the use of end plates for a cylinder in the sub …...ii the use of end plates for a cylinder in...
Post on 25-Apr-2020
0 Views
Preview:
TRANSCRIPT
The Use of End Plates for a Cylinder in the Sub-critical Flow Regime
by
Adam Douglas Blackmore
A thesis submitted in conformity with the requirements for the degree of Master of Applied Science in Engineering
Institute for Aerospace Studies University of Toronto
© Copyright by Adam Douglas Blackmore 2011
ii
The Use of End Plates for a Cylinder in Sub-critical Flow Regime
Adam Douglas Blackmore
Master of Applied Science in Engineering
Institute for Aerospace Studies
University of Toronto
2011
Abstract
Experiments were conducted in a free-surface, re-circulating water channel to determine the
dependence of spanwise flow uniformity in the near wake of a circular cylinder on the end
conditions using Particle Image Velocimetry. The Reynolds number was 10,000. The end
conditions consisted of plates with different leading edge geometries and configurations. A
cylinder bounded by two endplates with sharp leading edge geometry generated the most
uniform near wake. The horseshoe vortex dynamics in the cylinder/ wall and cylinder/endplate
junctions were also studied. Upstream flow separation significantly altered the behavior of the
horse shoe vortices. Periodic horse shoe vortex oscillation was found for experiments with the
upstream flow attached; this periodic oscillation was disrupted with the presence of upstream
flow separation. The endplate leading edge distance was also investigated. The oscillation
frequency of the horse shoe vortex system was found to decrease with increasing leading edge
distance.
iii
.Acknowledgments
I would firstly like to thank my supervisor Dr. Alis Ekmekci for her continuous support,
guidance and expertise throughout this project. I sincerely appreciate her patience and help
through the many hours spent analyzing data and solving problems. Being able to drop by her
office unannounced whenever something exciting…or problematic came up was extremely
helpful. Her willingness to be available during any time of day is a testament to both her work
ethic and kindness. I was also lucky to have a supervisor who found the time and money to bring
me to conferences where I was inspired to keep working hard and ultimately continue my studies
and pursue a PhD.
Secondly, I would like to thank my Research Assessment Committee members, Dr. Zingg, Dr.
Lavoie, Dr. Ekmekci and Dr. Martins for their guidance and advice during the course of this
investigation.
I would like to thank my Fiancée Danielle for her support throughout this degree. The late hours
spent in the lab or working on the computer would have been even harder were it not for her
constant guidance, and backing; I am truly lucky to have her.
I want to extend a special thank you for all of the students in the office who helped make the
experience at UTIAS extremely rewarding. In particular, I would like to thank Tayfun Aydin
and Ronald Hanson for all of their advice and guidance throughout this project. I can say with
great confidence that without their help and expertise I would not be where I am today.
Lastly I would like to thank my parents for all of their support and assistance throughout this
experience, I am truly grateful.
iv
Table of Contents
Acknowledgments ..................................................................................................................... iii
Table of Contents ...................................................................................................................... iv
Nomenclature .......................................................................................................................... xiv
INTRODUCTION ...................................................................................................................... 1
1.1 Background and Literature Review ................................................................................. 1
1.2 Flow past Cylinders: Terminology and Definition of Flow Regimes ............................... 3
1.3 Literature Review on the Design of an End Plate for Flow past a Cylinder ...................... 5
1.4 Review of Junction Flow Studies .................................................................................... 7
EXPERIMENTAL SYSTEM AND TECHNIQUES ................................................................. 11
2.1 Hot Film Anemometry .................................................................................................. 11
2.2 Particle Image Velocimetry ........................................................................................... 12
2.2.1 PIV Exposure Technique ................................................................................... 13
2.2.2 Computation of Particle Displacement ............................................................... 13
2.3 Volumetric 3-Component Velocimetry ......................................................................... 14
2.3.1 Using Particle Defocus to Calculate Out of Plane Velocity Components ............ 15
2.3.2 Tracking the Particles ........................................................................................ 16
2.4 Flow Facility ................................................................................................................. 17
2.4.1 Characterization of the Flow Facility ................................................................. 18
2.5 Experimental Setups ..................................................................................................... 18
2.5.1 Experimental Setups for the Measurements in the Symmetry Plane of the
Near-Wake Region along the Cylinder Span ...................................................... 21
2.5.2 Experimental Setups for the Measurements in the Upstream of Cylinder-Wall
and Cylinder-Endplate Junctions ....................................................................... 22
2.6 Significance of the Leading-Edge Geometry of the Endplates ....................................... 28
SPANWISE UNIFORMITY OF THE NEAR-WAKE OF A CYLINDER: SIGNIFICANCE
OF THE ENDPLATE CONFIGURATION ......................................................................... 32
v
3.1 The Near Wake along the Span of a Cylinder with Various End Conditions: Patterns
of Time-Averaged Streamwise Velocity ........................................................................ 34
3.1.1 No Endplate ...................................................................................................... 34
3.1.2 Cylinder Bounded by the Endplate with Sharp Leading Edge (SLE) and the
Free Surface ...................................................................................................... 35
3.1.3 Cylinder Bounded by the Endplate with Elliptical Leading Edge (ELE) and
the Free Surface ................................................................................................ 36
3.1.4 Cylinder Bounded by Two Endplates with Sharp Leading Edges ....................... 37
3.2 Global Autospectral Density of Streamwise Velocity in the Near Wake ........................ 38
3.3 Summary and Results of Measurements in the Near Wake: Demarcation Line Factor ... 39
HORSESHOE VORTEX DYNAMICS AT THE JUNCTION REGION .................................. 54
4.1 Unsteady Flow Characteristics at the Cylinder-Wall Junction: Temporal Evolution of
Vorticity Contours ........................................................................................................ 55
4.2 Unsteady Flow Characteristics Upstream of the Junction of a Cylinder with an
Endplate having Sharp Leading Edge Geometry: Temporal Evolution of Vorticity
Contours ....................................................................................................................... 57
4.3 Unsteady Flow Characteristics in the Upstream of the Junction of a Cylinder with an
Endplate having Elliptical Leading-Edge Geometry: Temporal Evolution of Vorticity
Contours ....................................................................................................................... 58
4.4 Frequency Characteristics of the Horseshoe Vortex Systems: Spectral Analysis of
Streamwise Velocity ..................................................................................................... 59
4.5 Summary of the Horseshoe Vortex Measurements ........................................................ 61
CHAPTER 5 ............................................................................................................................ 72
Conclusions and Future Work................................................................................................... 72
References ................................................................................................................................ 76
Appendix A – Uncertainty Analysis ......................................................................................... 79
vi
List of Figures
Figure 2.0: Schematic of a PIV light sheet and interrogation grid. The upper portion of the figure
describes a zoomed in region of the entire interrogation region with K horizontal grid points and
L vertical grid points. Particles within the grid can be seen displacing in time. .......................... 13
Figure 2.1: Schematic of a V3V setup. Three cameras are focused on the rear of the measurement
volume. A particle in front of the rear plane (within the measurement volume) will be out of
focus. The amount of de-focus is used to measure the out of plane velocity component. ........... 15
Figure 2.2: Schematic of the free-surface water channel, located at the University of Toronto
Institute for Aerospace Studies. Nd:YAG laser and the illumination plane used the PIV
measurements are also incorporated in the schematic. ............................................................... 17
Figure 2.3: Schematic of the components of the PIV and V3V systems. The V3V and PIV setups
are similar with the exception of the dual CLFC Frame grabbers required to transfer the V3V
images to the computer, and the cameras, which have twice the pixel resolution in the V3V
setup. Both systems utilized the same Nd:YAG laser unit with 200 mJ per pulse laser and the
same synchronizer during the experiments. ............................................................................... 20
Figure 2.5: Experimental setups used in the cylinder-wall and cylinder-endplate junction
experiments. For the experiments involving the use of an endplate, the cylinder was bounded by
the endplate at the bottom and by the free surface at the top. The cylinder position on the
endplate was varied by changing the distance to the leading edge, represented by λ=L/D, for
values of 1, 2.5 and 5. The leading-edge geometry of the second and third experiments was
different. A sharp leading-edge shape was used in the second one. This was determined to
produce significant upstream separation. An elliptical leading-edge was designed for the third
experiments from top and found to eliminate flow separation at the tip of the plate. The field of
view in all experiments was approximately 1D in the streamwise and 0.8D in the spanwise
direction. .................................................................................................................................. 26
Figure 2.6: V3V setup used to study the junction flow behavior of the cylinder-wall arrangement.
The Reynolds number based was 10,000. The streamwise position of the cylinder was 105 cm
downstream of the test section entrance, which gave approximately 25 mm (0.5D) of upstream
vii
junction region imaged within the volume. The volume height was reduced to 0.98D in order to
increase vector resolution. The final vector resolution, based on a starting grid of 0.15D with
75% overlap was 0.04D. This resolution made it possible to identify the primary horseshoe
vortex. ...................................................................................................................................... 27
Figure 2.7: On the left hand-side image, contours of time-averaged normalized vorticity
<>D/Uo are superposed over the time-averaged streamlines, demonstrating significant flow
separation for flow past the plate with sharp leading-edge geometry. The plate is exposed to flow
at ReD of 10,000 and no cylinder is placed in the flow. The right-hand side sketch shows the PIV
field of view. ............................................................................................................................ 28
Figure 2.8: Schematic showing the coordinate frame for the elliptical leading edge design in
Equation (3) where „a‟ and „b‟ are the major and minor axes of the ellipse chosen. ................... 29
Figure 2.9: Superposition of the time-averaged normalized vorticity <>D/Uo and the time-
averaged streamline patterns for the plates with sharp and super-elliptical leading edges. The
results for the sharp leading-edge design are shown in the left frame of the figure, which
demonstrates significant separation. The plate with super-elliptical nose, shown in the right
frame, successfully eliminates the separation. The field of view in these measurements was
approximately 1D in the streamwise and 0.8D in the spanwise direction, and the vector
resolution was approximately 0.006D. ...................................................................................... 31
. ................................................................................................................................................ 41
Figure 3.0: Contour patterns of time-averaged streamwise velocity <u>/Uo and spanwise
velocity<v>/Uo components in the near-wake of the circular cylinder without the use of
endplates. White rectangular boxes are used to remove the contours from regions that are close
to the free surface and the solid cylinder boundary, where considerable laser light reflection was
present, and to remove the discontinuous contours in from the mid-span vicinity, where the two
PIV images (the top and bottom halves of the near wake) were merged. Negative and positive
<u>/Uo are represented by dashed and solid lines respectively ................................................. 41
Figure 3.1: Contours of time-averaged streamwise velocity <u>/Uo in the near-wake of the
cylinder bounded by a single endplate with sharp leading edge for λ = 0.5, 1, 2, 2.5, and 3.0.
viii
White rectangular boxes are used to remove the contours from regions that are close to the free
surface and the solid cylinder boundary, where considerable laser light reflection was present,
and to remove the discontinuous contours in from the mid-span vicinity, where the two PIV
images (the top and bottom halves of the near wake) were merged. Negative and positive
<u>/Uo are represented by dashed and solid lines respectively. ................................................ 42
Figure 3.2: Time-averaged contours of streamwise velocity <u>/Uo in the near-wake of the
cylinder bounded by a single endplate with sharp leading edge for λ=3.5, 4, 5, 6, and 7. White
rectangular boxes are used to remove the contours from regions that are close to the free surface
and the solid cylinder boundary, where considerable laser light reflection was present, and to
remove the discontinuous contours in from the mid-span vicinity, where two PIV images (the top
and bottom halves of the near wake) were merged. Negative and positive <u>/Uo are represented
by dashed and solid lines respectively. ...................................................................................... 43
Figure 3.3: Time-averaged contours of normalized streamwise velocity component <u>/Uo in
the near wake of the cylinder bounded by a single elliptical endplate at the bottom and by the
free surface at the top for λ=1.5, 2, 2.5. Solid lines indicate positive streamwise velocity, and
dashed lines indicate negative streamwise velocity. White rectangular boxes are used to remove
the contours from regions that are close to the free surface and the solid cylinder, and where two
PIV images were merged at the mid span. ................................................................................ 44
Figure 3.4: Time-averaged contours patterns of normalized streamwise velocity component
<u>/Uo in the near-wake of a cylinder bounded by a single elliptical endplate at the bottom and
by the free surface at the top for λ=3.5, 4, 5, 6, 7. Solid Lines indicate positive streamwise
velocity, and dashed lines indicate negative streamwise velocity. White rectangular boxes are
used to remove the contours from regions that are close to the free surface and the solid cylinder
boundary, where considerable laser light reflection was present, and to remove the discontinuous
contours in from the mid-span vicinity, where the two PIV images (the top and bottom halves of
the near-wake) were merged. .................................................................................................... 45
Figure 3.5: Contour plots of time-averaged normalized streamwise velocity <u>/Uo in the near
wake for λ=2, 2.5, and 3. The cylinder was bounded at both ends by the endplates having sharp
leading-edge geometry. Solid lines indicate positive streamwise velocity, and dashed lines
indicate negative streamwise velocity. White rectangular boxes are used to remove the contours
ix
from regions that are close to the free surface and the solid cylinder boundary, where
considerable laser reflections occurred, and where two PIV images were merged at the mid span.
