UFL/COEL-99/012
TEMPORAL DEVELOPMENT OF CLEAR WATER LOCAL
SCOUR AROUND CYLINDRICAL PIERS
by
Edward Albada
Thesis
1999
TEMPORAL DEVELOPMENT OF CLEAR WATER LOCAL
SCOUR AROUND CYLINDRICAL PIERS
By
EDWARD ALBADA
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
1999
ACKNOWLEDGMENTS
First and foremost I would like to express my gratitude towards Dr. D. Max
Sheppard for his encouraging words and thoughts, without which this thesis would not have
developed into what it has, and for the countless opportunities given to me in furthering my
academic career. It has truly been a pleasure working with such a compatible, easy-going,
laid-back professor with whom I share so many common interests.
I am also grateful to Sterling Jones with the FHWA and Shawn McLemore and Rick
Renna with the FDOT for funding this work, especially Sterling Jones for the use of his data.
I must also thank the Massachusetts crew, the COE Laboratory crew, and all the
professors for all the time spent explaining various subjects, both inside and out of the
classroom, and of course, Becky and Helen, without whom we would all be lost.
Thanks also go to my roommates who lived with me at 1928 1st Ave, as well as my
fellow students from all over the world, for making my entire graduate education enjoyable.
Particularly, Tom, Thanasis, Lisa, Wendy, Bill, Matt, Roberto, Guillermo, Erica, Al, and
Nicolas. I intend to visit all of you in your countries eventually.
To Nicole you were the reason I returned to Gainesville. I thank you for the
wonderful times we have shared together whatever the future holds for us I will always
love you.
Finally to my family Mom and Dad, Laurens & Kelly, and my grandmothers -
thank you for the support and encouragement from near and afar.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ............................................................................................ ii
LIST OF TABLES ......................................................................................................... v
LIST OF FIGURES...................................................................................................... vi
KEY TO SYMBOLS ................................................................................................... ix
ABSTRACT............................................................................................................... xii
CHAPTERS
1 INTRODUCTION.................................................................................................... 1
2 BACKGROUND AND LITERATURE REVIEW..................................... ............. 7
2.1 B ackground.............................................................................. ................ 7
2.2 Scour Formation for Steady Flow .............................................................10
2.3 Effect of the Various Parameters..............................................................16
2.3.1 Effect of Velocity Ratio U/U .......................................... .............. 16
2.3.2 Effect of Aspect Ratio yo/b ..... ........................................... ............ 20
2.3.3 Effect of Sediment to Pier Size Dso/b ........................................ ..21
2.4 Literature Review .............................................................................22
3 EXPERIMENTAL PROCEDURES .........................................................................33
3.1 Introduction ............................................................................................... 33
3.2 E quipm ent.................................................... .............................................33
3.2.1 Flum e ................................................................................................ 34
3.2.2 Electronic Apparatus .....................................................................37
3.2.2.1 Acoustic transducers.................................... ....................37
3.2.2.2 Electromagnetic current meters ............. ...........................38
3.2.2.3 Digital thermometer.........................................................38
3.2.2.4 Cameras...........................................................................39
3.2.3 T est Setup ................................... .. ........................................39
3.2.4 Models ............................................... --........ ............................ 40
3.2.5 Sedim ent ........................................ ................. ............................40
3.3 Laboratory Test Procedure ................................... ......................................42
3.3.1 Bed Preparation ................................................ .......................... 42
3.3.2 Laboratory Procedure .................................................................... 42
4 DATA REDUCTION AND ANALYSIS ................................................................43
4.1 R aw D ata ................................................................................................. .43
4.2 Data Adjustment and Smoothing ..............................................................44
4.3 Experiment Summaries .............................................................................46
4.4 A analysis of D ata ....................................................................................... 49
5 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS...............................73
5.1 Sum m ary................................. ..................................... ........................... 73
5.2 Conclusions .............................................................................................. 74
5.3 Recommendations..................................................................................78
APPENDICES
A INDIVIDUAL CURVE FITS............................................... .................................82
B SEMILOG PLOTS OF SCOUR DEPTH VERSUS TIME....................................... 95
C SCOUR HOLE CONTOUR PLOTS ................................. ..............................99
REFEREN CES .................... ................................................... ........ ...........................103
BIOGRAPHICAL SKETCH ........................................................ 106
LIST OF TABLES
Table page
4.1 Summary of experiments completed at USGS Laboratory......................................47
4.2 Summary of experiments completed at FHWA Laboratory ....................................48
LIST OF FIGURES
Figure page
2.1 Diagram showing forces on a particle resting on the bed........................................8
2.2 Shields' diagram ................................................................................................ 9
2.3 Diagram showing flow vortices around pile in flow................................... ...12
2.4 Downflow variation with depth from Ettema (1980)................................. ....13
2.5 Dependence of nondimensionalized scour
depth on velocity ratio for two sediment sizes ...................................... .....17
2.6 Changes in bed features with increasing velocity from Snamenskaya (1969).........19
2.7 Dependence of equilibrium scour depth on aspect ratio for which
U/Uc and d50/b are held constant........................................................ ............20
2.8 Variation of scour depth on d50/b with U/Uc=1 and y0/b>3.0 ...............................22
3.1 Aerial schematic of flume.......................................................... ....................35
3.2 Schematic figure of the cross-section of flume.................................................35
3.3 Detail of MTA arrangement used in small-pier experiments ................................38
3.4 Photograph of cameras used for the 0.114 m pile in casing...................................39
3.5 Grain size distribution for sand #1 (do0 = 0.22 mm)..............................................41
3.6 Grain size distribution for sand #2 (dso = 0.80 mm)..............................................41
4.1 Scour depth versus time plots for the video and acoustic transponder data.
(b=0.305m, d50=0.80mm, y0=1.268m, and U=0.381m/s)...................................44
4.2 Comparison between raw and adjusted scour depth data ......................................46
4.3 Smoothed scour history plots for all UF/USGS-BRD Laboratory experiments......47
4.4 Smoothed scour history plots for all FHWA Laboratory experiments ..................49
4.5 Comparison of ds/dse versus time for USGS Laboratory experiments ..................50
4.6 Equilibrium scour depth versus time for equilibrium for USGS
Laboratory experiments ...................................................... ..................51
4.7 Plot of ds/dse versus t/te for USGS Laboratory experiments ................................54
4.8 Rate of scour (d/dt(ds/dse)) versus ds/dse for all experiments ..............................55
4.9 Power spectrum of turbulent bursts in the horseshoe vortex
in a wind tunnel from Baker (1978)........................................ .................. 58
4.10 Plot of rate of scour versus scour depth showing increasing pier diameters ..........62
4.11 Author's hypothesis of the relative contributions of the mechanisms on
the rate of scour...................................................................................................64
4.12 Diagrams representing early and later stages of scour............................................66
4.13 Plot of equilibrium scour depth versus t90 for all experiments............................69
4.14 Plot of equilibrium scour depth versus fl for all experiments..............................70
4.15 Plot of equilibrium scour depth versus f2 for USGS Laboratory experiments .......71
A-1 Curve fit of smoothed data for experiment 3......................................................82
A-2 Curve fit of smoothed data for experiment 4......................................................83
A-3 Curve fit of smoothed data for experiment 6......................................................84
A-4 Curve fit of smoothed data for experiment 9......................... ......................85
A-5 Curve fit of smoothed data for experiment 11.................................................86
A-6 Curve fit of smoothed data for experiment 12......................................................87
A-7 Curve fit of smoothed data for experiment 13......................................................88
A-8 Curve fit of smoothed data for experiment 57...................................................89
A-9 Curve fit of smoothed data for experiment 74............................................. .......90
A-10 Curve fit of smoothed data for experiment 86......................................................91
A-I Curve fit of smoothed data for experiment 126......................................................92
A-12 Curve fit of smoothed data for experiment 128......................................................93
A-13 Curve fit of smoothed data for experiment 133...............................................94
B-I Semilog plot of scour depth versus time for experiment 3 ......................................95
B-2 Semilog plot of scour depth versus time for experiment 4 ......................................96
B-3 Semilog plot of scour depth versus time for experiment 6 .....................................96
B-4 Semilog plot of scour depth versus time for experiment 9 .....................................97
B-5 Semilog plot of scour depth versus time for experiment 11 ...................................97
B-6 Semilog plot of scour depth versus time for experiment 12 ..................................98
B-7 Semilog plot of scour depth versus time for experiment 13 ..................................98
C-l Scour hole contour plots for experiment 3..........................................................99
C-2 Scour hole contour plots for experiment 4.........................................................100
C-3 Scour hole contour plots for experiment 6..........................................................100
C-4 Scour hole contour plots for experiment 9...................................... ........ 101
C-5 Scour hole contour plots for Experiment 11 .......................................................101
C-6 Scour hole contour plots for Experiment 12........................................................102
C-7 Scour hole contour plots for Experiment 13.............................................. 102
KEY TO SYMBOLS
Al particle area constant
A2 particle volume constant
b pile width (diameter)
B channel width
c exponent related to bed load
C constant
Cd drag coefficient
Cwr rectangular weir coefficient
d16 sediment size for which 16 percent of bed material is finer
dso median sediment diameter
d84 sediment size for which 84 percent of bed material is finer
ds scour depth
dse equilibrium scour depth
f frequency
fl function
f2 function
Ff frictional force
Fn normal force
Fr Froude number
Frc critical Froude number
g acceleration of gravity
H head over the weir
Ki constant
KI flow intensity factor
L length of pier
Ns sediment number
Nsc critical sediment number for sediment removal
P power
Pw weir height
qs discharge per unit width
Q discharge
r radial distance
Re* boundary Reynolds number
s relative sediment density
St Strouhal number
te time required for scour depth to reach equilibrium
t90 time required for scour depth to reach 90% equilibrium
U mean depth average velocity
Ue angle component of velocity
u*c critical bed friction velocity
u velocity near bed
u* bed shear velocity
u*c critical bed shear velocity
V pier downflow velocity
w width of flume
yO depth of flow
e rate at which particles in stagnation plane are transported
A dune height
X dune width
0 angle of attack of flow
v kinematic viscosity of water
p density of water
Pm density of mixture
T* Shields' parameter
to bed shear stress
co frequency
Ys specific weight of sediment
Ym specific weight of mixture
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
TEMPORAL DEVELOPMENT OF CLEAR WATER LOCAL
SCOUR AROUND CYLINDRICAL PIERS
By
Edward Albada
August 1999
Chairman: Dr. D. Max Sheppard
Major Department: Coastal and Oceanographic Engineering
When a bottom mounted structure such as a bridge pier is located in an erodible
sediment and is subjected to water flow, local scour will occur provided the flow velocity
is sufficiently large. The local scour processes are complex and vary in relative
magnitude as the scour hole develops. The rate at which scour occurs is equally complex
and difficult to predict. For a steady flow velocity the scour hole will deepen with time
until an equilibrium depth is reached.
Both the equilibrium scour depth and the rate at which the hole develops is
important in the design of structures that will experience scour. This thesis analyzes local
scour time history data for a number of long duration experiments. The structures are
circular piles with diameters ranging from 0.114m to 0.914m. The University of Florida
tests were performed at the USGS-BRD Laboratory in Turners Falls, Massachusetts. The
FHWA tests were performed at the Turner Fairbanks Laboratory in McLean, Virginia.
Attempts are made to determine the parameters affecting the rate of scour, the variation of
rate with scour hole depth, and the time required to reach equilibrium. Using
observations made of the dependencies of various parameters on the rate, a predictive
method for determining the scour depth at any time based on the conditions of the
experiment is presented.
CHAPTER 1
INTRODUCTION
1.1 Motivation
There is a long history in the use of rivers and bridges as means of transportation.
Initially, rivers were used to carry goods both downstream and upstream because of the
advantages of water-based transport. Eventually, over time, as more settlements
flourished inland away from both coasts and rivers, the need arose for land-based
transport and the very rivers that were once beneficial to transportation became
hindrances. Bridges were developed to connect the banks between which the rivers
flowed.
The development of bridge design has become very technical and involved, but
for most bridges support comes in the way of piles driven into the bed upon which the
bridge sits. The piles hinder the natural flow of the river, and as a result water is forced
to accelerate around the piles, and sediment surrounding the piles becomes scoured out,
creating a depression. This depression must be accounted for in assuring the structural
stability of the pile for a specified storm condition.
