated the statistics of the flow turbu lence. The influence of turbulence on the entrainment of a sediment particle from the bed into the flow was investigated through an analysis of the bursting processes and, in particular, the sweep event as applied in the stochastic-deterministic force balance model. The model was solved numerically for two cases, instantaneous and time averaged shear stresses, simultaneously. The instantaneous and time averaged particle velocities in two cases were computed for comparison. It was found that for a particular particle diameter, the initiation of particle motion under the influence of the instantaneous shear velocity would occur with lower flow rates than predicted using the time averaged model. An inherent assumption in the instantaneous force balance model is that sweep events are the primary process inducing particles into motion. The validity of this assumption was tested by comparing the statistics of particles in motion and instantaneous shear stresses in sweep events. An image processing technique was used to observe particle 201 Chapter 7 motion and to develop the necessary statistical data. In order to find the number of particles in motion at any instant in time, over a mobile bed, a subtraction technique was used to get the differences between a series of sequential video images. With this method a series of secondary images were produced containing the differences between images; these differences were due to particles moving from one location to another within the observation area. By counting the number of entrained particles the percentage of area eroded in an increment of time was obtained. Good agreement was found between the statistics of the area entrained and the instantaneous shear stresses in the sweep event with significant correlation coefficients. The statistical relationships considered were; • cross-correlation of the instantaneous shear stresses in sweep events with the area in motion, from which it was concluded that there was no significant lag. • exceedance probabilities of the area in motion and the instantaneous shear stresses in sweep events from which it was concluded that the exceedance probabilities of the entrained area of sediment was highly correlated with the exceedance probability of the instantaneous shear stresses in sweep events. Additionally, using a convolution technique and a cross-correlation tool with a sequence of images, the displacements of particles over the mobile bed were obtained. From analysis of these displacements, the instantaneous velocities of particles in a time increment (0.1 s) was obtained. Using these experimental data, the particle velocities were determined and compared with those predicted by the stochastic-deterministic 202 Chapter 7 force balance model. As shown in Figures 6.22 to 6.24, for a 50% exceedance probability, the predicted and measured particle velocities were found to be within the 95% confidence limits . Finally, using the exceedance probability of the entrained area from the image processing and the measured turbulent shear stresses near the bed within the flow, a modification to Shields diagram was proposed. This modification, as shown in Figure 6.17-6.19, indicates the probability of a particle being induced into motion. The present study was focused on defining the initiation of particle motion with the influence of bursting processes and in particular sweep events included. The study was carried out over a flat mobile bed with a uniform grain size distribution. Consequently, there are some limitations to this study and the conclusions developed. Effects which were not studied are variable sediment sizes, mobile beds with bed forms such as ripples and dunes, and different particle densities. The study of these items was considered to be beyond the scope of the present study but are natural extension of it and certainly warrant further investigations. 203 References REFERENCES Abbott, J.E. and Francis, J.D.R., 1977. Saltation and suspension trajectories of solid grains in a water stream. Phil. Trans, of the Royal Society of London, Series A, 284(1321):225-254. Ackers, P. and White, W.R., 1973. Sediment transport: New approach and analysis. ASCE, Journal of Hydraulics Division, Vol. 99, HY11. Ackers, R, 1991. The Ackers-White sediment transport function for open channel flow: A review and update. Technical Note, HR, Wallingford, UK. Anwar, H.O. and Atkins, R., 1982. Turbulence structure in an open channel flow. Euromech 156: Mechanics of sediment transport, Istanbul, pp. 19-25. Adrian, R.J., 1991. Particle-imaging techniques for experimental fluid mechanics. Annual Reviews Fluid Mechanics, 23:261-304. Bagnold, R. A., 1954. Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. of the Royal Society of London, Series A, 225:46-63. Bagnold, R.A., 1956. The flow of cohesionless grains in fluids. Phil. Trans, of the Royal Society of London, Series A, No. 249:291-293. Bagnold, R.A., 1966. An approach to the sediment transport problem from general physics. US Geological Survey, Prof. Paper, 422-1. Bagnold, R.A., 1973. The nature of saltation and bed-load transport in water. Proc. of the Royal Society of London, Series A, 332:473-504. Bagnold, R.A., 1979. Sediment transport by wind and water. Nordic Hydrology, 10:309-322. Ball, J.E. and Keshavarzy, A., 1995. Discussion on incipient sediment motion on non-horizontal slopes by Chiew and Parker. Journal of Hydraulics Research, LAHR, 33(5):723-724. 204 References Batchelor, G.K., 1965. The motion of small particles in turbulent flow. Proc. of2ndInt. Conf. on Hydraulics and Fluid Mechanics. University of Auckland, New Zealand Bauer, W.J., (1953). Turbulent boundary layer on steep slopes. Proceeding of American Society of Civil Engineers, Transactions, Paper No. 2719, pp 1212-1242. Bayazit, M, 1976. Free surface flow in a channel of large relative roughness. Journal of Hydraulics Research, 14(2): 115-126. Bennett, S.J. and Best, J., 1995. Mean flow and turbulence structure over fixed, two dimensional dunes: Implications for sediment transport and bed form stability. Sedimentology, 42:491-513. Best, J., 1992. On the entrainment of sediment and initiation of bed defects: insights from recent developments within turbulent boundary layer research. Sedimentology, 39:797-811. Best, J., 1996. The fluid dynamics of small-scale alluvial bedforms. In Advances in fluvial dynamics and stratigraphy, edited by Carling, P.A., and Dawson, M.R., John Wiley & Sons, Chichester. Billi, P, Hey, R.D., Thome, C.R. and Tacconi, R, 1992. Dynamics of gravel bed rivers. Chichester, John Wiley and Sons. Blinco, P.H. and Partheniades, E., 1971. Turbulence characteristics in free surface flows over smooth and rough boundaries. Journal of Hydraulics Research, IAHR, 9(1):43-71. Blinco, P.H. and Simon, D.B., 1974. Characteristics of turbulent boundary shear stress. ASCE, Journal of Eng. Mech. Div., 100:203-220. Blinco, P.H., Mahmood, K., and Simon, D.B., 1973. Stochastic structure of the turbulent boundary shear stress process. Proc. of 15th Congress of International Association Hydraulics Research, Vol. 1, pp. 371-381. Bowden, K.F. and Fairbum, L.A., 1956. Measurement of turbulent fluctuations and Reynolds stresses in a tidal current. Proc. of the Royal Society of London, A., 237. 205 References Box, G.E.P. and Cox, D.R., 1964. An analysis of transformation. Journal of the Royal Statistical Society, Series B, 26:211-252. Bridge, J.S. and Bennett, S.J., 1992. A model for the entrainment and transport of sediment grains of mixed sizes, shapes and densities. Water Resources Research, 28(2):337-363. Bridge, J.S., 1981. Hydraulics interpretation of grain-size distributions using a physical model for bed load transport. Journal of Sediment Petrology, 51(4): 1109-1124. Carling, P. A., and Dawson, M.R., 1996. Advances in fluvial dynamics and stratigraphy. John Wiley & Sons, Chichester. Cheng, E.D.H. and Clyde, C.G., 1972. Instantaneous hydrodynamic lift and drag forces on large roughness elements in turbulent open channel flow. In Sedimentation, edited by Shen H.W., Water Resources Publications, Littleton, Colo., pp. 3-1 to 3-20. Cheong, H.F. and Shen, H.W., 1976. On the motion of sand particles in an alluvial channel. Hydraulic Problems Solved by Stochastic Methods, Proc. 2nd Int. IAHR Symp. on Stochastic Hydraulics, Sweden, edited by Hjorth R, JÖnsson L. and Larsen P, Water Resources Publications, Littleton, Colo.. Chepil, W.S., 1958. The use of evenly spaced hemispheres to evaluate aerodynamic forces on a soil surface. Transactions, American Geophysical Union, Vol. 39, No. 3. Chiew, Y. M. and Parker, G., 1994. Incipient sediment motion on non-horizontal slopes. Journal of Hydraulics Research, IAHR, 32(5):649-660. Chow, Ven Te, 1959. Open channel hydraulics. McGraw-Hill Book Company. Christensen, B.A., 1972. Incipient motion of cohesionless channel banks. In Sedimentation, edited by H. W. Shen, Water Resource Publications, Littleton, Colo., pp. 4-1 to 4—22. Clifford, N.J., French, J.R. and Hardisty, J., 1993. Turbulence; perspectives on flow and sediment transport. John Wiley & Sons, Chichester. 206 References Coleman, N.L., 1967. Bed particle Reynolds modelling for fluid drag. Journal of Hydraulics Research, IAHR, 17(2):91—106. Coleman, S.E., and Melville, B.W., 1996. Initiation of bed forms on a flat sand bed. Journal of Hydraulic Engineering, 122(6):301—310. Corino, E.R. and Brodkey, R.S., 1969. A visual investigation of the wall region in turbulent flow. Journal of Fluid Mechanics, 37:1-30. Drake, T.G., Shreve, R.L., Dietrich, W.E., Whiting, RJ. and Leopold, L.B., 1988. Bed load transport of fine gravel observed by motion-picture photography. Journal of Fluid Mechanics, 192:193-217. Egiazaroff, I.V., 1957. L’equation generale du transport des alluvions non cohesives par un courent fluide. Paper No. D43, Proc. of the 7th Congress of International Association of Hydraulics Research, Lisbon, Portugal, pp. 1-10. Egiazaroff, I.V., 1965. Calculation of non-uniform sediment concentrations. Journal of Hydraulics Division, ASCE, 91(HY4):225-247. Einstein, H.A., 1937. Bed load transport as a probability problem. D.S. Thesis, Federal Inst of Tech., Zurich, Switzerland. Einstein, H.A., 1942. Formula for the transportation of bed load. Transactions, ASCE, Paper No. 2140, pp. 133-149. Einstein, H.A. and El-Samni, E. A., 1949. Hydrodynamics forces on a rough wall. Review of Modem Physics, 21:520-524. Einstein, H.A., 1950. The bed-load function for sediment transport in open channel flows. U.S. Dept, of Agriculture, Techn. Bulletin, Paper No. 1026. Einstein, H.A. and Li, H., 1958. The viscous sublayer along a smooth boundary. Transactions, ASCE, Vol. 123, Paper No. 2992, pp. 293-313. Engelund, E, and Hansen, E., 1967. A monograph on sediment transport in alluvial streams. Teknisk Vorlag, Copenhagen, Denmark. 207 References Engelund, F.„ 1970. A note on the mechanics of sediment suspension. Basic Research Program, Report 21, Hydraulic Lab., Tech. Univ. Denmark, pp. 7-14. Engelund, F. and Fredsoe, J., 1976. A sediment transport model for straight alluvial channels. Nordic Hydrology, 7(5):293-306. Fenton, J.D. and Abbott, J.E., 1977. Initial movement of grains on a stream bed: the effect of relative protrusion. Proc. of the Royal Society of London, Series A, 352:523-537. Fernandez Luque, R. and Van Beek, R., 1976. Erosion and transport of bed-load sediment. Journal of Hydraulics Research, 14(2): 127-144. Francis, J.D.R., 1973. Experiments on the motion of solitary grains along the bed of a water-stream. Proc. of the Royal Society of London, Series A, 332:443-471. Garcia, M., Lopez, F. and Nino, Y., 1995. Characterization of near-bed coherent structures in turbulent open channel flow using synchronized high-speed video and hot-film measurements. Experiments in Fluids, 19:16-28. Garde, R.J. and Ranga Raju, K.G., 1977. Mechanics of sediment transportation and alluvial stream problems. John Wiley and Sons, New York. Gessler, J., 1965. The beginning of bed load movement of mixture investigated as natural armoring in channels. Report No. 69, Laboratory of Hydraulic Research and Soil Mechanics, Swiss Federal Institute of Technology, Zurich, Switzerland. Translation T-5, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, Calif. (1967-68). Gessler, J., 1970. Self-stabilizing tendencies of alluvial channels. Journal of Waterways and Harbors Division, ASCE, 96(2):235-249. Gessler, J., 1971a. Beginning and ceasing of sediment motion. In River Mechanics, Volume I, Edited and Published by Shen H.W, Fort Collins, Colorado. Gessler, J., 1971b. Critical shear stress for sediment mixture. Proc. of 14th Congress of International Association of Hydraulics Research, Vol. 3, pp. Cl-1 to Cl-8. 208 References Gonzalez, R.G., and Woods, R.E., 1992. Digital image processing. Addison-Wesley Publishing Company, Inc.. Gordon, C.M., 1974. Intermittent momentum transport in a geophysical boundary layer. Nature, 248, pp. 392-394. Graf, W.H., 1984. Hydraulics of sediment transport. Water Resources Publications, Littleton, Colorado. Graf, W.H. and Pazis, G.C., 1977. Deposition and erosion in an alluvial channel. Journal of Hydraulics Research, 15:151-166. Grass, A.J., 1970. Initial instability of fine bed sand. Journal of Hydraulics Division, ASCE, 96(3):619-632. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries. Journal of Fluid Mechanics, 50(2):233-255. Grass, A.J., 1982. The influence of boundary layer turbulence on the mechanics of sediment transport. Euromech 156: Mechanics of sediment transport, Balkema, Rotterdam. Grass, A.J., Stuart, R.J. and Tehrani, M., 1991. Vortical structures and coherent motion in turbulent flow over smooth and rough boundaries. Phil. Trans, of the Royal Society of London, Series A, 336(1640): 35-65. Helsel, D.R. and Hirsch, R.M., 1992. Statistical methods in water resources. Elsevier Science, Amsterdam. Henderson, F. M., 1966. Open channel flow. The Macmillan Company, New York. Hey, R.D., Bathurst, J.C. and Thome, C.R., 1982. Gravel bed rivers. Chichester, John Wiley and Sons. Hipel, K.W., and Mcleod, A.I., 1994. Time series modelling of water resources and environmental systems. Development in Water Science, No. 45, Elsevier, Amsterdam, The Netherlands. 209 References Hjulstorm, F. 1935. The morphological activity of rivers as illustrated by river fyris. Bull, of Geological Institute of Uppsala, Sweden, 25, Ch. III. Hogg, A.J., Huppert, H.E. and Soulsby, R.L., 1993. The dynamics of particle-laden fluid elements. In Sediment mechanisms in coastal environments and rivers, Euromech 310, Edited by Belorgey, M., Rajaona, R.D. and Sleath, J.F.A., World Scientific. Ikeda, S., 1971. Some studies on the mechanics of bed-load transport. Proc. of Japan Society of Civil Engineers, 185:61-69. Ikeda, S., 1982. Incipient motion of sand particles on side slopes. Journal of Hydraulics Division, ASCE, 108(1):95-114. Jackson, R.G., 1976. Sedimentological and fluid-dynamic implications of the turbulent bursting phenomenon in geophysical flows. Journal of Fluid Mechanics, 77(3):531-560. James, C. S., 1990. Prediction of entrainment conditions for nonuniform, noncohesive sediments. Journal of Hydraulics Research. IAHR, 28(1):25-41. Jiang, Z., and Haff, P.K., 1993. Multiparticle simulation methods applied to the micromechanics of bed load transport. Water Resources Research, 29(2):399—412. Kalinske, A.A. and Hsia, C.H., 1945. Study of transportation of fine sediments by flowing water. Bulletin 29, University of Iowa. Keshavarzy, A. and Ball, J.E., 1995. Instantaneous shear stress on the bed in a turbulent open channel flow. HYDRA 2000, Proc. of XXVIth Congress of IAHR, London, Volume 1: Integration of research approaches and applications, edited by Ervine A., Paper No. 1A10, pp. 81-86. Keshavarzy, A. and Ball, J.E., 1996a. Characteristics of turbulent shear stress in open channel. Stochastic Hydraulics’96, Proc. of 7th IAHR International Symposium on Stochastic Hydraulics, edited by Kevin Tickle, A.A. Balkema Publisher, Rotterdam. 210 References Keshavarzy, A. and Ball, J.E., 1996b. The influence of turbulent shear stress on the initiation of sediment motion in an open channel flow. Stochastic Hydraulics’96, Proc. of 7th IAHR International Symposium on Stochastic Hydraulics, edited by Kevin Tickle, A. A. Balkema Publisher, Rotterdam. Keshavarzy, A. and Ball, J.E., 1996c. An investigation of entrainment of sediment particles in open channel flows. 23 rd of Hydrology and Water Resources Symposium, Water and Environment, Hobart, Tasmania. Keshavarzy, A. and Ball, J.E., 1997. An analysis of the characteristics of rough bed turbulent shear stresses in an open channel. Journal of Stochastic Hydrology and Hydraulics. Vol. 11, No 3. Kim, H.T., Kline, S.J. and Reynolds, W.C., 1971. The production of turbulence near smooth wall in a turbulent boundary layer. Journal of Fluid Mechanics, 50:133-160. Kironoto, B.A., and Graf, W. 1994. Turbulence characteristics in rough uniform open-channel flow. Proc. Inst. Civil Eng., Water, Marit. and Energy, 106:333-344. Kline, S.J., Reynolds, W.C., Schraub, F.A., and Runstadler, P.W., 1967. The structure of turbulent boundary layers. Journal of Fluid Mechanics, 30(4):741-773. Knapp, R. T, 1938. Energy balance in stream flows carrying suspended load. Am. Geophys. Union Trans., pp. 501-505. Krogstad, P.A., Antonia, R.A. and Browne, L.W.B., 1992. Comparison between rough and smooth wall turbulent boundary layers. Journal of Fluid Mechanics, 245:599-617 Kramer, H., 1935. Sand mixtures and sand movement in fluvial models. Transactions, ASCE, Paper No. 1909, 100:798-878. Lane, E.W., 1955. Design of stable channels. Transactions, ASCE, Paper No. 2776, 120:1234-1279. LeFeuvre, A.R., Altinbilek, H.D. and Carstens, M.R., 1970. Sediment pickup function. Journal of Hydraulics Division, ASCE, 96(10):2051-2063. 211 References Läufer, J., 1950. Some recent measurements in a two dimensional turbulent channel. Journal of Aeronautical Science, 17(5):277-287. Li, Z., and Komar, P.D., 1986. Laboratory measurements of pivoting angles for application to selective entrainment of gravel in a current. Sedimentology, 33:413-423. Ling, C.H., 1995. Criteria for incipient motion of spherical sediment particles. Journal of Hydraulic Engineering, ASCE, 121 (6):472—478. Looney, S.W. and Gulledge, T.R., 1985. Use of the correlation coefficient with normal probability plots. The American Statistician, 39:75-79. Lyn, D.A., 1986. Turbulence and turbulent transport in sediment-laden open channel flows. Report No. KH-R-49, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, Calif. Mantz, P. A., 1977. Incipient transport of fine grains and flakes by fluids-extended Shields diagram. Journal of Hydraulics Division, ASCE, 103(HY6):601-615. McQuivey, R.S., and Richardson, E.V., 1969. Some turbulence measurements in open-channel flow. Journal of Hydraulics Division, ASCE, 95(l):209-223. Miller, R.L., and Byrne, R.J., 1966. The angle of repose for a single grain on a flexible rough bed. Sedimentology, 6:303-314. Morsi, S. A. and Alexander, A.J. 1972. An investigation of particle trajectories in two-phase flow systems. Journal of Fluid Mechanics, 55(2): 193-207. Nakagawa, H., Nezu, I. and Ueda, H., 1975. Turbulence of open channel flow over smooth and rough beds. Proc. of the Japan Society of Civil Engineers, 241:155-168. Nakagawa, H., and Tsujimoto, T, 1976. On probabilistics of motion of individual sediment particles on stream beds. Hydraulic Problems Solved by Stochastic Methods, Proc. 2nd int. IAHR Symp. on Stochastic Hydraulics, Sweden, edited by Hjorth R, Jönsson L. and Larsen R, Water Resources Publications, Fort Collins. 212 References Nakagawa, H. and Nezu, L, 1978. Bursting phenomenon near the wall in open channel flows and its simple mathematical model. Mem. Fac. Eng., Kyoto University, Japan, XL(4), 40:213-240. Nakagawa, H., and Tsujimoto, T., 1980. Sand Bed Instability due to Bed-load Motion. Journal of Hydraulics Division, Vol. 106, No. HY12. Nelson, J.M., Mclean, S.R. and Wolfe, S. R., 1993. Mean flow and turbulence fields over two dimensional bed forms. Water Resources Research. 29:3935-3953. Nelson, J.M., Shreve, R.L., Mclean, S.R. and Drake, T.G., 1995. Role of near-bed turbulence structure in bed load transport and bed form mechanics. Water Resources Research. 31(8):2071-2086. Nezu, L, and Nakagawa, H., 1981. Structure of space-time correlations of bursting phenomena in an open-channel flow. Journal of Fluid Mechanics, 104:143. Nezu, I. and Nakagawa, H., 1993. Turbulence in open-channel ßows. IAHR Monograph, Balkema, Rotterdam, The Netherlands. Nielsen, P., 1992. Coastal bottom boundary layers and sediment transport. World Scientific Publisher, Singapore. Nino, Y. and Garcia, M., 1995. Sediment particle motion in the wall region of a turbulent boundary layer. Proc. of 1 stint. Symp. on Water Water Resources Engineering, edited by William H., Espey Jr. and Combs P.G., Texas, Aug. 14—18,1995 Vol.2, published by ASCE, New York. Novak, P. and Nalluri, C., 1984. Incipient motion of sediment particles over fixed beds. Journal Hydraulics Research., IAHR, 22(3): 181-197. Nychas, S.G., Hershey, H.C. and Brodkey, R.S., 1973. A visual study of turbulent shear flow. Journal of Fluid Mechanics, 61:513-540. Offen, G.R. and Kline, S.J., 1975. A proposed model of the bursting process in turbulent boundary layers. Journal of Fluid Mechanics, 70(2):209-228. 213 References Oppenheim, A., and Schafer, R.W., 1975. Digital image processing. Prentice-Hall Publisher. Petroff, C.M., 1993. The interaction of breaking solitary waves with an armored bed. Report No. KH-R-55, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, Calif. Philips, M, 1980. A force balance model for particle entrainment into a fluid stream. Journal of Physics, D: Applied Physics, 13:221-233. Rao, K.N., Narasimha, R. and Badri Narayanan, M.A., 1971. The bursting phenomenon in a turbulent boundary layer. Journal of Fluid Mechanics, 48:339-352. Raudkivi, A.J., 1963. Study of sediment ripple formation. Journal of Hydraulics Division, ASCE, 89(6): 15-33. Raudkivi, A.J. and Callander, R. A., 1975. Advanced fluid mechanics, An introduction. Edward Arnold Publishers, London. Raudkivi, A.J., 1990. Loose boundary hydraulics. 3rd Ed., Pergamon Press, Oxford. Raupach, M.R., 1981. Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers. Journal of Fluid Mechanics, 108:363-382. Reynolds, A.J., 1974. Turbulent flows in engineering. Wiley Interscience Publication. Robert, A., 1993. Bed configuration and microscale progress in alluvial channels. Progress in Physical Geography, 17(2): 123-136. Rouse, H., 1937. Modem conceptions of the mechanics of turbulence. Transactions, ASCE, 102:463-543. Rouse, H., 1946. Elementary mechanics of Fluids. John Wiley & Sons, Inc., New York,. Rubey, W. W, 1933. Equilibrium conditions in Debris-laden streams. Am. Geophys. Union Trans., 14th Ann. Mtg., pp. 497-505. 214 References Ruiter, J.C.C. de, 1982. The mechanism of sediment transport on bed forms. Euromech 156: Mechanics of sediment transport, Balkema, Rotterdam. Ruiter, J.C.C. de, 1983. Incipient motion and pick-up of sediment Report R 657-XI, Delft Hydraulics. Delft, The Netherlands. Saiedi, S., 1993. Experience in design of a laboratory flume for sediment studies. International Journal of Sediment Research, 8(3):89—101. Schlichting, H., 1968. Boundary layer theory. McGraw-Hill Co., New York, N. Y. . Schober, E, 1989. Towards a distinct treatment on incipient motion. Proc. of the International Symp. in Sediment Transport Modelling, edited by Wang S.Y., pp. 284-289. Shen, H.W., and Gessler, J., 1967. General report. Proc. 12th Congress of International Association for Hydraulic Research, Vol. 5, pp. 401-408, Fort Collins, Colorado. Shen, H.W. and Cheong, H.F., 1973. Dispersion of contaminated sediment bed load. Journal of Hydraulics Division, ASCE, Vol. 99, No. HY11, pp. 1947-1965. Shen, H.W. and Cheong, H.F., 1980. Stochastic sediment bed load models. In Application of stochastic processes in sediment transport., edited by Shen H.W. and Kikkawa H., Water Resources Publications, Littleton, Colorado. Shen, H.W. and Julien, P.Y., 1993. Erosion and sediment transport. In Handbook of Hydrology, edited by Maidment, D.R. Ch. 12, pp. 12.1-12.61, McGraw-Hill Inc.. Shen, H.W. and Tabios, G.Q., 1996. movement of sediment using a stochastic model with parameters varying as a function of discharge. Stochastic Hydraulics'96, Proc. of 7th IAHR International Symposium on Stochastic Hydraulics, edited by Kevin Tickle, A.A. Balkema Publisher, Rotterdam Shields, A. 1936. Anwendung der Ahnlinchkeitsmechanik und Turbulenzforshung auf die Gesciebebewegung. Mitteil Preuss. Versuchsant. Wasser, Erd, Schiffbau, Berlin, No. 26. 