................................................................................................................................................. 46
Figure 3.6: Global contour patterns of the autospectral density Su(f) of the streamwise velocity
component in the near-wake of the cylinder bounded by a water channel floor at the bottom and
the free surface at the top. As a result of a Strouhal number resolution of 0.018, Su(f) contours are
defined at two values of St = 0.196 & 0.214. ............................................................................. 47
Figure 3.7: Global contour patterns of the autospectral density Su(f) of the streamwise velocity
component in the near-wake of the cylinder bounded by an endplate having sharp leading edge at
the bottom and the free surface at the top. Leading edge distances are λ=0.5, 1, 2, 2.5, 3. As a
result of a Strouhal number resolution of 0.018, Su(f) contours are defined at two values of St =
0.196 & 0.214. .......................................................................................................................... 48
Figure 3.8: Global contour patterns of the autospectral density Su(f) of the streamwise velocity
component in the near-wake of the cylinder bounded by an end plate having sharp leading edge
at the bottom and the free surface at the top. Leading edge distance are λ=3.5, 4, 5, 6, 7. As a
result of a Strouhal number resolution of 0.018, Su(f) contours are defined at two values of St =
0.196 & 0.214. .......................................................................................................................... 49
Figure 3.9: Global contour patterns of the autospectral density Su(f) of the streamwise velocity
component in the near-wake of a cylinder bounded by an endplate having elliptical leading edge
at the bottom and the free surface at the top. Leading edge distance are λ=1.5, 2, 2.5, 3.5, 4. As a
result of a Strouhal number resolution of 0.018, Su(f) contours are defined at two values of St =
0.196 & 0.214. .......................................................................................................................... 50
Figure 3.10: Global contour patterns of the autospectral density Su(f) of the streamwise velocity
component in the near-wake of the cylinder bounded by an endplate having sharp leading edge at
the bottom and the free surface at the top. Leading edge distances are λ=5, 6, 7. As a result of a
Strouhal number resolution of 0.018, Su(f) contours are defined at two values of St = 0.196 &
0.214.. ...................................................................................................................................... 51
Figure 3.11: Global contour patterns of the autospectral density Su(f) of the streamwise velocity
component in the near-wake of a cylinder bounded by two endplates having sharp leading edge
x
geometry. Leading edge distances are λ=0.5, 1, 2, 2.5, 3. As a result of a Strouhal number
resolution of 0.018, Su(f) contours are defined at two values of St = 0.196 & 0.214. .................. 52
Figure 3.11: Variation of the demarcation line factor with the leading-edge distance λ of the
endplate for all experimental arrangements considered in the present investigation, and the
variation of the demarcation line factor along the span of the cylinder for a fixed value of λ=2.5.
The top graph shows the demarcation line factor for each λ value, tested in every end plate
arrangement, with 50% of the span used in the calculation. The solid line denotes the value when
no endplate is used, which is the basis case. The solid circle represents the case where a single
endplate with sharp leading edge is used, the solid triangle represents the case where a single
elliptical endplate is used and, the open circle shows the results when two sharp leading edge
endplates are used. The results demonstrate the clear advantage of using two endplates, as seen
for various λ values in the top graph. When 50% of the span is used in the calculation for the
optimum λ=2.5, it can be seen in the bottom graph that two endplates provide a uniform
demarcation line factor along the cylinder span. ....................................................................... 53
Figure 4.0: Time series evolution of normalized vorticity: upstream junction region – no
endplate. Individual time series are represented by each column of frames. The blue horizontal
rectangle shows the channel floor, and the blue horizontal rectangle shows the cylinder. The
evolution of the primary vortex can be seen as it approaches the cylinder, and begins to diminish
in size and strength. Eventually it is amalgamated into the secondary vortex. ........................... 63
Figure 4.1: Multi slice V3V measurements of time series evolution of vorticity magnitude:
upstream junction – no endplate. Contours of vorticity at multiple measurement planes show the
diminishing primary vortex seen in the „Y-X” plane, where PIV measurements were conducted.
This coincides with an increase in vorticity in the legs of the horseshoe vortex, seen in the „Y-Z‟
plane. ....................................................................................................................................... 64
Figure 4.2: Time series evolution of vorticity in the upstream junction of a cylinder-sharp leading
edge endplate λ=1. Each column of frames represents a time series evolution of vorticity. The
horizontal blue boundary represents the endplate, and the vertical blue boundary represents the
cylinder. It can be seen that the primary horseshoe vortex does not undergo a periodic decrease in
magnitude and strength. ............................................................................................................ 65
xi
Figure 4.3: Time series evolution of vorticity in the upstream junction of a cylinder-sharp leading
edge endplate λ=2.5. Each column of frames represents a time series evolution of vorticity. The
horizontal blue region represents the endplate, and vertical blue region represents the cylinder. It
can be seen that there is one steady primary horseshoe vortex which does not diminish in size
periodically due to the entrainment of upstream vorticity. Pockets of negative vorticity can be
seen surrounding the primary vortex, but do not significantly diminish its size over time due to
the addition of vorticity from the upstream separation, which is caused by using a sharp leading
edge.......................................................................................................................................... 66
Figure 4.4: Time series evolution of vorticity in the upstream junction of a cylinder-sharp leading
edge endplate λ=5. Each column of frames represents a time series evolution of vorticity. The
horizontal blue region represents the endplate, and the vertical blue region represents the
cylinder. It can be seen that there is one steady primary horseshoe vortex which does not
diminish in size periodically due to the entrainment of upstream vorticity. Pockets of negative
vorticity can be seen surrounding the primary vortex, but do not significantly diminish the size
over time, due to the addition of vorticity from the upstream separation. .................................. 67
Figure 4.5: Time series evolution of vorticity magnitude for upstream junction of elliptical
leading edge endplate: λ=1. Each column of frames represents a time series evolution of
vorticity. The horizontal blue region shows the endplate, and the vertical blue region represents
the cylinder. A periodic movement, and reduction in size of the primary vortex as it approaches
the cylinder can be seen clearly. The secondary vortex amalgamates with the reduced primary
vortex when the vortex diminishes in size, and is close to the larger oncoming secondary vortex.
................................................................................................................................................. 68
Figure 4.6: Time series evolution of vorticity magnitude for upstream junction of elliptical
leading edge endplate: λ=2.5. Each column of frames represents a time series evolution of
vorticity. The horizontal blue region shows the endplate, and vertical blue region shows the
cylinder. The primary vortex can be seen approaching the cylinder and reducing in size and
strength, when the secondary vortex reaches the primary vortex it amalgamates with the primary
vortex. ...................................................................................................................................... 69
Figure 4.7: Time series evolution of vorticity magnitude for upstream junction of elliptical
leading edge endplate: λ=5. Each column of frames represents a time series evolution of vorticity
xii
magnitude. The horizontal blue region shows the endplate, and the vertical blue region shows the
cylinder. The periodic reduction of vorticity magnitude of the primary vortex can be seen as it
approaches the cylinder. When the secondary vortex reaches the reduced primary vortex it
amalgamates with the primary vortex. ...................................................................................... 70
Figure 4.8: Plots of frequency spectra taken for all endplate configurations, and no end plate
case. Spectra were sampled at the location of time averaged maximum vorticity. Results of the
spectral analysis demonstrate a dominant frequency of 0.36, when no endplate is used. An
endplate with a sharp leading edge generated significant upstream separation, which caused the
primary vortex to retain its size and strength, and not periodically diminish. Therefore the spectra
appear broadband, and no dominant frequency is detected. The results presented for the elliptical
endplate with no upstream separation are shown in the bottom half of the figure. A dominant
frequency can be seen for each case. An inversely proportional relationship is seen between the
leading edge distance and the dominant frequency, with larger leading edge distances resulting in
smaller frequency values. ......................................................................................................... 71
xiii
List of Tables
Table 2.0: Summary of Water Channel Characterization ........................................................... 18
Table 2.1: Summary of Experimental Systems .......................................................................... 19
xiv
Nomenclature
ρ Density (kg/m3)
µ Dynamic Viscosity (kg/ms)
λ L/D
Ψ Stream Function
a Super Ellipse Major Axis (mm)
b Super Ellipse Minor Axis (mm)
CCD Charge Coupled Device
pbC Base Pressure Coefficient
CTA Constant Temperature Anemometry
D Cylinder Diameter (mm)
ELE Endplate with Elliptical Leading Edge
f Frequency (Hz)
F.O.V. Field of View
HSV Horseshoe Vortex
HWA Hot Wire Anemometry
L Distance from the Leading Edge to the Cylinder Axis (mm)
Lu Distance from the Cylinder to Demarcation Line (mm)
N Number of Samples
xv
Re Reynolds Number
S Span of the Cylinder (mm)
Scov Span of the Cylinder used in the Demarcation Line Factor Calculation (mm)
u Streamwise Velocity (mm/s)
Uo Freestream Velocity (mm/s)
u’ Streamwise Velocity Fluctuation (mm/s)
v Spanwise Velocity (mm/s)
x Streamwise Coordinate (mm)
y Spanwise Coordinate (mm)
< > Denotes Time Averaged Quantities
1
CHAPTER 1
INTRODUCTION
1.1 Background and Literature Review
Flow past bluff bodies has been the subject of intense research for many years. In particular, flow
past a circular cylinder has received great attention by the fluid mechanics community (Roshko
1954, Bloor 1964, Williamson 1989, Prasad 1996, Williamson 1996) because flow past this
simple geometry produces complicated fluid mechanic phenomena, such as flow separation,
wake dynamics, boundary layer transition etc. Flows around circular cylinders are also
representative of many practical and important scenarios of engineering, such as flow past tube
bundles in a nuclear heat exchanger, cables on a suspension bridge, offshore risers on an oil rig,
tall industrial chimneys, and towers.
Flow past a circular cylinder is often studied with the assumption that the vortices are shed
parallel to the span of the circular cylinder. This assumption would be valid if experiments were
conducted on an infinitely long cylinder; however, this is not the case in laboratory experiments,
where effects from the walls of wind tunnels or water channels cannot be underestimated in
comparison to the aspect ratio of the cylinder (Stansby 1974, Szepessy 1993, 1994). Several
studies have demonstrated that the use of plates mounted orthogonally to the span of the cylinder
at its ends minimizes the effect of the wall boundary layer by forming a new thin boundary layer
on the plate, and promotes spanwise flow uniformity in the near wake (Stansby 1974, Szepessy
and Bearman 1992, Szepessy 1993). Effectiveness of an end plate has been shown to depend on
many factors, including the aspect ratio of the cylinder (Norberg 1994), size of the end plates
(Stager et al. 1991), and flow regime (Williamson 1996). Furthermore, it is well known that the
junction region between an end plate and a cylinder or between a wall and a cylinder involves
complex flow physics. In this region, an adverse pressure gradient develops along the end plate
or the wall, causing the boundary layer to separate, roll up, and form a vortical structure or
system of vortices around the cylinder; these are commonly called horseshoe vortices (HSV)
(Baker 1979, Simpson 2001). The behavior of these vortices has been shown to depend on the
state of the approaching boundary layer and the bluntness of the object (Simpson 2001, Wei et al.
2
2008). The investigation which is presented in this thesis examines the effects of various end-
plate configurations for a circular cylinder and compares the spanwise flow uniformity in the
near wake to determine the end plate arrangement that attains very nearly-parallel shedding for a
cylinder at the subcritical Reynolds number of 10,000. Furthermore, the behavior of horseshoe
vortex systems at the junction region of these configurations is also studied. To sum up, the
purpose of the thesis is to investigate the following main unresolved issues:
The degree of spanwise flow uniformity in the near wake of a cylinder when various end
conditions are employed. The aim is to determine the end plate configuration that can
promote improved quasi-two-dimensionality.
Characteristics of the horseshoe vortex systems occurring at the junction region between
an end plate and a cylinder, and between a channel wall and a cylinder. Primary
emphasis is on the effects of the upstream flow conditions on the horseshoe vortex
dynamics, specifically, evaluating how the separation or no-separation of the approach
flow, and the leading-edge distance of the end plate from the cylinder will influence the
horseshoe vortex systems.
The thesis is organized into five chapters: 1-Introduction, 2-Experimental System and Methods,
3-Spanwise Measurements in the Near Wake, 4-Flow Characteristics at the Junction Region, and
5-Conclusions and Recommendations. The first chapter will address the purpose of the
investigation and introduce the scholarly literature pertinent to the present research. The second
chapter will provide the details of the experimental setups, the techniques employed during the
present investigation, and an overview of the underlying theory associated with each technique.
Characteristics of the flow structure in the spanwise symmetry plane of the cylinder near-wake
with various end conditions will be discussed in Chapter three. In Chapter four, the dynamics of
the horseshoe vortex systems upstream of the end plate/cylinder junction as well as channel
wall/cylinder junction will be presented. Chapter five will summarize the major findings derived
from this investigation and provide recommendations for future work.
3
1.2 Flow past Cylinders: Terminology and Definition of Flow Regimes
In this subsection, the terminology used to describe the regions of flow past a bluff body is
introduced in detail. When a flow encounters a bluff body, a boundary layer develops around the
body and could, at some point, separate, depending on the pressure gradient imposed by the flow
conditions and geometry of the body (Williamson 1996). With the separation of the boundary
layer, a region of high-speed flow forms. This region is typically called the shear layer (or
separated boundary layer). Under certain conditions, the region behind the cylinder (wake) may
contain coherent swirling structures called vortices. As intuition would dictate, the flow physics
is sensitive to the oncoming flow speed, which is related to the non-dimensional number called
the Reynolds number. It can be defined, based on the cylinder diameter, as:
DU o
D Re (1)
where ρ and μ denote the density and dynamic viscosity of the fluid respectively. Uo and D are
the freestream velocity and the diameter of the cylinder.