The importance of predicting the equilibrium depth of scour cannot be
understated. Underprediction can result in the collapse of the bridge, resulting in
expensive delays, loss of property and even fatalities. On March 10, 1995, the Interstate
5 bridge over Arroyo Pasajero in California collapsed as a result of scour during a flood
2
event, killing seven people (Truhlar & Telis, 1997). Brice and Blodgett (1978)
determined that approximately half of 383 bridge failures studied were caused by local
scour, most of them due to erroneous equilibrium depth predictions. The added cost for
ensuring the structural integrity of a bridge easily outweighs the expense of having to
replace the bridge once it has failed. However, overprediction can result in unnecessary
expense and can add to the construction time needlessly. For the proposed Bonner
Bridge over Oregon Inlet in North Carolina, a 20 percent reduction in the estimated
equilibrium scour depth can save close to a million dollars in construction costs (D.
Dock, Parsons Brinkerhoff, Raleigh, NC, personal communication, March 1998).
The complexities of the river flow, with chaotic turbulence and erratic bank
behavior make scour prediction extremely difficult. The very parameters with which the
engineers work are usually determined statistically for a specified storm condition which
may or may not occur during the designed lifetime of the structure. Thus, the success or
failure of the design is largely dependent on the probability of a storm event that is more
severe than the one used for design.
Streambed erosion in the vicinity of bridges can be divided into the following
additive components:
1) Long-term aggradation and degradation. Aggradation and degradation are
erosion or deposition of sediment in the riverbed due to a change in the
equilibrium of the river upstream and/or downstream of the bridge site. These
elevation changes occur independently from the presence of the bridge, and are
attributed to either natural or man-induced processes affecting the path of the
river, the sediment concentration, or the flow intensity. Factors that contribute to
long term aggradation and degradation include channelization, changes in
downstream hydraulic control, cutoffs or meander loops, regulation or diversion
of stream flow, changes in basin rainfall-runoff characteristics, climate changes,
gravel mining from the stream bed, dams, reservoirs, changes in basin land use,
and catastrophic floods (Landers & Mueller, 1996). Aggradation and degradation
are generally slow processes, but the long-term trend must be evaluated for the
lifetime of the structure.
2) Contraction Scour. Contraction scour is the erosion of sediment due to the
reduction in the channel flow area of the river, either by natural obstructions
created in the course of the river or man-made supporting structures such as
roadway embankments which project into the channel. In keeping with the law of
continuity, a decrease in the channel width will increase the velocity of the river
to maintain a constant water discharge. The increase in velocity correspondingly
increases the shear stress on the bed, and sediment is eroded. The removal of bed
material gradually increases the channel depth (thus increasing the cross-sectional
area of the river) until equilibrium is reached balancing the shear stress due to the
velocity of the stream and critical shear stress for the onset of particle mobility.
The increase in flow velocity caused by contraction scour can also affect local
scour around the bridge piers significantly once a transition is made from a clear-
water to a live-bed condition when a certain critical velocity is exceeded. The
differences between clear-water and live-bed scour will be discussed further in
this chapter.
3) Local Scour. Local scour is caused by vortices and eddies which are
created by the disturbance of the flow pattern around structures such as bridge
piers. The pier effectively becomes an obstruction to the flow, forcing the flow to
accelerate around it. This acceleration initiates the scour process. The main
scouring mechanism is controlled by a horseshoe vortex system, formed as a
result of flow separation from the leading edge of the scour hole. The mechanics
of local scour are more thoroughly described in Chapter 2.
Both equilibrium scour depth and the time rate of scour are dependent on the
velocity of the water flow. Unless specified otherwise, flow velocity in this thesis refers
to the depth-averaged velocity for fully developed flow. At low velocities, the shear
stress on the bed is not large enough to initiate sediment transport. As the velocity
increases past a critical value, transport of bed material is initiated. Clear-water scour is
defined as local scour which occurs when the flow velocity is below the value needed to
initiate sediment motion on the flat bed upstream of the structure. Note that the upstream
velocity need not exceed the critical velocity for clear-water scour to occur.
Live-bed scour is local scour that occurs when the flow velocity exceeds the value
needed to initiate sediment motion (suspended and/or bedload) upstream of the structure
(Sheppard, 1998). The critical velocity for the transition between the two conditions can
be obtained from the modified Shields curve and is dependent upon various parameters
such as the water density and viscosity, sediment density and size, bed roughness, and
water depth.
Keeping all other parameters constant, scour first begins at about 0.45 times the
critical velocity for a circular pile. Increasing the velocity results in a deepening of the
equilibrium scour hole, until a peak depth is reached at the critical velocity. Further
increases of the current velocity into the live-bed range initially reduce the equilibrium
depth of scour. The presence of the scour depth peak at the critical velocity has led
researchers to conduct the majority of scour experiments with velocities in the clear-
water range, so that data may be extrapolated to the peak depth at the critical velocity.
Most of the scour research completed to date has been in the prediction of
equilibrium scour depth, which is used in design computations. However, the time
required for the scour process to reach equilibrium can be much longer than the duration
of many storm events. Thus, for many bridge piers situated in locations with short flood
duration, the hydraulic calculations for the foundations may be in error.
The objective of this thesis is to devise a methodology for the prediction of the
temporal rate of clear-water scour around cylindrical piers in the hope that design scour
depths may become more accurately predicted. Data from laboratory experiments
performed over a range of conditions is used in the formulation.
Chapter 2 describes the mechanics of local scour, and details the various
nondimensional parameters that influence the scour processes. A literature survey on the
current knowledge and hypothesis concerning the time rate of scour is then presented.
A description of the experiments performed in the USGS-BRD Laboratory is
presented in Chapter 3. The methods for both the scour and hydrodynamic data that were
collected are described.
Chapter 4 is devoted to the interpretation of the data collected from the
experiments. A prediction equation for the time rate of scour is presented using the
observations obtained from the data.
6
A summary of the work and conclusions reached are described in Chapter 5. Also
contained in this chapter are recommendations for further study.
CHAPTER 2
BACKGROUND AND LITERATURE REVIEW
2.1 Background
A brief review of the most important local scour mechanisms is given prior to
discussing the research leading up to the work presented in this thesis. Also, a literature
review of known works associated with the time rate of local scour is reviewed.
Submerged particles initially at rest on the bed are brought into suspension as a
result of forces acted upon them by the action of the fluid flow. The resisting moment is
surpassed by the hydrodynamic moment of forces about the center of gravity of the
particle. The hydrodynamic forces include buoyancy, lift, and drag, while the resisting
forces are the weight, and normal and frictional forces from neighboring particles. A
force diagram is shown in Figure 2-1. Note the normal force (Fn) and frictional force (FF)
at each point of contact with neighboring particles. Equating the various moments, a
dimensionless shear stress called the Shields parameter can be obtained:
ro Pmu-.
TO = PU (2.1)
= s -m )d50 (s -m )50
Flow ABuoyaIcy
A Lf
Figure 2-1. Diagram showing forces on a particle resting on the bed.
The value of rc. which corresponds to the initiation of motion is dependent on the angle
of repose of the sediment, as well as the flow conditions, and flow regime. Shields
performed laboratory experiments measuring sediment transport for various values of r.,
and extrapolated back to the point of vanishing sediment transport (Julien, 1998). His
data led to the modified Shields curve, shown in Figure 2-2, where r. is plotted as a
function of the boundary Reynolds number Re..
Weight
I
0
II
U)
4)
4)
U,
U,
U,
'3
1.00
!!!!!
-I 1
I I I flll
- -....-.. -. - -J .. - T- -r
0.10
Figure 2-2.
0.1 1.0 10.0 100.0
Boundary Reynolds Number, Re,= ucd5v
Shields' diagram.
100
0.0
The critical velocity is therefore a function of the sediment, fluid and flow
characteristics. The sediment characteristics are the relative bed roughness (the
Nikuradse roughness length divided by the median sediment diameter), sediment density,
and median diameter. The fluid properties are viscosity and density. The flow
characteristics are the water depth and velocity (and acceleration for unsteady flows).
As mentioned in Chapter 1, the critical depth averaged velocity represents the
division between clear-water and live bed conditions. For live bed conditions, sediment
transport is occurring throughout the bed, and various bed features can be formed. These
bed features undergo significant metamorphosis with increasing velocity, from a smooth
bed to ripples to dunes to flat beds to antidunes. The primary variables that affect the
bedforms are the flow velocity, water depth, bed particle size, particle fall velocity and
the slope of the energy grade line. Numerous researchers have classified the bedform
transitions for various parameters (Julien, 1998). The bedforms may migrate into the
local scour hole and vary the scour depth with time. As a result, the equilibrium scour
depth for live bed scour is defined as the point where the sediment leaving the scour hole
is equal to that entering.
2.2 Scour Formation for Steady Flow
Clear water scour starts when the bed shear stress upstream of the structure is
sufficient to initiate sediment motion at the structure and ends when the critical shear
stress is reached. The lower limit is an approximate value that depends primarily on the
structure shape (Hancu, 1971 and Ettema, 1980). In clear-water scour, material
surrounding the pile is gradually scoured out at a decreasing rate until the hole is deep
enough not to permit any further removal, and the equilibrium scour depth is attained.
Local scour develops as a result of a number of flow mechanisms that combine to
remove sediment near a structure. The relative importance of the mechanisms changes as
the scour hole progresses thereby changing the rate of scour. These mechanisms can be
described as follows.
The scour process initiates on the sides of the pier, approximately 450 to the
direction of the incoming flow (Ettema, 1980). Shear stress measurements made by
Hjorth (1975) indicate that these locations correspond to local maximum bed stresses.
However, Melville (1975), in a separate but similar experiment, discovered the maximum
stresses to be at locations 1000 to the flow direction. The holes formed on either side of
the cylinder propagate upstream (as sediment cascades into the newly formed
depressions) to meet in the front of the pier, and downstream to meet behind the pier.
A blunt-nosed pier causes the formation of a variation of stagnation pressure
about its stagnation plane directly in front of the pier (Chiew, 1984). The velocity profile
of the approaching flow is such that at the bottom the velocity is zero, and increases with
elevation to the air-water interface. As the flow encounters the bridge pier, the
logarithmic velocity profile exerts a pressure at the top of the pier greater than near the
bed, thus inducing a downflow in front of the pier. Tison (1937) and Hjorth (1975) have
also shown that the downflow is also caused by the curvature of the streamlines around
the pier. The downflow impinges on the bed and is directed upstream. The approaching
flow meets this redirected downflow and a horseshoe vortex is formed, named after the
shape of the vortex (viewed from above) as it is swept downstream. A horseshoe vortex
is also formed when the oncoming flow separates at the edge of the depression created
initially by the downflow, inducing a circulation within the scour hole (Chiew, 1984).
The downflow and the horseshoe vortices are believed to be the primary mechanisms for
scour around bridge piers. A bow wave or rolling vortex is also formed at the head of the
flow obstruction. Vertical vortices called wake vortices are generated as a result of the
flow separation on the sides of the pier, effectively aiding the lifting of sediment into
suspension much like a tornado. A diagram showing the local scour mechanism is shown
in Figure 2-3.
Bow Wave
Stern Wave
/
A Upflow
W ake
Vortices
Horseshoe/
Vortex Downflow
Figure 2-3. Diagram showing flow vortices around pile in flow.
The downflow has been identified as the leading cause of scour (Shen at al, 1965),
acting as a vertical jet on the bed. Ettema (1980) measured the variation of the downflow
strength as the scour hole deepens, and found that the velocity of the jet initially increases
with the development of the hole, reaching a maximum when the scour depth ratio (ds/b)
is between 0.8 and 1 before decreasing (Figure 2-4). He also found that the strength was
dependent on the approach flow (flow velocity and water depth) and the diameter of the
cylinder.
.0
Y/b
1.0
The downflow and horseshoe vortex erodes the sediment and the hole expands in
width and depth. Avalanches occur as the angle of the side of the hole becomes too
great, widening the depression even more. Measurements have shown that the slope of
the equilibrium hole corresponds to the angle of repose of the sediment.
The horseshoe vortex is initially weak and small in cross-section. As the scour
hole develops, the downflow velocity increases and the vortex grows in size and strength.
Melville (1975) postulates that the circulation associated with the vortex increases as the
scour hole enlarges due to the expanding cross-section, fueled by the quantity of fluid
supplied by the downflow. The downflow, initially being a function of the upstream
flow, eventually attains a peak value and then decreases with further deepening of the
scour hole as described earlier. The strength of the circulation of the vortex
correspondingly grows weaker and weaker, decreasing the rate of erosion until
equilibrium is reached. Equilibrium can thus be defined as the depth for which the
downwardly directed jet in front of the pier is not strong enough to dislodge particles
resting on the bottom of the hole.
The flow around a circular pile encounters an increasing pressure as it progresses
around the sides (i.e. an adverse pressure gradient). This pressure gradient slows the flow
adjacent to the pile until at some point it comes to rest and "separation" occurs. The
location of the point of separation is not stable and oscillates in time. Vortices form
downstream of the point of separation and can either remain attached to the pile or
separate and flow downstream depending on the pile Reynolds number (Ub/v). The
shedding of the vortices from the pier occurs alternatively in a periodic manner, in what
is known as the Kdrman vortex trail.