215 References Smith, C.R., and Walker, J.D.A., 1990. A conceptual model of wall turbulence. In: Robinson, S. (Ed.) Proceeding of the NASA Langley Boundary Layer Workshop. Smith, C.R., and Walker, J.D.A., Haidari, A.H. and Sobrun, U., 1991. On the dynamic of near-wall turbulence. Philosophical Transactions of the Royal Society of London, Series A, 336:131-175. Streeter, V.L. and Benjamin Wylie E., 1979. Fluid mechanics. Seventh Edition, McGraw-Hill Kogakusha, Ltd., Tokyo. Sumer, B.M., and Oguz, B., 1978. Particle motion near the bottom in turbulent flow in an open channel. Part 1, Journal of Fluid Mechanics, 86:109-127. Sumer, B.M., and Deigaard, R., 1981. Particle motion near the bottom in turbulent flow in an open channel. Part 2, Journal of Fluid Mechanics, 109:311-337. Sutherland, A.J., 1966. Entrainment of fine sediments by turbulent flows. Report No. KH-R-13, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, California. Sutherland, A.J., 1967. Proposed mechanism for sediment entrainment by turbulent flows. Journal Geophys. Res., 72:6183-6194. Taylor, G.I., 1935. Statistical theory of turbulence. Proceedings of Royal Society of London, Series A151, pp. 421—454. Tennekes, H., and Lumley, J.L., 1972. A first course in turbulence. MIT Press, Cambridge, Terzaghi, K., Peck, R.B. and Mesri, G., 1996. Soil mechanics in engineering practice. 3rd edition, John Wiley & Sons, Inc. Thome, P.D., Williams, J.J. and Heathershaw, A.D., 1989. In situ acoustic measurements of marine gravel threshold and transport. Sedimentology, 36:61-74. Townsend, A. A., (1956). The structure of turbulent shear flow. Cambridge University Press, Cambridge. 216 References Townsend, A.A., (1976). The structure of turbulent shear flow. 2nd edition, Cambridge University Press, Cambridge. Van Rijn, L.C., 1984a. Sediment transport, Part I: Bed load transport. Journal of Hydraulic Engineering, ASCE, 110(10). Van Rijn, L.C., 1984b. Sediment pick-up functions. Journal of Hydraulic Engineering, ASCE, 110(10): 1494-1502. Van Rijn, L.C., 1986. Application of sediment pick-up function. Journal of Hydraulic Engineering, ASCE, 112(9):867-874. Van Rijn, L.C., 1993. Principles of sediment transport in rivers, estuaries and coastal seas. Aqua Publications, The Netherlands. Vanoni, V. A., 1964. Measurements of critical shear stress for entraining fine sediments in a boundary layer. KH-R-7, W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, California. Vanoni, V.A., 1966. Sediment transportation mechanics: Initiation of motion: Progress report of the Task Committee on particles of sedimentation manual, Journal Hydraulics Division, ASCE, 92(2):291-314. Vanoni, V.A., 1975. Sedimentation Engineering. ASCE, Manuals and Reports on Engineering Practice, No. 54. Wang, X., and AN, F., 1994. The fluctuating characteristics of hydrodynamic forces on bed particles. Int. Journal of Sediment Research, Vol. 9, Special issue, Dec.. White, C.M., 1940. The equilibrium of grains in the bed of a stream. Proc. Royal Society of London, Series A, 74:324—338. Wilcock, P.R., and Southard. J.B., 1988. Experimental study of incipient motion in mixed-sized sediment. Water Resources Research, 24(7): 1137-1151. Willert, C.E., and Gharib, M., 1991. Digital particle image velocimetry. Experiments in Fluids, 10:181-193. 217 References Williams, P.B., and Kemp, P. H., 1971. Initiation of ripples on flat sediment beds. Journal of Hydraulics Division, ASCE, 97(4):505-522. Williams, P.B., and Kemp, P. H., 1972. Initiation of ripples by artificial disturbances. Journal of Hydraulics Division, ASCE, 98(6): 1057-1070. Williams, J.J., Thome, C.R. and Heathershaw, A.D., 1989. Measurement of turbulence in the benthic boundary layer over a gravel bed. Sedimentology, 36:959-971. Williams, J.J., 1990. Video observations of marine gravel transport. Geo. Mar. Lett., 10:157-164. Williams, J.J., 1996. Turbulent flow in rivers. In Advances in fluvial dynamics and stratigraphy. Edited by Carling, P.A., and Dawson, M.R., John Wiley & Sons, Chichester. Yalin, M.S., 1972. Mechanics of sediment transport. Pergamon Press, Oxford. Yalin, M.S., 1977. Mechanics of sediment transport. Second edition, Pergamon Press, Oxford. Yalin, M.S., 1992. River mechanics. Pergamon Press, Oxford. 218 Appendix A Appendix A Publications 220 Appendix A 1. Keshavarzy, A. and Ball, J.E., 1997. An analysis of the characteristics of rough bed turbulent shear stresses in an open channel. Journal of stochastic hydraulics and hydrology. Vol. 11, No. 3. 2. Keshavarzy, A. and Ball, J.E., 1995. Instantaneous shear stress on the bed in a turbulent open channel flow. HYDRA 2000, XXVIth IAHR Congress, London, Volume 1: Integration of research approaches and applications, edited by Ervine A., Paper No. 1A10, pp. 81-86. 3. Keshavarzy, A. and Ball, J.E., 1997. An application of image processing in the study of initiation of sediment motion and transport in an open channel flow. 2nd International symposium on rainwater catchment systems, 21-25 April 1997, Tehran, Iran. 4. Ball, J.E. and Keshavarzy, A., 1995. Discussion on incipient sediment motion on non-horizontal slopes by Chiew and Parker. Journal of Hydraulics Research, IAHR, 33(5):723-724. 5. Keshavarzy, A. and Ball, J.E., 1996a. Characteristics of turbulent shear stress applied to the bed particles in an open channel flow. 7th IAHR International Symposium on Stochastic Hydraulics, Mackay, QLD, Australia. 6. Keshavarzy, A. and Ball, J.E., 1996b. The influence of turbulent shear stress on the initiation of sediment motion in an open channel flow. 7th IAHR International Symposium on Stochastic Hydraulics, Mackay, QLD, Australia. 7. Keshavarzy, A. and Ball, J.E., 1996c. An investigation of entrainment of sediment particles in open channel flows. 23th of Hydrology and Water Resources Symposium, Water and Environment, Hobart, Tasmania. 221 Appendix A 8. Keshavarzy, A. and Ball, J.E., 1997. A model for prediction of instantaneous particle velocity on a mobile bed. Accepted for presentation in 24th of Hydrology and Water Resources Symposium, Water and Environment, 24-28 November 1997, Auckland, New Zealand. 9. Keshavarzy, A. and Ball, J.E., 1997. A mathematical model for initiation of sediment motion considering the influence of turbulence. Journal of hydraulic research, ready for submission. 10. Keshavarzy, A. and Ball, J.E., 1997. An application of image processing in a study of sediment motion. Journal of hydraulic research, ready for submission. In addition to the above publications, I have done some cooperative study and published following papers; 11. Keshavarzy, A. and Erskine, W.D., 1995. Investigation of Dominant Discharge on South Creek, NSW, Australia. The 2nd international Symposium on urban Storm Water Management, Melbourne, Australia, pp. 261-266. 12. Erskine, W.D., and Keshavarzy, A., 1996. Frequency of Bankfull Discharge on South and Eastern Creeks, NSW, Australia. 23th of Hydrology and Water Resources Symposium, Water and Environment, Hobart, Tasmania. 13. Keshavarzy, A. and Erskine, W.D., and Eslamian S., 1995. River Management VS, Urban Development in the Hawkesbury-Nepean River Basin, Australia. WRM’ 95, Proc, Regional Conf on Water Resources Management, Isfahan University of Technology, Isfahan, Iran. 222 Appendix A Appendix A (A-1...A-9) 223 Appendix A Appendix A-l A Paper entitled: An analysis of the characteristics of rough bed turbulent shear stresses in an open channel. Published in Journal of stochastic hydraulics and hydrology. Vol. 11, No. 3,1997. 224 Appendix A AN ANALYSIS OF THE CHARACTERISTICS OF ROUGH BED TURBULENT SHEAR STRESSES IN AN OPEN CHANNEL Alireza Keshavarzy and James E. Ball Water Research Laboratory School of Civil Engineering The University of New South Wales ABSTRACT Entrainment of sediment particles from channel beds into the channel flow is influenced by the characteristics of the flow turbulence which produces stochastic shear stress fluctuations at the bed. Recent studies of the structure of turbulent flow has recognised the importance of bursting processes as important mechanisms for the transfer of momentum into the laminar boundary layer. Of these processes, the sweep event has been recognised as the most important bursting event for entrainment of sediment particles as it imposes forces in the direction of the flow resulting in movement of particles by rolling, sliding and occasionally saltating. Similarly, the ejection event has been recognised as important for sediment transport since these events maintain the sediment particles in suspension. In this study, the characteristics of bursting processes and, in particular, the sweep event were investigated in a flume with a rough bed. The instantaneous velocity fluctuations of the flow were measured in two- dimensions using a small electromagnetic velocity meter and the turbulent shear stresses were determined from these velocity fluctuations. It was found that the shear stress applied to the sediment particles on the bed resulting from sweep events depends on the magnitude of the turbulent shear stress and its probability distribution. A statistical analysis of the experimental data was undertaken and it was found necessary to apply a Box-Cox transformation to transform the data into a normally distributed sample. This enabled determination of the mean shear stress, angle of action and standard error of estimate for sweep and ejection events. These instantaneous shear stresses were found to be greater than the mean flow shear stress and for the sweep event to be approximately 40 percent greater near the channel bed. Results from this analysis suggest that the critical shear stress determined from Shield’s diagram is not sufficient to predict the initiation of motion due to its use of the temporal mean shear stress. It is suggested that initiation of particle motion, but not continuous motion, can occur earlier than suggested by Shield’s diagram due to the higher shear stresses imposed on the particles by the stochastic shear stresses resulting from turbulence within the flow. 1 INTRODUCTION The motion of sediment particles and the initiation of that motion is an important component of many engineering studies. For example, there is a need to quantify the threshold conditions for entrainment of sediment particles in the design of stable channels, bank protection works, and, in general, problems associated with the erosion of channels. Further problems where the initiation of sediment particle motion is important arise when motion of pollutants attached to sediment particles becomes an important component of managing water quality. There are many processes which influence the availability and entrainment of sediment particles; among these influences, the flow turbulence and the associated coherent structures within the flow are very important but, 225 Appendix A as yet, have not been defined completely. In this regard, Nezu and Nakagawa (1993) pointed out that the mechanism of turbulence production in flow over rough beds in rivers required investigation. Associated with this production of turbulence and the resultant momentum transfer are the processes by which constituents such as sediment particles are entrained and moved with the flow. Recent research has focused on developing an understanding of the processes influencing and the mechanisms related to the transfer of momentum between the laminar and turbulent regions of the boundary layer. Several alternative models have been developed for the description of turbulent flow within and near the boundary layer, see, for example, Grass (1971), Offen and Kline (1975), and Raupach (1981) for a description of some of these models. In general, these models considered the mean velocity profile within the flow and the statistical characteristics of the coherent turbulent structures within the flow. The concept of the bursting phenomenon was introduced by Kline et al. (1967) as a means of describing the transfer of momentum between the turbulent and laminar regions near the boundary. Four (4) quadrants or classes of events were identified. Each event was classified according to the quadrant associated with the velocity fluctuations occurring during the event. These velocity fluctuations (u^v') arc defined as the variations from the temporal mean velocities in the longitudinal and vertical directions. Algebraically, they are defined by u/ = u - u (l) and v = v - v (2) where u and v are the instantaneous velocities in the longitudinal and vertical directions respectively and u and v are the temporal mean velocities in the longitudinal and vertical directions. These temporal mean velocities are given by (3) and (4) where N is the number of instantaneous velocity samples. 226 Appendix A v' /\ (II) (I) ejection outward interactions (u' < 0,v' > 0) (u' > 0, v' > 0) \ S u inward interactions ^ \ sweep (u' < 0,v' < 0) (u' > 0,v' < 0) (DDL) (IV) Figure 1 - Four classes of bursting event and the associated quadrant. The four quadrants identified for the velocity fluctuations and the associated bursting event were • outward interaction or quadrant I (upward front), in which u1 > 0 and v/ > 0; • ejection event or quadrant II (upward back), in which u' < 0 and v' > 0; • inward interaction or quadrant HI (downward back), in which u/ < 0 and v1 < 0; and • sweep event or quadrant IV (downward front), in which u' > 0 and v1 < 0. Shown in Figure 1 is a phase diagram with the quadrant of each event class indicated. It is obvious from this figure that sweep events result in a velocity vector angled towards the bed. Therefore, bursting events in this quadrant will have a significant influence on the entrainment of particles into the flowing water. Of particular concern is the magnitude of the horizontal velocity component during quadrant IV events and the angle which the velocity vector impacts on the bed particles. In recent years, the contributions of coherent structures, such as the sweep (quadrant IV) and ejection (quadrant II) events, to momentum transfer have been extensively studied by quadrant analysis or probability analyses based on two-dimensional velocity information. Studies by Nakagawa and Nezu (1978) and Grass (1971, 1982) have indicated that just above the channel bed, the sweep event is more responsible than the ejection event for transfer of momentum into the boundary layer. In addition, Nakagawa and Nezu (1978), Thome et al. (1989), and Keshavarzy and Ball (1995) pointed out that the sweep and ejection events occur more frequently than the outward interaction (quadrant I) and inward interaction (quadrant HI) events. 227 Appendix A The four types of bursting events identified earlier have different influences on the rate, and mechanisms of sediment entrainment in a turbulent flow. In studies of sediment transport by, for example, Williams (1990) and Thome et al. (1989), it has been shown that sediment entrainment occurs from the bed most frequently during sweep events and only occasionally during outward interaction events, whereas transport of suspended sediment depends primarily on the ejection event Additionally, Keshavarzy and Ball (1995) pointed out that the magnitude of the instantaneous shear stress in a sweep event is greater than that which occurs during both outward and inward interaction events. These bursting events impose a rapid and significant pressure fluctuation on the bed; it is these fluctuations that are considered to have a significant influence on the entrainment of sediment particles from the bed with a resultant temporally variable sediment transport rate. These instantaneous pressure fluctuations lower the local pressure near the bed and hence particles may be ejected from the bed by hydrostatic pressure. Raudkivi (1990) postulated that even sheltered particles can be entrained by this mechanism. Entrainment of sediment particles, therefore, is substantially influenced by bursting events. Sediment particles move mostly by rolling and sliding with the sweep force and may saltate with higher sweep forces. Thus the impact of the instantaneous sweep force is very important for motion of sediment particles and even more important for the initiation of that motion. Ball and Keshavarzy (1995) pointed out the effect of the instantaneous, shear stress on incipient motion and consequent reasons for difficulties in defining sediment entrainment functions. Despite this importance of the bursting events in sediment transport, however, their characteristics have not been investigated in sufficient detail. In the study reported herein the characteristics of the bursting phenomenon in an open channel flow over a rough bed are investigated. The magnitude of the shear stress, the frequency and the angle of events was determined from experimental data. Also determined was the variation of shear stress, frequency and angle for an individual event with respect to depth. 2 EXPERIMENTAL APPARATUS The experiments undertaken during this study were carried out in a non-recirculating tilting rectangular flume of 0.61 m width, 0.60 m height and 35 m length. The side walls of the flume were made of glass, making it possible to observe and record the flow. The bed was constructed with movable concrete sheets which are covered by sand particles of 2 mm diameter. As a result, it is possible to perform experiments with different bed roughness simply by changing the concrete sheets. More detailed descriptions of the experimental flume are presented by Saiedi (1993). The longitudinal and vertical components of instantaneous velocity were measured using a small 228 Appendix A electromagnetic velocity meter which will be referred to as an EMC. One advantage of the EMC is that there are no movable parts which can be affected by particles in the water. The EMC probe has two pairs of 10 20 30 40 50 60 70 80 90 100 lit Time (Sec) Outward Interaction (I) Ejection (II) £ 100- r* f loo § 50- ts 50- 1 f\{\ _lA A ~ 0--J -V 0-üSAi y-'-y'ywry ' -50 • Vyv V\ \jv -50-V -1004 -100- 1 «' i ~ 50- - 50- I -tu a a I I''\ si.AJi L \ A v v w^wyyv y /Vflf -25- v Jr -50- Time (Sec) Time (Sec) Inward Interaction ( TTT ) Sweep (IV) r ? 100- ? 100- 1 50- I 5°-, ^ 0- . 1 if\ \i (\ hl\J\-AnrMaI 3 0- V__rvA/rw; lyi/ v v\V -50- \}J ynjby v ^ -50. V -100- i -100-1 , 1 er so- i t 1 S n f\_\ |\ ., An' 1 Time (Sec) Time (Sec) Figure 2 - Example of the velocity time series from the experiments 229 Appendix A electrodes and, hence, is able to measure two components of the flow velocity. Use of a trolley mounting device enabled the EMC to be moved to the desired location for the velocity measurement The measurements were performed in the centerline of the flume at a location of 7 m downstream from the inlet of the flow. An example of the time series of longitudinal and vertical velocity fluctuations obtained during an experiment is shown as Figure 2. Velocity components in longitudinal and vertical directions were measured and stored in a digital format for later analysis. This analysis determined the temporal mean velocity in the longitudinal and vertical directions, the temporal mean shear stress, the velocity fluctuations, the mean shear stress for events occurring in each quadrant, and the number of events in each quadrant during the sample period. The analysis also determined the mean angle from the horizontal of events in quadrants II (ejection events) and IV (sweep events). A summary of the flow parameters for each experiment is shown in Table l. It was expected that, with these experiments, the effects of the Froude number and the Reynolds number were indirectly considered; the validity of this assumption was assessed by considering non-dimensional velocity profiles determined during the experiments. Shown in Figure 3 are the measured velocity profiles normalised by the cross-section average longitudinal velocity as a function of height above the bed (d) normalised by the flow depth (H); not all the experimental profiles are shown in this figure but those shown are representative of all profiles obtained. The similarity in the non-dimensional velocity profiles shown in Figure 3 validates the assumption of implicit inclusion of the Reynolds and Froude number effects. Table 1 - Hydraulic conditions during experiments No. 1 2 3 4 5 6 7 8 9 10 11 12 Test E F G H J KL M N O P Q Q(l/s) 63.7 76.3 52 58 73.6 40.2 61 30.3 22 48.5 79.8 97.5 H(mm) 355 154 76 120 145 120 212 154 70 166 230 265 T(°C) 15 15 15 15.5 14 13.5 13.5 13 12.2 13 13 12.6 U(m/s) 0.29 0.81 0.31 0.79 0.83 0.55 0.47 0.32 0.51 0.48 0.57 0.60 U^Cm/s) 0.35 0.14 0.38 0.86 0.99 0.65 0.58 0.40 0.56 0.38 0.38 0.37 Fr 0.16 0.66 0.19 0.73 0.7 0.51 0.33 0.26 0.62 0.38 0.38 0.37 230 Appendix A oa •- oa ■■ 07 •• os •• 04 ■■ 03 ■■ Figure 3 - Normalised velocity profiles for experiments 3 ANALYSIS OF EXPERIMENTAL DATA Data obtained from the experiments were analysed to calculate the time-averaged velocity, velocity fluctuations, root mean square, and the Reynolds shear stresses. Relationships used for these calculations were The root mean square of the velocity (turbulence intensity) “ou. = V UÄ = (5) TF S ("• - and v revs = - E (v - v)2 (6) \ n h \1 1 The Reynolds shear stresses were calculated from the velocity fluctuations using ii 1 v-' ' ‘ (7) U V = N j-i Vi and t = -pu/v/ (8) 231 Appendix A where u^, and are the turbulence intensities in the longitudinal and vertical directions respectively, and t is the Reynolds shear stress. Shown in Figures 4 and 5 are the turbulence intensities normalised by the shear velocity as a function of the measurement height above the channel bed (d) normalised by the flow depth (H). The shear velocity was determined using u * = v/gRs; (9) where u* is shear velocity, r0 is shear stress at the bed, R is the hydraulic radius, g is gravitational acceleration and Sf is the friction gradient which, for steady flow conditions, is equal to the energy gradient. These normalised turbulence intensities were compared with previously published data; as shown in Figures 4 and 5 good agreement between the alternative datasets was obtained. Similar to the previously published data, it was found that the turbulence intensity in the longitudinal direction decreased from the bed to the water surface while the turbulence intensity in the vertical direction was not found to change significantly over the flow depth. In addition, a comparison of normalised Reynolds shear stress with previously published data was undertaken. Shown in Figure 6 arc the non-dimensional Reynolds shear stress as a function of the non-dimensional vertical location; the data obtained during this study, as indicated in this figure, compares favourably with that previously published. Horizontal Turbulence Intensity OTNK 2.5 -- x x: X <9 ot hHm* 1V7V 1.5 ▲ • Figure 4 - Normalised turbulence intensity in the longitudinal direction 232 Appendix A ATmM Vertical Turbulence Intensity QTm»« ilUiimtNmwi O +Ö o£ O d/H Figure 5 - Normalised turbulence intensity in the vertical direction QTmcO OT»*f AT««CQ XT««1 XT««. OTmwi -|-T«»c* ♦ N»JaRiwtJk Nero 1975 ^CBU otb □Uukctmoi Om cQuvryt R ttludaon CO*7l •LiyoC-UCB«7) ALjnC-*)®«7) ♦ LVnC-S) Figure 6 - Normalised Reynolds shear stresses 233 Appendix A 4 RESULTS AND DISCUSSION An analysis of the experimental data indicated that the characteristics of the event classes (sweep, ejection, outward and inward interactions) differed over the flow depth. The following discussion outlines the important aspects obtained from the analysis of the experimental data. 4.1 Instantaneous Shear Stress of the Event Classes The magnitude of the mean shear stress for each quadrant was found to be different, and to differ from the temporal mean shear stress for the flow. The instantaneous shear stress for each event was normalised by the total mean shear stress at that point within the flow. Expressed algebraically, the non-dimensional instantaneous shear stress was determined by (10) T where t is the instantaneous shear stress andr is the mean shear stress at that point in the flow. In order to transform the normalised data into a normally distributed parameter, a Box-Cox power transformation (sec Box and Cox, 1964) was used to transform the original data denoted by the symbol C to the transformed data denoted by B(C). A Box-Cox power transformation is defined, for non-zero values of A., by (C + k)1 - 1 B(C) (11) k and, for zero values of A., by B(C) = ln(C + k) (12) where k is a constant and A. is transformation power. If all of the values in the time series are greater than zero, then the constant k usually is set to zero (Box and Cox, 1964). Application of the transformation to the data set comprising more than 50 sets of quadrant IV events (sweep events) and 50 sets of quadrant II events (ejection events) resulted in a value of A. equal to 0.28. Hence, the first case of the transformation (equation 11) is relevant here. The inverse transformation which is the transformation B'lof B(C), for this situation, is given by C = [A.B(C) + l]k (13) 234 Appendix A Frequency Distribution of Shear Stress Frequency Distribution of Shear Stress Sweep Event Ejection Event Percent Percent C2 C2 Figure 7 - Frequency distributions of transformed data during sweep and ejection events Normal Probability Plot Normal Probability Plot (Sweep Event) (Ejection Event) t. . j . J .os ■ ■* - .001 * ; -3 -2 -t •3 *2 And*rson-O«rtng Normality Avar eg«0.364769 C2 Average: <1626845 C2 Ander son-Carling Normality Test Std. 0«v.