For a given smooth cylinder, flow regimes can be defined based on ReD. Herein, only a brief
introduction to the main features of these flow regimes will be given, and further detailed
characteristics can be found in the review paper of Williamson (1996). The first regime concerns
the flow where the ReD is less than about 49. This regime involves a steady wake comprised of
two symmetrical and fixed vortices behind the cylinder. The second regime is seen when the ReD
is between 49 and approximately 194. In this regime, the wake develops instabilities, the onset of
which is due to a Hopf bifurcation (Williamson 1996). These instabilities arise from the
downstream end of the recirculation region. As ReD increases, the base suction monotonically
increases, the peak amplitude of shear stress increases and the location of the peak of the shear
stress moves upstream towards the cylinder, i.e., decrease in formation length (Williamson
1996). The vortex shedding in this regime is laminar and therefore this regime is called the
laminar shedding regime. If the ReD is further increased to a value between ReD = 190 to 260, the
wake transition regime arises. This regime involves two discontinuous changes in the wake as
ReD is increased. These discontinuities are detected in the base suction-Reynolds number
variation and the Strouhal number – Reynolds number (St- ReD) curve (Williamson 1996). The
4
first discontinuity occurs approximately in the range of 180-194 depending on the experimental
conditions (Williamson 1996), and consists of the formation of vortex loops and streamwise
vortex pairs at a wavelength of 3 to 4 cylinder diameters (mode A instability) as a result of the
deformation of the primary vortices being shed from the circular cylinder. The second
discontinuity manifests itself over a range of ReD from 230 to 250 (Williamson 1996). Finer-
scale streamwise vortices arise, at a wavelength of one diameter (mode B instability).
(Williamson 1996). The base suction coefficient and the St continue to increase in this regime
with increasing ReD. As outlined in Williamson (1996), the St reaches a maximum at a ReD of
about 260. As ReD is increased from 260 until about 1,000, the 3-D fine-scale streamwise vortex
structures become increasingly disorded, and there is a drop in the base suction coefficient and
the Reynolds stress, and an increase in the vortex formation length (Unal and Rockwell 1988).
To this point, the flow regimes which are briefly overviewed above have involved transition and
instabilities in the wake region. As the ReD is increased further, the transition point moves further
upstream and eventually the transition occurs in the shear layers separating from the sides of the
cylinder (Williamson 1996). The shear-layer transition regime (ReD = 1,000-200,000) involves
an increase in base suction coefficient and a decrease in the vortex formation region. In this
regime, the shear layers become unstable, and small-scale vortical structures develop in the shear
layer due to the shear-layer instability, named also as the Kelvin-Helmholtz instability. The
shear-layer instability is principally two dimensional and therefore contributes to the increase of
two-dimensional shear stress (Williamson 1996). Bloor (1964) discovered that the small-scale
vortices associated with the shear-layer instability introduce frequencies scaling with ReD1/2
, and
Prasad and Williamson (1996) showed that these frequencies actually scale with ReD0.87
. In this
regime, the transition to turbulence has not yet occurred in the boundary layer. With an increase
of ReD to a value greater than 200,000, the transition point moves upstream to the boundary
layer. This regime is called the critical transition regime. This critical regime and the regimes
encountered at much higher Reynolds numbers, i.e., supercritical and post-critical regimes, will
not be discussed herein, as the thesis is concerned only with flows in the sub-critical regime.
5
1.3 Literature Review on the Design of an End Plate for Flow past a Cylinder
For experiments involving flow past cylinders, accomplishment of spanwise uniformity in the
wake region is important. Measurements of pressure made by Stansby (1974) downstream of a
cylinder wake showed that the use of plates at both ends of the cylinder attained constant base
pressure coefficient pbC across the cylinder span, unlike the case without end plates. It was,
therefore, postulated that end plates can be used to promote a more uniform wake compared to
the case without them.
Several researchers directed their attention to the size of the end plate and studied its influence
on shedding frequencies in the spanwise wake region. Hot-wire measurements of Gerich and
Eckelmann (1982) showed a decrease in shedding frequency in the vicinity of end plates for
flows in the laminar shedding regime. This spanwise region, near the end plates, was classified
as “the end-plate-affected region”, and outside of this region as “the unaffected region”. The
affected region became longer with increasing plate size. The measurements were made using
both square- and circular- shaped end plates, and the shape of the end plate was determined to
have no significant effect on these regions. The region affected by the end plate is observed to
stretch over longer spanwise distances in the laminar shedding regime compared to that in the
subcritical regime. Stager and Eckelmann (1991) studied effects of an end plate up the ReD =
5,000 in the subcritical regime and observed considerable reduction in the size of the affected-
region (in terms of spanwise length) as the Reynolds number increases. Szepessy (1988) found
the affected region even barely detectable at ReD = 10,000.
To accomplish nearly parallel shedding conditions via end plates, it is necessary to design the
end plate configuration carefully. The distance to the trailing edge of the end plates from the
cylinder axis and the value of the cylinder aspect ratio were shown to have crucial role in the two
dimensionality of the wake by Szepessy and Bearman (1991) through their measurements of
pressure. Their end-plate size had a length of 8D in the streamwise direction and 7D in the
transverse direction, and their ReD range was 800 to 130,000. According to their findings, the
static pressure increases slowly downstream of a bluff body and, if the distance from the bluff
body to the trailing edge of the plate is shorter than the distance of pressure recovery in the wake,
a cross flow into the wake arises. It was, thus, recommended that the end plates be designed with
6
a trailing distance long enough to allow for pressure recovery in the wake. In their experiments,
the distance from the cylinder axis to the trailing edge was 4.5D. This distance was not subject to
optimization as it was large enough to allow the full pressure recovery in the wake. Integrating
the pressure readings attained from pressure taps distributed over the circumference of the
cylinder, Szeppasy and Bearman (1991) calculated the fluctuating lift force. Based on the
premise that higher fluctuating lift correlates with more coherent vortex shedding, the calculated
value was used to compare the coherency of vortex shedding for a range of cylinder aspect ratios
and flow Reynolds numbers. Both aspect ratio and Reynolds number were found to have
significant influence on the vortex shedding. The main finding of importance was that the aspect
ratio had no effect on the fluctuating lift for ratios larger than 6. This was tested for ReD =
16,000.
The correlation of pressure signals computed by Szepessy (1994) along the span of a cylinder led
to interesting insights about the mechanisms of vortex shedding. At high sub-critical Reynolds
numbers, three dimensionalities were shown to arise not only from the end conditions but also
from turbulence in the shear layers. They also revealed that phase drifts in shedding frequency
can occur along the span when vortex shedding is disturbed, accompanied by spanwise pressure
gradients. The phase drift can become large (around 40° to 60°), but further larger drifts are
prevented by a spanwise lock on. It was postulated that real time variations in shedding
frequency could cause spanwise cellular structures of vortex shedding that are slightly out of
phase. A further study on the use of end-plates to promote parallel shedding was conducted by
Szepessy (1993). They investigated the effect of end-plate size on flow uniformity through the
measurements of pressure. The effect of the end plate was found to strongly depend on the
Reynolds number. Flow with lower Reynolds numbers produced larger spanwise variation of
base suction, with a continuous increase of base suction to a peak value at the cylinder mid-span.
Their results also indicated that the trailing edge distance of the end plate from the cylinder
centerline was more important than the leading edge distance of the plate. A minimum distance
of 3.5D was recommended as the trailing edge distance of the end plate from the cylinder
centerline to attain parallel flow conditions. Szepessy (1993) also found that separation bubbles
on the leading edge of the end-plate had no important effect on the development of vortex
shedding. For end plates with short leading edge distances (0.6D) from the cylinder centerline, it
was found that the cylinder aspect ratio had the greatest influence on the shedding conditions.
7
For end plates with small leading edge distances, there were certain aspect ratios at which the
vortex shedding could be attenuated. In addition, the pressure recovery in the wake was
measured and found to be qualitatively similar to the findings of Szepessy and Bearman (1991).
That is, the wake pressure initially recovers quickly, and by the time the flow reaches a
downstream location of x/D ~ 10 the wake recovery gradient is substantially smaller.
Szepessy (1993) summarizes the role end plates play in promoting parallel shedding by
confirming their importance in cutting off interference from disturbed flow regions outside the
plates, such as the wall boundary layers, in addition to aligning the flow along the plate and
suppressing spanwise cross flow resulting from phase drift in the vortex shedding along the span.
The author also states that with a trailing edge distance greater than the wake recirculation
region, end plates play a critical role in damping lateral velocity fluctuations resulting in a less
chaotic shear layer. Szepessy (1993) concludes by investigating the effect of the horse-shoe
vortex in the development of vortex shedding, for Reynolds numbers of 10,000 and 40,000.
Although the circulation of the horse-shoe vortices is not insignificant compared to the
circulation of Von Karman vortices (on the order of 10%), the orientation of the vortices is
orthogonal, and thus, it was suggested the horse-shoe vortices have little impact on the
development of parallel vortex shedding.
1.4 Review of Junction Flow Studies
A seminal work in the study of junction flows was conducted by Baker (1978). In his
experiments, the boundary layer forming along the floor of a low-speed wind tunnel was kept
laminar via the use of suction slots in the tunnel floor. A cylinder was placed in the wind tunnel
and mounted flush to the floor of the test section such that the laminar boundary layer would
encounter the cylinder and form a horse-shoe vortex system. The cylinder was placed at different
streamwise locations in the wind tunnel at various wind tunnel velocities. Smoke visualization
and hot wire measurements were performed to determine the flow physics occurring within the
horse-shoe vortex system. The results confirmed that the separation of the boundary layer
upstream of the cylinder results in the formation of the horse-shoe vortices. This was quantified
on the basis of separation lines viewed by smoke visualization. Measurements of velocity
occurring beneath the vortices showed that the shear stress at this location was very large. This
result intuitively makes sense because scour patterns observed around bridge piers or in the snow
8
around a telephone pole are most likely due to the horse-shoe vortices forming around the
obstacle in question. Depending on the Reynolds number of the flow, multiple horse-shoe
vortices were observed, and their behavior was either steady or unsteady. As the wind tunnel
speed was increased, regular periodicity of the vortices vanished, and they exhibited irregular
formation. Precise classification of regimes with different vortex formation behavior was not
noted by Baker (1978). However, assessment of the hot-wire data revealed the following
categories for the horse-shoe vortices as the flow speed increases:
1. Steady trace with no oscillation
2. A low frequency oscillation (St = 0.26)
3. A high frequency oscillation at St = 0.4 increasing to St = 0.6 for higher ReD
4. An irregular turbulent trace
Baker (1978) tested the effect of vortex shedding on the behavior of the horse-shoe vortex
systems. In these experiments, a splitter plate was used to prevent the formation of regular
Karman vortices. Disrupting the regular shedding of Karman vortices had no impact on the
development of the horse-shoe vortex regimes. Thus, it was concluded that the behavior of
horse-shoe vortex systems depends on the flow physics, but is not related to the Karman vortices.
A comprehensive review performed by Simpson (2001) described the physics of junction flows
emanating from both turbulent and laminar boundary layers. In all types of junction flows, it was
noted that all the primary vortices have the same direction of rotation to the vorticity of the
approach boundary layer, whereas the secondary vortices have opposite direction of rotation.
Simpson (2001) states that because of the identical sense of rotation with the vorticity of the
boundary layer, the primary vortices entrain high speed outer fluid, and in turn, enhance mixing
in the junction region. Hence, a potentially beneficial aspect of horse-shoe vortices is that they
can increase the rate of heat transfer in the junction. The factors that can influence the behavior
of a horse-shoe vortex system include the aspect ratio of the obstacle, free-stream turbulence,
Reynolds number based on the characteristic dimension of the obstacle, and the displacement
thickness of the approach boundary layer. As mentioned above, Baker (1978) showed that the
Karman vortex shedding and the development of the horse-shoe vortex systems are not related.
Simpson (2001) reinforces this finding by stating that the behavior of horse-shoe vortex systems
9
is due to the inherent stability breakdowns within the system, not the shedding in the wake, and
thereby, the mechanisms of horse-shoe vortices and the Karman vortices exhibit a certain
dichotomy in response to changes in the flow.
The strongest factor influencing a horse-shoe vortex system was deemed to be the shape of the
obstacle forming the junction. This is because the pressure gradient, which is responsible for the
separation of the boundary layer, is related to the shape of the obstacle. In particular, the degree
of obstacle bluntness is defined as the most important factor by Simpson (2001). According to
Simpson (2001), in general, the greater the bluntness of an object, the stronger the horse-shoe
vortex formed at the junction. Wei et al. (2008) studied the effect of the cross-sectional shape of
the obstacle on the development of the horse-shoe vortex systems employing flow visualization
and Laser Doppler Velocimetry measurements. A sharper cross-sectional shape, such as a
diamond shape, was found to suppress the strength of the horse-shoe vortices and result in the
positioning of the vortices closer to the obstacle. Wei et al. (2008) concluded that a decrease in
the bluntness of the obstacle, i.e., a weaker adverse pressure gradient, results in less pronounced
horse-shoe vortex strength. This concept was utilized to alter the strength of the horseshoe
vortices by Gupta (1986), where a delta-shaped wedge was inserted at the upstream junction of a
vertical pier. They modified the development of the horseshoe vortex system successfully, which
is postulated to reduce the shear stress occurring near the pier junction. Multiple simultaneous
horse-shoe vortices have been observed by Chou et al. (2000) in their experimental study, where
an obstacle with rectangular cross-section and very large aspect ratio was used.
Unsteady characteristics of laminar junction flows were studied by Thomas (1986). He found
that the frequency of formation and convection of vortices towards the cylinder increases with
Reynolds number, between ReD = 3,000 and 13,000. For the ReD value of 10,000, which is the
value of the ReD employed in the present study, he determined the dimensionless frequency as
approximately St = 0.32.