The Strouhal number, a measure of the ratio of inertial forces due to the
unsteadiness of the flow to the inertial forces due to velocity changes in spatial variations
of the flow field, can be used to describe the frequency of shedding. The Strouhal
number is given by
St = b (2.2)
U
where co = frequency of operation,
b = cylinder diameter, and
U = mean approach flow velocity.
Melville (1975) determined that vortex shedding occurs at a Strouhal number
varying from 0.229 to 0.238. He also noticed that the shedding frequency decreases with
increasing depth. The wake vortices, with their vertical axes and low-pressure center,
assist in the removal of sediment placed in suspension by the downflow and horseshoe
vortices in a manner similar to tornadoes.
Melville (1975) further theorized that the arms of the horseshoe vortex oscillate
laterally and vertically at the same frequency of the shedding wake vortices. He
describes the sequence of the coupling of the two vortex systems during one period:
The decreased pressure within an individual cast-off vortex draws up fluid from
the horseshoe vortex region, pulling the vortex arm with it. As this first wake-
vortex passes downstream, the arm of the horseshoe vortex recedes back into the
scour hole, while the other arm of the vortex is similarly affected by the second
wake vortex shed from the other side of the cylinder. (Melville, 1975, p.195)
Behind the structure there is a relatively calm region sheltered from the uncoming
flow. Sediment that is put in motion by the horseshoe scour vortex and not removed
from the scour hole by the wake vortices will deposit in this area to form a characteristic
mound behind the pier. Sediment interacting with the wake vortices will be carried with
them downstream, initially experiencing a vertical displacement due to the upward flow
associated with the low pressure in the center of the vortex system. As the wake vortices
progress downstream they lose energy and slow down and the sediment redeposits on the
bed.
A list of the variables associated with the complex mechanisms contributing to
erosion can be assembled. The temporal rate of scour around a cylindrical bridge pier in
a cohesionless, erodible bed is a function of the fluid and sediment properties, pier
characteristics, flow parameters and time. The fluid properties are mass density and the
dynamic viscosity. The sediment properties are mass density, size and size distribution,
and particle shape. Pier characteristics include pier shape and surface roughness. The
flow parameters are water depth, velocity, velocity distribution and acceleration (for
unsteady flows).
2.3 Effect of the Various Parameters
These variables can be combined to form independent dimensionless groups (the
number of which is given by the Buckingham n theorem). The subset of these groups
that best describe the local scour processes is still a subject of debate. Different
researchers use different dimensionless groups to present their results. Sheppard (1996)
has found that the equilibrium scour depth can be expressed in terms of the three groups
U/Uc, yO/b, and d50/b.
2.3.1 Effect of Velocity Ratio U/Uc
The velocity ratio U/Uc has been examined thoroughly. For circular cylinders,
the depth of the local scour hole increases almost linearly from a U/Uc value of
approximately 0.45 to the transition critical velocity, where a maximum clear-water peak
is attained. For the model tests that have been conducted for U/Uc greater than one (i.e.
in the live-bed range), the equilibrium scour depth is seen to decrease to a minimum
before reaching another peak at some higher value of U/Uc. The nature of live-bed scour
with increasing velocity appears to be dependent on the sediment to structure diameter
ratio (d50/b). As the ratio drops, the clear water peak first increases then decreases while
the live bed peak tends to remain relatively constant for fixed values of yO/b. A typical
plot of nondimensional scour depth dse/b versus U/Uc for two nondimensionalized
sediment particle sizes (d50/b) is shown in Figure 2-5.
Clearwater Live-bed
Dso/b = 0.016
dse/b
Yo/b = const.
U/Uc
Figure 2-5. Dependence ofnondimensionalized scour depth on velocity ratio for
two sediment sizes.
Scour initiates at a velocity ratio of about 0.45-0.5. This can be attributed to the
increase in velocity of the flow around the pier, in so much as doubling the magnitude of
the approach flow to surpass the critical threshold on either side of the pier. From
potential flow theory the 0 component of the velocity on the cylinder can be expressed as
U =-U( + (b/2) )sin, (2.3)
where U is the upstream flow velocity, and b is the cylinder diameter. At r=b/2 and
sin0=l (900 to the approach flow), Uo reaches a maximum value of 2U, or twice the
upstream velocity.
As the upstream velocity is increased to beyond the critical value for the incipient
motion, bedload transport occurs throughout the bed. The bedload and suspended
sediment transport affects the logarithmic velocity profile and decreases the stagnation
pressure gradient on the upstream face of the pier. Since the magnitude of the pressure
gradient supplies the energy to the downflow and horseshoe vortex, the rate of scour
decreases immediately after the velocity of the incoming flow changes from clear water
to live bed. Upon further increase in velocity the equilibrium depth reaches a minimum
before increasing once again. One possible explanation for the apparent minimum in
scour depth in the live-bed range is for at that velocity ratio where the minimum is
located, there is a transition in the sediment-carrying capacity as the flow is able to
actively transport suspended sediment as well as bedload.
The formation of the secondary live-bed peak is thought to occur at the point
where the bed flattens (Sheppard, 1998). The transition of the bed from dunes to a flat
bed can be seen in the graph by Snamenskaya (1969) in Figure 2-6. The Froude number
is plotted versus the flow velocity normalized by the sediment fall velocity. The Y
parameter is the dune height to length ratio and is a measure of the dune steepness. From
this graph, the transition of the bed with increasing velocity is divided into seven distinct
sections, with the live-bed peak occurring in region six.
In the live-bed range, the scour depth dependence on U/Uc relies heavily on a
third parameter, d50/b. Analysis done by Sheppard (1998) indicates that for larger water
depths (y0/b>2.5), although the location of the live bed maximum appears to be
independent of grain size (remaining relatively constant at dse/b 2.1), the maximum of
the scour depth at the critical velocity fluctuates with varying d50/b.
U
1.5
1.0
0.5
HI
'i A
lif
3
lit '
,1110
oaIF
1 10 100 1000 10000
Figure 2-6. Changes in bed features with increasing velocity, taken from
Snamenskaya (1969).
Most data collected on circular pile bridge scour have been performed in the clear
water range, as the maximum equilibrium scour depth was assumed to be attained at the
transition between clearwater and live bed conditions. The lack of controlled, accurate
experiments in the live-bed range is due in part to the generation of bedforms. In small
flumes, the height of the bed forms can be of the same order as the scour hole. Thus,
differentiating between local scour and bed forms can be difficult and the results
misleading.
UU
1. Flat Dunes
2. Ripples
3. Dunes
4. Steep Dunes
5. Dune Distribution
6. Smooth Bed
7. Antidunes
--
2.3.2 Effect of Aspect Ratio vO/b
The aspect ratio, yO/b, is a significant factor if the water depth is relatively
shallow. Laboratory data indicate the equilibrium scour depth increases, with increasing
values of yO/b, from zero to approaching a constant value when yO/b is about 2.5 to 3
(Figure 2-7), when the other factors U/Uc and d50/b are held constant.
2.00
.0
" o 1.00
0.00
U/U= constant
d50/b= constant
0.00 2.00 4.00
YO/b
Figure 2-7. Dependence of equilibrium scour depth on aspect ratio for which
U/Uc and d50/b are held constant.
The reasoning behind the shape of the curve is as follows. At low water depths,
the surface roller that forms at the free surface may interfere with the horseshoe vortex
formed at the base of the pier. These two vortices have opposite axes of rotation, so the
effect of their interaction yields a reduction in their individual strengths. On increasing
the water depth, the distance separating the center of rotation of the two vortices is
increased, until eventually there will be no interaction between them. This occurs at a
water depth ratio y0/b,2.5.
Effect of Sediment to Pier Size D50/b
The dependence of the scour depth on the third parameter d50/b is not fully
understood. Sheppard (1996) postulates that the parameter d50/b is actually a function of
the ratio of two different Reynolds numbers, one based on the sediment diameter and the
other on the structure diameter:
d,0 d50d
5C (2.4)
b bU '
where the velocity near the bed (u) is related to the depth averaged velocity (U) for a
fully developed flow.
There exists a limit as to the smallest size of the median diameter that can be used
in the laboratory. This is due to the cohesive behaviour of fine sediment (d50<0.06 mm).
Additionally, sediment finer than about 0.6 mm will form ripples as a result of the small
relative size of the particles to the viscous sublayer formed by the flow (Ettema et al.,
1998). These characteristics of smaller sediment behave differently to larger sediment
found in streambeds, and thus cannot be accurately used in modeling prototype situations.
As sediment used in model tests are frequently of the same order in size as found
in prototype piers, the data collected for d50/b is comprehensive for small diameter piles
from laboratory tests, and meagre for larger, prototype piles. From the tests that have
been completed, the dependence of d50/b on the equilibrium scour depth is shown in
Figure 2-8, for constant values of U/Uc and yO/b.
3.00
2.00
1.00 -
0.00 -
Figure 2-8.
0Ivc;=1.0
yo /., ,3. 0.
-5 -4 -3 -2 -1 0
ogl (d50 I b)
Variation of scour depth on d50/b with U/Uc=l and yO/b>3.0.
The graph indicates that for a given pier width, an increase in the sediment size will first
increase the scour depth until a maximum is reached, then decrease the depth.
2.4 Literature Review
Research on the local scour phenomenon to date has mainly concentrated on the
determination of equilibrium scour depth. The temporal rate of scour has not been given
the same attention since bridge designers have tended to seek an equation predicting the
final depth for design rather than understand the underlying processes that govern the
development of the scour hole. Additionally, reliable and unobtrusive methods for
measuring the scour depth over time have only recently been adopted.
Shen et at (1966) was one of the first to address the variation of scour with time.
They conducted tests with a single cylindrical pile and a single sediment size, and varied
the water depth and approach flow velocity. They plotted the ratio of scour depth to the
flow depth versus a parameter which incorporated a quantity that depends on the flow,
the Froude number, and time. They were, however, unable to observe any trends in their
data.
They then defined t?5 as the time required to reach 75% of the equilibrium scour
depth. Incorporating the work of Chabert and Engeldinger (1956), they plotted t75 versus
the velocity squared for all of the experiments. From this rather scattered plot, they
concluded that the time required to reach 75% of equilibrium increases with increasing
sediment size, and decreasing velocities.
Later in a separate paper, Shen et al (1966) fit the equation d, = K1 ln(K2t) to his
data set and arrived at the following equation:
d = 2.5Fr04 6 (lmE) (2.5)
where m=0.026e-2.932yo and (2.6)
/E = FrO 033 lnIUyJ. (2.7)
Nakagawa and Suzuki (1975) formulated a temporal rate model, using the
assumption that the horseshoe vortex is the principle scouring mechanism and the
strength and scale of it remains constant during the scouring process. They derived
equations representing circulation in the approach flow and in an area bounded by an
imaginary triangle on the lower half of the pier. By equating these, they found an
expression for the bottom velocity in terms of the depth averaged flow velocity. A
sediment transport rate was formulated using a probabilistic technique and volumetric
representation, which included the shear stress on the bed. They observed a small ridge
of sediment around the immediate vicinity of the pier from where most of the scour
seemed to be taking place. The radial distance to the edge of this sediment buildup was
observed to be constant throughout the process at one quarter of the pier diameter. By
assuming that all scour takes place in this region and is replenished by sediment sliding
down from the sides of the hole, they could represent the rate of sediment being supplied
to the hole. Using the continuity equation, they derived the following expression for t
and ds:
Ki? A,2dp d 1+2K, (d 2 +/ d (2.8)
1-,) Ab= b 2(1 + K, )tan 3(1 + K,) tan 2 b)
where Ki = 0.25,
AI = particle area constant,
A2 = particle volume constant and
= rate which particles in the stagnation plane are transported out the hole.
When the results of the equation are compared to laboratory data, the equation follows
the initial phase of scour well, but does not follow the asymptotically approaching
equilibrium phase.
Carstens (1966) tried to model the rate of scour using data of Chabert and
Engeldinger (1956). He assumptions included
1) the boundary layer thickness was negligible in the area of active scour,
2) the velocity distribution was a function of the geometry of the hole and the
structure, and
3) the hole was shaped like an inverted right cone with an angle equal to the
angle of repose of the sediment.
He equated the rate of sediment transport to the volumetric rate of change of the hole. By
using geometrical equations to define the volume of the hole, and determining an
appropriate sediment transport function, he arrived at the following equation:
4.14 E-6(N'- N C Y2 )lb + l
tan 0t16 24
( d ,(2.9)
tan d 2 (tan 0) (tan 4 d(2.9)
+ +b lb + (tan In +i
32 32 64 tan +
where N, = -- is defined as the sediment number, and Nsc is the sediment
number corresponding to the lowest value for which scour will occur. The derivation of
the constants in the equation was based on a small range of laboratory data, which did not
take into consideration the effect of varying sediment size.