-t.S2 Std. Dav.al .63 oorrcriation-0.993 Not data:3K Correlation -0.997 Ndd6ta;294 Figure 8 - Probability distributions of transformed data during sweep and ejection event Frequency distributions of the instantaneous shear stresses for events in quadrants IV (sweep) and II (ejection) after application of the transformation are shown in Figure 7. As shown in Figure 8, the transformed data are almost normally distributed. A good normality distribution can be concluded from Figure 8 and considerationof the correlation coefficients of 0.997 and 0.993 obtained for quadrant IV and quadrant H events respectively. As discussed by Looney and Gulledge (1985) and Helsel and Hirsch (1992), the critical correlation coefficients for acceptance of the normality hypothesis with a 0.05 level of significance are 0.993 and 0.991 for events in quadrants IV and II respectively; these values are less than those obtained suggesting that the normality hypothesis can be accepted. Mean values of the transformed data B(C) for each quadrant were calculated using 235 Appendix A \ B(C\ - it B(C,) (14) V n>-' k “ I ..4 The inverse Box-Cox transformation was applied to these mean values to enable determination of the mean instantaneous shear stress ratio for events in a particular quadrant. For events in quadrant IV (sweep events), it was found that the magnitude of the shear stress close to the bed of the channel was approximately 150% of the temporal mean shear stress while for events in quadrant II, it was found that the magnitude of the shear stress was approximately 135% of the temporal mean shear stress. Furthermore, as shown in Figures 9 and 10, it was found that the mean shear stress for events in quadrants II and IV increased with depth from the bed to the surface. Regression equations were developed for the ratio of the mean quadrant shear stress to the temporal mean flow shear stress; these relationships arc plotted with 95% confidence limits in Figures 9 and 10. Algebraically, the relationship for quadrant IV events was Q = 1.48 * 1.134 (15) ri and for quadrant II events was q * 1.35 * 1.83 4 (16) H Correlation coefficients and standard error of estimate for these relationships were 0.73 and 0.36 respectively for quadrant IV events, and 0.81 and 0.38 respectively for quadrant II events. Variation of shear stress in sweep Variation of shear stress in ejection event event 5- 4 4 X - - *MEE . . •♦MCE 3 IS 5 - -X- *yX’ * " * lO 2 ‘ ^ x*x* ----- x 2 1 ■ *...... * 1 0 0 01 02 03 04 OS 0£ 03 CS 0 01 02 0J 04 05 OS 0J OB d/H d/H » Figure 9- Normalised shear stress in quadrant IV Figure 10- Normalised shear stress in quadrant II events events 236 Appendix A A focus of this study was the analysis of bursting event characteristics and particularly sweep events in the region close to the channel bed This focus resulted from a need to consider the influence of the instantaneous shear stress magnitude on the entrainment of sediment particles into the flow. From the analysis described above, the magnitude of the instantaneous shear stress during quadrant IV events on average is approximately 1.40 times the temporal mean shear stress in the region near the channel bed. It is expected, therefore, that particles which would not move at the temporal mean shear stress are able to move during sweep, or quadrant IV, events due to the higher induced shear force. This does not mean, however, that sediment particles are induced into continuous motion but rather are dislodged from their current location. 4.2 Frequency of Events The instantaneous shear stress data was divided into different classes based on the quadrant or phase of the bursting process. The frequency of each quadrant was determined by (17) and (18) N - k-1E where P is the frequency of each event class (quadrant), nk is the number of occurrences of each event class, N is the total number of events, and the subscript represents the individual quadrants (k =1...4). 3S T 30 -- 25 -- 10 -- O eject! oa OLL Figure 11- Frequency of events in quadrants IV Figure 12- Frequency of events in quadrants II and I and m 237 Appendix A The frequency of events in each of the four quadrants was determined from the experimental data and plotted in relation to the normalised depth of flow. Shown in Figures 11 and 12 are the variations in the determined frequency with respect to the normalised depth for each of the four quadrants. It can be seen that the frequency of an individual event varies with the quadrant of the event and the normalised depth. From these figures, it can be seen also that the frequency of events in quadrants IV and II (sweep and ejection events), particularly close to the bed of an open channel, is higher than the frequency of events in quadrants I and III (outward and inward interaction events). Close to the bed, quadrant IV and quadrant II events have a frequency of approximately 30%, whereas that of events in quadrants I and III is approximately 20%. As a result, it can be concluded that sweep and ejection events should occur approximately 30 percent more frequently than outward and inward interaction events. Additionally, the frequency of the sweep and the ejection events decreases with depth from the bed to the water surface, whereas those of the outward and inward interactions increase with the depth. Furthermore, as shown in Figures 11 and 12, the frequency of ail events in all quadrants approaches a value of 25% near the free water surface which suggests that all events are equally probable. The relationship between the frequency of an event in a quadrant and the normalised depth was investigated with regression relationships developed for estimating the probability of events at a defined normalised depth in quadrants IV and II. The regression relationship for events in quadrant IV (sweep events) was (19) and that for events in quadrant II (ejection events) was (20) Correlation coefficients for these relationships were 0.77 and 0.83 respectively. 43 Angle of the events The force applied to sediment particles on the channel bed during events in quadrant IV depends upon the inclination angle of the force to the bed. The angle of the applied force for individual events was determined from the ratio of the turbulent velocity fluctuations; expressed algebraically, the angle of an event was determined from 238 Appendix A 0. = arctan — (21) where 0k is the angle of the event measured from the horizontal. Similar to the analysis of the shear stress magnitude, it was necessary to apply a Box-Cox transformation to the calculated angles. The mean angle of events in a particular quadrant after application of the transformation was determined then as a function of the normalised depth. The relationships developed for the mean event angle, for events in quadrants IV and II, were 84 - 21.6 * 22.3-1 (22) ri and 02 ■= 18.8 * 21.24 (23) H respectively. Correlation coefficients for these relationships were 0.67 and 0.71 respectively. Shown in Figures 13 and 14 are the mean angles of the applied force due to events in quadrants IV (sweep) and 13 ( ejection). It can be seen from Figure 13 that the mean angle of events increases with depth from the bed to the free water surface. Close to the bed, the mean angle of sweep events is approximately 20°-22° while, near the free water surface, the mean angle increases to nearly 45°. In a similar manner, the mean angle of events in quadrant II is shown in Figure 14 as a function of the normalised depth. Once again, the mean angle of the event increases with the normalised depth; this increase is from approximately 18° to nearly 45° at the free water surface. Figure 13 - Angle of applied force for Figure 14 - Angle of the applied force for events in quadrant IV (sweep events) events in quadrant II (ejection events) 239 Appendix A 5 RESULTS AND CONCLUSIONS The motion of sediment particles and, particularly, the initiation of that motion is an important component of many water quality problems. As the first stage of a study to investigate the entrainment of sediment particles into the flow, the influence of flow turbulence and the associated coherent structures on the shear stress applied to sediment particles was investigated and reported herein. The investigation was based on the analysis of the turbulence characteristics of flows in a laboratory flume with a mobile bed. From the analysis of the experimental data, it was concluded that • The measured data was consistent with previously published data. • The magnitude of the mean shear stress in each quadrant (sweep, ejection, outward and inward interactions) was different from the overall temporal mean shear stress at a defined depth within the flow. For quadrant IV events, the mean shear stress was approximately 140% of the overall temporal mean shear stress in the region near the bed and increased to approximately 200% near the free water surface. In quadrant II events, the shear stress ratio is about 1.4 near the bed and increases to approximately 2.5 at the water surface. The shear stress ratios of the quadrant I and III events are low compared to quadrant Iv and II events. • The frequency of each event was determined also. It was found that, near the bed, the frequency of quadrant IV and II events was approximately 30%, whereas that of quadrant I and ID events was only 20%. Consequently, the events in quadrants I and HI occur 30% less often than events in quadrants II and IV. • Also determined was the average angle of action for events in quadrants II and IV. This angle was found to vary with depth from the bed of the channel to the free water surface and to change from approximately 20° at the bed to 45° at the water surface for events in quadrant IV. Similar angles for events in quadrant II were found. The implication of these instantaneous shear stresses is that the instantaneous forces applied to a sediment particles arc higher than those suggested through application of Shield's diagram which is based on the overall temporal mean shear stress. Consequently, it is suggested that sediment particles will be induced into motion at lower flow rates than those obtained from Shield's diagram. This motion, however, need nqt be continuous. 240 Appendix A REFERENCES Anwar, H.O. and Atkins, R., (1982), Turbulence structure in an open channel flow, Euromech,, 156: Mechanics of sediment transport, Istanbul, pp. 19-25. Ball, J.E. and Keshavarzy A., (1995), Discussion on Incipient Sediment motion on non-horizontal slopes by Chiew and Parker, J. ofHyd Res., 33(5):723-724. Bridge, J.S., (1981), Hydraulics interpretation of grain-size distributions using a physical model for bed load transport, J. of Sediment Petrology, 51:1109-1124. Box, G.E.P. and Cox D.R., (1964), An analysis of transformation. Journal of the Royal Statistical Society, Series B, 26:211-252. Grass, A.J., (1971), Structural features of turbulent flow over smooth and rough boundaries, J. of Fluid Mechanics, 50(2):233-255. Grass, A.J., (1982), The influence of boundary layer turbulence on the mechanics of sediment transport, Euromech 156. Mechanics of sediment transport, Istanbul, pp3-17. Helsel, D.R. and Hirsch, R.M., (1992), Statistical methods in water resources, Elsevier Science, Amsterdam. Keshavarzy A. and Ball, J.E., (1995), Instantaneous shear stress on the bed in a turbulent open channel flow, Proc. of XXVIIAHR Congress, London. Kline, S J., Reynolds, W.C., Schraub, F.A., and Runstadler P.W., (1967), The structure of turbulent boundary layers, / of Fluid Mechanics, 30(4):741-773. Krogstad, P.A., Antonia, R.A. and Browne, L.W.B., (1992), Comparison between rough and smooth wall turbulent boundary layers, J. of Fluid Mechanics, 245:599-617 Läufer, J., (1950), Some recent measurement in a two dimensional turbulent channel, J. of Aeronautical Science, 17(5):277-287. Looney, S.W. and Gulledge, T.R., (1985), Use of the correlation coefficient with normal probability plots, The American Statistician, 39:75-79. Lyn, D.A., (1986), Turbulence and turbulent transport in sediment-laden open-channel flows, Report No. KH- R-49, W.M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Pasadena, Calif. McQuivey, R.S., and Richardson, E.V., (1969), Some turbulence measurement in open-channel flow, J. of Hydraulics Division, ASCE, 95(HYl):209-223. 241 Appendix A Nakagawa, HL, Nezu, L, and Ucda R, (1975), Turbulence of open channel flow over smooth and rough beds. Proceeding of the Japan Society of Civil Engineers, 241:155-168. Nakagawa, R, Nezu, L, (1978), Bursting phenomenon near the wall in open channel flows and its simple mathematical model, Mem. Fac. Eng., Kyoto University, Japan, XL(4), 40:213-240. Nezu, I. and Nakagawa, R, (1993), Turbulence in open-channel flows, IAHR Monograph, Balkema, Rotterdam, The Netherlands. Offen, G.R. and Kline, S.J., (1975), A proposed model of the bursting process in turbulent boundary layers, J. of Fluid Mechanics, 70(2):209-228. Raudkivi, A.J., (1990), Loose Boundary Hydraulics, 3rd Ed., Pergamon Press, Oxford. Raupach, M.R., (1981), Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers, J. of Fluid Mechanics, 108:363-382. Saiedi, S., (1993), Experience in design of a laboratory flume for sediment studies. International Journal of Sediment Research, 8(37:89-101. Thome, P.D., Williams, J.J. and Heathershaw, A.D., (1989), In-situ acoustic measurements of marine gravel threshold and transport, Sedimentology, 36.:61-74. Williams, J.J., (1990), Video observations of marine gravel transport. Geo. Mar. Lett., 10:157-164. Yalin, M.S., (1992), River Mechanics, Pergamon Press, Oxford. 242 Appendix A Appendix A-2 A Discussion on a Paper entitled: incipient sediment motion on non-horizontal slopes by Chiew and Parker published in Journal of Hydraulics Research, IAHR, 33(5):723-724,1995. 243 Appendix A Discussion INCIPIENT SEDIMENT MOTION ON NON-HORIZONTAL SLOPES Yee-Meng Chiew and Gary Parker, Journal of Hydraulic Research, Vol. 32, 1994, No. 5, pp. 649-660 Discussers: J.E. BALL and A. KESHAVARZY The authors in this paper have pointed out the difficulty of defining incipient sediment motion in a laboratory or natural channel. The writers agree that it is difficult to define the incipient motion and it is worthwhile, therefore, to investigate reasons for this difficulty. The most important reason for this is the turbulence within the flow and the variable instantaneous shear stress acting on the sedi ment particles. The effect of this instantaneous shear has rarely been considered. The mechanisms involved in entrainment and deposition of particles in a mobile bed depends on the hydrodynamic forces. In turbulent flows these forces are not constant but vary with time at fre quencies proportional to the velocity-pressure fluctuations. Currently, most of the existing critical shear stress relationships used in sediment transport models are based on time-average channel shear stresses, which are defined in terms of flow depth, fluid density and energy gradient. How ever, particle dynamics depend not only on the spatial distribution of mean boundary shear stresses but also on their instantaneous value and the probability distribution of these values. The importance of the bursting events and in particular, sweep event in the entrainment of sediment particles from a movable bed has been reported previously by many researchers. The phase diagram of bursting events is shown as Figure 1. Nakagawa and Nezu (1978) and Grass (1971, 1982) indi cated that just above the bed the sweep event is the main bursting event responsible for transfer of momentum into the boundary layer. In addition, studies of sediment transport, have shown that sed iment entrainment occurs from the bed primarily during sweep events (e.g. William 1990, Thome et al. 1989). These bursting events impose a significant pressure fluctuation on the bed. It is these fluctuations that are condidered to have a significant influence on the entrainment of sediment par ticles from the bed with a resultant, variable temporal sediment rate. Furthermore, these pressure fluctuations lower the local pressure near the bed and particles may be ejected from the bed or by rolling and sliding. Even sheltered particles can be entrained by this mechanism (Raudkivi 1990). Analysis of sweep events, therefore, may assist with defining incipient motion of sediment parti cles. As part of an ongoing investigation into the entrainment of sediment particles, the turbulence characteristics of flow in a laboratory channel were measured and has been reported by Keshavarzy and Ball (1995). It was found that the instantaneous turbulent shear stress applied at the bed for an individual sweep event may be as high as 2.3 times larger than the time average of the total turbu lent shear stress. The magnitude of this instantaneous turbulent shear stress for sweep events is shown as a function of flow depth in Figure 2 where RSS4 is defined by Average Shear Stresslsweep) RSS 4 (1) Time Average Turbulent Shear Stress (total) JOURNAL OF HYDRAULIC RESEARCH. VOL. 33. IV95, NO. 5 723 244 Appendix A The results obtained suggest that there are differences in the instantaneous shear stress and the mean shear stress. These differences result, in any instant, in a spatial variability of the shear stresses applied to the sediment particles with, consequently, some particles in motion and others stationary while consideration of only the mean shear stress would suggest general motion. There fore, definition of incipient motion requires consideration of the probabilistic nature the turbulent structure of the flow. v' h Variation of shear stress in sweep event ejection outward interactions 6 ■ («' < O.v' > 0) (a’ > O.v' > 0) 5 ■ . . +2SEE V / 4 ------> .• g 3. os / \ 2 • («' < 0. V- < 0) («' > O.v' < 0) 1 inward interactions sweep u * 1 i • * 1 * 0 0.1 02 03 0.4 0.5 0.6 0.7 0.8 d/H Fig. 1. Bursting events and their regions. Fig. 2. Normalised shear stress in sweep event. Appendix/References Grass, A. J. (1971), Structural features of tubulent flow over smooth and rought bed, Journal of Fluid Mechanics, Vol. 50, Part 2. GRASS, A. J. (1982), The influence of boundary layer turbulence on the mechanics of sediment transport, ' Euromech 156, Mechanics of sediment transport, Lnstanbul. Keshavarzy, A. and Ball, J. E. (1995), An investigation of the turbulence structure of flow in an open channel with a rough bed, Journal of Hydroscience and Hydraulic Engineering, JSCE, (submitted). Nakagawa, H. and NEZU, I. (1978), Bursting phenomenon near the wall in open channel flows and its simple mathematical model, Mem. Fac. Eng., Kyoto Uni., Japan, XL(4), Vol. 40. Raudkivi, A. J. (1990), Loose Boundary Hydraulics, 3rd Ed. Pergamon Press. Thorne, P. D., Williams, J. J. and Heathershaw, A. D. (1989), In situ acoustic measurements of marine gravel threshold and transport, Sedimentology, Vol. 36. Williams, J. J. (1990), Video observations of marine gravel transport, Geo. Mar. Lett., Vol. 10. JOURNAL DE RECHERCHES HYDRAULIQUES. VOL. 33. 1995. NO. 5 Reply By Authors: The authors are also appreciative of the comments of Ball and Keshavarzy. They are correct in pointing out that whether or not a grain moves is dependent upon instantaneous events near the bed rather than the mean flow. In a turbulent flow these events are in turn strongly influenced by the presence of coherent structures such as the burst-sweep cycle. The authors’ own treatment is implicitly based upon a similarity assumption, according to which relatively low-probability events associated with the mobilization on a bed grain are assumed to scale in an approximately universal way with some mean parameter of the turbulent flow near the boundary, i.e., the mean bed shear stress. Explicitly including the details of near-wall turbulence in a treatment of incipient sediment motion is a challenging task, and Ball and Keshavarzy thus deserve encouragement for their efforts. JOURNAL DE RECHF.RCHES HYDRAULIQUES, VOL. 33. 1995. NO. 5 Appendix A Appendix A-3 A Paper entitled: Instantaneous shear stress on the bed in a turbulent open channel flow HYDRA 2000, XXVIth IAHR Congress, London, Volume 1: Integration of research approaches and applications, edited by Ervine A., Paper No. 1A10, pp. 81-86,1995. 247 Appendix A lAlO Instantaneous shear stress on the bed in a turbulent open channel flow A. KESHAVARZY and JAMES E. BALL Water Research Laboratory, School of Civil Engineering The University of New South Wales, Sydney, NSW, 2093, Australia ABSTRACT The entrainment of sediment from a bed of an open channel is a function of mean shear stress and instantaneous shear stress. The importance of the sweep event on sediment motion from the bed has been reported by many researchers. In this study, the shear stress of the sweep event was investigated in open channel flow with a rough bed. The instantaneous velocities of flow were measured in a laboratory flume using a small electromagnetic velocity meter and the magnitude of forces on the bed determined. The analysis of data showed that in sweep events, the mean shear stress is approximately more than two times the total time averaged Reynolds shear stress. The mean angle of the sweep events with the bed was determined also and found to be about 28° from the main direction of the flow toward the bed. The results presented form part of a major study into the turbulence and sediment transport in open channels. INTRODUCTION Bed load entrainment by river and stream flow is recognised as one of the most important problems in the field of sediment transport. The mechanics of sediment entrainment is one of the process where turbulence imposes a dominant influence. The turbulent motion strongly influences the rate of entrainment, deposition and transport of sediment particles. The frequencies at which sand particles are entrained or deposited are associated with instantaneous hydrodynamic shear forces on the sediment particles. In turbulent open channel HYDRA 2000(Vol. 1). Thomas Telford, London, 1995 248 Appendix A RESEARCH APPROACHES AND APPLICATIONS flow, the hydrodynamic forces are not constant, but have a temporal variation. The entrainment and deposition of the particles not only depends on the mean shear stress at the bed but also on the instantaneous magnitude of the shear stress. One aspect of recent research in turbulence has been the study of the bursting process which is a sequence of events including sweep, ejection outward and inward interactions. Kline et ai (1967) introduced the concept of bursting phenomena as a process by which momentum transfers between the turbulent and the laminar region near the bed. The bursting process consists of four events such as: sweep (downward front of flow velocity) u'>0 and v'<0, ejection u'< 0 and v'>0, outward interaction u'> 0 and v'>0 and inward interaction w'<0 and v' <0. Grass (1971) used a hydrogen bubble technique to visualise the turbulent flow in open channel over smooth and rough bed. He concluded that the sweep and ejection events are responsible for most of the energy production and for the major contribution to Reynolds shear stress. He also found that beyond a certain distance from the boundary the turbulence intensity becomes independent of bed roughness. Thome et ai (1989) and Nakagawa and Nezu (1978) pointed out that the sweep and ejection events occur more often than outward and inward interactions. William (1990) and Thome et al. (1989) have shown that sediment entrainment occur from the bed mostly during the sweep event or flow with high velocity toward the bed. Most of the existing critical shear stress models in sediment transport model are based on an average channel shear stress, which is defined in terms of depth, density of the flow and energy gradient. The particles on the bed however are subjected to instantaneous shear stresses much higher than the critical shear stress base a time averaged values. Raudkivi (1963) examined the sediment entrainment along the ripples on the bed and found that while the average shear stress is zero, entrainment occurs due to instantaneous shear stress. Novak and Nalluri (1975) studied the incipient motion of isolated sediment particles in circular and rectangular channel. They indicated that the effect of channel shape results higher turbulence intensity and critical shear stress as compared rectangular to circular cross section. They indicated that the critical shear stress obtained for entrainment of particles was appreciably lower than those generally adopted due to higher turbulence intensity in rectangular channel. Previous investigations on bursting events have mostly concentrated on the study of the turbulent structure in air flow by means of wind tunnels. However, due to differences between air flow and water flow it is necessary to investigate the 249 • .