Kelso and Smits (1995) performed experiments where a transverse jet was used to create a
system of horseshoe vortices instead of a solid obstacle such as a circular cylinder. Depending on
the ReD being tested, the authors witnessed steady, oscillating or coalescing vortex regimes,
whose frequency characteristics compared well with the published findings from authors who
10
used solid obstacles. A result not seen in other experiments was the tendency of the wake to lock
on to the horseshoe vortex frequency, or a sub-harmonic where it becomes slightly out of phase.
11
CHAPTER 2
EXPERIMENTAL SYSTEM AND TECHNIQUES
The present chapter provides an overview of the experimental techniques and the setups used in
the present investigation. Descriptions of the measurement principles, the flow facility, and the
physical specifications of equipment are described in detail.
Three types of measurement techniques were used in the course of the investigation:
• Hot Film Anemometry
• Particle Image Velocimetry
• Volumetric 3-Component Velocimetry
Sections 2.1 to 2 .3 introduce the measurement techniques listed above. Section 2.4 provides
information about the flow facility where all the measurements were performed along with the
characterization data of this facility. Section 2.5 describes the experimental setups and the
models used during the present investigation. The chapter ends with section 2.6, where attention
is directed experiments conducted to search the effect of the leading-edge geometry of an
endplate on flow separation.
2.1 Hot Film Anemometry
Hot film anemometry is a point-based measurement technique that determines the speed of the
flow based on the relationship between the convective heat transfer rate of a film (or wire, if
measuring in air flows) and the velocity of the heating/cooling fluid within which the wire is
immersed. The wire is connected to a circuit containing a Wheatstone bridge. The particular
form of hot film anemometry employed in the present experiments was Constant Temperature
Anemometry (CTA), where the circuit maintains the wire temperature at a constant value by
adjusting the voltage depending on the flow speed. The technique relies on the relationship
between the convective heat transfer rate of the wire and the velocity of the fluid. This
relationship is determined by altering the speed of the fluid to pre-known values and recording
12
the changes in the voltage applied to the wire. This creates a calibration curve from which the
velocity of the fluid can be determined from the voltage of the CTA.
Spatial resolution of the measurements depends on the probe dimensions, which are on the order
of 2 mm in length and 70 µm in diameter. Temporal resolution of the measurements is related to
the response time of the probe, which was approximately 10,000 Hz (in accordance with the
Nyquist criterion, and a sampling frequency of 20,000 Hz). These characteristics make it
possible to resolve small temporal events such as fine-scale turbulence. The CTA used in the
present measurements consisted of a single-wire probe which was capable of measuring only a
single velocity component. Note that one can also find probes with different wire configurations,
which can compute multiple components of velocity and vorticity. A more thorough treatment of
the basic of hot wire anemometry can be found in Brunn (1995).
2.2 Particle Image Velocimetry
Particle Image Velocimetry (PIV) is a non-intrusive measurement technique that was used for
the majority of the experiments done in this study. The technique involves seeding the flow with
tracer particles, illuminating the particles with a laser, and capturing the images of the particles at
two instances in time to compute the velocity over a global flow field of interest. If the time
difference between the capture of the two successive images is small enough, then the
displacements of the particles will be small and the velocity can be computed by the simple
linear relationship:
dt
dXU (2)
In Equation (2), the displacement vector “X” is two dimensional and thus the velocity vector
calculated from “X” is two dimensional. Other Particle Image Velocimetry techniques such as
stereo Particle Image Velocimetry and Tomographic Particle Image Velocimetry use more than
one camera to calculate the out of plane velocity component. For more information, see Raffel et
al (2007). A typical Particle Image Velocimetry system consists of a digital camera, a laser
system, a synchronizer and a computer to operate the synchronizer and acquire the images from
the camera.
13
2.2.1 PIV Exposure Technique
The PIV exposure technique employed in the experiments was “Double Frame/Single
Exposure” in which a single frame is acquired at two instances in time. There are several
exposure techniques including “Single Frame/Double Exposure”, “Single Frame/Multi-
Exposure”, etc. The particular technique chosen by researchers depends on the equipment
available and the particular constraints of the measurement being attempted. A full treatment of
the variety of methods can be found in Raffel et al (2007).
2.2.2 Computation of Particle Displacement
Because the laser pulse timing is set by the user for a given experiment, the only quantity in
Equation (2) required to compute the velocity vector is the displacement of the particles in the
flow. To perform this calculation, the area to be investigated is divided into grids, as shown in
Figure 2.0.
Figure 2.0: Schematic of a PIV light sheet and interrogation grid. The upper portion of the figure describes a
zoomed in region of the entire interrogation region with K horizontal grid points and L vertical grid points.
Particles within the grid can be seen displacing in time.
14
The image intensity is mapped onto the grid, and the particles are located based on their image
intensity profiles, which are usually assumed to have Gaussian characteristics (see Raffel et al.
(2007) for further detail). The statistical displacement of the particles within one grid box is
computed by performing a discrete cross correlation:
),(),(),( ' yjxiIjiIyxRL
Lj
K
KiII
(3)
Where I and I’ are sample intensity values, K and L are the number of locations, and x and y are
image shifts. The image is shifted around the grid and the discrete cross correlation is performed
at each location. The shift that generates the maximum cross correlation peak is determined to be
the particle displacement. Therefore, only one vector is given for each grid unit, and the velocity
is determined in a statistical sense, since the algorithm tracks the greatest average shift of a group
of particles to generate one velocity vector per grid unit. The spatial resolution of PIV is thus the
size of grid spacing used in the experiment. This fact is important because of the spatial
averaging inherent in PIV; if the flow is expected to have important flow features smaller than
one grid unit, the measurement may not adequately represent those features (Raffel et al. 2007).
Therefore, the chosen grid size in a PIV measurement acts as a spatial filter, and important flow
features in the flow below this filter size will not be resolved.
A key assumption of PIV is that tracer particles are fully displaced within the flow being
investigated. A correctly sized particle will be neutrally buoyant, and this assumption is usually
valid. However, flows with regions of extremely high shear, such as shocks, can potentially
cause errors if the velocity gradient across a grid unit is large. A detailed analysis of this
potential error can be found in Raffel et al. (2007).
2.3 Volumetric 3-Component Velocimetry
Volumetric 3-Compenent Velocimetry (V3V) employs the similar concept of velocity vectors
being calculated via particle displacement as Particle Image Velocimetry and, in addition, uses
optical theory to calculate the out of plane velocity component to generate a three dimensional
velocity field within a volume. A thorough explanation of the measurement fundamentals of
Volumetric 3-Component Velocimetry can be found in a series of papers by Pereira and co-
15
workers (Pereira et al. 2000, 2006). The technique is briefly outlined below with a focus on
features that are most relevant to the thesis methodology.
2.3.1 Using Particle Defocus to Calculate Out of Plane Velocity Components
In Particle Image Velocimetry the camera is focused on the plane of measurement (i.e., the laser
sheet); therefore only particles in the laser sheet will be imaged, and the particles cannot move
out of the plane of measurement or they will be lost. As the name of the technique implies,
Volumetric 3-Component Velocimetry makes measurements of velocity over a volume instead
of a plane. The method involves seeding the flow with tracer particles and tracking their path
through the illuminated volume over time. The unique aspect of this measurement technique is
the use of particle de-focus. A measurement volume is chosen, and three apertures (cameras),
that all lie in the same plane, are focused on the rear of the measurement volume as seen in
Figure 2.1.
Figure 2.1: Schematic of a V3V setup. Three cameras are focused on the rear of the measurement volume. A
particle in front of the rear plane (within the measurement volume) will be out of focus. The amount of de-
focus is used to measure the out of plane velocity component.
Thus, particles flowing through the volume will be seen at from slightly different perspectives.
When a given particle is within the focal plane, the overlay of each of the three camera images of
that particle will create only one particle image, due to the fact that each aperture is focused on
16
the same plane. When the particle is outside of the focal plane within this volume, each camera
observes three, slightly offset records of the particle‟s location. This creates a triplet image
(equilateral triangle) of the particle, in which the triplet size represents the distance of the particle
from the focal plane, and the triplet center delimits the actual coordinates of the particle (Peireia
et al. 2006). Thus, a particle located at the front of the measurement volume would appear as a
very large equilateral triangle if each image was overlaid on top of each other due to the aperture
defocus. If at a later time the particle location shifts towards the rear of the volume, then the
triplet would be smaller and the change in triplet size can be related to the out of plane velocity
component.
2.3.2 Tracking the Particles
Particle Image Velocimetry tracks the statistical, average displacement of a group of particles
through a measurement plane. Volumetric 3-Component Velocimetry uses concepts from
Particle Tracking Velocimetry (PTV) to track individual particles through a volume, due to the
fact that Volumetric 3-Component Velocimetry requires the defocus of individual particles to
gather information on the out-of-plane component. The method uses a “Relaxation” algorithm to
match particles between frames. The algorithm estimates a particle‟s location based on the bulk
flow velocity and then calculates the probabilities of matching particles within the estimated
neighborhood. A thorough explanation of these concepts can be found in Peireia et al. (2006).
17
2.4 Flow Facility
Experiments were conducted in a recirculating, free-surface water channel, located at the
University of Toronto Institute for Aerospace Studies. Its schematic is provided in Figure 2.2
Figure 2.2: Schematic of the free-surface water channel, located at the University of Toronto Institute for
Aerospace Studies. Nd:YAG laser and the illumination plane used the PIV measurements are also
incorporated in the schematic.
The channel consists of a 5 m long test section with a 0.68 m x 0.76 m cross section. Flow
conditioning is accomplished through a set of honeycombs and 3 screens upstream of a 6:1
contraction. The maximum attainable speed in the channel (at a water height of 0.67 m) is 0.78
m/s.
18
2.4.1 Characterization of the Flow Facility
The water channel was characterized using hot film anemometry, which revealed the presence of
a low-frequency fluctuation at approximately 0.1 Hz in the velocity signal. Upon further
inspection, it was decided to install temporary covers on the contraction, flow conditioning and
return plenum sections at the top in an attempt to eliminate this undesirable low-frequency
fluctuation in the velocity signal. These covers significantly decreased the spectral amplitude of
the low frequency. After these preliminary experiments, permanent top covers were carefully
manufactured and installed eliminating the free-surface at these sections. Further experiments
showed that the use of these permanent covers completely eliminated the low-frequency signal in
the test section of the channel. A full account of the characterization of the channel flow
characteristics can be found in the internal laboratory report, “Blackmore, Aydin, and Joshi –
The Efficiency of Covers on the Flow Quality of the UTIAS Water Channel”. The results of
the characterization are summarized in the Table 2.0 below.
Turbulence Intensity (o
RMS
U
u'
) < 1% for all channel flow speeds
Flow Uniformity ~ 0.3 %
Maximum Velocity 780 mm/s
Table 2.0: Summary of Water Channel Characterization with the Permanent Top Covers on the Contraction,
Flow Conditioning and the Return Plenum Sections
2.5 Experimental Setups
The present study experimentally examined the use of endplates for flow past a finite-length
cylinder at the sub-critical ReD value of 10,000 via quantitative visualization techniques.
Emphasis was directed to two regions of flow. Firstly, to investigate how the end conditions of a
cylinder affect the spanwise flow uniformity in the near wake, the near-wake flow region along
the cylinder span in the symmetry plane of the wake was studied. Secondly, to investigate the
19
horse-shoe vortex dynamics, the flow region upstream of the junction of the cylinder and a
bounding surface (in the form of either an endplate or the channel floor) was studied. As such,
the experiments can be divided into two parts:
The flow characterization in the near wake of the cylinder along the span at the symmetry
plane; and
The flow characterization in front of the cylinder-wall or cylinder-endplate junction.
This section presents the experimental setups used to perform these measurements.
Particle Image Velocimetry and Volumetric 3-Component Velocimetry were used to measure the
global velocity fields. Details of these techniques were given in Sections 2.2 and 2.3. Both
systems were provided by TSI Inc., and their relevant specifications can be seen below in Table
2.1. A schematic of the systems can be seen below in Figure 2.3.
Component 2D – Particle Image
Velocimetry System
Volumetric 3-Component
Velocimetry System
Camera 2MP Powerview Plus 4MP Powerview Plus (3)
Laser Newwave 200 mJ/Pulse Newwave 200 mJ/Pulse
Synchronizer Model 610035
Model 610035
Frame Grabber 64 bit Frame Grabber 2 x DVR Express CLFC
Software TSI Insight 3G TSI Insight V3V
Table 2.1: Summary of Experimental System Components
20
Figure 2.3: Schematic of the components of the PIV and V3V systems. The V3V and PIV setups are similar
with the exception of the dual CLFC Frame grabbers required to transfer the V3V images to the computer,
and the cameras, which have twice the pixel resolution in the V3V setup. Both systems utilized the same
Nd:YAG laser unit with 200 mJ per pulse laser and the same synchronizer during the experiments.
21
2.5.1 Experimental Setups for the Measurements in the Symmetry Plane of the Near-Wake Region along the Cylinder Span
The flow in the near wake of a circular cylinder with a diameter of 50.8 mm was investigated via
the technique of Particle Image Velocimetry (PIV) for four different end conditions to assess the
performance of a given configuration in promoting spanwise flow uniformity in the near-wake.
Experimental setups for these end conditions are sketched in Figure 2.4. They involve the
following:
1- Cylinder bounded by the channel floor at the bottom and the free surface at the top (in
Figure 2.4, the first experimental sketch from left),
2- Cylinder bounded by an endplate at the bottom and the free surface at the top (in
Figure 2.4, the second sketch from left). The endplate had a sharp leading edge with a
bevel angle of 23.6°.