In his dissertation, Ettema (1980) identified three distinct stages of local scour
over time. The initial phase is from a planar bed to the erosional phase. This phase is
dominated by the acceleration of the flow around the pier. When the erosional phase
initiates, the downflow and horseshoe vortex become the leading mechanisms. The
horseshoe vortex, rapidly growing in size, eventually becomes submerged in the scour
hole. The final phase, the equilibrium phase, represents the stage at which equilibrium is
reached for the given conditions.
Ettema (1980) looked at the temporal rate problem in a mathematical approach.
He approximated the half of the upstream scour hole by half an inverted frustum of a
right circular cone with side slopes equaling the angle of repose of the sediment. From
this, volume and surface area equations for a cone were used. He described an
entrainment zone, from the cylinder to about midway up the scour hole slope, from which
sediment is observed to be removed by the horseshoe vortex. He assumed that when a
layer of particles is removed from the entrainment zone, complete transport away from
the scour hole takes place, and the surface of the scour hole is immediately replenished
by particles sliding down from above. Also, bed particles were assumed to be spherical,
and sediment porosity was neglected.
Using these assumptions, Ettema (1980) described the rate of sediment transport
out of the scour hole as:
q, = E (# particles entrained) x (volume/particle) x entrainmentt probability). (2.10)
The effective number of particles entrained per unit area is a product of the total number
of particles per unit area by the ratio of particles on the surface of the semi-circular scour
hole to the particles on the surface of the entrainment zone.
Regarding the surface area of the entrainment zone, two simplifying assumptions were
considered:
1) the surface area is proportional to the pier diameter and varies with the
depth of the scour hole
2) the surface area is proportional only to the pier diameter.
Ettema explains the first assumption is valid for the initial phase of scour, while the latter
assumption is valid for the later erosional phase. By assuming the probability function is
independent of particle size, and that the angle of repose is constant, Ettema arrived at the
following relationship:
a(ds / b) DsD (2.11)
or the rate at which the normalized scour depth increases with time is proportional to the
mean particle size divided by the cube of the pier diameter.
Ettema then fitted two straight-line segments for the principle erosional phase,
each having the form
d = K, In (D,(u~(5 'b ) i)}+lnK2 (2.12)
where K1 and K2 are functions of U/Uc, d50/b, and the corresponding phase of scour.
While some insight regarding the scour rate problem was gained by Ettema's
model, more work is needed in relating the dependent parameters to the equation
coefficients. Also, while the erosional phase of scour is described, there is no mention of
the scour development in the initial or equilibrium phases.
Yanmaz and Altmbilek (1991) formulated an equation using a similar derivation
to Carstens (1966). They used a more recent sediment pickup function, and performed 71
flume tests to determine coefficients for the sediment transport out of the hole. Their
final equation for the rate of scour was
dd, 300000Fudo(2d, + btan ) (2.13
dt rN C (tanf))54 5 (d +db tan )
D/ ,50 S
Later in a separate paper, the authors state that the equation is good only within the limits
of the flume data they tested, because of the extreme sensitivity of the drag coefficient
determination.
Johnson and McCuen (1991) assumed the scour rate could be represented as a
function of the difference of the strength of the downflow velocity and the critical
velocity. Using experimental data, they fitted an equation relating the downflow to the
cylinder diameter, water depth, and approach velocity. The critical velocity was found by
equating the shear stress on the bed to the resistive force (component of weight along the
slope of the hole). Adjustments were made to account for sediment gradation and
turbulence. Their final equation took the form:
Ad,_ = C, 1 C4i -e-c' by C 2p dsi +CGe-c (2.14)
At 1 + e-cd,'-c3b I 3pJ 2(1- sino)'I2
The constants CI through C8 were found using a nonlinear least squares method and
laboratory data. The scour depth can be solved for each chosen incremental time step.
The shortcomings of this formulation lie in the negligence of other scour mechanisms
that contribute to the erosional rate.
Sumer et al. (1992) used live-bed scour data to analyze the scour rate in steady
currents. Their final equation includes the equilibrium scour depth, and is of the form:
d, = d (1 e-IT)
T b2yo u2 -2.2 (2.15)
2000b g(s-l)d3 ,g(s-1)d)
The authors recommend a value of 1.0-1.5 times the pier diameter for dse if the
equilibrium scour depth is unknown. The equation is based entirely on live-bed scour
data.
Kothyari et al. (1992) provided an algorithm to outline a procedure for calculating
the time required to scour an incremental depth. They incorporated the work of Paintal
(1971) to determine the time required for a single particle to move. Assuming the main
scouring mechanism is the horseshoe vortex, they used experimental data to devise a
relationship for the diameter of the vortex as a function of the pier diameter and water
depth. Following the work of Melville (1975), they represented the increase in area of
the horseshoe vortex as it descends into the scour hole as a function of the scour hole and
initial vortex area. Coefficients were determined using flume data, which was obtained
by the authors for a wide range of conditions. The algorithm basically computes the
initial vortex size, shear stress on the bed, and probability of movement for one particle.
The time required to move the particle is then added to the cumulative time, and the scour
depth is increased by one sediment diameter. Shear stress is again calculated and if it
falls below the initial value, the loop begins a new cycle. However, the authors admit
that their estimation of the size of the horseshoe vortex using the thickness of the
separated boundary layer may not be precise.
Gosselin (1997) conducted experiments using both erodible and fixed beds. He
measured velocity vectors in and around the scour hole using an Acoustic Doppler
Velocimeter. Using results from the experiments, he simulated the development of the
scour hole over time from a three-dimensional Navier-Stokes equation solver program.
The simulations were in turn used to develop a method to predict the scour depth over
time based on volumetric changes as well as energy considerations of the scour hole. His
analysis of the velocity vectors of the horseshoe vortex is comprehensive, however the
apparatus used was unable to pick up the higher-frequency turbulent fluctuations.
Chiew and Melville (1996) performed 35 experiments, varying pier diameters,
sediment sizes, flow depths, and approach velocities. Motivated by the lack of
consistency of test durations completed by previous researchers, they defined a 'time to
equilibrium' (te) as the time taken for which the rate of development of the scour hole
does not exceed five percent of the pier diameter in a 24 hour period.
Graphical analysis of their results show interesting trends. A plot of scour depth
normalized by equilibrium scour depth versus time normalized by time for equilibrium
shows a family of curves with increasing velocity ratio in the direction of increasing
scour depth (or decreasing time), demonstrating the significance of time in the
measurement and estimation of scour depth. On a separate plot, time to equilibrium was
plotted versus pier diameter, holding the water depth constant. For a given velocity ratio,
time to equilibrium increases with increasing pier diameter. Also from the same graph, it
can be seen that the time to equilibrium increases with the velocity ratio for a given pier
diameter. The reasoning behind this observation is somewhat unclear. An increase in the
velocity ratio correspondingly increases the energy of the flow and its capacity to
transport sediment, which should decrease the time required for equilibrium. However,
the increased velocity ratio would result in deeper equilibrium scour depths, which would
increase the time required to reach equilibrium. The authors explain that the increased
transport capability of the flow has a lesser effect on the time to reach equilibrium, and
the effect of the increased time to develop the deeper scour hole is dominant.
The authors also plotted a non-dimensional parameter t.=Ut/b versus U/Uc, from
which they determined the empirical relation:
t,= 3 x 10o4(bU )4. ), (2.16)
which is valid for 0.5__
__
velocity ratio, a time required for equilibrium can be obtained. Then for any given time,
a scour depth can be read off the previous graph described earlier. The authors, however,
admit their results require further research and are only based on their limited laboratory
data.
In a later paper by the same authors, Melville and Chiew (1999) use additional
data and the same parameters to develop an equation for the scour depth as a function of
equilibrium scour depth and time. They found that both the equilibrium scour depth and
the time to reach equilibrium depended on the same parameters. Plotting the parameters
ds/dse versus t/te as in the previous paper, the authors arrived at the following expression
for the temporal development of the scour depth:
d s =exp{- 0.03[U Uln( .6 (2.17)
A plot of the relation of the nondimensional parameter t* and the aspect ratio yO/b show
striking similarities to a similar plot of dse/b versus yO/b, in that there is an apparent limit
to the influence of the water depth on t*. The following equation for the dependence was
given:
S/\0.25
t. = 1.6x 106 Y~ 0 for y0/b<6 and (2.18)
t. = 2.5 x106 for y0/b>6. (2.19)
The dependence of t* on U/Uc was plotted and an equation given:
t = 4.17 x 106 (U 0.4) for 0.4__
__
Finally the dependence of t* on the sediment size is presented:
t. =9.5xlI' ( 50^21 for b/d50<100 and (2.21)
t. = 2.5 x106 for b/d50>100.
(2.22)
By combining the effects of the water depth and the flow intensity, the resulting
equations can be written for te:
t = 48.26 (U U- -0.4 for yO/b>6 and (2.23)
t, = 30.89U(U 0.4J[ for y0/b<6. (2.24)
If the various flow and structure variables are known, the equilibrium scour depth and
time to reach equilibrium can be calculated. The scour depth at any point in time can
then be computed.
CHAPTER 3
EXPERIMENTAL PROCEDURES
3.1 Introduction
This chapter contains a description of the flume, equipment, and procedures that
were used in obtaining the data for analysis. The descriptions include precautionary
procedures so as to ensure accurate replication, thus minimizing errors due to differences
in test conditions.
The choices of the dimensions of the piers as well as the sediment size used were
made due to the lack of data of the independent parameter d50/b. As mentioned earlier,
using flume data to relate to prototype situations encounters the problem of scaling the
sediment size. There exists a minimal limit for which the sediment used in the model
cannot exceed, due to the cohesive nature of small diameter particles. Thus, sediment
used in model situations often has to be of the same order as that in the prototype. Most
data collected on the d50/b parameter fall in the range 0.0006
boundary has been limited by the size of the pier that flumes can accommodate. The test
data analyzed in this thesis used pile diameters as large as 0.914 m that extends and
sediment diameters as small as 0.22 m (yielding d50/b values as small as 0.0002).
The quantities measured during the experiments were water temperature and
elevation, flow velocity, and scour depth. The scour depth was measured using two
different instruments during the test and the scour hole was point gauged on completion
of each test after the water was drained. Two methods were used so as to provide a
backup in case one of the systems malfunctioned.
3.2 Equipment
3.2.1 Flume
The flume is located at the USGS-BRD S.O. Conte Anadromous Fish Research
Laboratory in Turners Falls, Massachusetts. The flume itself has three separate channels,
of which only the main center channel was used. That channel has a width of 6.1 m, a
length of 38.6 m, a maximum depth of 6.4 m and a zero bed slope. Schematic drawings
of the flume and sediment arrangement for the experiments are shown in Figures 3-1 and
3-2.
Water used in the flume comes from a reservoir on the Connecticut River that was
built to supply a hydro electric power plant a short distance from the laboratory. The
water is discharged into the Connecticut River downstream of control structures. The
head difference between the reservoir and river below the structures (approximately
10.67 m) drives the flow in the flume that can be as large as 9.91 m3/s.
Originally designed for studying the migration of anadromous fish, various
modifications were necessary to adapt the flume for use in bridge scour experiments. A
vertical weir was constructed at the end of the channel with variable height so as to
Flow Intake from Reservoir
NOT TO SCALE
All dimensions in feet
Flow Discharge
To Connecticut
River
Clearwater Scour Test Setup
Figure 3-1. Aerial schematic of flume.
NOT TO SCALE
All dimensions in feet
Test Sediment 6
Section A-A
126 -1
-- I
Filter Material Base Seciment Test Sediment Base Sediment
Section B-B
Figure 3-2. Schematic figure of the cross-section of flume.
_
control the water depth and discharge inside the flume. Also, a flow straightener was
installed at the entrance of the flume to ensure uniformity of the flow.
Only a section in the middle of the channel was completely filled with test
sediment. This section had a length of 9.75 m, and was located a distance 19.5 m from
the channel entrance. Gravel (d50 9.5 mm) was used as filler below the test sediment for
the remainder of the flume. The bed was lowered and covered with gravel for the first
8.5 m from the entrance in order to promote a faster development of the velocity profile
and to prevent the development of dunes that would have resulted from the jetting of
water through the flow straightener. The test sand was separated from the gravel by a
filter material with a 0.1 mm opening. In total, 356 m3 of gravel and 212 m3 of test
sediment were used.