-.-V'V Appendix A TURBULENT FLOW AND EXCHANGE PROCESSES: 1A10 characteristics of the bursting process in water flow and particularly in open channels with natural bed roughness. In the study presented here, the characteristics of the bursting process have been investigated by measurements of instantaneous velocity within the flow. In this paper the magnitude of shear stresses and angles during the sweep event will be presented and discussed. EXPERIMENTAL DETAIL An experimental investigation was carried out in a non-recirculating tilting flume used for turbulence and sediment studies which consisted of a channel 0.61 m wide, 0.60 m high and 35 m in length. The bed was covered with 2 mm sand panicles while the side walls were made of glass. The longitudinal and vertical velocity components were measured by means of a small electromagnetic velocity meter. These measurements were performed at a distance of 7 m from upstream end of the flume. The velocity components of the flow were recorded and analysed. More details of laboratory flume are presented by saiedi (1993). RESULT AND DISCUSSION The magnitude of shear stress in the sweep event is different from other events and also differs from mean shear stress of the flow in open channel. The event shear stress was determined after categorising the experimental data into the four regions. The shear stress of the sweep event was calculated and then normalised by the total shear stress at that point of the flow. This analysis is shown algebraically as equation 1. The variation of normalised shear stress (RSS) for the sweep event is shown in Figure 1. It can be seen that the shear stress in the sweep event increases with the depth from the bed to the water surface. u'v' (event) t = -pu'v' and RSS (1) |wV|(f0/fl/) From the analysis of the experimental data, it was found that the magnitude of the shear stress in the sweep event was more than twice that the total shear stress near the bed. The normalised shear stress for the sweep event was determined as equal as 2.3-2.4 for d/H=0.07-0.15 as expressed in equation 2 . Consequently it would be expected that some particles which are not moved at the total shear stress are, in fact, able to move during the sweep event. RSS(Sweep) == 23-2.4 for d/H=0.07-0.15 (2) where; u' , v' = velocity fluctuations in horizontal and vertical direction of flow. 250 Appendix A RESEARCH APPROACHES AND APPLICATIONS Using regression analysis, the magnitude of the shear stress during a sweep event as a function of normalised depth is: RSS=2.4+1.72(dlH), (r=0.57, SEE=0.41, n=57, P<0.001) (3) Where; r=correlation coefficient. Variation of shear stress in sweep event . . • 6 ' ö % V° Fig. 1. Normalised shear stress in sweep event with depth The applied force on the bed particles depends on the inclination angle of the sweep force to the bed. The instantaneous angle of the sweep and the ejection events was determined from the experimental data using equation 4. The mean angle of the events was calculated by taking the average at a defined depth of flow within the sample period of time. v'(sweep) 9{sweep) = tan * (4) u'(sweep) Where; 0 = angle of the events from horizontal direction of the flow and u' y = velocity fluctuations in horizontal and vertical direction. Shown in Figure 2 are the average angles determined in this manner as a function of the normalised depth of flow for all experimental runs. It can be seen in Figure 2 that the angle of the sweep event in an open channel increases with depth from the bed to the free water surface. The angle of the sweep event is about 28° close to bed defined as d/H=0.07-0.15 and increases to 45° at the water surface. 251 Appendix A TURBULENT FLOW AND EXCHANGE PROCESSES: IA10 Using regression analysis, the sweep angles as a function of normalised depth was determined as; (Y=29+15.7X, x = 0.76 ) (5) which is very significant. This equation is plotted in Figure 2. Variation of sweep angle in depth < 20 Fig. 2. Variation of the sweep angle with depth in bursting process CONCLUSION The magnitude of the shear stress in each event including; sweep, ejection, outward and inward interactions is very different from the mean shear stress at a certain point of the flow. In the sweep event the shear stress is approximately 2.3-2.4 times of the overall shear stress in the region near the bed. These temporal variations of the shear stress, have a distinct impact on the motion of sediment particles due to the high instantaneous force applied to sediment particles. The average angle for the sweep events from horizontal direction of the flow was determined and found to be approximately 28° at the bed and 45° at the water surface. REFERENCES Antonia, R. A. and Krogstad P. A., (1993), Scaling of the bursting periods in rough wall boundary layers, Experiments in Fluids, 15, pp. 82-84. Bridge, J. S., (1981), Hydraulics interpretation of grain-size distributions using a physical model for bed load transport, J. of Sediment Petrology, Vol. 151,4. 252 Appendix A RESEARCH APPROACHES AND APPLICATIONS Bridge, J. S. and Bennet, S. J.t (1992), A model for entrainment and transport of sediment grains of mixed sizes, shapes, and densities, Water Res. Res., 28, 2. Grass, A. J., (1971), Structural features of turbulent flow over smooth and rough boundaries, J. of Fluid Mechanics, Vol. 50, Part 2. Grass, A. J., (1982), The influence of boundary layer turbulence on the mechanics of sediment transport, Euromech 156, Mechanics of sediment transport, Istanbul. Kline, S. J., Reynolds W. C., Schraub F. A. and Runstadler P.W., (1967), The structure of turbulent boundary layers, J. of Fluid Mechanics, Vol. 30, Part 4. Nakagawa, H., Nezu, I., (1978), Bursting phenomenon near the wall in open channel flows and its simple mathematical model, Mem. Fac. Eng., Kyoto University, Japan, XL(4), Vol. 40, pp. 213-240. Nezu, I. and Nakagawa, H., (1993), Turbulence in open-channel flows, IAHR- Monograph, Rotterdam: Balkema. Novak, P. and Nalluri, N. (1975), Sediment transport in smooth fixed bed channels, Proc. ASCE, Vol. 101, No. HY9. Raudkivi, A. J., (1990), Loose Boundary Hydraulics, 3rd Ed. Pergamon Press. Raudkivi, A. J., (1963), Study of sediment ripple formation, ASCE, J. of Hydraulics Division, Vol. 89, HY6. Saiedi, S., (1993), Experience in design of a laboratory flume for sediment studies, International Journal of Sediment Research, Vol. 8, No 3, Dec.. Thome, P. D., Williams, J. J. and Heathershaw, A. D., (1989), In situ acoustic measurements of marine gravel threshold and transport, Sedimentology, 36. Williams, J.J., (1990), Video observations of marine gravel transport, Geo. Mar. Lett., 10, pp. 157-164. Yalin, M. S., (1992), River Mechanics, Pergamon Press, Ltd. Inc.. 253 Appendix A Appendix A-4 A Paper entitled: The influence of the turbulent shear stress on the initiation of sediment motion in an open channel flow Published in 7th IAHR International Symposium on Stochastic Hydraulics,96 29-31 July 1996 Mackay, QLD, Australia. 254 Appendix A Stochastic Hydraulics 96, Tickle. Goulter, Xu, Wasimi & Bouchart (eds) © 1996 Balkema, Rotterdam. ISBN9064106177 The influence of the turbulent shear stress on the initiation of sediment motion in an open channel flow Alireza Keshavarzy & James RBall Deportment of Water Engineering, School of Civil Engineering, The University of New South Wales, Sydney, NSW., Australia ABSTRACT: The initiation of sediment motion is an important component in the study of sediment transport Most existing critical shear stress models used in sediment transport models are based on an time- averaged channel shear stress, which is defined in terra of depth, density of the flow and energy gradient whereas the particles on the bed sustain instantaneous shear stresses which differ from the temporal average shear stress. These differences are derived from the turbulent nature of the flow. Since particle entrainment into the flow occurs as a result of the applied shear stress at the bed, it is necessary to consider the temporal variation in the shear stress in the development of a sediment entrainment model. Presented herein are the results obtained from a study on the effect of flow turbulence on the entrainment of sediment particles from the bed of an open channel. The turbulence characteristics of the flow were measured in a flume with a rough bed under differing flow conditions. The instantaneous turbulent shear stresses were applied in a simple force balance model to define an entrainment function for the prediction of incipient sediment motion. It was found that particle entrainment but not general motion occurred at a lower critical shear stress than that predicted using an average channel shear stress model, v. 1. INTRODUCTION the shear stress was very small or close to zero, the particles were entrained by the influence of turbulent The bed load sediment in a river comprises those agitation. In a similar manner, Graf (1971) pointed particles which travel downstream more slowly than out that incipient motion of similar sized particles the flow whereas suspended sediment are those under a given flow condition was statistical in nature particles which travel downstream with the flow. due to the turbulence of the flow. The studies of Bed load sediment in a river channel moves in a Cheng and Clyde (1971), Christensen (1972), Blinco downstream direction by the rolling, sliding and and Simon (1974), and Grass (1970, 1982) have saltation of the sediment particles. The continuing found that fluctuations of the instantaneous shear motion of these particles requires the application of stress about the temporal mean are the result of the forces which exceed the critical force for that flow turbulence. This turbulence of the flow is also particle which is defined as the force required to the main mechanism resulting in the entrainment of induce motion of the particle. An estimation of the particles from the bed. Furthermore, Grass (1982) critical force is obtained the resistance force or the and Thome et al (1989) noted that the mode and immersed weight of the particle in the initial rate of sediment transport changes as a function of condition of motion. When the applied force turbulence. exceeds the resistance force, the particle commences Due to the temporally variable nature of turbulence, movement along the bed; this movement may be by there is a difficulty in defining initiation of motion rolling, sliding or saltating depending on the for a particle resting on the channel bed. This arises magnitude of the available shear stress. For from the need to consider the instantaneous situations near, the threshold of movement, Le. the turbulent shear stress which is sometimes lower and initiation of motion, rolling is the dominant mode of sometimes higher than the critical shear stress for bed load transport for sand and gravel particles. that particle. Schober (1989) indicated that even for Raudlrivi (1990) observed that, particularly where a plane bed there are difficulties in defining incipient 255 Appendix A motion because of the random interference of 2. INITIATION OF SEDIMENT MOTION panicle stability and flow behaviour. Similar conclusions have been noted by, for example, Chiew Row in most naturally occurring channels varies and Parker (1994) who pointed out the difficulty of temporally. Consequently, the sediment particles on sediment entrainment in open channel flow, and Ball the bed of a channel will be subjected to periods and Kcshavarzy (1995) who discussed the difficulty where they will be in motion and periods where they of defining incipient motion of sediment particles. will be stationary. Long-term analysis of sediment The mechanism of the bursting process which is a motion, therefore, requires investigation of the process or a mechanism by which momentum is interface between the periods of sediment motion transferred into the boundary layer and its influence and the periods where the sediment is stationary. on the entrainment and movement of particles has This requires the investigation of the initiation of been a focus of many previous studies. The bursting sediment motion. process was introduced by Kline et aL (1967) as a Yalin (1972) states that in steady uniform flow of process which consists of four categories of events: water and sediment particles seven basic parameters these categories are the sweep (u’>0, v‘<0), ejection are needed to define the flow conditions; these (u'<0, v'>0), outward interaction (u>0, v’>0) and parameters arc: inward interaction (u'<0, v'<0) with each event having a different phase of action. Where u' and v' density of water (p ), are velocity fluctuations about temporal average in density of sediment (Pj), horizontal and vertical directions, respectively. dynamic viscosity (v), Bridge and Bennet (1992) noted that these four particle size (D), alternative types of bursting events have different flow depth (H), effects on the mode and rate of sediment transport channel slope (S), and Studies by Thome ct aL (1989), William (1990), acceleration of gravity (g). Nelson et aL (1995), Bennet and Best (1995) and These seven basic parameters can be reduced to a Drake it al. (1988) indicate that close to the bed set of three dimensionless parameters which are: most of the sediment entrainment occurs during the sweep event; this is particularly the case for coarse a mobility parameter u*2/(s-l)^D. sands and gravel sized particles. Shown in Figure 1 a particle Reynold’s number Re=u*D/v and is a schematic representation of how the sweep a specific density parameter pj/p . ‘event applies force towards the bed and induces motion of sediment particles. The forces applied to a sediment panicle on the bed of a horizontal plane are the tractive shear stress, a In this study, the instantaneous shear stress of the resistance force and a horizontal drag force due to a flow was applied in a force balance model in order pressure differential between the upstream and to develop a stochastic entrainment function that downstream sides of the particle. This drag force included the influence of turbulence. induces an agitation force for coarse sand and gravel sBSISSmsi sweep Fig. 1: Sweep event in a bursting process (after Yalin, 1992) 256 Appendix A Fig. 2. Applied Forces on Sediment Particles at the bed bed sediments which causes panicles to move by Engelund and Fredsoe (1976) introduced a model rolling and sliding. for sediment transport modeL They used a force balance model for estimation of bed load and From theoretical considerations, the forces particle velocity at the bed of open channel. This influencing entrainment into the flow of sand and model is given as foUow; gravel from the bed of an open channel are the drag force and the resistance force. These applied forces |c0pA;,(cu/ where u* is average shear velocity, Up is the average F0=ic„pAv,! (1) particle velocity, ß=tan where; The initiation of motion for sediment particle resting at the bed is controlled by the applied instantaneous F^ = ^KZ>3(pi-p)g (3) shear stress at the bed. These forces are generated by bursting processes and impinge on the boundary where; Fq is the drag force, Fr is the resistance layer. The most important event for sediment motion force, Fg is the gravitational force or weight of the in the bursting process is the sweep event which has sediment particle, ß is friction coefficient Cp is the a dominant role in entrainment of sediment particles drag coefficient for the sediment particle, Ap is the at the bed. The sweep event applies shear in the cross sectional area of the grain exposed to drag direction of the flow and provides additional forces which is given by lMrcD2, p is the density of to the viscous shear stress. Keshavarzy and Ball water p, is the density of sediment particle, D is the (1995) reported that the magnitude of instantaneous particle diameter, g is the acceleration due to shear stress in sweep event is much larger than gravity and Vr is relative velocity which is given by temporal mean shear stress. Vr-(iu- up) (4) During a sweep event, the impact of the flow In which u* is the shear velocity and Up is the velocity at the bed causes an initiation of particle particle velocity. Figure 2 shows the particles and motion due to the applied instantaneous force on the applied forces on the bed particles. particle. The applied shear stress resulting from a sweep event depends on the magnitude of the turbulent shear stress and its probability distribution. 257 Appendix A This magnitude of the shear stress in a sweep event gravitational coefficient, R is hydraulic radius and Sf is particularly important close to the bed. In Figure 3 is energy gradient of the flow. it is shown that how the magnitude of normalised instantaneous shear stress in a sweep event varies in The magnitude of the shear stress also depends on different point in depth (d) which is normalised by the impinging angle towards the bed. If the flow total depth of flow (H). The magnitude of the velocity in a sweep event is more tangential with the instantaneous shear stress in a sweep event was bed, then the applied force on particle movement is found to be approximately 1.4 times of the temporal more effective in entrainment. If the velocity in the mean shear stress for a region close to the bed. sweep event tends to be normal to the bed, the applied force for sediment entrainment has a minimum effect on entrainment. Thus the angle of Variation of shear stress in sweep velocity in sweep event has an important influence event on sediment entrainment. It is shown in Figure 4 how the angle of the sweep event influences the applied force on a sediment particle. * X* ' . . us« . - X • X' * Flow direction Fig. 3. Instantaneous shear stress in sweep event If the instantaneous shear stress in the sweep event Fig. 4. Inclination angle of the sweep event is x and the time averaged shear stress at each point of flow is x, the ratio of shear stress at a point of flow for sweep event is: The angle of sweep events (0) was calculated from the experimental data. It was found to be about 22° « = ö, or x = a.x (6) from the horizontal for a region very close to the x bed. The variation of this angle with flow depth is and shown in Figure 5. x =x +x' (7) where V is turbulent shear stress and is given by: Another important parameter in the entrainment of sediment particles is the probability of a sweep event x' = p.i42 . (8). (p). The probability of occurrence of sweep event was calculated from the experimental data and is Therefore, the value of turbulent shear stress is shown in Figure 6. It was found that the frequency calculated as follow; Variation of sweep angle In depth x' =x -x (9) If equation 6 is substituted in equation 9, the turbulent shear stress is calculated by x'=x(a-l) (10). At the bed the time-averaged shear velocity is given as «, = (t./p>m«(«AS )M (11) where u* is the shear velocity, x0 is the shear stress at the bed, p is the flow density, g is the Fig. 5. Variation of the sweep angle in depth 258 Appendix A 4 u *ß or kn /k (19) 21 3X.C, in which, X= shield's parameter (S - [)gd 4. EXPERIMENTAL DETAIL Experimental tests for the analysis described above were carried out in a sediment transport flume of 30 A O 04. m length, 0.60 ra width and 0.60 m height at the Water Research Laboratory, UNSW. The velocity Fig. 6. frequency of Sweep event in bursting process components in two directions of the flow were measured using a very small electromagnetic of sweep events was 30 percent of the total time for velocity meter. The bed of the flume was covered by bursting process. concrete sheet and a sediment particles with D5q=2 mm size diameter. More detail of the experimental The effect of instantaneous turbulent shear stress is equipment are described by Keshavarzy and Ball investigated as those parameters are applied in a (1995). The velocity components were recorded for force balance model in order to define the initiation subsequent analysis to calculate time averaged of sediment particles with the influence of velocity in two directions and the instantaneous turbulence. This model is based on the following shear stress for bursting events. The turbulence relationships which include the effects of the intensity of the flow determined and compared with instantaneous shear stress. previous data. -C„ pA (au -u )2 =—J3('p 5-p)*.ß(l2) 2 Dw pK * p} 6 ‘ 5. RESULTS AND DISCUSSIONS \ From the experimental study of the turbulence it was k^au.-up) = k2 (13) found that the magnitude of the shear stress in a sweep event is much higher than that of time- (a£*-üpj =k2/kl (14) averaged shear stress in a point of the flow. The magnitude, probability, and angle of instantaneous tt* = U* + uZ (15) shear velocity in the sweep event are reported by Keshavarzy and Ball (1995, 1996a). These bursting uZ - (da-l.cosQ.p)u* (16) parameters influence the particle entrainment in the bed of an open channel flow. The magnitude and where; u* is the total instantaneous shear velocity, Up is the particle velocity along the bed, uZ is the 0.10 * IllUlUMMI a * instantaneous sweep shear velocity, 0 is the sweep 0.09 A Um« tnra|«4 4 * « • angle and p is the probability of occurring sweep ^ 0.0* ■ •4(1 ^ * • event in a bursting process. « 0.07 4 . • => a" .* £ 0.04 The values of kt and k2 in equation 13 and 14 are !l A • ’S 0.05 A. • defined as; • * • J 0.04 A • ^ 0.03 A • 1 , „ A | 002 *1 “ 2 C° (17) 0.01 *2 = ^-rf3(pj-p)y.ß 0.00 0 2 4 6 I to 12 14 16 II 20 Partlda d&amcUr. i (mm) (18) 2 1 3 CD Fig. 7. Comparison of Critical shear velocity 259 Appendix A 0.010 velocity are shown in Figure 8. The normalised difference in shear stress is shown in Figure 9. As a result, the initiation of particle influence by instantaneous shear stress could start earlier than time average shear stress. This agitation which occurred into the initiation of motion caused is due O.OOJ # to the turbulent shear stress which imposed to the ♦ particles on the bed by turbulence. 0.000 1 ...... —...... 0 2 4 « ■ 10 12 14 1« 1« io 6. CONCLUSION Factld« 4«m«Wr, 4 (mm) ♦ mverrnf* modtL, (/’••■Mkamwu The influence of turbulence on the entrainment of sediment particle from the bed was investigated Fig. 8. Differences in critical shear velocity through an analysis of the bursting process and in particular the sweep event as applied in a force 0.1 balance model. The effect of sweep in the entrainment of sands and particles has been reported o.u ♦*♦♦♦♦♦♦♦♦♦♦♦•♦♦♦♦♦ by many researchers, however no attempt was made to predict the actual sediment rate with the influence of turbulence. In this study the influence of sweep it event on initiation of motion resulted an indication £> of sediment motion with the effect turbulence. It 0.03 was found that due to instantaneous turbulent shear stress in sweep event particles start to move earlier than predicted by time averaged shear stress models. 0 14 6 I 10 12 M 16 t« 20 fmrtld« duaaUr, d (mm) V ♦ V* Fig.9. normalised differences for two models Ball, J.E. and Keshavarzy, A., 1995. Discussion on Incipient Sediment motion on non-horizontal slopes by Chiew and Parker. J. of Hydraulics probability of the instantaneous shear stress is the Research, 33(5):723-724. most important factor for particle entrainment in a Bennett, S.J. and Best, J.L., 1995. Mean flow and mobile bed. Those parameters were derived by turbulence structure over fixed, two dimensional experimental data in a flume over a rough bed. The dunes: Implications for sediment transport and instantaneous applied forces on the particles bed form stability. Sedimentology, 42:491-513. influence on the estimation of sediment rate in open Blinco, P.H. and Partheniades, E., 1971. Turbulence channel. Those turbulence parameters arc applied in Characteristics in Free Surface Flows Over a force balance model to calculate the actual particle Smooth and Rough Boundaries. J. of Hydraulics velocity in the mobile bed. Research, 9(1):43-71. Blinco, P.H. and Simon, D.B., 1974. Characteristics After solving equations 12 and 5, the velocity of of Turbulent Boundary Shear Stress, ASCE, J. of particles for different shear velocity for two different Eng. Mech. Div., 100:203-220. cases (with and without the influence of turbulence) Bridge, J.S., 1981. Hydraulics interpretation of were calculated. The magnitude of critical shear grain-size distributions using a physical model for velocity for instantaneous shear stress and time- bed load transport, J. of Sediment Petrology, averaged shear stress were compared as shown in 51(4): 1109-1124. Figure 7. For a particular particle diameter, it could Bridge, J.S. and Bennet, S.J., 1992. A Model for the be seen that the shear velocity for initiation of Entrainment and Transport of Sediment Grains of particle motion in proposed model is lower than time Mixed Sizes, Shapes and Densities, Water averaged model. For different particle diameter Resources Research, 28(2):337-363. (1mm-20mm) these difference between critical shear Cheng, E.D.H. and Clyde C.G., 1971. Instantaneous 260 hydrodynamic lift and drag forces on large roughness elements in turbulent open channel flow, in Sedimentation edited by Shen E.W., pp. [3-IH3-20], Fort Colin, Colo. Drake, T.G., Shreve, R.L., Dietrich, W.E., Whiting, P.J. and Leopold, L.B., 1988. Bed load transport of fine gravel observed by motion-picture photography, / of Fluid Mechanics 192:193-217. Engelund, F. and Freds0e, J., 1976. A Sediment Transport Model for Straight Alluvial Channels, Nordic Hydrology, 7:293-306. Graf, W.H., 1971. Hydraulics of Sediment Transport, McGraw-Hill, New York. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries, J. of Fluid Mechanics, 50(2):233-255. Grass, A.J., 1982. The influence of boundary layer turbulence on the mechanics of sediment transport, Euromech 156, Mechanics of Sediment Transport, Balkeraa, Rotterdam,. Keshavarzy, A. and Ball, J.E., 1995. Instantaneous shear stress on the bed in a turbulent open channel flow, Proc. of XXVIIAHR Congress, London. Keshavarzy, A. and Ball J.E., 1996a. Characteristics of turbulent shear stress in open channel, accepted, 7th IAHR International Symposium on Stochasic Hydraulics, Mackay, QLD, Australia. Nelscn, J.M., Shreve, R.L., Mclean, S.R. and Drake, T.G., 1995. Role of near-bed turbulence structure in bed load transport and bed form mechanics, Water Resources Research. 31(8):2071-2086. Raudkivi, A.J., 1990. Loose Boundary Hydraulics, 3rd Ed., Pergamon Press, Oxford. Thome, P.D., Williams, J.J. and Heathershaw, A.D. 1989. In situ acoustic measurements of marine gravel threshold and transport, Sedimentology, 36:61-74. Williams, J.J., 1990. Video observations of marine gravel transport. Geo. Mar. Lett., 10:157-164. Yalin, M.S., 1972. Mechanics of Sediment Transport, Pergamon Press, Oxford. Yalin, M.S., 1992. River Mechanics. Pergamon Press, Oxford. Appendix A Appendix A-5 A Paper entitled: Characteristics of turbulent shear stress applied to bed particles in an open channel flow Published in 7th IAHR International Symposium on Stochastic Hydraulics,96 29-31 July 1996 Mackay, QLD, Australia. 262 Appendix A Characteristics of turbulent shear stress applied to Bed particles in an open channel flow Alireza Keshavarzy and James E. Ball Department of Water Engineering, School of Civil Engineering The University of New South Wales Sydney, 2052, Australia ABSTRACT: The entrainment of sediment particles is influenced by many characteristics of the flow and particularly the turbulence which produces stochastic shear stress fluctuations at the bed. Recent studies of the structure of turbulent flow have recognised the importance of bursting processes for description of the turbulent processes. Of these process, the sweep event has been suggested to be the most important turbulent event for entrainment of particles from the bed. It imposes variable forces in the direction of the flow and assist movement of particles by rolling, sliding and occasionally saltating. In this study, the characteristics of bursting processes and, in particular, the sweep event were investigated in a flume with a rough bed. The velocity fluctuations of the flow were measured in two-dimensions using a small electromagnetic velocity meter and the turbulent shear stresses were determined. The instantaneous shear stress applied to sediment particles on the bed resulting from sweep events depends on the magnitude of the turbulent shear stress and its probability distribution. From a statistical analysis of the experimental data, it was found that the magnitude of the turbulent shear stress in sweep and ejection events had a normal distribution after application of a Box-Cox transformation. This enabled determination of the mean shear stress, angle of action and standard error of estimate for the bursting events. It was found that the mean turbulent shear stress for a sweep event was 40 percent higher than for the time-averaged shear stress. phase where the event occurs; shown in Figure 1 is a 1. INTRODUCTION phase diagram with the zone of each event indicated. As illustrated in Figure 1 the sweep event results in The initiation of motion or threshold of movement of the shear stress being angled towards the bed of the sediment particles at the bed of open channels is an channel. important component of the management of river systems. Among the many processes which influence Recently, the contributions of coherent structures, the availability and entrainment of sediment particles, such as the sweep and ejection events, to momentum the turbulence of the flow and the associated transfer have been extensively studied by quadrant coherent structures are important but, as yet are not analysis or probability analyses based on two- completely understood. In this regard, Nezu and dimensional velocity information. Studies by Nakagawa (1993) pointed out that the mechanism of Nakagawa and Nezu (1978) and Grass (1982, 1971) turbulence in flow over rough beds in rivers required have indicated that just above the channel bed, the investigation. Associated with turbulence and sweep event is more responsible than the ejection momentum transfer are the processes by which event for transfer of momentum into the bed layer. sediment particles are entrained and transported with Nakagawa and Nezu (1978), Thome et al. (1989) the flow. and Keshavarzy and Ball (1995) all concluded that sweep and ejection event occurred more frequently The concept of the bursting phenomenon was than outward and inward interaction events. introduced by Kline et al. (1967) as a means of describing the transfer of momentum between the The four types of bursting events identified earlier turbulent and laminar regions near the boundary. The have different influences on the rate, and mechanisms phenomenon was considered to consist of four of sediment entrainment in a turbulent flow. In separate events which are defined by the zone in a studies of sediment transport by, for example, Thome et al. (1989) and Williams (1990) it has been found 263 Appendix A that sediment entrainment occurs from the bed more channel flow over a rough bed will be investigated. frequently during sweep events and occasionally The magnitude of the shear stress, the frequency and during outward interaction events whereas transport the angles of all events will be determined from of suspended sediment depends on the ejection event experimental data. The variation of shear stress, Keshavarzy and Ball (1995) also pointed out that the frequency and angle for each individual event with magnitude of the instantaneous shear stress during depth will be presented also. sweep event is greater than that which occurs during outward and inward interactions events. Bursting 2. EXPERIMENTAL APPARATUS events impose a rapid and significant pressure fluctuation on the bed; it is these fluctuations that are The experiments undertaken during this study were considered to have a significant influence on the carried out in a non-recirculating tilting rectangular entrainment of sediment particles from the bed with a flume of 0.61 m width, 0.60 m height and 35 m resultant variable temporal rate of sediment length. The side walls of the flume were made of entrainment. Ball and Keshavarzy (1995) pointed out glass, making it possible to observe and record the the effect of instantaneous shear stress on incipient flow. The bed was constructed with movable motion and the consequent reasons for the difficulty concrete sheets covered by sand particles of in defining the sediment entrainment function. These D5o=2mm for this study. As a result, it is possible to instantaneous pressure fluctuations lower the local perform experiments with different bed roughness pressure near the bed and hence particles may be simply by changing the concrete sheets. ejected from the bed by hydrostatic pressure. The longitudinal and vertical components of the Raudkivi (1990) postulated that even sheltered instantaneous velocity were measured using a small particles can be entrained by this mechanism. Despite electromagnetic velocity meter (henceforth referred the importance of the bursting events in sediment to as EMC). Use of a trolley mounting device transport, the statistical characteristics of bursting enabled the EMC to be moved to any location for events have not been investigated in sufficient detail measuring of the velocity fluctuations. The during previous studies. measurements were performed along centerline of the flume at a location of 7 m downstream from the v' inlet of the flow. 1 These recorded velocities were analysed to calculate ejection outward interactions the time averaged velocity in the horizontal and vertical directions, the overall mean shear stress, (u' < 0,v' > 0) (u' > 0,v' > 0) turbulent velocity fluctuations, the mean shear stress \ / — for each event, and to count the number of bursting events during the sample period. The analysis also / determined the angle from the horizontal for the Cu' < 0, v' < 0) (u‘ > 0, v' < 0) sweep and ejection events. inward interactions sweep Shown in Table 1 are details of flow conditions for a number of experimental tests. The velocity at *For definition of u' and v' see eqs 3 and 4. different point of flow normalised by cross sectional mean velocity (U) for some tests and shown in Figure Figure. 1: Bursting events and their associated regions 2 as a normalised depth (d/H). As expected, the normalised velocity profile are similar suggesting In the study reported herein the statistical that the effect of variation of flow conditions are characteristics of the bursting phenomenon in an open minimal. 264 Appendix A Table 1: The hydraulic conditions of the flow in the experiments Test E F G H J K L MN 0 P Q Q(l/s) 63.7 76.3 52 58 73.6 40.2 61 30.3 22 48.5 79.8 97.5 H(mm) 355 154 276 120 145 120 212 154 70 166 230 265 T(°Q 15 15 15 15.5 14 13.5 13.5 13 12.2 13 13 12.6 U(m/s) 0.29 0.81 0.31 0.79 0.83 0.55 0.47 0.32 0.51 0.48 0.57 0.60 i a 1 0.35 0.14 0.38 0.86 0.99 0.65 0.58 0.40 0.56 0.38 0.38 0.37 Ft 0.16 0.66 0.19 0.73 0.7 0.51 0.33 0.26 0.62 0.38 0.38 0.37 V4l>, (2) N w where u( and v; are instantaneous velocities, U and V are the local temporal mean velocities in the • For the velocity fluctuations in two components 04 about the mean value were given by 03 u'i =u.-U (3) 02 - v[ = v, - V (4) where u ( and are the velocity fluctuations in the longitudinal and vertical directions, respectively. • For the root mean square value of the velocity rn v(u‘-er)3 (5) Fig. 2. Normalised velocity profile of the experimental runs “"“=VU =# N 3. ANALYSIS OF EXPERIMENTAL DATA Ft U(v,-Vf V~=VV =1|£ * (6) As previously noted, data obtained from the experiments were analysed to calculate the following For the Reynolds shear stress characteristics of the flow: time-averaged velocity, x = —pu'v' (7) velocity fluctuations, root mean square and the Reynolds shear stress. Relationships used were: “'v'=j:'Zuyi (8) N i=l • For the time averaged mean velocity of the The experimental data were employed also to longitudinal and vertical components calculate the turbulence intensity of flow in the horizontal and vertical directions. The turbulence y=-^Zu. (i) intensities in the horizontal and vertical directions JY M 265 Appendix A normalised by the flow shear velocity (u*), are shown 4. RESULTS AND DISCUSSION in Figures 3 and 4 as a function of the normalised depth. To validate the data collected, the turbulence The analysis of the experimentally determined data intensities in the present experiments were compared resulted in the characteristics of the bursting events with previously published data. As shown in Figures differing with the normalised flow depth. Important 3 and 4 close agreement was obtained. aspects of the analysis are presented in the following discussion. The Reynolds shear stress normalised by the shear velocity is shown in Figure 5. Values obtained during this study were compared with previously published 4.1 Box-Cox transformation data and a good agreement was found in this The magnitude of the mean instantaneous shear stress parameter also. for each category of event is different, and it differs from the overall mean shear stress for the flow. The mean normalised shear stress for each event was determined for alternative normalised flow depths. At««-k Horizontal Turbulence Intensity A T«*t-K O ToU. Vertical Turbulence Intensity + Te»l-M + Tort-M □ Tcw-N □ Te*t-N O Tctt-O O Tcfl-O Xt««-p X Te»t-P | • Lain • L»«fcr.l950 ■ Gnu ♦ McQwvcy* Fig. 3. Turbulence intensity in horizontal direction Fig. 4. Turbulence intensity in vertical direction • Nakagaw* et «L 1975, Grass. 1971, Läufer, 1950, McQuivey & Richardson 1969 * H> ,i I «an»« *a— Fig. 5. The normalised Reynolds shear stress compared with previous data 266 Appendix A Prior to undertaken any statistical analysis, a Box- If the transformed data is represented by C2, then the Cox transformation was used to transform the reverse transformation from C2 to the original data is original data set into a normally distributed data set. defmed as As presented by Box and Cox (1964), the transformation is defined by Zt=e(."(^>)A )=eK m (zf+c)x-l .r , .n where ------— ‘f X*° (9) [ln(z,+c) if X = 0 KJn(QA+_l) (U) A where 2^ are the original data, and C is a constant. If all of the values in the time series are greater than The frequency distributions after transformation of zero, then the constant C usually is set to equal zero shear stress in the sweep and ejection events are (Box and Cox, 1964). Application of the shown in Figures 6 and 7. The probability distribution transformation to the data set comprising more than of the transformed shear stress for sweep and ejection 50 tests of sweep and 50 tests for ejection events events is shown also in Figures 8 and 9. resulted in a value of X equal to 0.28. Frequency Distribution of Shear Stress Frequency Distribution of Shear Stress Sweep Event Ejection Event 20 - C2 Fig. 6: Frequency Distribution of shear stress in sweep event Fig. 7: Frequency Distribution of shear stress ejection event Normal Probablity Rot tS*»««p Evrt) _ l_ _ i _ f - n1111 1---- « •__ ( « I I - • "1" r Av«fM«:*-****4* Fig. 8: Probability Distribution of shear stress in sweep event Fig. 9: Probability Distribution of shear stress in ejection event 267 Appendix A 4.2 Instantaneous shear stress • Ejection RSS2=U5+1.83(d/H), (r=0.81 ,SEE=0.38), (14) The mean values of the transformed data (C2) were calculated and reverse transformation applied. It was where RSS is shear stress ratio and r is correlation found that the magnitude of the shear stress for a coefficient. sweep event was approximately 1.4 times the mean shear stress close to the bed. Shown in Figures 10 4.3 Frequency of the events and 11 are the variation of the normalised mean shear The instantaneous shear stress data was divided into stress for each event in relation to normalised depth different groups based on phase of the bursting of flow for all of the experimental runs. The process. The frequency of each event was normalised mean shear stress for each event is determined by determined from equation 12. It can be seen that the mean shear stress for an individual event increases pk=~.7 “d n = 5>* (15) with depth from the bed to the free surface. /V * = 1 where P is the probability of an event occurring, nk is r< u'v' the number of event in each category, and N is total = (12) * «»Wi number of events. The probability of each category of event was where RSS is the ratio of shear stress in the event calculated for all of the experimental data and is shown as a function of normalised depth of flow in A focus of this study was analysis of the Figures 12 and 13. It can be seen that the probability characteristics of the bursting events and particularly, of an individual event varies with depth. From these the sweep event in the region close to the bed. This figures, it can be seen also that the occurrence of the focus resulted from a need to consider the influence sweep and the ejection events, particularly close to of the shear stress magnitude during a sweep event the bed of an open channel, are higher than outward on particle entrainment. Due to the instantaneous and inward interactions. Close to the bed, the sweep shear stress being greater than the mean shear stress, and the ejection events have a probability of it is expected, that particles which would not move at occurrence of approximately 30%, whereas those of the mean shear stress are able to move during a the outward and the inward interactions are sweep event due to the higher induced shear force. approximately 20%. As a result, the sweep and ejection events occur approximately 30 percent more Using the experimental data, regression equations frequently than the outward and the inward were developed for the ratio of the mean interactions. instantaneous shear stress to the mean flow shear stress. These relationships as plotted with 95% Additionally, the probability of the sweep and the confidence limits in Figures 10 and 11, are ejection events decreases with depth from the bed to the water surface, whereas those of the outward and • Sweep RSS4=1.48+1.13(dM), (r=0.73 ,SEE=0.36) (13) Variation of shear stress in sweep Variation of «hear stress in ejection event event Fig. 10. Normalised shear stress in sweep event Fig. 11. Normalised shear stress ih ejection event 268 Appendix A the average after original data transformed by Box- inward interactions increase with the depth. Cox transformation at a particular depth within the Furthermore, as shown in Figures 12 and 13, the flow and a period of time. probability of the events approaches a value of 25% v'(sweep) near the free water surface which suggests that all 9 (sweep) = tan 1 (18) events are equally probable. _ u!(sweep). v'(ejection) The relationship between the probability of the event 0 (ejection) = tan"1 (19) and the normalised depth was investigated also. A . u!(ejection). linear regression was performed between these two where 9 is the angle of the event from the horizontal variables resulting in , for a direction (clockwise). Angles determined from the experimental data are • Sweep event shown in Figure 14 and 15 in relation to the Y=319-7.2X (n=0.77) (16) and normalised depth of flow for all experimental runs. It • Ejection event can be seen that the angle of the sweep event Y=31.3-8.9X). (p=0.83) (17) increases with depth from the bed to the free water 4.4 Angle of the events surface. The angle of the sweep event is about 20° The applied force on the bed particles depends on the close to bed and increases to 45° at the free water inclination angle of the sweep force to the bed. The surface. In a similar manner, the angle of ejection instantaneous angle of the sweep and the ejection events is about 20° near the bed and increases to events was determined from the experimental data approximately 45° at the free water surface. using equations 18 and 19. The mean angle of the events was calculated by taking 25 •• fM" O ° O O O !• A imp O o.I. O o.I. Fig. 12. frequency of the Sweep and O.I. events Fig. 13. frequency of the ejection and I.I. events Variation of ejection angle in depth Variation of sweep angle in depth d/H d/H Fig. 14. Variation of the sweep angle in depth Fig. 15 Variation of ejection angle with depth 269 Appendix A Keshavarzy, A. and Ball, J.E., 1995. Instantaneous 5. RESULTS AND CONCLUSIONS shear stress on the bed in a turbulent open channel From the analysis of the bursting process, the flow, Proc. of XXVIIAHR Congress, London. following observations were made; Kline, S.J., Reynolds, W.C., Schraub, F.A. and Runstadler, P.W., 1967. The structure of turbulent • The magnitude of instantaneous shear stress in an boundary layers, J. of Fluid Mechanics, 30(4):741- event differed from the mean shear stress. In the 773. sweep event the shear stress was approximately 1.4 Läufer, J., 1950. Some recent measurement in a two times of the mean shear stress in the region near the dimensional turbulent channel, J. of Aeronautical bed and increased to approximately twice the mean Science, 17(5):277-287. shear stress near the water surface. For the ejection Lyn, D.A., 1986. Turbulence and turbulent transport events, the shear stress ratio is about 1.4 near the bed in sediment-laden open-channel flows, Report KH- and increases to approximately 2.5 at the water R-49, W. M. Keck Laboratory of Hydraulics and surface. Water Resources, California Institute of Technology, Pasadena, Calif. • The frequency of each event was determined from McQuivey, R.S., and Richardson, E.V., 1969. Some the experimental data. It was found that, near the turbulence measurement in open-channel flow, J. bed, the probability of occurrence for the sweep and of Hydraulics Division, ASCE, 95(HYl):209-223. ejection events was 30%, whereas that of outward Nakagawa, H., Nezu, L and Ueda, H., 1975. interaction and inward interaction events were 0.20. Turbulence of open channel flow over smooth and Thus, the outward and inward interactions occur rough beds, Proceeding of the Japan Society of 30% less often than sweep and ejection events.' Civil Engineers, 241:155-168. • The average angle for the sweep and ejection events Nakagawa, H. and Nezu, L, 1978. Bursting from horizontal in the clockwise direction was phenomenon near the wall in open channel flows determined also. This angle was found to vary with and its simple mathematical model, Mem. Fac. depth from approximately 20° at the bed to 45° at the Eng., Kyoto University, Japan, XL(4),40:213-240. water surface for both the sweep and ejection events. Nezu, L and Nakagawa, H., 1993. Turbulence in open-channel flows, IAHR Monograph Series, • These temporal variations in the instantaneous A.A. Balkema, Rotterdam. shear stress, have a distinct impact on the incipient Raudkivi, AJ., 1990. Loose Boundary Hydraulics, motion of sediment particles and it is suggested that 3rd Ed., Pergamon Press, Oxford. these temporal variations are one of the reasons that Thome, P.D., Williams, J.J. and Heathershaw, A.D., it is difficult to define incipient motion of sediment 1989. In situ acoustic measurements of marine particles. gravel threshold and transport. Sedimentology, 36:61-74. 6. REFERENCES Williams, J.J., 1990. Video observations of marine gravel transport, Geo. Mar. Lett., 10:157-164. Ball, J.E. and Keshavarzy, A., 1995. Discussion on Incipient Sediment motion on non-horizontal slopes by Chiew and Parker, J. of Hydraulics Research, 33(5):723-724. Box, G.E.P. and Cox, D.R., 1964. An analysis of transformation, J. of the Royal Statistical Society, Series B, 26: 211-252. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries, J. of Fluid Mechanics. 50(2):233-255. Grass, AJ., 1982. The influence of boundary layer turbulence on the mechanics of sediment transport, Proc. Euroraech 156: Mechanics of Sediment Transport, Istanbul, pp. 3-17. 270 Appendix A Appendix A-6 A Paper entitled: An investigation of entrainment of sediment particles in open channel flows 23th of Hydrology and Water Resources Symposium, Water and Environment, 21-24 May 1996 Hobart, Australia 271 Appendix A An Investigation of Entrainment of Sediment Particles in Open Channel Flows Alireza Keshavarzy, Postgraduate Student James E. Ball, ME., PhD, MIE AusL, Senior Lecturer Dept, of Water Eng., School of Civil Eng., UNSW, Sydney, Australia Abstract: The movement of sediment particles and the sorbed constituent material in an open channel has a significant impact on the environmental degradation of natural waterways and streams. Entrainment of these particulates and the sorbed constituents arises from erosion processes. This paper concentrates on the influence of flow turbulence on these entrainment processes. The shear stress applied to the sediment particles on the bed depends on the magnitude and probability distribution of the turbulent shear stress. The instantaneous shear stress and frequency of the bursting processes were determined from experimental data. It was found that sweep events occur more frequently and with a larger shear stress than other bursting events. 