3- Cylinder bounded by an endplate at the bottom and by the free surface at the top (in
Figure 2.4, the third sketch from left). This endplate arrangement had a super-
elliptical leading edge geometry, designed specifically as outlined in section 2.6.
4- Cylinder bounded by endplates at both ends (in Figure 2.4, the fourth sketch from
left). These endplates had a sharp-leading edge, beveled at an angle of 23.6°.
To perform the PIV measurements, the plane along the cylinder span in the near wake was
illuminated with two short-duration laser-sheet pulses, continuously generated by the double-
pulsed Nd:YAG laser system; and the fluid motion was made visible by seeding the flow with
tracer particles, which had a specific density of 1.08 and a mean diameter of 10 microns. The
images of the flow field were captured by a CCD camera, which was facing the illuminated
region perpendicularly. The field of view (F.O.V.) of these PIV experiments was approximately
6.75D in the spanwise direction and 4.5D in the streamwise direction; yielding a vector
resolution of 0.07D. Images were acquired at 14.5 Hz, giving a temporal resolution of 7.25 Hz
based on the Nyquist criterion. As the entire spanwise region in the near-wake along the cylinder
length could not be acquired with adequate spatial resolution at the same experimental run, the
spanwise region in the near-wake was acquired in two separate experiments, each one with a
field of view (F.O.V) of 6.75D in the spanwise and 4.5D in the streamwise direction. Because
22
the span of the cylinder was slightly different for the various end conditions, for example, the
overlap region was slightly larger in the experiments with two endplates. The entire span of the
cylinder was constructed from these two experiments as shown in Figure 2.4.
The ReD was kept at the sub-critical value of 10,000 for all the experiments. 105 cm downstream
of the test-section entrance, the cylinder was mounted with a vertical orientation and equidistant
from the channel side walls. Without a cylinder at this location, the thickness of the boundary
layer on the channel floor is determined to be 0.25D for a ReD of 10,000. In experiments
involving the use of an endplate, which are described above and in the sketches of Figure 2.4,
endplates were placed 1.25D above the channel floor, i.e., well above the boundary layer
forming along the channel wall. All endplates had 7.5D length in the streamwise direction and
12D width in the lateral direction with respect to the approach flow. The distance between the
leading-edge of the endplate from the cylinder centerline (D
L ) was varied for each
experimental arrangement that involved the use of an endplate (second, third and fourth sketches
from left in Figure 2.4) to test its effect of on the spanwise uniformity of the near wake.
The measurements were taken with 200 samples and time averaged results of these
measurements are presented in Chapter 3.
2.5.2 Experimental Setups for the Measurements in the Upstream of Cylinder-Wall and Cylinder-Endplate Junctions
The flow in front of the junction for a cylinder with the channel wall or an end plate was studied
in three different experimental configurations, sketches of which are given in Figure 2.5. These
configurations were as follows:
Cylinder-wall junction: The cylinder was mounted flush to the channel (see the first
sketch from top in Figure 2.5)
Cylinder-endplate junction, where the endplate had sharp leading-edge geometry with a
bevel angle of 23.6° to the horizontal. In this configuration, the cylinder was mounted on
the end plate and the air-water type free-surface bounded the cylinder at the top (see the
second sketch from top in Figure 2.5).
23
Cylinder-endplate junction, where the endplate had an elliptical leading-edge geometry.
In this configuration, the cylinder was bounded by the end plate at the bottom and the
free-surface at the top ( see the third sketch from top in Figure 2.5),
Similar to the experiments outlined in section 2.5.1 above, the cylinder in these experiments had
a diameter of 50.8 mm and was placed 105 cm downstream of the test section entrance of the
water channel. At this location, at a ReD of 10,000, the boundary layer was determined to be
laminar with approximately 0.25D thickness. The physical dimensions of both endplates were
7.5D long in the streamwise direction, and 12D wide in the lateral direction. In cylinder-endplate
experiments, the endplate was mounted on brass runners which made the height of the endplate
from the channel floor to be 1.25D. This is well above the thickness of the boundary layer
forming along the channel floor.
In all cases where Particle Image Velocimetry (PIV) was used to investigate the flow upstream
of the junction of the cylinder, the field of view was held constant at approximately 0.8D in the
spanwise direction by 1D in the streamwise direction. This field of view provided a vector
resolution of approximately 0.006D. Images were acquired at a rate of 14.5 Hz, enabling the
experiments to resolve time dependent events up to 7.25 Hz based on the Nyquist sampling
criterion.
Horseshoe vortex dynamics upstream of the cylinder mounted flush to the wall were investigated
further through the use of the Volumetric 3-Component Velocimetry (V3V) technique. For
clarity, a schematic of the V3V setup used in these measurements is given in Figure 2.6. The
spatial resolution of velocity vectors in V3V is determined differently compared to that of PIV,
as the former method relies on a particle-tracking technique. The number of vectors in a given
volume could theoretically be determined as the number of particles in that volume, however,
practically this is not the case. The algorithm used in the V3V technique first identifies the
particles and then applies a relaxation probability search criteria to identify the pairs in the next
frame, thereby generating the velocity vectors. However, after reaching a particle count of
approximately 120,000, the yield in vectors decreases. Therefore, increasing the vector
resolution requires addition of more particles to the flow, so that approximately 120,000 particles
can be identified, and simultaneously, decreasing the volume being investigated, so that the ratio
of particles to volume increases. The vectors identified by this relaxation method matching stage
24
are called “random vectors” by TSI and must be interpolated onto a rectangular Cartesian grid.
This interpolation is the final “control” on vector resolution. The final grid spacing chosen is
decided by an initial grid size and an overlap percentage, much like the conventional Particle
Image Velocimetry technique. Decreasing the starting grid size and increasing the overlap
percentage yields higher vector resolution. However, the many interpolations required to perform
this operation typically increase noise in the data, and a tradeoff must be made between the
vector resolution and the data noise. In the cylinder-wall experiments, it was determined that a
starting Cartesian grid size of 0.16D (8 mm) with a 75% overlap allowed for the greatest vector
resolution with acceptable levels of noise; this meant the final vector resolution was 0.04D (2
mm). The Volumetric -3-Component Velocimetry system acquires data at 7.5 Hz. As a result,
frequencies up to 3.25 Hz could be resolved, based on the Nyquist sampling criterion.
25
Figure 2.4: Sketch of experimental setups for the measurements conducted in the symmetry plane of the near wake along the cylinder span. Four different end conditions
were tested, each with several different cylinder positions measured from the leading edge of the endplate. In the table above, this distance is represented by λ=L/D and all
the values tested for each experiment are listed. The field of view was kept constant during each measurement, however, the cylinder span changed depending on the end
condition, which resulted in different overlap sizes for each end condition. The top and bottom portions of the flow field were captured separately and merged, as shown in
the sketches above, to construct the images of entire near wake region along the span (S) of the cylinder. The ReD, in all cases, was held constant at a value of 10,000.
26
Figure 2.5: Experimental setups used in the cylinder-wall and cylinder-endplate junction experiments. For the experiments
involving the use of an endplate, the cylinder was bounded by the endplate at the bottom and by the free surface at the top.
The cylinder position on the endplate was varied by changing the distance to the leading edge, represented by λ=L/D, for
values of 1, 2.5 and 5. The leading-edge geometry of the second and third experiments was different. A sharp leading-edge
shape was used in the second one. This was determined to produce significant upstream separation. An elliptical leading-edge
was designed for the third experiments from top and found to eliminate flow separation at the tip of the plate. The field of
view in all experiments was approximately 1D in the streamwise and 0.8D in the spanwise direction.
27
Figure 2.6: V3V setup used to study the junction flow behavior of the cylinder-wall arrangement. The Reynolds number based was 10,000. The streamwise position of the
cylinder was 105 cm downstream of the test section entrance, which gave approximately 25 mm (0.5D) of upstream junction region imaged within the volume. The volume
height was reduced to 0.98D in order to increase vector resolution. The final vector resolution, based on a starting grid of 0.15D with 75% overlap was 0.04D. This resolution
made it possible to identify the primary horseshoe vortex.
28
2.6 Significance of the Leading-Edge Geometry of the Endplates
Experiments were performed to quantify whether or not flow separation was present at the tip of
the endplate with the sharp, beveled leading-edge geometry. Details of this endplate arrangement
were given in sections 2.5.1 and 2.5.2. In Figure 2.7, the time-averaged streamline <> patterns
are superposed over the time-averaged contour patterns of normalized vorticity <>D/Uo around
the tip of the plate when no cylinder was used, that is, the plate was placed into the flow by itself,
at ReD of 10,000. This figure clearly demonstrates flow separation at the leading edge. In order to
avoid repetition of images, only the pattern for the case with no cylinder use is presented here.
However, our tests revealed the presence of significant unsteady flow separation at the tip of the
plate also for the cases where the cylinder was placed on the same plate at various positions (λ=
1, 2.5, 5).
Figure 2.7: On the left hand-side image, contours of time-averaged normalized vorticity <>D/Uo are
superposed over the time-averaged streamlines, demonstrating significant flow separation for flow past the
plate with sharp leading-edge geometry. The plate is exposed to flow at ReD of 10,000 and no cylinder is
placed in the flow. The right-hand side sketch shows the PIV field of view.
Flow separation occurs because the stagnation point does not rest perfectly along the leading
edge of the endplate. This is because the free stream flow beneath the endplate travels faster, due
to the flow being constrained between the bottom of the endplate and the developing channel
boundary layer. The high speed flow beneath the endplate causes a lower pressure region which
moves the stagnation point below the leading edge. The sharp leading edge will, thus, act as a
29
forced separation point. In flat plate boundary layer experiments, the plate is usually designed
with a trailing-edge flap, in which the angle can be varied relative to the channel floor to alter the
pressure and move the stagnation point, until the desired location is achieved (Tropea et al.2007).
In cylinder flow studies, it is not possible to use a trailing-edge flap because researchers are often
interested in the near wake of the cylinder, and the trailing edge would interfere with the wake
mechanics.
The presence of flow separation was confirmed experimentally through PIV measurements for
the plate with the sharp leading edge geometry, and measurements indicated that the flow
separation was sensitive to cylinder position. The separation bubble was seen to monotonically
increase in relation to λ. It was decided that a new endplate should be constructed with a
specially designed leading edge that can avoid flow separation at the tip of the plate. A super-
elliptical shape was adopted for the leading edge of the plate following the research of
Narasimha and Prasad (1994). These authors modified various parameters and computationally
tested a number of experimental conditions to measure the effect of the leading edge shape on
the development of the laminar boundary layer along the leading edge of a flat plate. They found
that a leading edge based on the equation of a cubic super-ellipse with an aspect ratio of 6 or
greater produces the best results. Accordingly, Equation (4) was chosen to describe the shape of
the leading edge:
n
n
a
xaby
/1
1
(4)
Figure 2.8: Schematic showing the coordinate frame for the elliptical leading edge design in Equation (3)
where ‘a’ and ‘b’ are the major and minor axes of the ellipse chosen.
30
A super-elliptical nose with an aspect ratio of a/b = 6 was chosen for our experiments as the new
plate leading-edge shape. Figure 2.9 compares the time-averaged flow patterns for the plate with
sharp leading edge and the plate with the new super-elliptical nose. It is clear from the time-
averaged streamline <> patterns that this new super-elliptical nose design successfully
eliminated flow separation at the leading edge of the plate.
31
Figure 2.9: Superposition of the time-averaged normalized vorticity <>D/Uo and the time-averaged streamline patterns for the plates with sharp and
super-elliptical leading edges. The results for the sharp leading-edge design are shown in the left frame of the figure, which demonstrates significant
separation. The plate with super-elliptical nose, shown in the right frame, successfully eliminates the separation. The field of view in these measurements
was approximately 1D in the streamwise and 0.8D in the spanwise direction, and the vector resolution was approximately 0.006D.
32
CHAPTER 3
SPANWISE UNIFORMITY OF THE NEAR-WAKE OF A CYLINDER: SIGNIFICANCE OF THE ENDPLATE CONFIGURATION
One of the major objectives of this study was to determine the effect of various endplate
configurations on the spanwise uniformity of subcritical flow past a cylinder in the near wake
region. To this end, a technique of Particle Image Velocimetry was used to characterize the
unsteady and time-averaged flow features in the symmetry plane of the near-wake region along
the cylinder span. This chapter of the thesis focuses on these measurements and discusses the
impact of end-plate leading edge geometry, cylinder end conditions, and cylinder position on
two-dimensionality of the flow in the near wake.
Throughout the entire investigation, ReD had a value of 10,000, which was produced through a
free stream velocity of about 200 mm/s on a cylinder with a diameter of D = 50.8 mm.
Depending on the laboratory conditions on the day of the experimentation, the temperature of
the flow was determined to vary between 17 and 23 °C. The flow speed was slightly adjusted for
experiments conducted on different laboratory conditions (on a different day with a difference in
the ambient room temperature) to accommodate these temperature changes to get the desired
ReD.
Measurements were conducted on experimental setups with four different end conditions, as
discussed in Section 2.5.1 and summarized via the sketches of Figure 2.4. In these setups, the
ends of the cylinder involved the following boundaries:
1. Channel floor and the free surface (no end plate): In this configuration, the cylinder in
fluid flow was mounted flush to the channel floor, and was bounded by the free surface at
the top (in Figure 2.4, the first experimental sketch from left).
33
2. Endplate with sharp leading-edge geometry (SLE) and the free surface: This
configuration involved a cylinder bounded by an endplate with sharp leading edge at the
bottom and the free surface at the top.