The discharge in the flume was calculated using the equation:
Q=Cr23 2gwH3, (3.1)
where Cwr is the rectangular weir coefficient, w is the flume width, and H is the head over
the weir. For most flume channel situations, the viscous and surface tension effects are
negligible compared to the geometric effect, and the weir coefficient can be computed
using
C,, = 0.611+ 0.075 H (3.2)
/ -w
where Pw is the height of the weir (Rouse, 1946 and Henderson, 1966).
3.2.2 Electronic Apparatus
The electronic apparatus used for monitoring the scour depth were acoustic
transponders and video cameras mounted inside the piers. Electromagnetic current
meters were used to measure the velocity of the flow. A digital thermometer was used to
measure the temperature of the water.
3.2.2.1 Acoustic transducers
Acoustic transducers were used to record scour depth as a function of time. By
emitting pressure pulses directed at the hole, the distance from the transducer to the scour
hole can be calculated by measuring the time it takes for the pulses to travel from the
sensor to and from the bed. Twelve sensors were used, arranged in 3 arrays of 4 sensors
each, as shown in Figure 3-3. The Multiple Transducer Arrays (MTA) were positioned at
the front and at angles of 830 on either side of the structure. The MTAs were mounted a
few centimeters below the water surface so as not to interfere with the dynamics of the
scour development. The distance from the arrays to the bed directly under the crystals
was measured continuously and recorded at predetermined intervals throughout the
experiments.
Different MTA arrays were used for small and large pile diameters. For the small
piles, the sensors 2.5 cm in diameter and spaced 4 cm apart. For the large pile, the
sensors were 4 cm in diameter and spaced 8 cm apart. The frequency of the acoustic
system used was 2.25 MHz.
MTA for small pile diameters (0.5 to 1 ft. diameter)
side view (cross-section) front view (cross-section)
mounting ring (1.5 cim)
1 57 in (4 cm)--- 0.375 n
(0.95 cm)
2.25 MHz transducers
top view
-2.50 In.
(6.3 cm)
[1SEATEK
64 in (16.3 cm) Drawn by: Chris Jette' 3-24-97
Figure 3-3. Detail of MTA arrangement used in small-pier experiments.
3.2.2.2 Electromagnetic current meters
Two electromagnetic velocity meters (Marsh-McBirey Models 523 and 511)
were used to measure the velocity of the flow. These meters accurately measure the
horizontal components of velocity. The sensors have no moving parts, and are therefore
less susceptible to fouling or clogging by foreign matter.
3.2.2.3 Digital thermometer
A digital thermometer was used to periodically check the temperature of the water
during the experiments.
3.2.2.4 Cameras
Two miniature video cameras were placed inside the piles on a vertically
traversing devise. The speed of the traversing mechanism was adjusted to match the
scour rate so that the cameras were directed at the water-sediment interface in the scour
hole. A photograph of the video recording equipment used in the smaller pile is shown in
Figure 3-4.
3.2.3 Test Setup
The velocity meters, water level indicator, and digital thermometer were all
connected to a personal computer with a 486 processor. The computer was programmed
to take one-minute data samples every half-hour throughout the tests.
Figure 3-4. Photograph of cameras used in the 0.114 m pile in casing.
-M -
The acoustic transducers were connected to a SeaTek Control Box, which
converts the signals from the acoustic sensor into digital numbers representing the
distances from the sensors to the bed. The Control Box was connected to a second 486
personal computer. Communication between the computer and Control Box was
accomplished with a communication software package (CrosstalkM). Ten second
samples of data were recorded every ten minutes throughout the experiments.
3.2.4 Models
Cylindrical piers of varying diameter were used in all of the experiments. The
first pier had a diameter of 0.92 m, the second 0.31 m, and the third 0.114 m. The largest
pier stood 5.5 m high, and was made out of Polypropylene with two Plexiglas windows.
Measuring tapes were located inside the pile near the windows in view of the cameras.
Steel channels secured the pier from deformation under the water pressure. The other
piles were made out of Plexiglas, and stood 3.35 m tall. In order to reduce the forces on
the large pile it was flooded during the experiments and the cameras placed inside a
waterproof case.
3.2.5 Sediment
Tests were performed with two different sediment diameters. The sizes used were
0.22 mm and 0.80 mm. The sand was sieved by the supplier to provide as near a uniform
grain size as possible. The sigma (o = 84 ) for the two sands were 1.51 and 1.29,
respectively. Graphs showing grain size distributions are given in Figures 3-5 and 3-6.
dg4 = 0.32 mm
d5o= 0.22 mm
d16 = 0.14mm
diameter (mm)
Figure 3-5. Grain size distribution for sand #1 (d50=0.22 mm)
d,4 = 1.07 mm
d5o = 0.80 mm
d16= 0.64 mm
0.1 1
diameter (mm)
Figure 3-6. Grain size distribution for sand #2 (d50=0.8 mm)
3.3 Laboratory Test Procedure
A description of the experimental procedure including preparation of the bed prior
to each test is given below.
3.3.1 Bed preparation
First, the gravel was placed in the flume, using buckets attached to an overhead
crane. After leveling, the filter material was placed on top of the gravel. The test
sediment was then placed on top of the gravel followed by the installation of the structure
and the placement of sediment in the test area. The sediment was compacted every 20-30
cm with a diesel compactor and a hand tamper next to the structure.
3.3.2 Laboratory Procedure
The electronics and computers were positioned on a wooden observation platform
that spanned the width of the flume at the test section. The platform had dimensions of
3.05 m x 6.10 m with an opening for the pile. All electronics were run through a series
of tests to ensure everything was working before the experiment was started. The flume
was then filled with water. Care was taken not to disturb the leveled bed as the flume
filled. When the water overflowed the weir at the end of the flume, the experiment was
ready to start.
The sluice gates were opened slowly so as to increase the flow of water to the
desired water level and flow velocity. Adjustments made to both gates were necessary so
as to achieve the correct velocity reading on both velocity meters.
The tests were run approximately 24 hours beyond the point where scour was no
longer detected. Then the gates were shut, and the water was slowly drained. A mapping
of the scour hole was then performed with a point gauge.
CHAPTER 4
DATA REDUCTION AND ANALYSIS
Time dependent, local scour data collected from clearwater experiments
performed at the USGS-BRD Laboratory in Turners Falls, Massachusetts are described
and analyzed in this chapter. Data collected by J. Sterling Jones at the FHWA Turner
Fairbanks Laboratory in McLean, Virginia are also used in the analysis. Certain trends
were observed and are depicted graphically.
The USGS Laboratory experiments are somewhat unique in that
1) Their duration was such that there was no question about their equilibrium scour
depths and
2) They include one of the largest test structures used in steady flow laboratory
experiments.
This analysis has revealed a number of interesting correlations and the formulation of a
predictive relationship for the rate of scour.
4.1 Raw Data
The development of the scour hole was measured by acoustic pingers as well as
video cameras set up inside the piers connected to a VCR. Both methods allowed for
unobtrusive measurements to be performed, however, they both had limitations as well.
For the experiments where the acoustic transponders were working properly there was
good agreement between the results from the two methods, as shown in Figure 4-1.
--+-- Camera 1
Camera2
x Pinger
0 20 40 60 80 100
Time (hrs)
Figure 4-1. Scour depth versus time plots for the video and acoustic transponder
data. (b=0.305m, d50=0.80mm, y0=1.268m and U=0.381m/s)
4.2 Data Adjustment and Smoothing
Local scour is not a smooth, continuous process but rather a sequence of events
involving the local removal of sand followed by an avalanche of the surrounding
sediment into the scour hole. In order to enhance the analysis of the data it was smoothed
both visually and mathematically.
Most of the data was collected from the video recordings, with the pinger data
mainly used as a check. Missing data due to the cameras dipping below recordible levels
were connected as accurately as possible. The lack of scour deepening due to the
sediment sticking to the cylinder was corrected by adjusting the scour history to achieve a
smooth curve. Finally, minor discontinuous jumps in the acoustic readings were assumed
to be due to foreign particles carried in by the river and the corresponding data points
were rejected.
Another adjustment had to be made to the scour data so that the deepest
equilibrium scour depth was attained. Neither the acoustic data nor the video data
achieved the deepest scour depth, found after point gauging the bed on completion of the
experiment. This was because the location of measurement did not coincide exactly with
the deepest point. It was assumed that the time taken for equilibrium to be reached at the
measured site was the same as for the point of maximum scour.
For this adjustment, a Matlab program was written. First, the equilibrium scour
depth had to be determined from the raw data. Because the experiments were run for
such a long period of time, the level as to where the hole had stopped scouring was
obvious. The time for equilibrium was then determined by finding the time at which the
deepest scour occurred first on the scour time history plot. Dividing the difference in the
maximum scour depth and the measured equilibrium depth by the time for equilibrium
yields an incremental scour addition for each time step. By this method, the time for
equilibrium stays the same, while the maximum equilibrium scour depth is attained.
Figure 4-2 shows the original data and the adjusted data for the deepest equilibrium point
for experiment #12.
...... Adjusted Data Raw Camera Data
45
40
S35
30
S 25
, 20
g 15
8 10
5
0
0 20 40 60 80
time (hrs)
Figure 4-2. Comparison between raw and adjusted scour depth data.
4.3 Experiment Summaries
As mentioned earlier, a total of seven experiments involving circular piles were
completed. The experiments differ mainly in cylinder diameter, sediment size, and water
depth, with minor variations in the U/Uc parameter. Table 4-1 below shows the
conditions of each experiment performed.
Table 4-1. Summary of experiments completed in the USGS Laboratory
Exp # b (m) Dso(mm) yO (m) U(m/s) Uc (m/s) T ( C) U/Uc dse (m)
3 0.114 0.22 1.186 0.290 0.316 18.0 0.918 0.1336
4 0.305 0.22 1.190 0.305 0.315 18.0 0.968 0.2586
6 0.914 0.22 2.268 0.325 0.326 17.1 0.996 0.3683
9 0.914 0.80 2.402 0.454 0.464 1.0 0.979 0.9271
11 0.914 0.80 0.866 0.335 0.411 1.0 0.815 0.6375
12 0.305 0.80 1.268 0.381 0.431 1.2 0.884 0.4039
13 0.114 0.80 1.280 0.388 0.429 3.9 0.904 0.1885
The scour depth versus time plots for all experiments are given in Figure 4-3.
--Exp 6 -- Exp12 ---Expll -e--Exp9
-awExp3 -- Exp4 -+- Expl3
0 50 100 150 200 250 300
Time (hrs)
Figure 4-3. Smoothed scour history plots for all UF/USGS-BRD Laboratory
experiments.
All the curves show a high initial rate of scour, which slowly decreases until an
equilibrium depth is reached.
Incorporating the FHWA Laboratory data provides valuable information about the
effects of U/Uc and the sediment particle size. Jones used the same structure with the
same water depth for all of his experiments. The effects of an individual parameter could
be investigated without having to account for changes in the other variables. Table 4-2
summarizes Jones' data.
Table 4-2. Summary of experiments completed at FHWA Laboratory
Exp # b (m) Dso (mm) YO (m) U (m/s) Uc (m/s) T (0C) U/Uc Dse (m)*
57 0.152 1.2 0.267 0.340 0.455 ? 0.747 0.181
74 0.152 2.4 0.267 0.427 0.649 ? 0.658 0.124
86 0.152 2.4 0.267 0.429 0.649 ? 0.661 0.127
126 0.152 5.0 0.267 0.713 0.919 ? 0.776 0.267
128 0.152 2.4 0.267 0.544 0.649 ? 0.838 0.236
134 0.152 1.2 0.267 0.422 0.455 ? 0.927 0.225
* Estimated from extrapolation of data.
The scour plots for the Jones data are shown in Figure 4-4. Although the duration
of the experiments are quite long (up to 6 days) and the scour seems to be leveling off, a
finite equilibrium scour depth cannot be accurately determined from the raw data
presented.
49
I-+ Exp74 -- Exp126 -~- Exp57 -- Exp134 -+-Exp86 -a- Exp128
30
25
S20
8 o10
5
0 50 100 150 200
Time (hrs)
Figure 4-4. Smoothed scour history plots for all FHWA Laboratory experiments
4.4 Analysis of Data
Analyzing Jones data proved to be challenging due to the lack of equilibrium
depths. Scour plots have always taken the shape of approaching the maximum scour
depth asymptotically. The equilibrium depth and time had to be estimated by manually
continuing the scour plots until the rate of scour leveled off. Time for equilibrium was
especially difficult to predict as the rate of scour decreases to minimal values as a
maximum scour depth is approached. Defining t90 as the time required for 90% of the
maximum scour depth to be completed, both the USGS Laboratory data and the data
received from Jones could be compared with better accuracy. However, due to the
uncertainties in the equilibrium depth, analysis of Jones data was limited to the initial
scour rates.