1 INTRODUCTION sediment particles and the difficulties in defining a model of sediment entrainment are discussed by Ball and Keshavarzy Water quality issues currently arc a major concern to the general (1995). population. Consequently, the process influencing the motion and treatment of quality constituents has been a focus of many recent To ameliorate these difficulties in defining a model of sediment studies. One conclusion developed from these studies has been entrainment, a statistical analysis of the bursting processes and the an enhanced understanding of the need to be able to describe the induced instantaneous shear stress was undertaken with the entrainment and deposition of particulates and, in particular, intention of investigating the entrainment of sediment particles sediment particles. from the channel bed. Reported in this paper arc some results from this study. The initial or threshold motion of a particle due to the action of fluid flow is defined as a condition for which the applied forces 2 EXPERIMENTAL EQUIPMENT AND DATA of fluid drag and lift, which cause the particle to move, exceed the stabilising force or resistance force. Previous studies on sediment Measurements of the turbulence were undertaken using a small movement in turbulent flows have postulated that particles are electromagnetic velocity meter in the centre of the sediment flume entrained into the flow by the effect of the bursting processes at the Water Research Laboratory, UNSW. More details of the induced by the natural turbulence within the flow. This flow experimental equipment are described by Keshavarzy and Ball turbulence imposes an instantaneous shear stress on the sediment (1995). The velocity components of flow in two dimensions were particle and, as a result, the particles are entrained into the flow measured and recorded for subsequent analysis to calculate the from the bed. The concept of the bursting process was introduced time averaged velocity in the two dimensions monitored, the by Kline at al. (1967) as a process by which momentum is instantaneous shear stress for each bursting event, the angle of transferred to the laminar region of the boundary layer. Four action for each bursting event, the time-averaged shear stress, and bursting processes have been identified with each process relating the mean shear stress and angle of action for each category of to the phase of the induced shear, these processes are the sweep, bursting event The turbulence intensity of the flow determined ejection, outward and inward interactions. Studies by Grass from the experimental data was compared with published data, as (1982) and Nakagawa and Nezu (1978) have concluded that shown in Figures 1 and 2; since similar characteristics were found immediately above the bed, the event which is most responsible thus validating the experimental procedure. for the transfer of momentum is the sweep event Furthermore, Thome et al. (1989) pointed out that the sweep and ejection 3 FREQUENCY OF EVENTS events occur more frequent than the outward and inward interaction events while Keshavarzy and Ball (1995) pointed out The instantaneous shear stress data was subdivided based on the that the magnitude of the instantaneous shear stress during a phase of the bursting process. Using the subsets of the recorded sweep event is greater than that which occur during outward and data, the frequency of each category of bursting event was inward interaction events and the time averaged shear stress determined as a function of the normalised flow depth. Shown in typically used for defining the sediment transport capacity of flow Figures 3 and 4 are the variation of frequency with depth for each in channels. In addition to the increased instantaneous shear category of bursting event It was found that the frequency was a stress, Raudkivi (1990) postulated that the instantaneous pressure function of the normalised flow depth and that close to the fluctuations lower the local pressure close the bed and particles channel bed, the frequency of the ejection and sweep events is may be entrained from the bed by hydrostatic pressure. The effect higher than that of the outward and inward interaction events. As of the instantaneous shear stress on the incipient motion of a result the sweep and ejection events occur more frequently. 272 Appendix A Vertical TerfcekKt ImUntHy ar«< ♦ O ’«O Xr«# X:m A ■ Ow4.l«*l • >«.IK Figure 1 - Horizontal Turbulence Intensity Figure 2 - Vertical Turbulence Intensity ♦ O |.I. Figure 3 - Frequency of Sweep and Outward Interaction Events Figure 4 - Frequency of Ejection and Inward Interaction Events This has implications for the entrainment of sediment as the Kline, SJ, Reynolds WC, Schraub FA and Runstadlcr PW, (1967), The sweep event is directed towards the bed. structure of turbulent boundary layers, J. of Fluid Mechanics, 30(Part 4). 4 CONCLUSION Läufer, J. (1950), Some recent measurement in a two dimensional turbulent channel, J. of Aeronautical Science, 17(5)^77-287. It was found that the sweep and ejection events occurred more frequently than the outward and inward interactions. This has McQuivey, RS, and Richardson. EV, (1969), Some turbulence implications for models of sediment entrainment. measurements in open-channel flow, ASCE, J. of Hyd. Div., 95(HY1). REFERENCES Nakagawa, H., Nczu, L and Ucda R, (1975), Turbulence of open channel flow over smooth and rough beds, Proc of the Japan Society Ball, JE and Keshavarzy A. (1995), Discussion of Incipient Sediment of Civil Engineers, No. 241. motion on non-horizontal slopes by Chicw and Parker, J. of Hyd. Res., 33(5). Nakagawa, R, Nezu, L, (1978), Bursting phenomenon near the wall in open channel flows and its simple mathematical model, Mem. Fac. Grass, AJ, (1982), The influence of boundary layer turbulence on the Eng., Kyoto University, Japan, XL(4), 40:213-240. mechanics of sediment transport, Proc Euromech 156, Mechanics of sediment transport, Istanbul, Turkey. Raudkivi, AJ., (1990), Loose Boundary Hydraulics, 3rd Ed. Pergamon Press. Keshavarzy A. and Ball JE, (1995), Instantaneous shear stress on the bed in a turbulent open channel flow, Proc. of XXVI LA.HR Congress, Thome, PD, Williams, JJ and Heathcrshaw, AD., (1989), In-situ acoustic Sept, London, UK, VoL A, pp 81 - 86. measurements of marine gravel threshold and transport. Geo Mar. Lett., 10:157-164. 273 Appendix A Appendix A-7 A Paper entitled: Investigation of Dominant Discharge on South Creek, NSW, Australia Published in The 2nd international Symposium on urban Storm Water Management, July 1995 Melbourne, Australia 274 Appendix A Investigation of Dominant Discharge on South Creek, NSW, Australia by Alireza Keshavarzy School of Civil Engineering, University of NSW, Sydney, 2052, Australia and Wayne D. Erskine School of Geography, University of NSW, Sydney, 2052, Australia Summary: Flood mitigation schemes and urban and industrial development on the flood plains of rivers and natural channels require a detailed understanding of flood characteristics due to the social and economic impacts of flooding. There is a complicated interaction between the geometry of the channel and the flows that pass through it. The channel shape is formed by a specific discharge called “Dominant Discharge”. In this study, different definitions of dominant discharge, such as that discharge forming the meander wavelength and that discharge coresponding to an inflection of the rating curve were^applied to South Creek, NSW, Australia. For this purpose, 36 years of flood data (1956-1991) were employed. Rood frequency analysis using several probability distributions was applied to the flood data. The results showed that the dominant discharge on South Creek has an average recurrence interval (ARI) of 1.89 to 2.40 years on the partial or annual exceedance series. 1. INTRODUCTION In this study, the flood data for South Creek at Mulgoa Road gauging station (No. 212320) (one of the tributaries of The Hawkesbury-Nepean River valley is one of the most Hawkesbury-Nepean River) for 36 years of record important rivers in Australia with an area of 22000 Km2 (1956-1991) are analysed to determine the dominant and a population of 500,000. It is significant for urban water supply, agricultural production, extractive industries discharge. South Creek has an area of about 88 Km12 and and many other urban and industrial purposes. A large the data were recorded by the NSW Department of Water proportion of the future population growth for the Sydney Resources. South Creek has a sinuosity of 2.04, a mean conurbation will be concentrated in this area and it will meander wavelength of 580 m, a mean width-depth ratio impose much stress on the riverine environment. To of 7.1 and a bankfiill width of 16m. The flood plain determine the morphological impacts of future averages 200m in width and the channel rarely impinges on urbanisation on the rivers of the Hawkesbury-Nepean the valley sides. From detailed work by Riley (1967) and drainage basin, it is necessary to know what is the current the authors the flood plain consists of fine-grained dominant discharge and how it will change. This paper sediments ranging from silt loams to silty clay loams to silty addresses the former question. clays to light-median clays (texture groups of Northcote, 1984). Riley (1967) found that the flood plain is formed by oblique accretion whereby sediment is deposited on the 275 Appendix A face of the bank in layers of almost constant thickness and transports either the most suspended sediment (Benson and grain size composition. Sand and gravel are largely Thomas, 1966), or the most bed load (Pickup and Warner, confined to the channel bed. Pickup and Warner (1976) 1976) or the most total sediment load (Andrew, 1980) over have also found that up to 0.6 m of overbank deposits time. This discharge is also called the effective discharge containing European artefacts have been laid down on for sediment transport; ii) the natural bankfull discharge; many streams in this area in recent years. iii) a flood of fixed frequency such as the 1.58 yr or 2.33 yr flood; and iv) the discharge that exhibits the best statistical According to Inglis (1949) the dominant discharge (Qd) of correlation or relationship with various channel a natural stream is that discharge representative of the morphological characteristics (characteristic discharge) whole range of discharges that pass through the channel and such as meander wavelength, channel width, etc. that forms the channel morphology. It is usually defined in one of four ways: i) it is equated to that range of flows which flood plain flood plain "main channel level of bankfull discharge Fig. 1:Topical cross section of a natural stream Bankfull discharge is also equated to a flood of fixed 2. MEANDER WAVELENGTH frequency, to the effective discharge for sediment transport as well as to the flow determining channel planform and Meander wavelength varies with the square root of bankfull cross sectional morphology (Dury 1965,1976; Benson and discharge (Dury, 1965). Any empirical relationship Thomas 1966; Andrew, 1980). Therefore, bankfull between wavelength (L) and bankfull discharge (Qbf) may discharge is usually equated to dominant discharge be statistical rather than causal, although this distinction is although this is not always the case (Pickup and Warner, perhaps a fine one. Leopold and Wolman (1957) concluded 1976; Erskine, 1994). Bankfull discharge is that flow which that bed width is determined directly by discharge, whereas just fills the channel without overtopping the banks (Figure wavelength depends directly on width and thus only 1). It is often argued that bankfull discharge is a flow of indirectly on discharge. Then; moderate magnitude which occurs relatively frequently L = K. qb (2) and which is competent to erode the banks and construct the flood plain. where L *= meander wavelength* Wolman and Leopold (1957) found that bankfull discharge q = dominant discharge* (Qbf) is equal to a flood with a frequency of 1-2 years on K = coefficient and the annual series. Leopold, Wolman and Miller (1964) and b = exponent. Woodyer (1968) also proposed that dominant discharge * The above parameters are in FPS unit. (Qd) is equal to bankfull discharge If the annual maximum flood series confirms to the theory of extreme values (Gumble, 1954, 1958), the mean annual Qä = Qbf (1) flood has a return period of 2.33 years and the most The aim of this study is to investigate the relationships probable annual flood has a return period of 1.58 years. between dominant discharge, bankfull discharge and characteristic discharge for South Creek in the Hawkesbury Inglis (1941) found a relationship between channel River basin. meander wavelength (L) and width (W) and also between width (W) and discharge (q): 276 Appendix A 3. STAGE-DISCHARGE CURVE L = 6.06W°" - 6.06W W = 4.88<7°^ (4) Another definition of bankfull discharge is that point at which the rating curve exhibits an abrupt flattening in Combining equations of (3) and (4) gives slope. Wolman and Leopold (1957) recommended that L = 29.6q°s <5) bankfull discharge has an Average Recurrence Interval Inglis (1949) found that (ARI) of 1-2 years (Qbf=Qi-2 years) on the annual series. Dury (1976) suggested that bankfull discharge is a L = 3 6q05 (6) discharge with ARI of 1.58 years or Qbf= 0.97Qi^g. which agrees closely with equation (5). Dury (1965) used a very large data set and found that Knight and Demetriou (1983) concluded that there is a linear relationship for the “ in-bank ” and “out-bank” L = 30 qos (7) flow in eq. (8) with a hinge point at the bankfull level. This relationship has been applied to South Creek. Wavelengths was measured on 1:25000 topographic maps. LogH = y . log Q + <3 (8) The number of meanders (n) measured were 18 and the where y and <5=constants whose values are determined average meander wavelength (L) was 578 metres. by linear regression, H=depth (mm) and Q=discharge Therefore, bankfull discharge calculated by the Dury (1/s) method (equation 7) is about 113 m3/s or 9765 ML/d. B h Fig. 2 : Typical cross section of open channel with flood plains A typical cross section of an open channel with a flood plain 4. FLOOD FREQUENCY ANALYSIS is shown in Figure 2. An abrupt flattening of the rating curve occurs when overbank flow first occupies the flood Flood frequency analysis is a statistical method for analysis plain because of the massive increase in discharge for a of recorded floods. It can estimate annual exceedance relatively small increase in head. probability from the peak instantaneous discharge. Rood frequency analysis may be done analytically or graphically. Sellin (1964) has discussed that the presence of some For the analytical method a probability distribution is fitted interactions along the vertical interface between the mathematically to the recorded data. For the graphical shallow and deep sections which results in the transfer of method the recorded data is plotted on appropriate linear momentum between the main channel and the flood probability paper and a frequency curve is drawn by eyes. plain. Bankfull discharge can be derived from the point of There are many advantages of the analytical method of abrupt change in the rating curve. The slope of the rating flood frequency analysis such as: curve just for channel flow is steeper than that for channel . for each annual exceedance probability, the respective and overbank flow. flood peak discharge is directly derived; . it is not necessary to make an assumption between From the rating curve (logarithmic or normal scales) for probabilities of rainfall and runoff; South Creek in Figure 3 , the value of bankfull discharge . the estimation of rainfall losses is not required; and is 10,000 ML/d or 115.7 m3/s. This agrees closely with the . confidence limits can be fitted to indicate the accuracy of characteristic discharge determined from meander the predicted discharge for various annual exceedance wavelength (9765 ML/d). probabilities. 277 Appendix A Stjage-Discluqge Curve I------|.------:------i- 10000 100000 Discharge (ML/d) Fig. 3: Rating Curve for South Creek at Mulgoa-Rd Weir 4.1 Data Collection and Analysis 5. RESULT AND DISCUSSION In this study the partial series was selected for flood The ARI’s of bankfull discharge in Table 1 are close to those frequency analysis. In this analysis (partial series) the reported in the literature. However, morphological bankfull maximum peak discharge per month were considered. The discharge is much smaller than that reported by Pickup and number of floods (K) normally differs from the number of Warner (1976) for other streams in this area (4 to 7 years on years of record (N) and depend on the selected threshold the annual series in their cases). It would seem that their flood. If (K) is equal to (N), it gives the best results (Pilgrim, Cumberland Basin streams did not have the broad flood 1987). For this analysis, the 36 largest floods during the 36 plain found at South Creek gauging station. This conclusion years of record were chosen (K=N). This series is also is also supported by the close agreement between bankfull called the annual exceedance series. and characteristic discharge on the basis of meander wavelength, on South Creek. Page (1988) has also found 4.2 Procedure for Flood Frequency Analysis the rivers with broad floodplains have ARI’s of bankfull A log-Pearson type m distribution was fitted to the annual discharge within the range reported by Wolman and exceedance series by the method of Pilgrim and Doran Leopold (1957), Leopold etal. (1964) and Woodyer (1968). (1987). Furthermore, extreme value 1, log normal and normal distributions were fitted by both the method of Rood variability was proposed by Erskine (1994) as one moments and method of maximum likelihood to the same factor which partially accounts for large departures of series. The log-Pearson HI distribution is shown in Fig. 4. ARI’s for bankfull discharge from the range of 1-3 years on the annual series. Rivers with high flood variability have The annual exceedance probability (AEP) is inversely steep annual series flood frequency curves. The standard related to the average recurrence interval (ARI). Table 1 deviation of the log-transformed annual series on rivers shows both AEP and ARI for bankfull discharge (10,000 with high flood variability exceeds 0.7 (Erskine, 1993, ML/d) for all distributions. The choice of distribution does 1994). Such rivers experience large floods more frequently not significantly affect the ARI of bankfull discharge. The than those with flat flood frequency curves. If a AEP’s for the bankfull discharge were derived from flood catastrophic flood is defined as an event with a flood peak frequency curves with four distributions and seven methods discharge at least 10 times greater than the mean annual and the results are shown in Table 1. flood (Erskine, 1994 ), then such events only occur within 278 Appendix A FLOOD FREQUENCY PLOT PARTI ALSERIES log—Pearson III by indirect miment 10000. 100000 1000000 10000000 10000000 Peak Discharge (ML/d) Fig. 4: Flood frequency analysis using log-pearson iii by indirect moment the time period needed to repair the flood damage on rivers floodplain and protect the channel from flood damage. with high flood variability. The in-channel benches Such rivers are not only insensitive to the impacts of large described by Woodyer (1968) are often present a floods, but are also likely to have an ARI of bankfull flood-enlarged channel (Erskine, 1994). Broad flood plains discharge between 1 and 3 years on the annual exceedance tend to dissipate flood power over a well-vegetated series. Table 1: Results of flood frequency analysis with some distributions No. Types of Distribution AEP ARI (Years) 1 Extreme Value 1 (Gumble) using Max. Likelihood 0.462 2.165 2 Extreme Value 1 (Gumble) by moment 0.485 2.060 3 Log-Pearson III by indirect moment (3-parameter) 0.520 1.920 4 Log-Normal Max. Likelihood (2-Parameter) 0.446 2.240 5 Log-Normal Direct Moment (2-Parameter) 0.417 2.400 6 Normal Distribution by Max. Likelihood 0.528 1.890 7 Normal Distribution by Moment 0.528 1.890 279 Appendix A 6. CONCLUSION Page, KJ., 1988, Bankfull discharge frequency for the Murrumbidgee River, NSW, in R.F. Warner (ed.)Ruvial In this investigation the characteristic discharge and rating Geomorphology of Australia. Academic Press, Sydney, curve methods were used to determine bankfull and pp. 267-281. dominant discharge. The discharge determined (9765 vs Pickup, G. and Warner, R.F., 1976, Effects of hydrologic 10,000 ML/d) agreed closely and had ARI’s between 1.89 regime on magnitude and frequency of dominant discharge, Journal of Hydrology. 29, pp. 51-75. and 2.4 years on the annual exceedance series. These Pilgrim, D.H. and Doran, D.G., 1987, Flood Frequency discharges are larger than the most probable annual flood Analysis, Australian Rainfall and Runoff, pp. 197-235, of Dury (1965, 1976), as was also found in this area by ed. by D.H. Pilgrim. The Institution of Eng., Australia. Pickup and Warner (1976), but cover the span of ARI’s for Public Works Department, 1987, Assessment of the mean annual flood. Rivers with broad floodplains often Hydrologic data. Hawkesbury River Hydraulic and have a bankfull discharge with an ARI close to the mean sediment transport process, report No. 4. annual flood. South Creek in no exception. Riley, S. J., 1967, The morphological significance of sediment accretion along South Creek, Unpublished 7. ACKNOWLEDGEMENT B.Sc. (Hons.) Thesis, Department of Geography, University of Sydney. The authors thank the NSW Department of Water Sellin, R. H. J., (1964), A laboratory investigation into the Resource, River Gauging Branch for the rating curve and interaction between flow in the channel of a river and that hydrological data for the South Creek station. Wolman, M.G. and Miller, J.P., 1960, Magnitude and 8 . REFERENCES frequency of forces in geomorphic processes, Journal of Andrews, E.D., 1980, Effective and bankfull discharges of Geology. 68, pp. 54-74. streams in the Yampa River basin, Colorado and Wolman, M.G. and Leopold, L.B., 1957, River flood Wyoming. Journal of Hydrology. 46: pp.311-330. plains: some observations on their formation, U.S. Geol. Benson, M. A. and Thomas, D. M., 1966, A definition of dominant discharge. Hydrological Science Bulletin. 11, pp. 76-80. Dury, G.H., 1965, Theoretical Implications of underfit Streams, U.S. Geol. Sur.. Prof. Paper, No. 452-C. Dury, G.H., 1976, Discharge prediction, present and former from channel dimensions, Journal of Hydrology. 30, pp. 219-245. Erskine, WX)., 1993, Erosion and deposition produced by a catastrophic flood on the Genoa River, Victoria, Australian Journal of Soil and Water Conservation. 6(4), pp. 35-43. Erskine, WD., 1994, Sand slugs generated by catastrophic floods on the Goulbum River, NSW, International Association of Hydrological Sciences. Pub. No. 224, pp. 143-151. Inglis, C. C., (1949), The behavior and control of rivers and canals, Central Water-Power Irrig, and Navigation Research Station, Poona (India), Research Publ. 13, 2v. Knight, D.W. and Demetriou, J.D., 1983, Rood Plain and Main Channel flow Interaction, Journal of Hydraulic Engineering Vol. 109 No.8, p. 1073. Knight, D.H., Demetrio, JD. and Hamed, M.H., 1984, Stage Discharge Relationships for Compound Channels, Proceeding of 1st International Conference on Channel and Channel Control Structures, p. 4.21. Northcote, K. H., 1984, A factual kev for the Recognition of Australian Soils. Rellim, Adelaide. 280 Appendix A Appendix A-8 A Paper entitled: Frequency of Bankfull Discharge on South and Eastern Creeks, NSW, Australia. Published in 23th of Hydrology and Water Resources Symposium, Water and Environment, 21-24 May Hobart, Australia 281 Appendix A Frequency of Bankfull Discharge on South and Eastern Creeks, NSW, Australia by Wayne D. Erskine School of Geography, University of NSW, Sydney, 2052, Australia and Alireza Keshavarzy School of Civil Engineering, University of NSW, Sydney, 2052, Australia Summary: Estimation of the frequency of flood plain inundation on natural streams is necessary for urban planning, the design of flood mitigation works and the determination of flood damages. Flood plain inundation starts when the capacity of the channel is exceeded. The bankfull discharge of South Creek at Mulgoa Road and Eastern Creek at Great Western Highway gauging stations was determined by the following methods: a) rating curve inflection point, b) flood plain surface level by field survey, and c) discharge responsible for producing the measured meander wavelength. The bankfull discharge so determined ranged between 9765 and 10000 ML/d on South Creek and between 3478 and 5850 ML/d on Eastern Creek. The associated average recurrence intervals were determined by fitting a log Pearson type HI distribution to the annual exceedance series (partial series of the N largest floods in N years of records). The average recurrence intervals of bankfull discharge ranged between 1.9 and 2.4 years on South Creek and between 1.08 to 3.18 years on Eastern Creek. The range of average recurrence intervals is within that reported in the literature for bankfull discharge. However, these results are much lower than those reported for the same area by others because there is a broad flood plain present at our sites. 