3. Endplate with super-elliptical leading-edge geometry (ELE) and the free surface: The
cylinder, in this setup, was bounded by an endplate having an elliptical leading edge at
the bottom and the free surface at the top.
4. Two endplates with sharp leading-edge geometry: The cylinder was bounded, at both
ends, by endplates having sharp leading-edge geometry.
The arrangement with no endplate was chosen as a “basis” case, according to which a
comparison of the effectiveness of endplates in promoting spanwise uniformity in the near wake
could be made for a given experimental arrangement. For experiments involving the use of an
endplate, the leading-edge distance of the endplate from the cylinder center, denoted as L, was
varied to evaluate the significance of D
L for the promotion of uniformity in the spanwise
near-wake region, as previously indicated in Section 2.5.1.
To capture the near-wake region along the entire cylinder span, two separate PIV experimental
runs were performed; one was covering the upper portion and the other was covering the lower
portion of this region with some overlap between the two regions (see Figure 2.4 for
clarification). The effective grid region covered the total cylinder span region, and the near-wake
region measurement length was 4.5D, with a spatial grid resolution of 0.06D. Through a
continuous PIV record, a sequence of 200 image pairs was acquired at a rate of 14.5 frame pairs
per second. To determine all the time-averaged and spectral characteristics of the flow, 200
snaphots of the flow, determined through PIV, were used.
The first section of this chapter presents and discusses the contour patterns of time-averaged
streamwise velocity component <u>/Uo in the near-wake of the cylinder for all four end
conditions that were tested in the present investigation. The contours of constant streamwise
velocity <u>/Uo show a sharply definable “demarcation line” located between the positive and
negative streamwise velocity levels. This line represents the border of the recirculation bubble,
i.e., the formation length, along the span of the cylinder. In a perfectly two-dimensional flow,
34
time-averaged plots of the demarcation line would be completely parallel to the cylinder. In
qualitative flow visualization via the injection of dye particles into the flow, vortex filaments are
visualized to evaluate the spanwise flow uniformity in the near-wake, as seen in the flow
visualizations of Williamson (1989). Because Particle Image Velocimetry is a quantitative flow
visualization tool, the orientation of the time-averaged demarcation line is particularly suitable as
a quantitative indicator of the degree of spanwise uniformity in the near wake. Let <Lu/D>
represent the time-averaged normalized distance of the demarcation line from the base of the
cylinder, i.e., the time-averaged length of the recirculation zone or in other words time-averaged
formation length, at a given spanwise location. The value of <Lu/D> was evaluated over the
spanwise region with a spatial resolution of 0.06D. Let <Lu/D>AVG indicate the spatially
averaged value of <Lu/D>, and <Lu/D>RMS indicate the root-mean-square deviation from
<Lu/D>AVG. The following equation can be used to quantitatively indicate a measure of the
degree of parallelism of the near wake to the span of the cylinder, and thereby the degree of
spanwise uniformity:
AVGU
RMSU
DL
DL
/
/ (5)
The third section of this chapter summarizes the effectiveness of different end conditions based
on the value calculated from Equation (5) and discusses the implications of these results on the
design of endplates to promote nearly parallel shedding conditions in the near wake.
3.1 The Near Wake along the Span of a Cylinder with Various End Conditions: Patterns of Time-Averaged Streamwise Velocity
3.1.1 No Endplate
As indicated above, experiments were conducted on a circular cylinder bounded by the channel
floor at the bottom and the free surface at the top (in Figure 2.4, the first experimental sketch
from left). Results are plotted in Figure 3.0, which shows the time-averaged contour patterns of
normalized streamwise velocity <u>/Uo and normalized spanwise velocity <v>/Uo. White boxes
were used to remove the contours from the regions close to the free surface and the solid cylinder
boundary, where laser light reflection occurred, as well as to remove the discontinuous contours
35
from the vicinity where two PIV images (the top and bottom halves of the spanwise region) were
merged. In the contour plots, negative values are represented by dashed lines and the positive
values by solid lines.
The <u>/Uo contours in Figure 3.0 show that even without the use of endplates, the demarcation
line seems qualitatively parallel over the span of the cylinder with the exception of a slight
decrease of the time-averaged length of the recirculation bubble <Lu/D> towards the free
surface. The value of AVGU
RMSU
DL
DL
/
/is calculated as approximately 9%.
3.1.2 Cylinder Bounded by the Endplate with Sharp Leading Edge (SLE) and the Free Surface
Time-averaged contour plots of streamwise velocity <u>/Uo in the near wake of a cylinder
mounted on an endplate with a sharp leading edge and bounded by the free surface on the top are
shown in Figure 3.1 for λ values ranging from 0.5 to 3.0 and in Figure 3.2 for λ values ranging
from 3.5 to 7.0. From these results, it is clear that the leading-edge distance of the endplate
significantly influences the spanwise uniformity of the near wake of the cylinder. When the
cylinder was placed very close to the leading edge of the endplate (λ = 0.5, 1), the time-averaged
length of the recirculation bubble <Lu/D> varies substantially along the span of the cylinder
apparent from the inclination of the demarcation line. This result could potentially be due to the
fact that the leading edge of the endplate was not long enough to straighten the oncoming flow,
an important characteristic noted previously by Szepessy (1993). For λ = 0.5 and 1, the patterns
of <u>/Uo show that the recirculation region decreases significantly towards the free surface. A
similar observation was also noted in the case where no endplate was used, however, the change
of recirculation-region length over the entire span (from bottom to top) is significant herein.
Among the intermediate values of leading-edge distances (λ = 2 – 4), the parallelism of the
demarcation line to the cylinder appears somewhat improved for some λ values. Visual
inspection of the patterns in Figures 3.1 and 3.2 reveals that the best λ value (for the end
condition discussed in the present section) was λ = 2.5. This value coincides with the range of
optimal values described by Stansby (1974) and Szepessy (1993). Inspection of the demarcation
line along the entire span also showed significant changes in the length of the recirculation
region, the most significant example being the case of λ=4. The bottom half of the cylinder-span
36
exhibited a very large recirculation bubble; however, the top half showed a major decrease of the
length of the recirculation region towards the free surface.
An endplate with a large leading-edge distance (λ = 5 -7) produced interesting results. For the
case where λ =5, the top half of the recirculation region exhibits a marked decrease in
recirculation length. When the λ=6 case is considered, the inspection of the demarcat ion line
along the span suggests that it is one of the most two-dimensional near-wake flows in this series
of experiments. This is counterintuitive and would seem to differ in comparison with results from
other researchers because short trailing edges have been shown to promote three-dimensionality
in the near wake via spanwise pressure gradients (Szepessy 1993). In the following subsequent
section, where contour patterns of spanwise velocity are compared for the range of λ values
considered here, significant levels of upward-oriented spanwise flow from the trailing edge of
the plate in the near wake will be revealed for λ = 6. The reason of the relatively improved
situation in terms of the spanwise two-dimensionality of the demarcation line for λ = 6 may be
related to the bounding of the recirculation bubble with this upstream oriented flow.
Nevertheless, the presence of an upstream oriented flow is already an introduction of further
three-dimensionality into the flow. The case with λ = 7 corresponds to the case with no trailing
edge at all. The <u>/Uo contour pattern, presented in Figure 3.2 for this case, clearly indicates
that the recirculation bubble is affected significantly by the flow passing beneath the endplate
and hence λ = 7 produces the worst of all λ values in terms of spanwise uniformity in the near
wake.
3.1.3 Cylinder Bounded by the Endplate with Elliptical Leading Edge (ELE) and the Free Surface
Time-averaged contour patterns of streamwise velocity <u>/Uo for a cylinder bounded by an
endplate having an elliptical leading edge at the bottom and the free surface at the top are
provided in Figures 3.3 and 3.4 for a range of λ values. It can be seen that the demarcation line,
generally, shows a discernable improvement in terms of the parallelism to the cylinder over
various λ values, when compared to the case with the use of an endplate having sharp leading
edge as well as to the case with no endplate.
In terms of the general shape of the demarcation lines, the case of a small leading-edge value
(λ=1.5), for the use of an endplate with elliptical leading edge, show a near wake qualitatively
37
similar to the case where a sharp leading edge was used, that is, the length of the recirculation
decreases towards the free surface,, although the difference between the top and the bottom half
of the spanwise region is not as significant as the situation where a sharp leading edge was used.
For the intermediate leading-edge distances (λ=2-4), <u>/Uo contour patterns in Figures 3.3 and
3.4 qualitatively demonstrate excellent demarcation line profiles, in terms of the general
parallelism to the cylinder span. These profiles also appear significantly improved, towards the
promotion of parallelism in the near wake region, compared to the demarcation line profiles of
the sharp leading edge geometry. The decrease of the length of the recirculation region towards
the top end of the cylinder, which was also seen in the sharp leading-edge geometry experiments,
is seen in the present end condition only for λ=2.5 and 3.5.
Careful examination of the demarcation lines for both small and intermediate leading-edge
distances (λ=1.5-3.5) reveals, in the close vicinity of the endplate over a small distance normal to
the endplate, rapid extension in the size of the recirculation bubble as the distance normal to the
endplate decreases. Such a rapidly altered recirculation region was not present for the case where
an endplate with sharp leading edge was used (compare with the patterns in Figure 3.1). This
subtle discrepancy between endplates with different leading-edge geometries suggest that this
region near the endplate could be significantly influenced by the separation or no-separation of
the approach flow at the leading edge of the endplate.
The contour patterns of streamwise velocity for larger leading-edge distances of λ = 5 and 6
show recirculation regions which are larger at the base of the cylinder near the endplate, with the
recirculation region becoming slowly thinner near the free surface. The case at λ=7 shows similar
demarcation line profile compared to the corresponding case where one endplate with a sharp
leading-edge geometry was used at λ=7. That is, the recirculation region is distorted significantly
due to the large spanwise flow present from the trailing edge of the plate.
3.1.4 Cylinder Bounded by Two Endplates with Sharp Leading Edges
Three λ values (2, 2.5, 3) were tested using the cylinder bounded by two endplates, each with a
sharp leading-edge geometry beveled at an angle of 23.6°. In Figure 3.9, the contour plots of
time-averaged streamwise velocity <u>/Uo are given. Qualitatively speaking, these plots show a
demarcation line that is nearly parallel to the cylinder axis for all three leading-edge distances
38
tested here. Compared to the other end conditions, presented in the preceding, the use of two
endplates with sharp leading edges produced the best situation in achieving parallelism of the
demarcation line to the cylinder in the spanwise near wake region. The time-averaged plots of
streamwise velocity in Figure 3.5 also show that the recirculation region is more consistent along
the span compared to the case with one endplate with sharp leading edge, given in section 3.1.2,
as a pronounced decrease in the length of the recirculation region is not observed towards the
free surface when two endplates bound the cylinder.
3.2 Global Autospectral Density of Streamwise Velocity in the Near Wake
Global contour patterns of the autospectral density Su(f) of the streamwise velocity component
for all endplate arrangements tested in the present investigation are provided in Figures 3.6 to
3.10 at two dimensionless frequencies. Either one of these frequencies were determined to be
predominant for a given point in the global near-wake field in pointwise spectral analyses. The
difference in these two frequencies was due to the frequency resolution value, which is
dependent on the image pairs acquired in a continuous PIV data acquisition sequence, and the
PIV image acquisition rate, which for the present investigation, is:
∆f =1/(total number of image pairs× acquisition rate) = 1/(200×(1/14.5))=0.0725Hz
Consequently, the resolution of the (dimensionless frequency) Strouhal number is:
∆S =(∆f)D/Uo = 0.0725×50.8/200 ≅ 0.018
Therefore, we achieved either 0.196 or 0.214 as the predominant Strouhal number in the domain.
The contours of constant amplitudes of Su(f1) at S1 = 0.196 are presented on the left-hand side
column and the contours of constant amplitudes of Su(f2) at S2 = 0.214 on the right hand side for
several λ values in Figures 3.6 to 3.10.
Overall assessment of Su(f1) and Su(f2) for the cylinder with an endplate having both the sharp
leading-edge and the elliptical leading-edge geometries in Figures 3.6 and 3.7 reveals generally a
decreased autospectral amplitude at the peak frequency near the endplate for all λ values. This
attenuation in Karman frequency amplitude (defined as either f1 or f2 due to frequency resolution)
near the endplate is probably due to the spanwise velocity component which is thought to
39
originate from the base vortex as explained in the preceding. For the use of two endplates with
sharp leading edge geometries, combined consideration of Su(f1) and Su(f2) levels in Figure 3.10
indicate a continuous presence of the Karman shedding amplitudes along the span of the cylinder
for λ = 2 and 2.5. However, a significant decrease in autospectral amplitudes was found in the
bottom half of the near-wake for λ=3. Findings are not completely explainable when compared
with the corresponding profiles of the demarcation line. This might suggest the need for a larger
number of flow samples to increase the convergence of the statistical and spectral analyses.
3.3 Summary and Results of Measurements in the Near Wake: Demarcation Line Factor
Equation (5) yielded a factor, which we name as “demarcation line factor”, calculation of which
gave a quantitative basis of comparison for spanwise near-wake uniformity for different cylinder
end conditions. Figure 3.11 displays two graphs and a table. The table therein displays the list of
different end conditions of the cylinder (e.g., case 2 represents cylinder bounded by the endplate
having elliptical leading-edge geometry at the bottom and the free surface at the top, case 1
represents the cylinder bounded by the endplate having sharp leading-edge geometry at the
bottom and the free surface at the top, etc). The uppermost graph displays the demarcation line
factor for varying values of λ for the cases displayed in the table when 50% of the span is
considered in the calculation of this factor. The straight line in this graph represents the
demarcation line factor when no endplates were used. A clear advantage of using elliptical
leading edge geometry over sharp leading edge geometry is apparent in the plot. Furthermore,
the use of two endplates with sharp leading edge geometries displays a definite improvement
over the use of one endplate, having either sharp or elliptical leading-edge geometry. The results
show a curve with optimal leading edge distance values around ranging around λ = 2 to 3.