The plot shown below is of the nondimensionalized parameter ds/dse (scour depth
divided by the max scour depth) versus time. For this plot, all curves approach ds/dse=l
at varying times. What is interesting is that the experiments reach ds/dse=l in order of
increasing equilibrium scour depth. In other words, smaller equilibrium scour depths are
attained in less time than larger scour depths. This indicates a dependence of the rate of
scour on the maximum scour depth.
Exp 6 -- Expl2 -a- Expll -x-Exp9
Exp3 ---Exp4 -+- Expl3
1.2 -
1-
0.8
0.6
0.4
0.2
0
0 50 100 150 200 250 300
Time (hrs)
Figure 4-5. Comparison of ds/dse versus time for USGS experiments
This dependence is accentuated in the comparison of equilibrium scour depth
versus time required to reach equilibrium. An almost linear relationship can be seen
joining the individual points in the plot shown in Figure 4-6.
300
I-
E 200
= 150
100
S 50 4
0
0 20 40 60 80 100
Equilibrium Scour Depth (cm)
Figure 4-6. Equilibrium scour depth versus time for equilibrium for USGS
experiments.
The basis for the time to equilibrium's dependence on the equilibrium scour depth
comes from the mechanisms that govern the scour process. The maximum
nondimensional scour depth (dse/b) has been shown to be dependent on the parameters
d50/b, U/Uc, and yO/b. If the two phenomena are related, as can be deduced from the
above plots, then the same parameters must also influence the time to equilibrium. This
makes intuitive sense as the time required for one particle to move a certain distance must
be dependent on such variables as the water velocity, the size of the particle and the water
depth.
As the number of data points used for the formulation of the equation describing
the deepest scour depth far exceeds the number of experiments containing scour time
history data, finding a relationship between the maximum scour and the time required for
equilibrium is valuable. Including the maximum scour depth equation in the analysis
carries with it a greater sense of confidence due to the amount of research concentrated
on the subject. It also alleviates the need to model the time required for equilibrium
based on just seven experiments, an impossible task considering the numerous factors
involved in this complicated process. However, the basis of the relationship between the
time and maximum equilibrium depth must be determined from the very same seven
experiments, and although the relationship is simplified by having just two independent
variables (dse and te), the lack of a more comprehensive data set requires that the results
be used with caution.
Additionally, considering the time required to reach equilibrium alone effectively
ignores changes in the rate of scour as the scour hole develops. In comparing the
maximum scour depth to the equilibrium time, valuable information on the initial rates is
lost as the time required to reach equilibrium effectively averages the variations in scour
rate. Substantial differences in the behavior of each experiment that are important in
understanding this complex phenomenon are ignored.
Empirical curve fits to the data were obtained with the help of a 2D curve fit
program (TableCurve 2DTM). The time history data for each of the experiments are
therefore individually described by an equation. The benefit of this is that either the
scour depths or their respective rates can be plotted and compared on the same axis. All
experiments at the USGS Laboratory were fitted with the equation:
d, a +b(- e-ct)+ d(-e-t), (4.1)
dsc
where unique values for the coefficients a,b,c,d,and f were obtained for each experiment.
This equation was chosen for its ability to fit the raw data for every experiment, and for
its simplicity in form and ease of differentiation. Jones' data were also fitted with
individual equations describing the scour depth versus time. Equilibrium scour depth for
each experiment had to be estimated from the scour depth versus time plots. Due to
differences in the general shape of the scour history plots, a different equation was used
for Jones' data:
d (a+ct)(4.2)
d,,= + btO.5) (4.2)
See Appendix A for plots of the individual curve fits.
Variations in the initial rates for each experiment can be seen in Figure 4-7 of the
normalized parameters ds/dse versus t/te. The advantage of using the normalized
parameters is that all data must pass through both the origin and the point (1,1). With the
exception of experiments 13 and 6, the plots indicate a high rate of scour for the initial
time period (until about t/te=0.1), and then a smaller rate until an equilibrium depth is
achieved. Experiment 13 and 6 make the transition somewhat later, at around t/te=0.2.
These transitional features have been noticed and identified by other researchers
such as Ettema (1976). In his dissertation, he describes the initial phase as the transition
from a planar bed to the principal erosion phase. The erosion during this phase occurs as
a result of general sediment transport caused by the accelerating fluid around the
structure. The erosional phase is dominated by the downflow and corresponding
horseshoe vortex that descends into the increasing scour hole. As the downflow
decreases, the rate of scour diminishes, and ultimately an equilibrium depth is reached.
Ettema suggests the stages can best be viewed on semilog plots of scour depth versus
time, which can be seen in Appendix B.
1-
0.8
0.6
- Exp3
I Exp 4
0.4 --- Exp 6
A--- Exp 9
D----- Exp 11
/ --- Exp 12
0.2 -- Exp 13
0 -I
0 0.2 0.4 0.6 0.8 1
t/te
Figure 4-7. Plot of ds/dse and t/te for USGS experiments.
An investigation into the factors contributing to the initial rate of scour was then
attempted. Jones data was included, since the primary focus was on the initial rate and
not on the unfinished latter stages of the scour process. By differentiating the respective
equations from the curve fitting, an equation describing the rate of scour was formulated.
When this was plotted versus the normalized parameter ds/dse, the plot in Figure 4-8
resulted.
A
\ Exp3
- Exp 4
--- Exp 6
0.8 Exp9
A-- Exp ll
--- Expl2
0.6 \ Exp 13
S*\ Exp 57
A Exp 74
SExp 86
S0.4 Exp 126
\< Exp 128
Se Exp 134
0.2
0 -
0 0.2 0.4 0.6 0.8
ds/dse
Figure 4-8. Rate of scour d/dt(ds/dse) versus ds/dse for all experiments
A wealth of information can be gained from plots of rate of scour versus ds/dse.
The Jones data, which have much larger initial rates, all are for larger sediment. This
indicates a dependence on the absolute particle diameter. Looking at the individual
experiments in the Jones data, the general trend is for an increased initial scour rate with
increasing particle diameter. The same general trend can be seen in the USGS
Laboratory data.
This noticeable trait can be explained by considering the volume of the scour hole
as scour is initiated. Larger particle diameters correspond to a steeper angle of repose.
As the angle of repose dictates the slope of the scour hole and the volume of sediment
removed for a given scour hole, a steeper slope would have a smaller surface area and
reduced volume. The smaller surface area and volume would erode faster, hence
resulting in a larger scour rate. Calculations for a 5 cm scour hole show a 11% decrease
in erodible surface area in going from a 2.4 mm to a 5 mm sediment. Using the rate plot
above, for the same two sediments an increase of 25% of the respective initial rates is
observed. Other factors such as the difference in U/Uc were not considered in these
calculations.
The increase in the initial scour rates with increasing sediment size appear at first
to disagree with observations made by other researchers. Shen et al (1966) observed that
t75, or the time required for the scour hole to reach 75% of the maximum scour depth,
tends to be greater for larger sediment sizes. Compared to USGS Laboratory data the
time plots of Jones' data, which are for significantly larger sediment sizes than those used
in the USGS experiments, show a faster scour rate followed by a slower rate which
results in a long time to reach equilibrium. The same is noticed in comparing experiments
6 and 9, in which only the sediment diameter has been changed. Experiment 9, using the
larger sediment, approaches equilibrium much slower than experiment 6.
In accordance with Shields' curve, an increase in sediment diameter requires a
larger critical shear stress for the onset of particle motion. Towards the end of the scour
process, turbulent bursts in the horseshoe vortex near the bed become instrumental in
moving the sediment out of the scour hole. A stronger burst is required to move the
larger sediment. In addition to requiring a faster flow for the initiation of sediment
movement, the increase in sediment size increases the scour depth, so the sediment at the
bottom of the scour hole must travel further up the face of the slope before the hole
deepens. It can be assumed that the higher energy bursts required to remove the sediment
occur less frequently, hence the slower rate of scour.
Baker (1978), in his dissertation, made detailed measurements of vortex flow
around the base of a cylinder. He sectioned his analysis of the horseshoe vortex into that
formed by a separating laminar or turbulent boundary layer. His experiments, however,
were conducted in a wind tunnel, but the basis of the horseshoe vortex system should be
similar to that formed in water.
With regards to the oscillations of the turbulent 'bursts' in a laminar boundary
layer, he concluded they were caused by disturbances in the vortex system downstream of
the model, but correlation was found with the frequency of the wake vortex shedding was
found. For a turbulent boundary layer, which is the more likely situation for the scour
experiments considered here, Baker concluded the distribution of turbulent energy with
frequency within the horseshoe vortex system is determined by the energy distribution in
the upstream boundary layer. He also deduced that the wake flow does not affect the
spectra of the velocity fluctuations in the horseshoe vortex. A plot of Baker's horseshoe
vortex spectrum is reproduced in Figure 4-9. Note that the form of the graph is consistent
with the earlier hypothesis that the higher energy turbulent bursts occur less frequently
than those with lower energy. It should be noted that the frequency of Baker's
measurements is higher than what is noticed in the experiments as Baker's experiments
were conducted in a wind tunnel.
0 f(hz) 100
Figure 4-9. Baker's plot of the power spectrum of the turbulent bursts of the
horseshoe vortex in a wind tunnel
Visual observations of the bursts for the 0.305m diameter pier also support the
above hypothesis. Measurements of the frequency of the bursts were made by counting
the number of bursts that occurred during a given period of time. Using the videotape
observations were made mid-way through the scour process (ds/dse=0.5) and towards the
end (ds/dsewl). No bursts were observed at the onset of the experiments when the bed
was flat and the horseshoe vortex is small in size and intensity. Experiment 4, with the
smaller 0.22mm sediment, had similar burst frequencies of 0.29 Hz midway through
scour, and 0.30 Hz when equilibrium was approached. However, midway through the
scour, sediment swept up by the vortex action was eroded away from the hole
completely, whereas towards equilibrium the sediment was merely lifted up from the
base of the pier and deposited on the slope, eventually cascading down the slope into the
base of the scour hole. This supports the theory noted by a number of previous
researchers (Baker, Ettema, and others), that as the scour hole deepens, the strength of the
horseshoe vortex decreases while its size increases.
It is also apparent from looking at the Jones experiments that the
nondimensionalized parameter U/Uc also has a direct effect on the scour rate. This can
be seen from experiments 57 and 134, in which the sediment size, water depth, and
structure size are all held constant while the velocity is increased. Experiment 134,
which has a higher initial rate correspondingly has the greater velocity and U/Uc ratio.
The exception from the preceding argument that the scour depth is directly
dependent on the sediment size is experiment 134, in which a high rate is noticed while
the sediment size is a relatively small value of 1.2 mm. However, this experiment also
has a high U/Uc ratio of 0.93. The rate, which by the above argument should be less due
to the smaller size sediment, is comparable to the adjacent experiments 86 and 128
having the larger sediment diameter of 2.4 mm. The effect of the difference in the
particle size is offset by the increase in U/Uc. Experiments 86 and 128 both have a
smaller U/Uc value in the range 0.65-0.66.
The dependence of scour rate on U/Uc appears to be more obvious. A higher
U/Uc increases the shear stress on the bed, resulting in a greater capacity for sediment
removal. Bedload transport increases as particles are transported faster along the bed;
consequently the rate of sediment removal from the scour hole increases. Baker (1978)
describes the increase in velocity to a shift from the low to a higher frequency of the
power spectra density. In other words, the greater the flow speed, the more the
occurrence of higher energy turbulent bursts.
In looking at the dependence of the characteristics of the scour depth on an
increased velocity ratio, we see that although the scour depth deepens marginally, a larger
change is noticed in the time required for equilibrium. This is more apparent in the Jones
data, as only minor variations in U/Uc were used for the USGS experiments. Comparing
similar conditions, experiment 57 and 134 used the same sediment, water depth, and pier
size, the only change being with the U/Uc parameter. Experiment 57 had a U/Uc of 0.75,
while experiment 134 had a U/Uc of 0.93. The average rate, computed by dividing the
equilibrium scour depth by the time required for equilibrium, increased from 0.17 cm/hr
to 1.13 cm/hr. A similar comparison can be made with experiments 74 (U/Uc = 0.66)
and 128 (U/Uc = 0.84), where the average rates increased from 0.15 cm/hr to 0.79 cm/hr.
Thus it can be concluded that not only does the initial rate increase, the average rate also
increases with higher velocity ratios.