1. INTRODUCTION The Hawkesbury-Nepean River valley is one of the most flows determining the shape of the bed. Bates and Pilgrim important rivers in Australia with an area of 22000 km12 and (1983) also found that the bankfull discharge at the Eastern a population of 500,000. It is significant for urban water Creek gauging station had a return period of approximately 4.5 supply, agricultural production, extractive industries and years on the annual maximum series. However, Keshavarzy many other urban and industrial purposes. A large proportion and Erskine (1995) found that bankfull discharge at the South of the future population growth for the Sydney conurbation Creek gauge was much smaller than reported by the above will be concentrated in this area and it will impose much stress authors and had a return period of 1.89 to 2.4 years on the on the riverine environment. To determine the morphological annual exceedance series, depending on the method used to impacts of future urbanisation on the rivers of the define bankfull and depending on the distribution used to fit Hawkesbury-Nepean drainage basin, it is necessary to know the annual exceedance series. The results did not vary greatly what is the current dominant discharge and how it will change. between different distributions and between different curve This paper addresses the former question. fitting procedures. The purpose of this paper is to resolve this discrepancy by reanalysing the South Creek data and by Previous research in the Cumberland Basin near Sydney by presenting new data for Eastern Creek. Pickup and Warner (1976) found that bankfull discharge had return periods of 4 to 7 years on the annual maximum series. 2. STUDY SITES However, the most effective discharge or that discharge which transported the most bed load over time had return periods In this study, the frequency of bankfull discharge at the South much less than 1.5 years. They argued that there were two Creek at Mulgoa Road (No. 212320) and Eastern Creek at groups of channel forming discharge with bankfull flow Great Western Highway (No. 212340) gauging stations are determining the size of the channel and with most effective compared. South Creek at Mulgoa Road gauge has a 282 Appendix A catchment area of 88 kmr and a continuous flood record of 36 ü) that discharge corresponding to the top of the lowest years (1956-1991). South Creek has a sinuosity of 2.04, a surveyed bank which forms part of the floodplain (surveyed mean meander wavelength of 580 m, a mean width-depth ratio bankfull discharge); and of 10.7 and a bankfull width of 45 m. The flood plain averages iii) that discharge at which the rating curve exhibits an abrupt flattening of slope (rating curve determined bankfull 200m in width and the channel rarely impinges on the valley discharge). sides. Eastern Creek at the Great Western Highway gauge has a catchment area of 24.9 km2 and a continuous flood record of Characteristic discharge was determined by measuring years (1970 -1991). Eastern Creek has a sinuosity of 1.5, a between 14 and 18 complete meander wavelengths at each mean meander wavelength of 344m, a mean width-depth ratio gauge and then calculating bankfull discharge from Dury’s of 13.2 and a bankfull width of 56 m. The flood plain is more (1965) equation. Surveyed bankfull discharge was equated to than 200 m wide and the channel does not impringe on the the top of the lowest bank corresponding to the floodplain on valley sides in the study area. The location of the rivers a surveyed cross section close to the staff gauges and recorder. Only one cross section was used at each gauge because there investigated here is shown in Figure 1. was little variation in this level throughout the gauge pooL 3. METHODS Figures 2 and 3 show the surveyed cross section and bankfull level at each gauge. The rating curve determined bankfull Bankfull stage is that discharge which just fills the channel discharge corresponds to the discharge shown in Figures 4 and without overtopping the banks and inundating the flood plain. 5. While a number objective definitions of bankfull have been proposed (Riley, 1972), they are no more reliable than A partial or annual exceedance series (N largest floods in N subjectively determined assessments (Williams, 1978). For years of record) was constructed for each gauge. A log Pearson this paper, three definitions of bankfull discharge have been type 3 distribution was fitted to each annual exceedance series adopted: by the method of Pilgrim and Doran (1987). i) that discharge which has formed the meander pattern (called the characteristic discharge); Sydney HAWKESBURY-______RIVER Figure 1; Hawkesbury River and tributaries (South and Eastern Creeks) 283 Appendix A Floodpalin Bankfull Bench Top of Weir Chainage (m) Fig. 2: Cross section at Eastern Creek at Great Western Highway gauge 43.0-T 42.0- Floodplain RanJcfull Top of Weir Chainage (m) Fig. 3: Cross section for South Creek at Mulgoa Road Weir gauge 284 Appendix A Stage-Dijcharge Curve 100000 Discharge (ML/d) Fig. 4: Rating Curve for Eastern Creek at Great Western Highway gauge 10.0-, Stjage-Dischajfge Curve 10000 100000 Discharge (ML/d) Fig. 5: Rating Curve for South Creek at Mulgoa-Road Weir gauge 285 Appendix A 4. RESULTS discharge being approximately equal to or less than the bankfull discharge. Page (1988) found that rivers with broad Table 1 lists the results of the bankfull discharge floodplains have average recurrence intervals of bankfull determinations and Figures 6 and 7 show the annual discharge within the range found by Wolman and Leopold exceedance series flood frequency curves. At South Creek (1957) and Woodyer (1968). there is a very close agreement between all of the bankfull discharge determination methods. As a result, the associated None of the bankfull discharge estimates in Table 1 have recurrence intervals only vary between 1.72 and 1.92 years on recurrence intervals close to 1.58 years which Dury (1965) the annual exceedance series. At Eastern Creek, the surveyed proposed as bankfull discharge. The simplistic equation of and rating curve determined bankfull discharges again agree bankfull discharge to a flood of fixed frequency is not but the characteristic discharge is much lower. As a result, the supported by the present data. It is immaterial whether an associated recurrence intervals exhibits a much greater range average recurrence interval of 1.58 or 2.33 years is used, of 1.08 (characteristic discharge) to 3.18 years on the annual because neither conform to the present results. Furthermore, exceedance series (bankfull discharge). Both results are less it must be stressed that it is impossible to compare average than Bates and Pilgrim’s (1983) estimate of 4.0 years on the recurrence intervals for different distributions. The 1.58 year partial series. flood is only the modal annual flood for an extreme value distribution. Similarly, the 2.33 year flood is only the mean annual flood for an extreme value distribution. The modal and Table 1: Results of the bankfull discharge determination mean annual floods do not have fixed recurrence intervals for a log Pearson type 3 distribution because of the variable Rating Curve Gaug Characteristic Surveyed coefficient of skewness. ing Sta Discharge Bankfull determined Bank- tion Discharge full Discharge (ML/d) Flood variability was proposed by Erskine (1994) as one factor (ML/d) (ML/d) which partially accounts for large departures of ARI’s for South 9765 10 000 10 000 Creek bankfull discharge from the range of 1-3 years on the annual series. Rivers with high flood variability have steep annual Eastern 3470 5850 5850 Creek series flood frequency curves. The standard deviation of the log-transformed annual series on rivers with high flood Keshavarzy and Erskine (1995) used 4 distributions and 5 variability exceeds 0.7 (Erskine, 1993, 1994). Such rivers methods of curve fitting to the annual exceedance series at experience large floods more frequently than those with flat South Creek. They showed that the results from the log flood frequency curves. If a catastrophic flood is defined as an Pearson type 3 distribution agree closely to the other methods event with a flood peak discharge at least 10 times greater than and distributions. the mean annual flood (Erskine, 1994 ), then such events only occur within the time period needed to repair the flood damage 5. DISCUSSION on rivers with high flood variability. The in-channel benches Although Pickup and Warner (1976) reported that benches are described by Woodyer (1968) are often present a generally absent from the channels of the Cumberland Basin, flood-enlarged channel (Erskine, 1994). Broad flood plains Fig. 3 shows that Eastern Creek does have at least one bench tend to dissipate flood power over a well-vegetated floodplain present. This bench is very extensive both up and downstream and protect the channel from flood damage. Such rivers are not of the gauge. Characteristic discharge on Eastern Creek is best only insensitive to the impacts of large floods, but are also correlated with benchfull rather than bankfull discharge. likely to have an ARI of bankfull discharge between 1 and 3 The range of recurrence intervals for bankfull discharge years on the annual exceedance series. reported above for Eastern and South Creeks is within the We do not support Bates and Pilgrim’s (1983) result for range found by Wolman and Leopold (1957) and Woodyer Eastern Creek. Cumberland Basin streams with broad (1968) but is less than the range found by Pickup and Warner floodplains should not be expected to conform to Pickup and (1976) and Bates and Pilgrim (1983). It would seem that their Warner’s (1976) results. As a result, a bankfull average Cumberland Basin streams do not have the broad, extensive recurrence interval of less than 4 years should have been floodplains found at the Eastern and South Creeks gauging expected. stations. This conclusion is also supported by the characteristic 286 Annual Exceedance Prob; 212320.mxd Fig. Fig. 4 1 7: 6: ..nL ■Ui L u Si i i £L Flood Flood ___ — / L j frequency frequency L / *. FLOOD * analysis analysis FREQUENCY Peak Peak using using Discharge log-Pearson Discharge log-pearson log-pearson (ML/d) (ML/d) 287 PLOT iii iii III by by by indirect indirect indirect moment moment moment PART1ÄLSERJES Appendix A Appendix A Given the close to correspondence between the variously Pickup, G. and Warner, R.F., 1976, Effects of hydrologic defined bankfull discharges, we concluded that it is the same regime on magnitude and frequency of dominant discharge, as dominant discharge on the floodplain rivers of the Journal of Hydrology. 29, pp. 51-75. Cumberland Basin. We further recommend that bankfull Pilgrim, D.H. and Doran, D.G., 1987, Flood Frequency Analysis, Australian Rainfall and Runoff, pp. 197-235, ed. discharge be used for design purposes for works such as those by D.H. Pilgrim. The Institution of Engineers, Australia, recently completed on Eastern Creek at the Railway by the Barton. Department of Land and Water Conservation. Riley, S. J., 1972. A comparison of morphometric measures of 6. CONCLUSION bankfull. Journal of Hydrology. 17, pp 23—31. Williams, G. P, 1978. Bankfull discharge of rivers, Water The dominant discharge on the flood plain reaches of Eastern Resources Research. 14. pp 1141-1154. and South Creeks has a recurrence interval of less than 3.18 Woodyer, K. D., 1968, Bankfull frequency in rivers. Journal of years on the annual exceedance series. This is less than Hydrology. 6, pp 114-142. previously reported for this area by Pickup and Warner (1976). Wolman, M.G. and Leopold, L.B., 1957, River flood plains: Similarly, the recurrence intervals reported here do not equate some observations on their formation, U.S. Geol, Survey. Prof. Paper. No. 282-C. to Dury’s (1965) 1.58 years. While bankfull and dominant discharge are sensibly the same for flood plain reaches of Cumberland Basin streams, they do not conform to a flood with a fixed or constant recurrence interval. 7. ACKNOWLEDGEMENT The authors thank the NSW Department of Land and Water Conservation for providing rating curve and hydrological data, J. Tilley, University of NSW for providing cross sectional data and also Mr Saeid Eslamian for helping to provide Figure 6. 8. REFERENCES Bates, B. C. and Pilgrim, D. H., 1983, Investigation of Storage-Discharge Relations for River Reaches and Runoff Routing Models, Civil Engineering Transaction. The Institution of Engineers. Australia. CE 25:153-161. Dury, G.H., 1965, Theoretical Implications of underfit Streams, U.S. Geol. Sur.. Prof. Paper, No. 452-C. Erskine, WJD., 1993, Erosion and deposition produced by a catastrophic flood on the Genoa River, Victoria, Australian Journal of Soil and Water Conservation 6(4), pp. 35-43. Erskine, W£>., 1994, Sand slugs generated by catastrophic floods on the Goulburn River, NSW, International Association of Hydrological Sciences. Pub. No. 224, pp. 143-151. Keshavarzy A. and Erskine W. D„ 1995, Investigation of Dominant Discharge on South Creek, NSW, Australia. The Second International Symposium on Urban Stormwater Management, Melbourne. 11—13 July 1995. The Institution oI^EngineeiS^Auslialia, National Conference Publication No. 95/03, pp 261-266. Page, KJ., 1988, Bankfull discharge frequency for the Mumimbidgee River, NSW, in R.F. Warner (ed.) Fluvial Geomorohology of Australia. Academic Press, Sydney, pp. 267-281. 288 Appendix A Appendix A-9 A Paper entitled: An application of image processing in the study of initiation of sediment motion and transport in an open channel flow 2nd International symposium on rainwater catchment systems, 21-25 April 1997, Tehran, Iran. 289 Appendix A An application of Image processing in the study of initiation of sediment motion and transport in an open channel flow Alireza Keshavarzy and James E. Ball School of Civil Engineering, The University of New South Wales Sydney, Australia 1. ABSTRACT The entrainment of sediment particles from a plane mobile bed is investigated using an experimental investigation of particle motion. Image processing techniques were used to analyse sequence of images. A subtraction method was used to derive the number of particles entrained in an increment of time in a specified area over a mobile bed at the centre of the flume. A cross correlation was applied to investigate the relation between the number of entrained particle and the instantaneous shear stress. A significant correlation with no lag was found for this relation. The results of this investigation were used to modify Shields diagram. The modified Shields diagram enables one to find the probability distribution of particles in motion and also the initiation of motion. A convolution technique was also used to detect instantaneous velocity of the particles at the bed. These results were applied to verify a proposed stochastic-deterministic model. Good agreement was found between observed and predicted instantaneous velocity of particles at the bed. 2. INTRODUCTION Despite the importance of the initiation of individual sediment particles motion, no general definition yet exists. Also there is a difficulty in definition of incipient particle motion over a flat bed. This difficulty arises from the influence of turbulence on sediment motion which has been discussed by Ball and Keshavarzy (1995) and Keshavarzy and Ball (1996, 1997). One reason for this is the difficulty of observing sediment particles at the initiation of motion. Recently, attention has been focused on the use of image processing techniques for observing the motion of sediment particles. In several studies, for example those by Nelson et al. (1995), image processing techniques were used as a tool to investigate the intermittent nature of particle entrainment upstream and downstream of dunes. The application of image processing techniques has the potential to assist in understanding the processes influencing incipient of particle motion. This technique can show the entrainment of particles from the bed, settlement of particles, speed of particle movement, transport mode, and resting periods of a particle on the bed. With the capturing and collection of these data, it is possible to develop a statistical description of particle entrainment or particle initiation at the bed. In this study attention is paid to define the initiation of sediment particle motion. An experimental test was undertaken over a mobile bed in a flume. In order to understand this process an image processing technique was used to observe the particle movement in an instant of time over an desired area. 3. ANALYSIS OF IMAGES OF PARTICLE MOTION In order to analyse the captured images, two different techniques were used in this study. These techniques were • a probability analysis of the entrained particles determined by counting the number of particles in motion at an instant This approach was useful to obtain an exceedance probability of particles in motion in time, respective to the exceedance probability of shear stresses of the bursting process at the bed. • application of some statistical tools to determine the displacement of particles between images. Cross correlation and Fourier transforms were used for these convolution techniques. 290 Appendix A The difference between two images can be obtained using in subtraction technique. Two images can be compared by computing the difference between the light intensities at all pairs of corresponding pixels from image one and image two. Here, this technique was used and a sequence of images were compared to find the number of particles which entrained and deposited over specified area and in a given time increment. For analysis of the images, each image was digitised into an array of 384 by 288 pixels. Two different types of format were selected in digitising process, a BMP format in colour (24 b/p) and a PGM grey scale (8 b/p) subformat. The PGM grey format was selected for its lower storage requirements and the ease of processing and file transfer between different computers. The images were digitised using a 486, 50 MHZ computer using Image Maker® software driving a frame grabber in order to convert video film to a series of separate frames and hence to store them as a group of files in the computer. Initially this software captures a sequence of frames as a large file and then in another process converts this file to a series of frames with the desired format. It was these series of images that were analysed to investigate the initiation of sediment motion. In this part of this study the purpose is to derive the difference between two sequential images in order to ascertain the number of particles entrained in an instant of time. To meet this aim and in order to analyse the images a specially written computer program was required to read the binary files and to process them for analysis. This program was developed in the C"1- language due to its utility and capability for image processing. 3.2 The Convolution and Cross Correlation Technique The displacement of two images in an increment time can be obtained using convolution technique. Using a cross correlation technique between two images, the displacement of the two images can be found. Here this method was used to obtain the dispacement of particles at the bed of an open channel flow. This method is useful only for small image sizes e.g. 32x32 or 64x64. In this study, this method mostly was used to estimate the displacement of particles. A frame size of 64 x 64 was selected from the centre of the flume with a frequency of 25 frames per second. 4. RESULTS AND DISCUSSION 4.1 Relation of Number of Particle in Motion and Instantaneous Shear Stresses The entrainment of particles from a mobile bed in an open channel flow has been investigated in several studies; for example by Einstein and Li (1958), where it was pointed out that this process is completely stochastic in nature due to the effect of turbulence. The number of entrained particles over a specified area will vary with time. The entrained particles at any time depends on the instantaneous turbulent shear stress arising from the velocity fluctuations and the instantaneous shear stresses at the bed. Shown in Figure 1 are the velocity fluctuations of the flow with time and the corresponding instantaneous shear stress at the bed. Also shown in Figure 1 is the number of particles in motion, at an instant of time and consequently how the entrained number of particles number varies with time. This relation was investigated using a cross correlation analysis between instantaneous shear stress in sweep events and the number of particles entrained. The number of entrained particles were counted in a sequence of produced images derived from subtraction of sequential recorded images. The dimensionless shear stress in sweep events was computed also from a time series of the velocity fluctuations recorded at the same timescale of the images. A cross correlation analysis was undertaken between the number of entrained particles and instantaneous shear stress in sweep event. Good correlation was found between the number of particles in motion and the instantaneous shear stress in a sweep event. 291 Appendix A 150- , 50- *\Ar\ aA/1 jirt MlV f\ aAa h/s/sAV A-a. /va jV±. MfJ. u -50- v V V* yjv yf4 “ wlAfvv\AP^ V \J v* *\AJ T v -150- v' 58 -50 2000- u v o- -2000- g 20- fl io; o-- o- z, ) 5 10 15 20 25 30 35 Time (Sec) Figure 1 An example of velocity of the flow with respective shear stresses and particles in motion 4.2 Modification to the Shields Diagram Gessler (1971) pointed out that the Shields diagram is inconvenient to use and suggested a modification to the Shields diagram. He stated that the probabilistic nature of the entrainment process depends only on the statistical nature of turbulence. In Gessler’s approach, attention was paid to deriving the probability that grains of a specific size would stay as an armour coat material. Other studies such as those by Grass (1971) and Mantz (1977) suggest some extensions to the Shields diagram at low Reynolds number. Mantz (1977) used some collected experimental data for fine cohesionless grains (e.g. 15, 30, 45, and 66 pm) and used regression analysis of the collected data. Mantz’s extension of Shields diagram was for low particle Reynolds number (Rn<10). In their extensions it was shown that the number of particles entrained was lower than predicted by Shields diagram. An investigation was carried out on the turbulent characteristics of flow and bursting process to get a probability density function for intermittent instantaneous shear stresses. Also an investigation was made for intermittent particle entrainment from the bed and a probability distribution was derived. These simultaneous probabilities were used to modify the Shields diagram through addition of the probability density function of particle motion. A statistical analysis of the experimentally observed particle entrainment from the bed was undertaken and an exceedance probability distribution obtained. As a result, for each exceedance probability between 0% to 100%, the percentage of area entrained can be obtained. In order to apply the instantaneous shear stresses and entrained area (percentage) to the Shields diagram, a 50% exceedance probability was selected and employed. These results were applied to Shields diagram to add information relevant to the initiation of motion. This modification to the Shields diagram enables the entrainment of particles with a defined probability of occurrence to be obtained from Shields diagram. As shown in Figure 2, at each point for 2 mm particle size, there is a probability associated for entrainment process. The associated probability density function at each point varies from 0% to 100% exceedance probabilities. As an example, the percentage of particle motion shown in this figure is derived for a probability density function for 50%. The entrainment of particles also varies from zero percent to 100% in a line on Shields diagram which is depicted as a dashed line in this figure. 292 Appendix A dm - 2mm ...... * • * • t I I percentage of «re« entrained percentage of area entrained Figure 2 Modified Shields diagram 5. CONCLUSION A statistical analysis of the experimentally observed particle entrainment from the bed was undertaken and an exceedance probability distribution obtained. As a result, for each exceedance probability between 0% to 100%, the percentage of area entrained can be obtained. These results were applied to Shields diagram to add information relevant to the initiation of motion. Using exceedance probability of the obtained area entrained from the image processing and the measured turbulent shear stresses near the bed within the flow, a modification to Shields diagram was proposed. This modification, indicates the probability of a particle being induced into motion. The modification to the Shields diagram enables the entrainment of particles with a defined probability of occurrence to be obtained from Shields diagram. 