Similar λ values have been seen to produce spanwise uniformity in the near wake for endplate
arrangements of Stansby (1974) and Szepessy (1993). For the optimum leading edge distance of
λ = 2.5, the variation of the demarcation factor along the span of the cylinder is shown in the
bottom graph in Figure 3.12. For comparison, this “optimum” λ value of 2.5 was chosen to
demonstrate the consistency of the demarcation line along the span of the cylinder for each
endplate arrangement. The calculation of the demarcation line factor in terms of cylinder span
was performed with regions very close to the endplate, merged regions or areas of large
reflection omitted. In general, this meant that approximately ninety percent of the span was used
40
in the calculation. The results show a clear advantage in using two endplates. The bottom graph
demonstrates that the endplate appears to lose its effect in the near-wake with increasing distance
from the endplate when only one endplate is used at the bottom end of the cylinder and the free
surface is present at the top. This is not the case when two endplates were used; that is, the
demarcation line factor is constant over the entire span, which would indicate that the endplates
promote uniformity along the span. Based on these results, the optimum configuration was
decided to be an arrangement that employed two endplates with sharp leading edges with a
cylinder placed λ = 2.5D downstream of the leading edge. The demarcation line factor in this
case was calculated to be on the order of 2%.
41
.
Figure 3.0: Contour patterns of time-averaged streamwise velocity <u>/Uo and spanwise velocity<v>/Uo components in the near-wake of the circular
cylinder without the use of endplates. White rectangular boxes are used to remove the contours from regions that are close to the free surface and the solid
cylinder boundary, where considerable laser light reflection was present, and to remove the discontinuous contours in from the mid-span vicinity, where
the two PIV images (the top and bottom halves of the near wake) were merged. Negative and positive <u>/Uo are represented by dashed and solid lines
respectively
42
Figure 3.1: Contours of time-averaged streamwise velocity <u>/Uo in the near-wake of the cylinder bounded by a single endplate with sharp leading edge
for λ = 0.5, 1, 2, 2.5, and 3.0. White rectangular boxes are used to remove the contours from regions that are close to the free surface and the solid cylinder
boundary, where considerable laser light reflection was present, and to remove the discontinuous contours in from the mid-span vicinity, where the two
PIV images (the top and bottom halves of the near wake) were merged. Negative and positive <u>/Uo are represented by dashed and solid lines respectively.
43
Figure 3.2: Time-averaged contours of streamwise velocity <u>/Uo in the near-wake of the cylinder bounded by a single endplate with sharp leading edge
for λ=3.5, 4, 5, 6, and 7. White rectangular boxes are used to remove the contours from regions that are close to the free surface and the solid cylinder
boundary, where considerable laser light reflection was present, and to remove the discontinuous contours in from the mid-span vicinity, where two PIV
images (the top and bottom halves of the near wake) were merged. Negative and positive <u>/Uo are represented by dashed and solid lines respectively.
44
Figure 3.3: Time-averaged contours of normalized streamwise velocity component <u>/Uo in the near wake of the cylinder bounded by a single elliptical
endplate at the bottom and by the free surface at the top for λ=1.5, 2, 2.5. Solid lines indicate positive streamwise velocity, and dashed lines indicate negative
streamwise velocity. White rectangular boxes are used to remove the contours from regions that are close to the free surface and the solid cylinder, and
where two PIV images were merged at the mid span.
45
Figure 3.4: Time-averaged contours patterns of normalized streamwise velocity component <u>/Uo in the near-wake of a cylinder bounded by a single
elliptical endplate at the bottom and by the free surface at the top for λ=3.5, 4, 5, 6, 7. Solid Lines indicate positive streamwise velocity, and dashed lines
indicate negative streamwise velocity. White rectangular boxes are used to remove the contours from regions that are close to the free surface and the solid
cylinder boundary, where considerable laser light reflection was present, and to remove the discontinuous contours in from the mid-span vicinity, where the
two PIV images (the top and bottom halves of the near-wake) were merged.
46
Figure 3.5: Contour plots of time-averaged normalized streamwise velocity <u>/Uo in the near wake for λ=2, 2.5, and 3. The cylinder was bounded at both
ends by the endplates having sharp leading-edge geometry. Solid lines indicate positive streamwise velocity, and dashed lines indicate negative streamwise
velocity. White rectangular boxes are used to remove the contours from regions that are close to the free surface and the solid cylinder boundary, where
considerable laser reflections occurred, and where two PIV images were merged at the mid span.
47
Figure 3.6: Global contour patterns of the autospectral density Su(f) of the streamwise velocity component in the near-wake of the cylinder bounded by a
water channel floor at the bottom and the free surface at the top. As a result of a Strouhal number resolution of 0.018, Su(f) contours are defined at two
values of St = 0.196 & 0.214.
48
Figure 3.7: Global contour patterns of the autospectral density Su(f) of the streamwise velocity component in
the near-wake of the cylinder bounded by an endplate having sharp leading edge at the bottom and the free
surface at the top. Leading edge distances are λ=0.5, 1, 2, 2.5, 3. As a result of a Strouhal number resolution of
0.018, Su(f) contours are defined at two values of St = 0.196 & 0.214.
49
Figure 3.8: Global contour patterns of the autospectral density Su(f) of the streamwise velocity component in
the near-wake of the cylinder bounded by an end plate having sharp leading edge at the bottom and the free
surface at the top. Leading edge distance are λ=3.5, 4, 5, 6, 7. As a result of a Strouhal number resolution of
0.018, Su(f) contours are defined at two values of St = 0.196 & 0.214.
50
Figure 3.9: Global contour patterns of the autospectral density Su(f) of the streamwise velocity component in
the near-wake of a cylinder bounded by an endplate having elliptical leading edge at the bottom and the free
surface at the top. Leading edge distance are λ=1.5, 2, 2.5, 3.5, 4. As a result of a Strouhal number resolution
of 0.018, Su(f) contours are defined at two values of St = 0.196 & 0.214.
51
Figure 3.10: Global contour patterns of the autospectral density Su(f) of the streamwise velocity component in
the near-wake of the cylinder bounded by an endplate having sharp leading edge at the bottom and the free
surface at the top. Leading edge distances are λ=5, 6, 7. As a result of a Strouhal number resolution of 0.018,
Su(f) contours are defined at two values of St = 0.196 & 0.214..
52
Figure 3.11: Global contour patterns of the autospectral density Su(f) of the streamwise velocity component in
the near-wake of a cylinder bounded by two endplates having sharp leading edge geometry. Leading edge
distances are λ=0.5, 1, 2, 2.5, 3. As a result of a Strouhal number resolution of 0.018, Su(f) contours are
defined at two values of St = 0.196 & 0.214.
53
Figure 3.12: Variation of the demarcation line factor with the leading-edge distance λ of the endplate for all
experimental arrangements considered in the present investigation, and the variation of the demarcation line
factor along the span of the cylinder for a fixed value of λ=2.5. The top graph shows the demarcation line
factor for each λ value, tested in every end plate arrangement, with 50% of the span used in the calculation.
The solid line denotes the value when no endplate is used, which is the basis case. The solid circle represents
the case where a single endplate with sharp leading edge is used, the solid triangle represents the case where a
single elliptical endplate is used and, the open circle shows the results when two sharp leading edge endplates
are used. The results demonstrate the clear advantage of using two endplates, as seen for various λ values in
the top graph. When 50% of the span is used in the calculation for the optimum λ=2.5, it can be seen in the
bottom graph that two endplates provide a uniform demarcation line factor along the cylinder span.
54
CHAPTER 4
HORSESHOE VORTEX DYNAMICS AT THE JUNCTION REGION
One of the major objectives of this study was to determine the effect of various endplate
configurations on the spanwise uniformity of subcritical flow past a cylinder in the near wake
region. Design of proper endplates, according to the specific experimental conditions, is a
significant issue that concerns experimentalists whose research involve flow past cylinders.
Endplates are used to promote parallel shedding in the near wake, and it is also believed that
these endplates suppress the intensity of the horseshoe vortices formed at the base of the
cylinder. It is postulated that this is achieved by growing a new, thinner boundary layer at the tip
of the endplate, thus creating a smaller horseshoe vortex system upstream of the junction. In
order to explore the fluid dynamics of horseshoe vortex systems in flow past cylinders, subjected
to various end conditions, quantitative flow visualization was conducted, in the present, upstream
of a cylinder-channel wall junction and cylinder-endplate junctions, with endplates having sharp
and elliptical leading-edge geometries.
This chapter presents and discusses the results of different experimental arrangements on the
dynamics of the horseshoe vortex system. Particle Image Velocimetry was used in all the
investigations to characterize the velocity field upstream of the cylinder-wall, and cylinder-
endplate junctions in the spanwise region equidistant from the channel sidewalls (centerline of
the cylinder and endplate). The field of view was kept constant for all experiments, and measured
approximately 0.8D in the spanwise direction and 1.0D in the streamwise direction, which
resulted in a vector resolution of 0.006D (0.3 mm). The Volumetric 3-Component Velocimetry
(V3V) measurements were also performed at the junction of the cylinder and channel floor to
provide additional insight into the horseshoe vortex dynamics. The vector resolution of the V3V
measurements, discussed in Section 2.5.2, was 0.04D (2 mm). This was spatially less resolved
than the Particle Image Velocimetry measurements, and therefore certain small-scale flow
features could not be measured with the same accuracy as the two-dimensional measurements.
55
Nevertheless, V3V gave the three-dimensional look into the global flow field. The ReD was kept
constant during all the experiments at a value of 10,000. The cylinder diameter was also constant
at a value of 50.8 mm, which required a free stream velocity of approximately 200 mm/s,
depending on the daily temperature of the fluid, which was determined to vary at most between
17-23°C.
In summary there were three (3) experimental arrangements tested:
Particle Image Velocimetry measurements in the upstream region of the cylinder-channel
wall junction. Volumetric 3-Component Velocimetry measurements of the upstream and
downstream region in the cylinder-wall junction.
Particle Image Velocimetry measurements at the upstream region of the cylinder-endplate
junction, where the endplate had sharp leading-edge geometry, for λ values of 1, 2.5, and
5.
Particle Image Velocimetry measurements at the upstream region of the cylinder-endplate
junction, where the endplate had elliptical leading-edge geometry, for λ values of 1, 2.5,
and 5.
Section 4.1 of this chapter discusses the findings of Particle Image Velocimetry and Volumetric
3-Component Velocimetry measurements at the cylinder-wall junction. Section 4.2 presents
Particle Image Velocimetry measurements of the junction flow occurring between the cylinder
and an end plate with sharp leading edge, at which there was significant unsteady flow
separation, as discussed in Section 2.6. The results of the horseshoe vortex dynamics at the
junction of a cylinder-endplate having elliptical leading edge geometry was investigated with
Particle Image Velocimetry, and the results are presented in Section 4.3. Section 4.4 focuses on
the frequency characteristics of various junction configurations, and Section 4.5 provides a
summary of the findings.
4.1 Unsteady Flow Characteristics at the Cylinder-Wall Junction: Temporal Evolution of Vorticity Contours
Measurements conducted at the junction region between the cylinder and water channel wall
were made with Particle Image Velocimetry and Volumetric 3-Component Velocimetry, details
of which were discussed in Section 0. Temporal evolution of the contour patterns of normalized
56
vorticity D/Uo, calculated from the Particle Image Velocimetry measurements, is given in
Figure 4.0. In each image, the right side shows the cylinder boundary and the bottom side shows
the channel wall boundary. These contour plots clearly show a well-defined periodicity of the
horseshoe vortex system. Each column in Figure 4.0 shows one full period, during which a
primary vortex (the vortex closest to the cylinder body) is observed to be stationary at a certain
location upstream of the cylinder, and a secondary vortex (the vortex on the left of the primary
vortex) approaches the primary vortex (and cylinder). As the secondary vortex approaches the
primary one, the magnitude and size of vorticity of the primary vortex decreases, whereas the
vorticity magnitude of the secondary vortex increases. Eventually, at a certain critical upstream
location from the cylinder, the original primary vortex totally diminishes and the remaining
vortex core amalgamates with the secondary oncoming vortex. At this instant, the secondary
vortex becomes the new primary vortex at around the same station upstream of the cylinder (see
the fifth image from top in each column in Figure 4.0). The first four images from the top in each
cycle also reveal the formation and then a continuous movement of a third vortex, which grows
its vorticity magnitude and size while moving towards the cylinder; this vortex becomes the new
secondary vortex after the amalgamation of the first two vortices. Negative vorticity is also
present behind the primary and secondary vortex cores, as seen in Seal et al. (1995).