However, the observation that an increase in U/Uc increases the rate and
consequently decreases the time required for equilibrium is opposite to that observed in
the USGS Laboratory data, and reported by other researchers (Shen et al. (1966), Baker
(1978), Melville and Chiew (1999)). Baker observed that the time required for
equilibrium increases with a higher velocity ratio in the clear water range. However, the
number of measurements made was few, and his methods of data collection were
somewhat crude. In a more recent publication, experiments completed by Melville and
Chiew (1999) agreed with the observations made by Baker. The authors note, however,
that higher velocities effectively increase the potential for sediment removal, which may
reduce the time required for an equilibrium depth to be reached. They also explain that
the increase in velocity ratio leads to a deepening of the scour hole, which consequently
takes longer to scour. They conclude that the effect of the increased scour depth
dominates over the greater capacity for sediment removal of the flow, hence the increase
in the time required for equilibrium.
Focussing on the USGS Laboratory data, in experiments 3,4, and 6 the same
sediment size is used, with comparable U/Uc values (0.92, 0.97, and 0.99, respectively).
However, the size of the structure increases (0.114m, 0.305m, and 0.914m). From the
graph, experiment 6 has the slowest rate, while experiment 3 has the largest. Thus it can
be deduced that the initial scour rate varies inversely with the size of the structure.
Isolating the experiments on a separate plot in Figure 4-10, it can be seen that the
smaller pier starts off with a larger rate, decreases rapidly until the rate is about 0.15 s ,
then maintains a slower relatively constant rate until equilibrium is reached. The same
shape can be seen for the medium-sized pier, except the initial rate is not as high, and the
transition between rates is not as abrupt. The larger pier, however, takes on a different
shape in that two separate rates cannot be defined. The variation in the change of rate as
scour progresses is minimal. A similar comparison can be seen from experiments 13, 12
and 9, in which d50 is also held constant and the pier diameter changes.
Another difference noted for the larger pile was the location of the maximum
scour depth. For the 0.114m and 0.305m piers, the maximum scour depth was located
directly in front of the piers. For the larger 0.914m pier, the deepest scour occurred at
locations between 450 and 600 to the front of the pier. See appendix C for the individual
scour hole profiles.
1-
0.8
-- Exp3-b=0.114m
--\---- Exp 4 b=0.305m
-0.6-- Exp 6 b=0.914m
0.4 -
0.2 -
0 4 8 12
ds
Figure 4-10. Plot of rate of scour versus scour depth showing increasing pier
diameter.
The general thinking behind the dynamics of the horseshoe vortex system agrees
with observations made with the smaller diameter piers, in that a high scour rate precedes
a lower scour rate until equilibrium is approached. The high scour rate is produced from
the increased shear stress as a result of the acceleration of the flow around the structure.
The scour hole develops, growing deeper as well as wider as sediment cascades down
from the sides of the hole. Once the scour hole deepens, the streamlines become spread
apart as the flow enters the hole signifying a reduction in velocity. The effect of the
acceleration on the scour process becomes smaller as the scour hole deepens. Eventually,
the dominant mechanism switches from the accelerated flow to the horseshoe vortex
system. The lower scour rate is produced by the horseshoe vortex system. The
horseshoe vortex evolves as the scour hole develops. A smaller, weaker vortex forms
initially when the bed is plane as a result of the variation in stagnation pressure along the
leading edge of the pile. The vortex formed in this case is localized and concentrated at
the base of the pier. As the scour hole develops from the accelerated flow and the
surrounding sediment avalanches into the scour hole, the flow separates at the upstream
edge of the newly formed hole. The flow in the hole downstream of the point of
separation is such that it reinforces the strength of the horseshoe vortex. The horseshoe
vortex formed descends into the hole, and then is transported by the flow around the
structure, dissipating the further it travels due to both adverse pressure gradients and
viscous effects. Thus in front of the cylinder, the accelerated flow weakens and the
horseshoe vortex strengthens as the scour hole develops. Further development of the
scour hole results in an enlargement and weakening of the horseshoe vortex until an
equilibrium depth is reached. Figure 4-11 shows the author's conceptions of the
contributions of the acceleration and horseshoe vortex on the overall rate of scour.
The lack of a clear break in the scour rate versus scour depth for the larger
diameter pier as well as the location of the maximum scour depth can be explained as
follows. The angle of repose for the sediments used in the USGS Laboratory
experiments was such that the scour hole produced by the accelerated flow on the sides of
the pile had only a small overlap in front of the structure. This reduced scour hole depth
in front of the pile impacted flow separation on the leading edge of the scour hole and the
energy that it feeds into the horseshoe vortex. The decrease in energy in the horseshoe
vortex reduces its role in the scour process and not only affects the equilibrium scour
depth but the rate of scour as well. It is anticipated that these effects (more uniform scour
rate with scour depth and further separation of maximum scour hole locations) will
increase with increased pile diameters. Changes in the scour rate, which can be seen with
the smaller piles, are anticipated until the pile diameter is large enough so that the scour
holes formed from the accelerated flow do not overlap in front of the pile. For larger
diameter piles the change should be minimal. For 0.2 mm diameter sand the pile
diameter where this occurs is estimated to be 0.9 m.
Combined effects
Rate of Scour i ^ ... 1->, ;
Time
Figure 4-11. Author's hypothesis of the relative contributions of the mechanisms
on the rate of scour.
Baker (1978) writes on the increase of the parameter b/d* (where d* is described
as the displacement thickness or j(1 ')dY, where u is the velocity inside the
0
boundary layer and U is the velocity outside the boundary layer) that the large energy
carrying eddies within the boundary layer upstream of the vortex system become smaller
and hence the frequencies associated with them become larger. If this is true, it provides
an explanation as to why the rates decrease with an increase in pier diameter. If the
pulses that drive the horseshoe vortex become smaller, even though they occur more,
there will be a limit as to which mechanism becomes dominant.
The acute variation in the rate of scour after a certain depth has been achieved
may also be due to a cutoff of the horseshoe vortices from the incoming flow. It may be
possible for the streamlines of the incoming flow to be diverted away from the horseshoe
vortices at the bottom of the hole for scour hole having steep side slopes. This concept is
based on the dynamics of a flow over a rectangular cavity (a limiting situation to an
increasing slope of the scour hole), in which a trapped rotating eddy is generated inside
the cavity, and streamlines of the incoming flow show only a slight deflection into the
cavity as they pass over the depression. The diagrams shown in Figure 4-12 illustrate the
concept.
The water depth dependence could not be comprehensively analyzed as all but
one of the experiments were run at yO/b values large enough so as not to affect the
maximum scour depth. It has been shown that the yO/b parameter has little effect on the
maximum depth above values of around 2.5. The exception was experiment 11, having a
yO/b of 0.95. Comparable to that experiment (having the same d50 as well as b),
experiment 9 had a yO/b of 2.62. Experiment 9 also had a higher U/Uc value (0.98 as
compared to 0.82 for experiment 11). Had the yO/b parameters been equal, experiment 9
should have had a higher initial scour rate. Surprisingly, the converse was found,
experiment 9 had a slightly smaller initial scour rate.
Scour approaching. .
Seqtiiin '. '
Figure 4-12. Diagrams representing early and later stages of scour showing less
contribution of the incoming flow to the horseshoe vortex system as equilibrium
is approached.
The general reasoning behind the effect of the yO/b parameter is that for values
smaller that about 2.5, the horseshoe vortex at the base of the pier interacts with the
surface roller vortex on the upstream edge of the pile and at the surface. The surface
roller and horseshoe vortex have opposite rotations, the horseshoe vortex having a
downward motion next to the structure. The interaction causes a decrease in the strength
of both vortices, hence producing a lesser maximum scour depth. Considering this
reasoning, the initial rate of scour should decrease with smaller yO/b. A decrease in yO/b
reduces the strength of the horseshoe vortex, thus inferring a weaker mechanism for
sediment removal. The opposite, however, was noticed. Not only did the initial rate
disagree with the hypothesis; the time-averaged rate over the respective scour depth also
showed an unfavorable difference. The average scour rate, found by dividing the
equilibrium depth by the time required for equilibrium, was 0.38 cm/hr for experiment 9
(having a greater water depth), while experiment 11 had an average rate of 0.54 cm/hr.
Concrete explanations as to why this happened cannot be attempted without
additional data. The variation in the rates between the two experiments was indeed small,
so it could be that some unforeseen factor affected the scour process, not an unreasonable
suggestion considering the number of variables involved.
The data examined in this thesis indicate a dependence of the rate of scour on the
parameters b, U/Uc, and d50 as well as the equilibrium scour depth dse. The range of
parameters investigated to date is not sufficient to establish the precise dependence but
the trends for rate dependence on many of these parameters can be established. The
equilibrium scour depth has already been shown to be dependent on the parameters U/Uc,
d50/b, and yO/b. The USGS Laboratory data shows a good correlation between the time
for equilibrium and the equilibrium scour depth. Could it be that the time required for
equilibrium is solely a function of the equilibrium scour depth?
Based on the results taken from the USGS Laboratory experiments alone, this
appears to be true. However, the range of variables used in the experiments is extremely
limited. All experiments were completed at high U/Uc values, and only two types of
relatively small sediment sizes were used. Further experimental data are needed before a
definitive relationship can be established.
Including the estimated maximum scour depth and the time required to reach 90%
of this depth, the findings shown in Figure 4-13 below indicate that the time for
equilibrium cannot be expressed solely as a function of the equilibrium depth. Although
most of the variables are the same, the dependence of the equilibrium time on these
variables may be different.
160
120
87
84
S80 6 41
40 4A26
J34
d 43
0 1 1 1 1 1--
0 20 40 60 80 100
dse (cm)
Figure 4-13. Equilibrium scour depth versus t90 for all experiments.
69
Using the estimated dse and t90 for Jones data in conjunction with the UF/USGS-
BRD data, an effort was made to find a way of expressing the data points so that a linear
relationship could be found. In this way, a predictive equation can be used to estimate a
scour depth for any time. Using the relationships deduced earlier from the scour rates,
the expression in Equation 4.3 was obtained. A plot of this function along with the data
is given in the logarithmic plot in Figure 4-14.
fl = (UI (log d50 X1000d5' (t9 )05, (4.3)
where 0.45__
__
100 -
O
o
C :
D,
---
1
0.1 1
dse (m)
Figure 4-14. Plot of dse versus fl using data from all experiments.
The line fitted through the data was calculated to be
ln(fl) = 0.6703 In dse + 2.9666. (4.4)
However, the validity of the method for predicting t90 using the equations above
is dependent on the accuracy of the estimations for the equilibrium scour depth and
corresponding time required for equilibrium made for Jones' data. More confidence can
be placed on equations developed from data taken from complete, accurate experiments
such as those completed in the USGS Laboratory. Equations based on the USGS data
alone are developed below.
A major fault of the dse versus te relationship for data taken from the USGS
Laboratory experiments shown earlier was that the line joining the points did not pass
through the origin. Earlier in the analysis it was shown that for the USGS experiments,
as the parameter U/Uc increased, the time for equilibrium increased. It was also shown
that although the initial rate increased with increasing d50, the average rate decreased, so
it took longer for equilibrium to be reached. Using these observations, the following
equation was proposed:
t9o = k(d5o)oo(YUU dse (4.5)
Equation 4.5 is shown in Figure 4-15 along with the data, where t90 is in hours, and d50
and dse are in meters. The constant k is computed to be 218.55 hr/m'05.
250-
200
^ 150-
g) 100 -
0- 50 <
50 -
0' I I
0 0.2 0.4 .0.6 0.8 1
dse (cm)
Figure 4-15. Plot of dse versus f2 for data collected from USGS experiments
only.
The equilibrium scour depth has been researched by a number of scientists over
the last 5 decades. The most recent and accurate calculation has been that of Sheppard,
which is generally accurate to within 10% of the corresponding laboratory results. Using
the predicted equilibrium scour depth and the results from the graph above, an
equilibrium time can be deduced. For a comparison of scour prediction equations, see
Pritsivelis (1999).
Finally, using the earlier plot shown for ds/dse versus t/te, a curve was fitted for a
conservative (fast rate) experiment. The following equation represents the development
of the nondimnsionalized scour depth with nondimnsionalized time:
S( 3 '5(4.6)
d = a+b (4.6)
dS/dse a+ ( +C( ) W I
72
where a=-0.0017, b=-1.1755, c=0.0745, d=2.1075
Using the equation above with the estimated scour depth and equilibrium time, a
scour depth can be computed for any given time, as described in the following steps:
1) Calculate the dse using Sheppard's equation
2) Compute t90 from Equation 4.5
3) For any given t, ds can be calculated using dse and te from Equation 4.6.
CHAPTER 5
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
5.1 Summary
The complex nature behind the mechanisms of local bridge scour has yet to be
fully understood. This thesis attempts to gain useful insight into these mechanisms by
analyzing the temporal development of scour at a cylindrical pile in a steady current.
Through examination of the changes in the rate as scour, a better grasp of the driving
mechanisms that control the process is achieved. Observations made in the analysis are
used to construct a predictive equation to estimate local scour depths as a function of
time.