6. REFERENCES Ball, J.E. and Keshavarzy, A., 1995. Discussion on incipient sediment motion on non-horizontal slopes by Chiew and Parker. Journal of Hydraulics Research, IAHR, 33(5):723-724. Einstein, H.A. and Li, H., 1958. The viscous sublayer along a smooth boundary. Transactions, ASCE, Vol. 123, Paper No. 2992, pp. 293-313. Gessler, J., 1971a. Beginning and ceasing of sediment motion. In River Mechanics, Volume I, Edited and Published by Shen H.W, Fort Collins, Colorado. Grass, A.J., 1971. Structural features of turbulent flow over smooth and rough boundaries, Journal of Fluid Mechanics, 50(2):233-255. Kesha varzy, A. and Ball, J.E., 1996. The influence of turbulent shear stress on the initiation of sediment motion in an open channel flow. Stochastic Hydraulics'96, Proc. of 7th IAHR International Symposium on Stochastic Hydraulics, edited by Kevin Tickle, A.A. Balkema Publisher, Rotterdam. Keshavarzy, A. and Ball, J.E., 1997. An analysis of the characteristics of rough bed turbulent shear stresses in an open channel. Journal of Stochastic Hydrology and Hydraulics. Vol. 11, No 2. in press. Mantz, P. A., 1977. Incipient transport of fine grains and flakes by fluids-extended Shields diagram. Journal of Hydraulics Division, ASCE, 103(HY6):601-615. Nelson, J.M., Shreve, R.L., Mclean, S.R. and Drake, T.G., 1995. Role of near-bed turbulence structure in bed load transport and bed form mechanics. Water Resources Research. 31(8):2071-2086. Shields, A. 1936. Anwendung der Ahnlinchkeitsmechanik und Turbulenzforshung auf die Gesciebebewegung. Mitteil Preuss. Versuchsant Wasser, Erd, Schiffbau, Berlin, No. 26. Van Rijn, L.C., 1993. Principles of sediment transport in rivers, estuaries and coastal seas. Aqua publications, The Netherlands. 293 Appendix B Appendix B ELECTROMAGNETIC VELOCITY METER 294 Appendix B ELECTROMAGNETIC VELOCITY METER 1 History The measurement of the flow velocity by electromagnetic induction was first carried out by Michael Farady (1832) using the earth’s magnetic field. Later, Guelke and Schoute-Vanneck (1948) described the first use of an artificial magnetic field for measuring flow in the sea. A brief description of the electromagnetic and measuring of turbulence and water movement with the wave is given by Bowden and Fairbaim (1956). Schercliffe (1962) published a full description of an electromagnetic velocity meter. 2 Principle of operation The device works on the principle of the dynamo. A coil of wire with a vertical axis is fixed in a housing. Shown in Figure 1 is the principle of operation of the electromagnetic velocity meter. Electric current is passed through the coil and produces a magnetic field, and the conducting water flowing through this field produces a potential gradient at the right angle to both the field and the direction of flow. The principle of electronic circuits is shown in Figure 2. 3 Specifications The Marsh-McBimey Electromagnetic velocity meter (Model 523) with 2 sets of electrode (X-axis, Y-axis) was designed for use in laboratory flume. It is a portable and 295 Appendix B has rechargeable batteries and can be operated on AC and DC power. It consists of • a transducer with cable and • a signal processor. The signal processor of the Marsh-McBimey Electromagnetic velocity meter (Model-523) is housed in a portable case. The velocities in the X and Y directions can be monitored on the two panel meter at the top of it. The signal processor is powered by ±6VDC and is charged with 110 VAC. The probe is installed on 3.17 mm diameter rod with 12.5 mm of diameter. 4 Accuracy Due to electromechanical reactions at the sensing electrodes, as well as electronic drift in the components, the output signals of the electromagnetic velocity meter sensors would drift away from zero volts if the probe were immersed in still water for long periods of time. This model of the electromagnetic velocity meter (Model-523) incorporates a feature which reduces long term zero drift to less than 2.13 cm/sec. Therefore, the long term zero drift for the electromagnetic velocity meter (Model-523) is reported to be equal or less than ±2.13 cm/sec. The output signals may deviate from exactly linear to increasing water velocity due to minor variations in flow streamlines at increasing velocity. The linearity of the response is ± 2% for full scale of velocity range. The gain accuracy of the electromagnetic velocity meter is ± 2%. In order to reduce the noise low pass filters are used on the 296 Appendix B output signals and this results in a peak to peak noise content of 9 mm/sec at the output jack with a one second time constant. References Bowden, K.F. and Fairbum, L.A., 1956. Measurement of turbulent fluctuations and Reynolds stresses in a tidal current. Proc. of the Royal Society of London, Series A., 237. Faraday, M., 1832. Experimental researches in electricity. Phil. Trans. Royal Society of London. 15, 75. Guelke, R.W. and Schoute-Vanneck, C.A., 1947. The measurement of sea-water velocity by electromagnetic induction. Journal of Inst. Electrical Engineering, 94, Part II, P.71. Shercliffe, J. A., 1962. The theory of electromagnetic flow measurement. Cambridge University Press. 297 Appendix B Magnetic field Water flow Electrode Induced voltage fig. i. The principle of operation of the log Electronic switch Measuring head II2 c.p.s. coil 15 ms delay 15 ms delay One pair of Differential [?Oms |?Oms pulse gen I < ///* nmrJifi+r i * I * High -Impedance Electronic switches buffer Op. amp Figure 1 The principle of operation of electromagnetic velocity meter 298 Appendix B Sensing Electrode----- > ( Marsh-McBimey, Model 523 Configuration ) Figure 2 Electromagnetic velocity meter (transducer probe ) 299 Appendix C Appendix C Velocity Profiles 300 Appendix C The time-averaged local velocity at each point in the flow was computed from the recorded velocity fluctuations in the flow direction. The velocity at each point in the flow is normalised by the cross-sectionally averaged flow velocity. Shown in Figures 1 to 12 are some of the velocity profiles for experimental tests. In these figures the velocity profiles are shown with relative depth ratio. In Table C-l the relative depth at which measurements were made are shown for each test. Table C-l Depth ratios for measurements of flow characteristics of the tests Test-E Test-F Test-G Test-H Test-J Test-K Test-L Test-M Test-N Test-O Test-P Test-Q dm d/H dm d/H d/H d/H d/H d/H d/H d/H d/H d/H 0.03 0.06 0.04 0.08 0.1 0.08 0.09 0.06 0.14 0.05 0.04 0.04 0.04 0.09 0.05 0.14 0.17 0.17 0.19 0.13 0.21 0.09 0.07 0.08 0.05 0.13 0.07 0.2 0.28 0.33 0.28 0.19 0.29 0.13 0.15 0.15 0.06 0.16 0.09 0.26 0.38 0.5 0.38 0.26 0.43 0.18 0.22 0.23 0.08 0.19 0.11 0.32 0.48 0.67 0.47 0.32 0.57 0.24 0.28 0.3 0.09 0.23 0.13 0.38 0.59 0.83 0.57 0.39 0.71 0.3 0.35 0.38 0.12 0.26 0.14 0.43 0.69 0.66 0.45 0.79 0.36 0.41 0.45 0.15 0.29 0.16 0.49 0.76 0.75 0.52 0.42 0.48 0.53 0.18 0.32 0.18 0.55 0.83 0.85 0.58 0.48 0.54 0.6 0.21 0.37 0.22 0.61 0.65 0.54 0.61 0.68 0.23 0.42 0.25 0.67 0.71 0.6 0.67 0.75 0.29 0.47 0.29 0.73 0.78 0.66 0.74 0.83 0.35 0.52 0.33 0.84 0.72 0.8 0.91 0.4 0.56 0.36 0.91 0.78 0.87 0.94 0.46 0.62 0.42 0.84 0.93 0.52 0.66 0.47 0.9 0.57 0.71 0.53 0.63 0.76 0.58 0.68 0.81 0.63 0.74 0.72 0.80 0.78 0.85 0.83 0.91 0.89 0.92 301 Appendix C 1.00 T 0.80 -■ 0.60 -• 0.40 ~ 0.20 •• 0.40 0.60 0.80 Figure 1 Normalised velocity profiles from experiment (Test-H) 1.00 T 0.80 •- 0.60 ■- 0.40 «- Figure 2 Normalised velocity profiles from experiment (Test-N) 302 Appendix C 0.80 •• 0.60 0.40 0.00 0.20 0.40 0.60 Figure 3 Normalised velocity profiles from experiment (Test-O) 1.00 t 0.80 -- 0.60 0.40 - Figure 4 Normalised velocity profiles from experiment (Test-B) 303 Appendix D Appendix D Box-Cox Transformation of Instantaneous Shear Stresses in Sweep and Ejection Events 304 Appendix D Al-I: Some Examples of Shear Stress for Sweep Events After Box-Cox Transformation C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-O-8) Normal Probability Plot test-06 (Sweep) .999-' r .95-- .05 * - t- .01 - - ^ Average: 0.639116 Anderson-Darling Normality Test St d Dev: 1.60184 A-Squared: 0.772 Not data: 292 p-value: 0.044 Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-8) C2=B(C) 305 Appendix L Frequency Distribution of Shear Stresses in Sweep Event (Test-O-15) Normal Probability Flot test 0-15 (SWEEP .899------1 - J_____ - J .05- ' Average: 0.373629 Anderson-Darling Normality Test St d Dev: 1.44986 A-Squared: 1.257 Nofdala:313 p-value: 0.003 Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-15) C2=B(C) 306 Appendix D -2.6 -1.6 -0.6 0.6 16 2.6 3.6 4.6 6.5 C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-O-22) Normal Robability Plot test 0-22 (SWEEF) L _ -.1 '«------i «J------i .001-’ r -2.6 -16 -0.6 0.6 Average: 0.324077 Anderson-Darling Normality Test StdDev: 1.37364 A-Squared: 0.779 Nofdata:315 p-value: 0.043 Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-22) C2=B(C) 307 Appendix D C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-O-30) Normal Robability Ftot test 0-30 (SWEEP) i------I r ~ i <____ f „ (------! Average: 0.436814 Anderson-Darling Normality Test StdDev: 1.4727 A-Squared: 0.332 Nofdala:313 p-value: 0.611 Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-30) C2=B(C) 308 Appendix D C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-O-40) Normal Ftobability Fkot test 0-40 (SWEEF) -3.5 -2.5 -1.5 -0.5 0.5 3.5 4.5 Average: 0.654364 Anderson-Dar ling Normality Test St d Dev: 1.54516 A-Squared: 0.311 Nofdata:307 p-value: 0.551 Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-40) r?=R(r) 309 Appendix D C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-O-60) Normal Probability Plot test 0-60 (SWEB) _ t___ _t— -l— .001------» Average: 0.575725 Anderson-Darling Normality Test St d Dev: 1.48307 A-Squared: 0.684 Not data:294______p-value: 0,073 Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-60) C2=B(C) 310 Appendix L C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-O-70) Normal Probability Plot test 0-70 (SWEEP .999 ’ - “ r ~ ------L ~ -t - -1.5 -0.5 Average: 0.878667 Anderson-Dar II ng Nor mal it y Test St d Dev: 1.46059 A-Squared: 0.873 Notdata:266______p-value: 0,025 Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-70) C2=B(C) 311 Appendix D 40- C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-O-80) Normal Probability Rot test 0-80 (SWEEF) ., t__ 1____f Average: 0.708607 Anderson-Dar ling Normality Test St d Dev: 1.41575 A-Squared: 0.629 No(data:310 p-value: 0.101 Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-80) C2=B(C) 312 Appendix D 70-f C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-O-90) Normal Probability Rot test 0-90 (SWEET) _ i— Average: 0.688441 Anderson-Darling Normality Test St d Dev: 1.53916 A-Squared: 1.101 Not dal tu207______p-value: 0.007______ Normal Probability Plot of Shear Stresses in Sweep Event (Test-O-90) C2=B(C) 313 Appendix L Frequency Distribution of Shear Stresses in Sweep Event (Test-P-10) Normal Probability Rot test Pt) (SWEEF) Average: 0.851703 Anderson-Dar ling Normality Test St d Dev: 1.64077 A-Squared:0.411 Not data:273 p-value: 0.339 Normal Probability Plot of Shear Stresses in Sweep Event (Test-P-10) C2=B(C)______ 314 Appendix L C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-P-20) Normal Robability Rot test R-20 (SWEB) I------«. .05------1 -3.6 -2.6 -1.6 -0.6 Average: 0.383768 Anderson-Darling Normality Test St d Dev: 1.50454 A-Squared: 0.889 Not data:305 p-value: 0.023 Normal Probability Plot of Shear Stresses in Sweep Event (Test-P-20) C2=B(C) 315 Appendix D 60- 50- C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-P-35) Normal Probability Flot test F35(SV\EEF) .99- - * " r ■8 .50 .05-- Average: 0.460608 Anderson-Dar ling Normality Test Std Dev: 1.48318 A-Squared: 0.587 Nofdata:297 p-value: 0.125 Normal Probability Plot of Shear Stresses in Sweep Event (Test-P-35) C2=B(C)______ 316 Appendix D C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-P-50) Normal Robability Rot test P-50 (SMEEF) .999 - 'S 1____ f .60*1 .05 -« Average: 0.612608 Anderson-Darling Normality Test St d Dev: 1.44052 A-Squared:0.358 Not data-283______p-value: 0.451______ Normal Probability Plot of Shear Stresses in Sweep Event (Test-P-50) C2=B(C) 317 Appendix D C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-P-65) Normal Probability Plot test P-65 (SWEEp .001------> Average: 0.63953 Anderson- Dar li ng Normal It y Test St d Dev: 1.48615 A-Squared: 0.736 No Normal Probability Plot of Shear Stresses in Sweep Event (Test-P-65) C2=B(C) 318 Appendix L C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-P-95) Normal Probability Hot test R-95 (SWEEF) .99-' -3.5 -2.5 -1.5 -0.5 0.5 Average: 0.673886 Anderson-Darling Normality Test St d Dev: 1.69285 A-Squared: 1.220 Not data:309______p-value: 0.003 Normal Probability Plot of Shear Stresses in Sweep Event (Test-P-95) C2=B(C) 319 Appendix L 70 - 60- 20- C2 Frequency Distribution of Shear Stresses in Sweep Event (Test-Q-10) Normal Probability Rot test Q-10 (SWEEF) .95 • j — -|- i T ~i------X j20 - i ~ — i. » ( jg .05 ■ < " " l~T«yr .01 ■ J J 1 < .001 - j-----j~ - j- “ T ~ Average: 0.335615 Ander son-Dari Ing Normality Test St d Dev: 1.51925 A-Squared:0.914 Not data:310 p-value: 0.020 Normal Probability Plot of Shear Stresses in Sweep Event (Test-Q-10) C2=B(C) 320 Appendix L Al-II: Some Examples of Shear Stress for Ejection Events After Box-Cox Transformation 60- C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-k-20) Normal Ftobabifity Plot test k-20 _ _ »______I« - 4. - . .05- ' .05------___L. J----- A „ _ i.-----u. .001 - ~ Average: 0.151925 Anderson-Darling NormaJity Test St d Dev: 1.66650 A-Squared:0.621 No(dala:339 p-value: 0.105 Normal Probability Plot of Shear Stresses in Ejection Event (Test-k-20) 321 Appendix L C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-k-40) Normal Probability Plot test k-40 .099 * 1 r ~ T r ~ r ~ .50 - 'j T ~ “f .05 f------1 Average: 0.336857 Anderson-Darling Normality Test St d Dev: 1.6008 A-Squared: 0.326 Not data:319______p-value: 0.620______ Normal Probability Plot of Shear Stresses in Ejection Event (Test-k-40) 322 Appendix D C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-k-60) Normal Probability Plot tost k-60 L r ~ "i" .50 — i------i \___ ) i _ ____ » Average: 1.33079 Anderson-Darling Normality Test St d Dev: 1.62161 A-Squared: 0.729 Nofdala:246 p-value: 0.057 Normal Probability Plot of Shear Stresses in Ejection Event (Test-k-60) 323 Appendix I C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-k-80) Normal Robability Rot test k-80 Average: 0.887738 Anderson-Darling Normality Test St d Dev: 1.70469 A-Squared: 0.711 Not data:274 p-value: 0.063 Normal Probability Plot of Shear Stresses in Ejection Event (Test-k-80) 324 Appendix L C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-k-100) Normal Probability Fkot test k-100 .99 -j ~ ~ .05 Average: 1.22327 Anderson-Darling Normality Test St d Dev: 1.68919 A-Squared: 0.842 Nof data:276 p-value: 0.030 Normal Probability Plot of Shear Stresses in Ejection Event (Test-k-100) 325 Appendix L -3-2-10 1 2 3 4 C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-1-20) Normal Ftobability Rot test L-20 ~ ~ “ — i. Average: 0.179000 Anderson-Dar ling Normality Test St d Dev: 1.37472 A-Squared: 1.389 Not data:320______p-value: 0.001______ Normal Probability Plot of Shear Stresses in Ejection Event (Test-1-20) 326 Appendix L 90 —| C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-1-40) Normal Ftobability Plot test L-40 r •* & __ i------.50-' .01 - • ~ .001 * ' Average: 0.168847 Anderson-Darling Normality Test St d Dev: 1.38944 A-Squared: 0.986 Not data:332______p-value: 0.013______ Normal Probability Plot of Shear Stresses in Ejection Event (Test-1-40) 327 Appendix D -3-2-10 1 23 C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-1-60) Normal R-obabiOty Rot t«st L-60 .01 ■ -*■ Avarag«:0.0350082 Andaraon-Darllng Normality T«s( StdD«v:l.U487 A-Squar*d:0.S20 No(data:3S6 p-vaiu«: 0.135 Normal Probability Plot of Shear Stresses in Ejection Event (Test-1-60) C2=B(C) 328 Appendix L I £ ii- 02 Frequency Distribution of Shear Stresses in Ejection Event (Test-1-80) Normal Robabifity Rot tost L-80 i------1 r------«■ i------r Average: 0X71633 Anderaofl-Darlfng NormalKy Test 6tdD«v: 165001 A-Squared; 1.413 N of dal a: 303 p-value: 0.001 Normal Probability Plot of Shear Stresses in Ejection Event (Test-1-80) C2=B(C) 329 Appendix D C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-1-100) Nbrmal Rrobabflity Rot test L-tX) I___ « L ~ Ji r “ “i i------1 16 -0.6 0.6 2.6 3.6 4.6 Andaraon-Darling Normality Ta«t St d Dav: 162735 A-Squared: 0.074 Nofdata:302 p-value: 0.0U Normal Probability Plot of Shear Stresses in Ejection Event (Test-l-lOO) C2=B(C) 330 Appendix D Frequency Distribution of Shear Stresses in Ejection Event (Test-1-120) Normal Robabifity Rot tost L-120 (EJECTION) .05*4 .001 1------\ t------1 -2.6 -1.6 -0.6 0.6 2.6 3.6 4.6 6.6 Avarag«: 103666 Andaraon-Darllng Normality Taat StdDav: 166004 A-6qu«rad: 0.400 Nofd*ta:2S4 p-valu«: 0.350 Normal Probability Plot of Shear Stresses in Ejection Event (Test-1-120) C2=B(C) 331 Appendix D Frequency Distribution of Shear Stresses in Ejection Event (Test-M-10) Normal Probability Rot tost M-TD (EJECTION) Avaraga: 0.665217 Andaraon-Darllng Normality Tast StdDav: 160027 A-Squar ad: 0.667 Not data:207 p-valua: 0.t26 Normal Probability Plot of Shear Stresses in Ejection Event (TestM-10) C2=B(C) 332 Appendix L eo 80 70 60 60 r 40 g 30 20 K) 0 -4 -3 -2 -1 0 1 2 3 4 6 6 C2 Frequency Distribution of Shear Stresses in Ejection Event (Test-M-20 Normal frobability Rot test M-20 (EJECTION) .800*1-----» l---- L _ 1____) r ~ r f ~ f 1' ( .05 * -1-----I .001* v* **» r-r-T-T’-r r - r •4 -3 Avaraga: 0.64127 Andaraon-Darltng Normality Ta*t StdDav: 161686 A-Squarad:0.460 Nof data:307 p-valua: 0247 Normal Probability Plot of Shear Stresses in Ejection Event (TestM-20 C2=B(C) 333 Appendix E Appendix E Sample Model Output File 334 Appendix E A /A d(mm) e/u (m/s) Rn Mp/W* Up/u+ 0.002 0.758 0.0145 25.43 — — 0.002 0.784 0.0150 26.31 — — 0.002 0.810 0.0155 27.19 — — 0.002 0.836 0.0160 28.06 — — 0.002 0.862 0.0165 28.94 0.112 — 0.002 0.888 0.0170 29.82 0.453 — 0.002 0.914 0.0175 30.69 0.774 — 0.002 0.940 0.0180 31.57 1.078 — 0.002 0.966 0.0185 32.45 1.365 — 0.002 0.993 0.0190 33.32 1.637 — 0.002 1.019 0.0195 34.20 1.896 0.182 0.002 1.045 0.0200 35.07 2.141 0.428 0.002 1.071 0.0205 35.95 2.374 0.661 0.002 1.097 0.0210 36.83 2.597 0.883 0.002 1.123 0.0215 37.70 2.808 1.095 0.002 1.149 0.0220 38.58 3.011 1.297 0.002 1.175 0.0225 39.46 3.204 1.491 0.002 1.201 0.0230 40.33 3.389 1.676 0.002 . 1.227 0.0235 41.21 3.566 1.853 0.002 1.254 0.0240 42.09 3.736 2.022 0.002 1.280 0.0245 42.96 3.898 2.185 0.002 1.306 0.0250 43.84 4.055 2.341 0.002 1.332 0.0255 44.72 4.205 2.491 0.002 1.358 0.0260 45.59 4.349 2.636 0.002 1.384 0.0265 46.47 4.488 2.775 0.002 1.410 0.0270 47.34 4.622 2.908 0.002 1.436 0.0275 48.22 4.751 3.037 0.002 1.462 0.0280 49.10 4.875 3.162 0.002 1.488 0.0285 49.97 4.995 3.282 0.002 1.515 0.0290 50.85 5.111 3.397 0.002 1.541 0.0295 51.73 5.223 3.509 0.002 1.567 0.0300 52.60 5.331 3.617 0.002 1.593 0.0305 53.48 5.435 3.722 0.002 1.619 0.0310 54.36 5.537 3.823 0.002 1.645 0.0315 55.23 5.635 3.921 0.002 1.671 0.0320 56.11 5.730 4.016 0.002 1.697 0.0325 56.99 5.822 4.108 335 Appendix E Continued; A /A d(mm) 0ß U+ (m/s) Rn Mp/M* Up/Ü* 0.002 1.776 0.0340 59.61 6.082 4.368 0.002 1.802 0.0345 60.49 6.163 4.450 0.002 1.828 0.0350 61.37 6.242 4.529 0.002 1.854 0.0355 62.24 6.319 4.606 0.002 1.880 0.0360 63.12 6.394 4.681 0.002 1.906 0.0365 64.00 6.467 4.754 0.002 1.932 0.0370 64.87 6.538 4.825 0.002 1.958 0.0375 65.75 6.607 4.894 0.002 1.984 0.0380 66.63 6.674 4.961 0.002 2.011 0.0385 67.50 6.740 5.026 0.002 2.037 0.0390 68.38 6.803 5.090 0.002 2.063 0.0395 69.26 6.866 5.152 0.002 2.089 0.0400 70.13 6.926 5.213 0.002 2.115 0.0405 71.01 6.985 5.272 0.002 2.141 0.0410 71.88 7.043 5.329 0.002 2.167 0.0415 72.76 7.099 5.386 0.002 2.193 0.0420 73.64 7.154 5.441 0.002 2.219 0.0425 74.51 7.208 5.494 0.002 2.245 0.0430 75.39 7.260 5.547 0.002 2.272 0.0435 76.27 7.311 5.598 0.002 2.298 0.0440 77.14 7.361 5.648 0.002 2.324 0.0445 78.02 7.410 5.697 0.002 2.350 0.0450 78.90 7.458 5.744 0.002 2.376 0.0455 79.77 7.505 5.791 0.002 2.402 0.0460 80.65 7.551 5.837 0.002 2.428 0.0465 81.53 7.595 5.882 0.002 2.454 0.0470 82.40 7.639 5.925 0.002 2.480 0.0475 83.28 7.682 5.968 0.002 2.507 0.0480 84.15 7.724 6.010 0.002 2.533 0.0485 85.03 7.765 6.051 0.002 2.559 0.0490 85.91 7.805 6.092 0.002 2.585 0.0495 86.78 7.845 6.131 0.002 2.611 0.0500 87.66 7.884 6.170 0.002 2.637 0.0505 88.54 7.922 6.208 0.002 2.663 0.0510 89.41 7.959 6.245 336 Appendix E Continued; A /A d(mm) eß U+ (m/s) rn Wp/M* Mp/«* 0.002 1.723 0.0330 57.86 5.911 4.197 0.002 1.750 0.0335 58.74 5.997 4.284 0.002 2.689 0.0515 90.29 7.995 6.281 0.002 2.741 0.0525 92.04 8.066 6.352 0.002 2.768 0.0530 92.92 8.100 6.387 0.002 2.794 0.0535 93.80 8.134 6.420 0.002 2.820 0.0540 94.67 8.167 6.454 0.002 2.846 0.0545 95.55 8.200 6.486 0.002 2.872 0.0550 96.42 8.232 6.518 0.002 2.898 0.0555 97.30 8.263 6.549 0.002 2.924 0.0560 98.18 8.294 6.580 0.002 2.950 0.0565 99.05 8.324 6.610 0.002 2.976 0.0570 99.93 8.354 6.640 0.002 3.002 0.0575 100.81 8.383 6.669 0.002 3.029 0.0580 101.68 8.412 6.698 0.002 3.055 0.0585 102.56 8.440 6.726 0.002 3.081 0.0590 103.44 8.468 6.754 0.002 3.107 0.0595 104.31 8.495 6.781 0.002 3.133 0.0600 105.19 8.522 6.808 0.002 3.159 0.0605 106.06 8.548 6.835 0.002 3.185 0.0610 106.94 8.574 6.860 0.002 3.211 0.0615 107.82 8.600 6.886 0.002 3.237 0.0620 108.69 8.625 6.911 0.002 3.264 0.0625 109.57 8.650 6.936 0.002 3.290 0.0630 110.45 8.674 6.960 0.002 3.316 0.0635 111.32 8.698 6.984 0.002 3.342 0.0640 112.20 8.722 7.008 0.002 3.368 0.0645 113.08 8.745 7.031 0.002 3.394 0.0650 113.95 8.768 7.054 0.002 3.420 0.0655 114.83 8.790 7.076 0.002 3.446 0.0660 115.71 8.812 7.098 0.002 3.472 0.0665 116.58 8.834 7.120 0.002 3.498 0.0670 117.46 8.856 7.142 0.002 3.525 0.0675 118.33 8.877 7.163 0.002 3.551 0.0680 119.21 8.898 7.184 0.002 3.577 0.0685 120.09 8.918 7.204 0.002 3.603 0.0690 120.96 8.938 7.224 337 Appendix E Continued; A /A d(mm) 0/0 U* (m/s) *n Wp/M* Up/u* 0.002 2.715 0.0520 91.17 8.031 6.317 0.002 3.629 0.0695 121.84 8.958 7.244 0.002 3.655 0.0700 122.72 8.978 7.264 0.002 3.681 0.0705 123.59 8.998 7.283 0.002 3.707 0.0710 124.47 9.017 7.303 0.002 3.733 0.0715 125.35 9.036 7.321 0.002 3.759 0.0720 126.22 9.054 7.340 0.002 3.786 0.0725 127.10 9.072 7.358 0.002 3.812 0.0730 127.98 9.091 7.377 0.002 3.838 0.0735 128.85 9.108 7.394 0.002 3.864 0.0740 129.73 9.126 7.412 0.002 3.890 0.0745 130.60 9.143 7.429 0.002 3.916 0.0750 131.48 9.161 7.446 0.002 3.942 0.0755 132.36 9.177 7.463 0.002 3.968 0.0760 133.23 9.194 7.480 0.002 3.994 0.0765 134.11 9.211 7.497 0.002 4.021 0.0770 134.99 9.227 7.513 0.002 4.047 0.0775 135.86 9.243 7.529 0.002 4.073 0.0780 136.74 9.259 7.545 0.002 4.099 0.0785 137.62 9.274 7.560 0.002 4.125 0.0790 138.49 9.290 7.576 0.002 4.151 0.0795 139.37 9.305 7.591 0.002 4.177 0.0800 140.25 9.320 7.606 0.002 4.203 0.0805 141.12 9.335 7.621 0.002 4.229 0.0810 142.00 9.350 7.636 0.002 4.255 0.0815 142.87 9.364 7.650 0.002 4.282 0.0820 143.75 9.379 7.664 0.002 4.308 0.0825 144.63 9.393 7.679 0.002 4.334 0.0830 145.50 9.407 7.693 0.002 4.360 0.0835 146.38 9.421 7.706 0.002 4.386 0.0840 147.26 9.434 7.720 0.002 4.412 0.0845 148.13 9.448 7.734 0.002 4.438 0.0850 149.01 9.461 7.747 0.002 4.464 0.0855 149.89 9.474 7.760 0.002 4.490 0.0860 150.76 9.487 7.773 0.002 4.516 0.0865 151.64 9.500 7.786 0.002 4.595 0.0880 154.27 9.538 7.824 0.002 4.621 0.0885 155.14 9.550 7.836 338 Appendix E Continued; A /A d( mm) eß U* (m/s) Rn Mp/W* up/ü+ 0.002 4.647 0.0890 156.02 9.562 7.848 0.002 4.673 0.0895 156.90 9.574 7.860 0.002 4.699 0.0900 157.77 9.586 7.872 0.002 4.725 0.0905 158.65 9.598 7.884 0.002 4.751 0.0910 159.53 9.610 7.895 0.002 4.778 0.0915 160.40 9.621 7.907 0.002 4.804 0.0920 161.28 9.633 7.918 0.002 4.830 0.0925 162.16 9.644 7.930 339 Appendix F Appendix F A sequence of images with differences 340 Appendix F 341 Appendix F 342 Appendix F 343 Appendix F ... güHSHi - v^;' - • I r* ft 'ÜP15' SH I * f :■ »•%ar ' iff- 2- S * « - / - ,, >’ H ü äsJäSSjB k-- . v ; £ / M0 I 4? *•-_- »: • r^_j 344 345 Appendix F I- s X* >■ ■ ' - : • . • ■ * Mi-p{fS0T. W • ' ;v.. %y ■ ;4>%# % * «* ' .*/ ,y - ' ".C 1 ■jS I, ’V ' 4' . i " - „X-A V , v.' ’ - s “V . • . • ” * - 346 Appendix F 347 Appendix F V ' , IV • ■* m a '4 4! ■f y V- *•• >\ ■». %I* ■ *# # >*■ ft--Y'/tv-v; ■ #'< ■- . .*• ■ •>' y'j r “■ ■:*•f: v3 H ‘ff/- 348 Appendix F 349 Appendix F 350 Appendix G Appendix G Cross correlation of entrained particles and shear stresses during sweep events Appendix G R5-R1 .2 0.7- U 0.4- R7-R1 R5-R2 R7-R2 Lag O (sec) Figure G-l Cross Correlation of instantaneous shear stress in sweep event and entrained particles number from the bed Appendix G Test R5-R3 G O c3 8 o U 00 oo O Wh u Test R7-R3 c o \Ö c3 *03 fc o U oo oo O Wh u Test R5-R6 c o 'S 13 fc o U oo co O Wh u Test R7-R4 c _o cd 13 fc o U oo ooO Lag (sec) Figure G-2 Cross Correlation of instantaneous shear stress in sweep event and entrained particles number from the bed