Data were collected from experiments conducted at the USGS-BRD Laboratory in
Turners Falls, Massachusetts. A total of seven experiments were run. The experiments
varied mainly in cylinder diameter and sediment size, with minor variations in water
depth to pile diameter ratios and the nondimensionalized velocity U/Uc. The
experiments were allowed to run until a definite equilibrium depth could be identified.
The scour depth was monitored by video cameras located inside the piers, and by
acoustic transponders located near the water surface. Water velocities were measured
using two Marsh-McBimey velocity meters. A digital thermometer and a water level
indicator were also used. On completion of each experiment, a survey was made of the
scour hole by point gauging. Additional data used in the analysis came from experiments
completed at the FHWA Turner Fairbanks Laboratory in McLean, Virginia under the
supervision of J. Sterling Jones.
The raw data were first corrected to attain the maximum scour depth recorded on
completion of each experiment by point gauging the bed. The data was then checked for
inconsistencies and smoothed to assist in the analysis. For each experiment, a separate
curve was fitted to the scour history data. This was done to achieve a better comparison
for the initial rates of the experiments.
5.2 Conclusions
The experiments conducted in the USGS Laboratory were not originally intended
for use in the investigation of time rate of scour. They were designed to investigate the
dependence of the parameter d50/b on the maximum scour depth, in order to check and
improve the existing predictive equations. Previously, the small size of the flumes put a
restriction on the maximum size of the structure that could be tested. The expanse of the
flume in the USGS Laboratory made it possible for large near prototype size piles to be
tested.
Because of this, the main variations from one experiment to another were in the
size of the sediment and the diameter of the pile. All were conducted at high U/Uc values
(above 0.8), and all had a yO/b parameter greater than 2. Of the three dimensionless
parameters believed to describe local scour (U/Uc, D50/b and yO/b), only D50/b was
varied in the USGS-BRD Laboratory experiments. It was therefore not possible to
examine the effects of U/Uc and yO/b on the rate at which clearwater local scour occurs
using this data.
A correlation between the time required to reach equilibrium and the equilibrium
scour depth was observed. However, the only time history data available in the literature
for experiments with sufficient duration to reach equilibrium scour depth are those
conducted at the USGS-BRD Laboratory. Jones' experiments were of long duration but
were all short of equilibrium. Since the scour depth versus time plot is very flat as
equilibrium depths are approached it is difficult to establish the time when equilibrium is
reached. Nevertheless, Jones' data were extrapolated to equilibrium and estimates made
for both equilibrium depth and time. Using these approximate values it became apparent
that additional factors, other than just the equilibrium depth were responsible for
determining the time required to reach equilibrium.
Using a plot of the nondimesionalized parameters ds/dse versus t/te, two distinct
scour rates could be identified for each experiment. These rates can be explained by a
transition of the dominant mechanism controlling the scour process. The first, stronger
rate is caused by the action of the acceleration flow around the sides of the pile. Scour
initiates at locations 45-600 to the front of the cylinder. The hole develops, deepening
and widening as sediment cascades down into the hole periodically due to slope
stabilization. Flow separation occurs at the upstream end of the hole, strengthening the
horseshoe vortex. The horseshoe vortex is initiated by the downflow due to the
stagnation pressure gradient on the front of the cylinder. This flow is redirected to meet
the incoming flow when it encounters the bed. As the scour hole deepens, the horseshoe
vortex enters the hole, and grows in strength and size. Streamlines of the accelerated
flow are spread apart, signifying a reduction of the velocity. After a certain depth, the
dominant mechanism controlling the scour process switches from accelerated flow to the
horseshoe vortex. The effect of the horseshoe vortex is considered to be weaker than the
accelerated flow and hence the transition in the rate of scour. Further deepening of the
hole results in the vortex expanding in size but weakening in strength. Equilibrium is
reached when the vortex is not strong enough to transport dislodged sediment out of the
hole
All data from the experiments were then fitted with individual curves for better
analysis of the initial rates. By differentiating the resultant equations for the normalized
scour depth versus time, another equation was obtained that described the time rate of
scour. Comparing this rate to ds/dse produced a number of interesting results.
An increase in particle diameter d50 increases the initial scour rate. This is
explained by a decrease in area of sediment available for scour as compared to sediment
of smaller diameter because the angle of repose, which determines the slope of the scour
hole, is greater. As the scour hole progresses, a larger sediment, which has a larger angle
of repose, will have a steeper side slope. Thus the erodible area will be less, and the
scour greater.
However, even though the initial rate increases, the time required for equilibrium
increases with increasing particle diameter. The scour plots for the larger sediment are
seen to approach equilibrium much slower. It is thought that at this stage of scour,
turbulent bursts are instrumental in achieving equilibrium. A larger particle requires a
stronger burst to be completely removed from the hole. Stronger bursts occur less
frequently, hence the slower rate of scour.
Increase in the nondimensionalized parameter U/Uc will increase the initial scour
rate. Greater values of U/Uc will have a larger capacity for sediment transport, and the
rate of scour increases. The effect on the overall time required for an equilibrium scour
depth to be reached is inconclusive. The FHWA experiments were included in the
analysis as they were performed over a wider range of velocity ratio. Using estimates for
both equilibrium depth and the time required for 90% of the scour to be completed, the
FHWA data indicates the time required for equilibrium decreases with increasing U/Uc.
However, this observation disagrees with both the data from the USGS-BRD
experiments, as well as findings made by previous researchers (Shen, Baker, Melville et
al.). This data shows increasing times to equilibrium with increasing U/Uc.
The scour rate plots show differences in the shape of the curves for the different
size piers. The smaller piers have two distinct scour rates, whereas the larger pile has a
scour rate that generally does not change significantly during the scour process.
Coincidentally, the location of the deepest scour depth changes, being directly in front of
the pier for the smaller cylinders and at angles 450-60 for the larger cylinder. These
differences are thought to be due to a difference in the extent of the scour initiated by the
accelerated flow around the sides of the pile. For the larger pile, if the scour spreading
from the points of initiation on either side of the pile do not meet, a section of the hole
directly in front of the pile is left relatively unscoured. This affects the flow separation,
the horseshoe vortex and the energy fed into the system, which is apparent in the lack of
variation in the scour rate. The flow is directed around the unscoured sediment, hence
the difference in the location of the maximum scour depth. The smaller piers seem to
agree with the hypothesis presented earlier.
From the preceding observations, an equation was derived so that a linear
relationship between it and the maximum scour depth could be fitted through the data.
Two equations.were presented one for all the data and another for just the data taken
from the USGS Laboratory experiments. Using one of the faster rate experiments for
conservative reasons, a separate equation was fitted relating the nondimensionalized
parameters ds/dse and t/te. Using a predictive equation to find the maximum scour depth,
the time for equilibrium could be estimated. Finally for any time, a scour depth can be
computed using the conservative scour plot equation.
5.3 Recommendations
The lack of data for a range of all of the important parameters affecting the rate at
which local scour occurs places limits on the confidence in the descriptions and
conclusions given in this thesis. In spite of these limitations a number of interesting
observations were made. These include the dependencies of the sediment diameter, pile
diameter, and velocity ratio on the scour rates, as well as variations in the rate changes
from one experiment to another as scour progresses.
Various uncontrollable aspects of the experiments, such as the variability of the
river water used and the relative inefficiency of the gates controlling the flow may have
resulted in small inaccuracies in the data collected. The time rate, more so than the
equilibrium scour depth, is especially sensitive to the subtle irregularities encountered.
Firstly, the initial rates were dependent on the speed at which the target velocity
was reached. The gates controlling the flow of water into the flume were tricky to use.
Several minutes would elapse before a steady velocity was attained. Although efforts
were made to set the target velocity as quickly as possible, the time elapsed before this
was accomplished must have had a bearing of the initial scour.
The source of the flow came from the Connecticut River. Several filters were
used to stop large obstacles from entering the test area. Inevitably, small particles such as
silt suspended in the water column were transported into the flume. This cohesive
sediment settled on the bed when the water flow had been shut off on completion of each
experiment, and was carefully removed before the bed was point-gauged.
Extreme temperatures of the river water were recorded during the winter months.
For two of the experiments, for example, the average temperature was a meager 10 C.
The temperature is known to affect both the density and the viscosity of the water, thus
changing the critical velocity for the onset of sediment motion. This was accounted for in
the calculation of the critical velocity for each experiment, but the effect of extreme
temperatures on other factors was not. It could be that the temperature also affects
cohesive attraction of individual particles. It also seems likely that because both density
and viscosity are affected, the dynamics of the horseshoe vortex could be altered as well,
either in magnitude or shape.
The validity of the rate of scour equation derived in this thesis is not only
restricted by the narrow range of experimental data but also dependent on the accuracy of
the predictive equation of the equilibrium scour depth. Although substantial research has
been conducted in determining the maximum depth of scour, the possibility of error
multiplies in using one empirically derived equation for use in another.
The nondimensionalized plot of ds/dse versus t/te allows the rates of the
experiments to converge at both the origin and at the point (1,1). However, in between
these points there are small differences in the rates in the individual experiments. The
slower experiments were used to devise the fitted equation to which a depth can be
estimated for any time. However, a better situation for estimating the scour depth using
the maximum depth and equilibrium time would be to incorporate the factors that
determine the rate of the scour process as it evolves.
Clearly a detailed series of experiments dedicated to the investigation of the
temporal rate of scour is needed. The experiments have to be completed in their entirety
to ensure both an equilibrium depth and time can be recorded. Non-intrusive methods of
data collection such as those used in the USGS-BRD Laboratory are recommended. The
experiments have to be varied singly in water velocity, sediment diameter, water depth,
and pier size, while holding the other variables constant.
Additionally, accurate measurements of velocity fluctuations of both the
accelerated flow and the horseshoe vortex need to be completed. A laser Doppler
anemometer would be the most ideal instrument for this purpose. Additionally, a study
into the intensity and frequency of the turbulent bursts that dictate the final stages of
scour would also be desired. Using this information, a complete understanding of the
mechanism behind the scour process could be formulated.
A better understanding of the dynamics of scour can be used in devising a
mathematical computer model. The very nature of the process, with the apparent
turbulent bursts of the vortex removing sand periodically, appears to be very similar to a
typical sediment transport model using an incremental time step. By representing the
contribution of both the acceleration and the horseshoe vortex as they vary in the
development of the hole, and taking the volumetric transport of the eroding sediment as
well as the neighboring cascading sediment into consideration, the design of a model
estimating the scour development over time is quite feasible.
Page 81
is missing
from original
APPENDIX A
INDIVIDUAL CURVE FITS
The following curve fits were accomplished using the 2-D empirical curve-fitting
program TablecurveM.
Exp 3
y=a+b(1-exp(-cx))+d(1-exp(-ex))
a=-0.00045214718 b=0.19307718 c=4.2084458
d=0.82592905 e=0.1728253
1.25 ---------- --------1.25
0.75 --- --- .75------ 0.5
0
0^s-I-f.~-----i-----.-------------------------...
0.25 ------------------------- -0.25
O __ _i i ____ -- --- .0. .
Time (hrs)
Figure A-1. Curvefit of data of experiment #3.
83
Exp4
y=a+b(l-exp(-cx))+d(l-exp(-ex))
a=0.00068531296 b=0.41187992 c=1.2901773
d=0.59838058 e=0.050830295
---- ---- .0.9
0.8
S0.7
0.6
0.5
Figure A-2. Curvefit of data of experiment #4.
84
Exp6
y=a+b(l-exp(-cx))+d(l-exp(-ex))
a=-0.0013300124 b=0.1254428 c=0.5870551
d=0.9123429 e=0.05407349
1.25 i f 1 1.25
0.75------------------------5-----------
0.75 .'----J-------,------------ .0.75
0.5 -0.5
0.25 .....-- 0.25
Time (hrs)
Figure A-3. Curvefit of data of experiment #6.
Exp9
y=a+b(l-exp(-cx))+d(1-exp(-ex))
a=0.0016169758 b=0.17848818 c=0.99698604
d=0.82649869 e=0.013839237
1.25 1 T 1.25
1 -------------- ---1-----1
0.75 ----- -75
0.5 --0.5
0.25-- 0.25
Time (lirs)
Figure A-4. Curvefit of data of experiment #9.
Exp11
y=a+b(l-exp(-cx))+d(l-exp(-ex))
a=-0.0021784218 b=0.33460626 c=0.63915118
d=0.72608262 e=0.020879841
1.25 I I 1.25
1. 25------------------------------- --1
0.75 -------- ------ ------- 0.75
0.5 -- 0.5
0.25 ------0.25
Time (hrs)
Figure A-5. Curvefit of data of experiment #11.
__
__ |