Volumetric 3-Component Velocimetry measurements of the cylinder-wall junction were
conducted in a volume of 140 mm length in the streamwise, 140 mm width in the lateral and 50
mm height in the spanwise direction. The cylinder, which measured 50.8 mm in diameter, was
placed in the volume so that approximately 25 mm (0.45D) of upstream junction region was
visualized. These volumetric measurements enable the investigation of the horseshoe vortex
behavior to be investigated as it wraps around the cylinder. The results presented in Figure 4.1
show that the vorticity increases as the secondary vortex approaches the junction region, and then
decreases until the vorticity is diminished. In Figure 4.1, temporal evolution of the contour
patterns of normalized vorticity D/Uo are given on multiple planes, sliced from the volumetric
data. The stationary vortex seen in the symmetry plane, upstream of the junction, reveals the
primary vortex, which changes its magnitude and scale periodically, i.e., at t=1/6T, its scale and
vorticity magnitude are both small. However, until t=1T, the size and vorticity magnitude of the
primary vortex increases, at which point it reaches its maximum . This observation is consistent
with the two dimensional Particle Image Velocimetry measurements, where we saw that the
scale and vorticity magnitude of the primary vortex change periodically, such that as the
57
secondary vortex approaches the primary one, the magnitude of vorticity and the scale of the
primary vortex drop gradually, and when the secondary vortex becomes the new primary vortex,
the size and vorticity magnitude suddenly increase. The resolution of the Volumetric-3
Component Velocimetry measurements was too coarse to fully capture the small-scale vortex
structures seen in the Particle Image Velocimetry measurements. The measurement volume was
also kept small to increase the vector resolution in volumetric measurements. As a result of this,
the secondary vortex was outside of the measurement volume; however the overall
characteristics of the primary vortex could still be described. Overall consideration of the contour
patterns in multiple slices of the volume in Figure 4.1 shows that when the primary vortex
upstream of the cylinder diminishes, an increase in vorticity magnitude in the legs of the
horseshoe vortex arises. This indicates that the vorticity magnitude periodically sweeps around
the cylinder.
4.2 Unsteady Flow Characteristics Upstream of the Junction of a Cylinder with an Endplate having Sharp Leading Edge Geometry: Temporal Evolution of Vorticity Contours
PIV measurements were performed upstream of a cylinder-endplate junction with the endplate
having a sharp leading edge to assess the effect of significant upstream separation on the system
of horseshoe vortex dynamics. Figures 4.2, 4.3 and 4.4 show the temporal evolution of the
patterns of normalized vorticity D/Uo for λ values of 1, 2.5 and 5, respectively. When the
cylinder is at λ =1, as seen in Figure 4.2, the horseshoe vortex system consists of one primary
vortex which is stationary in time. The time series of vorticity contours shows that the magnitude
of the primary horseshoe vortex is essentially constant over time. This is hypothesized to be
related to the entrainment of upstream vorticity into the primary vortex, preventing a change in
its scale and vorticity magnitude, as opposed to what was seen in Figure 4.0 where no endplate
was used.
Contour patterns of instantaneous D/Uo are presented in Figure 4.3 for the case when the
cylinder is moved further downstream on the endplate to λ=2.5. The influence of the flow
separation upstream of the junction (at the leading edge of the plate) can be seen more
significantly in this case. The primary horseshoe vortex is stationary at a certain location
upstream of the cylinder. The flow structures in the time series appear more chaotic, and patches
of positive and negative vorticity can be seen throughout the junction. As a result of a continuous
58
vorticity supply from the separated flow at the leading edge to the primary vortex, the scale and
magnitude of vorticity of the primary vortex is conserved. In contrast to the case when the
cylinder did not have an endplate, there exists no periodicity in the horseshoe vortex. For λ=2.5,
increased disorder of the vorticity patterns and entrainment of upstream vorticity by the primary
vortex are all postulated to be related to the increase of the length of the separation bubble, as a
result of positioning the cylinder further downstream from the leading edge of the endplate
compared to the λ = 1 case.
When the cylinder was moved even further downstream from the leading edge of the plate, to a
value of λ=5, the separation bubble was observed to become larger than the bubble at λ=2.5, in
PIV experiments conducted at the leading edge of the endplate, as outlined in Section 2.6. For
the λ=5 case, the primary horseshoe vortex entrains large amounts of upstream vorticity. This is
shown in the time series of vorticity D/Uo contour patterns presented in Figure 4.4. The
primary horseshoe vortex is also seen to consistently have large regions of negative vorticity
surrounding it between the endplate and the primary horseshoe vortex. Similar to other cases
where upstream separation was present, there was no periodicity in the temporal evolution of
these patterns because the primary horseshoe vortex preserves its size and vorticity magnitude.
The temporal evolution of the vorticity patterns, shown in Figures 4.2, 4.3, and 4.4, become
gradually disordered as λ is increased. From the measurements it is apparent that this increase in
disorganization of the flow structure must be associated with the increase in length of the
separation bubble with λ as discussed in Section Error! Reference source not found..
4.3 Unsteady Flow Characteristics in the Upstream of the Junction of a Cylinder with an Endplate having Elliptical Leading-Edge Geometry: Temporal Evolution of Vorticity Contours
An endplate with a super elliptical leading edge was designed (see Section 2.6 for further details)
to eliminate the unsteady flow separation at the leading edge of the endplate. This section will
focus on the dynamics of horseshoe vortices forming upstream of the junction between a cylinder
and this type of an endplate. The cylinder was initially placed close to the leading edge at λ=1,
then the distance between the cylinder and the leading edge was increased to λ=2.5 to establish a
representative case for an intermediate leading edge distance, and then a large leading edge
distance of λ=5 was investigated. Temporal evolution of vorticity D/Uo contours for λ=1, 2.5
59
and 5 are given in Figures 4.5, 4.6, and 4.7, respectively. Horseshoe vortex systems for all three λ
values demonstrate analogous dynamics. First of all, time series of all the horseshoe vortex
systems clearly demonstrate periodicity. The primary vortex (the vortex closest to the cylinder
body) travels downstream towards the cylinder, until it reaches a critical distance from the
cylinder. At this point, the magnitude and scale of the primary vortex decreases, and the
secondary vortex (the vortex on the left of the primary vortex), which is also moving
downstream towards the cylinder, amalgamates with the diminishing primary vortex, and reaches
a vorticity maximum. This newly formed primary vortex then begins to decrease in size and
magnitude, and this process repeats itself. A subtle point revealed by a comparison of the
vorticity patterns in Figures 4.5, 4.6, and 4.7 is that with increasing distance λ between the
cylinder and the leading edge, the scale of the horseshoe vortex slightly increases. Thus, the
smaller the distance λ, the smaller the horseshoe vortices are.
Overall consideration of all the findings discussed so far show that upstream separation has a
major impact on the spatial and temporal dynamics of the horseshoe vortex systems. This can be
seen when the results for the experiments with and without upstream separation are compared,
i.e., comparison of experiments involving the endplate with the sharp-leading edge and the
elliptical leading edge. The significant increase in upstream vorticity originating from the
unsteady separation at the sharp leading edge of an endplate causes the primary vortex to entrain
additional vorticity and prevent the periodic decrease in magnitude and size associated with the
regular behavior of the system. The vortex structures occurring near the junction of the cylinder
and endplate for a separated upstream flow appear larger than the corresponding vortices
generated from an attached flow for consistent λ values. Results from the Volumetric 3-
Component Velocimetry measurements for the junction flow of a cylinder mounted flush to the
channel wall show that the vorticity in the legs of the horseshoe vortex increases when the
primary horseshoe vortex upstream of the cylinder decreases in size and magnitude. This
indicates that the vorticity is swept around the cylinder, and then into the downstream region of
the flow periodically.
4.4 Frequency Characteristics of the Horseshoe Vortex Systems: Spectral Analysis of Streamwise Velocity
This section discusses the unsteady features of the horseshoe vortex systems by analyzing the
pointwise spectra Su(f) of the streamwise velocity component. The spectra were sampled at the
60
location of time-averaged maximum vorticity. This location was chosen because this represents
the region where the primary horseshoe vortex is present during the majority of the cycle, and
will thus generate frequency spectra related to its evolution over time.
The results are shown in the plots of Figure 4.8, where in the top row of plots, Su(f) for the no
endplate case and the case with an endplate having a sharp leading edge at leading edge distances
of λ = 1, 2.5 and 5 are given. In the bottom row, Su(f) for the case with an endplate having an
elliptical leading edge are provided for λ = 1, 2.5 and 5. The plots show autospectral amplitude of
streamwise velocity vs. Strouhal number. It is clear that, with no endplate, the horseshoe vortex
system shows periodicity at a dominant Strouhal number of approximately St = 0.36. The plots
of frequency spectra for the experiments with upstream flow separation (with the sharp leading-
edge geometry endplate) demonstrate that the frequency distribution is broadband, and thus the
spectral power in the horseshoe vortex system is not concentrated within one frequency, which
indicates that the system is not periodic. When the upstream flow separation was eliminated
(with the elliptical leading-edge-geometry endplate), the frequency spectra plots are very
different. For the endplate with an elliptical leading edge, λ=1 shows a clear spectral peak at a St
of approximately 1.When the leading edge distance is increased to λ=2.5 the spectral peak is
concentrated at St = 0.8, and when the leading edge distance is even further increased to λ=5 the
dominant frequency of the spectral peak decreases to St = 0.7.
The results for the case without upstream flow separation (endplate with elliptical leading edge)
show a clear trend, which indicates that as the leading edge distance λ is increased, the
predominant frequency of velocity fluctuations of the horseshoe vortex system decreases. These
results can be compared with the frequency measurements of Thomas (1986), who found that
horseshoe vortex frequency increases with increasing Reynolds number for a laminar boundary
layer junction flow. In other words, his findings indicate that the dominant horseshoe-vortex
frequency increased with decreasing boundary layer thickness. The primary effect of altering λ is
the change in the boundary layer formation length. Thus, with a decrease of λ, the boundary layer
thickness is expected to decrease, although measurements were not performed to quantify this
assumption. The increase of the dominant frequency with decreasing λ compares well with the
results of Thomas (1986), but additional measurements of boundary layer thickness would help
quantify this finding. The experiments performed with upstream flow separation did not
demonstrate a dominant frequency in the spectra, reflecting the time series analysis done in
61
Section 4.2 where system periodicity was not observed upon inspection of the temporal evolution
of the vorticity contours.
4.5 Summary of the Horseshoe Vortex Measurements
The measurements conducted upstream of the junctions of cylinder-wall and cylinder-endplate
demonstrate that end conditions of the cylinder have a major impact on the horseshoe vortex
dynamics. Measurements performed employing a technique of Particle Image Velocimetry in the
upstream region of a cylinder-wall junction show a periodic system, in which the primary
horseshoe vortex is stationary at a location upstream of the cylinder; however, the secondary
horseshoe vortex moves downstream towards the primary vortex. During this move, the
secondary vortex grows in magnitude and size until a critical distance from the cylinder, while
the magnitude and size of the primary vortex decreases. At this stage, the primary vortex is
amalgamated into the oncoming secondary horseshoe vortex, and this process repeats itself
periodically. Volumetric 3-Component Velocimetry measurements of the cylinder-wall junction
showed that the vorticity is being swept downstream around the cylinder at an instant when the
vorticity of the primary horseshoe vortex at the upstream of the junction region diminishes.
The use of an endplate with sharp leading-edge geometry was shown to have a significant effect
on the dynamics of the horseshoe vortex system upstream of the junction of the cylinder-
endplate, when examined using Particle Image Velocimetry. The leading edge geometry of this
endplate was found to result in flow separation at the leading edge of the plate upstream of the
junction. The system consisted of a large primary horseshoe vortex that did not diminish in size
over time due to the entrainment of vorticity from the oncoming separated flow. This effect has
been shown to be independent of the leading-edge distance used in the experiments, i.e.,
independent of the λ tested.
The results of measurements using Particle Image Velocimetry to investigate the horseshoe
vortex system at the junction of a cylinder-endplate without upstream separation (through the use
of an elliptical leading edge) revealed a periodic horseshoe vortex system with qualitatively
similar aspects to the dynamics seen in the system without an endplate. Measurements of
pointwise spectra showed that the dominant frequency of the system decreased when the leading-
edge distance λ was increased, i.e., the dominant frequency was seen to be inversely proportional
62
to the λ value used in the experiment. Furthermore, the value of λ is also found to affect the size
of the vortical structure. The smaller the distance λ, the smaller the horseshoe vortices are.
63
Figure 4.0: Time series evolution of normalized vorticity: upstream junction region – no endplate. Individual time series are represented by each column of
frames. The blue horizontal rectangle shows the channel floor, and the blue horizontal rectangle shows the cylinder. The evolution of the primary vortex can
be seen as it approaches the cylinder, and begins to diminish in size and strength. Eventually it is amalgamated into the secondary vortex.
64
Figure 4.1: Multi slice V3V measurements of time series evolution of vorticity magnitude: upstream junction – no endplate. Contours of vorticity at multiple
measurement planes show the diminishing primary vortex seen in the ‘Y-X” plane, where PIV measurements were conducted. This coincides with an
increase in vorticity in the legs of the horseshoe vortex, seen in the ‘Y-Z’ plane.
65
Figure 4.2: Time series evolution of vorticity in the upstream junction of a cylinder-sharp leading edge endplate λ=1. Each column of frames represents a
time series evolution of vorticity. The horizontal blue boundary represents the endplate, and the vertical blue boundary represents the cylinder. It can be
seen that the primary horseshoe vortex does not undergo a periodic decrease in magnitude and strength.
66
Figure 4.3: Time series evolution of vorticity in the upstream junction of a cylinder-sharp leading edge endplate λ=2.5. Each column of frames represents a
time series evolution of vorticity. The horizontal blue region represents the endplate, and vertical blue region represents the cylinder. It can be seen that
there is one steady primary horseshoe vortex which does not diminish in size periodically due to the entrainment of upstream vorticity. Pockets of negative
vorticity can be seen surrounding the primary vortex, but do not significantly diminish its size over time due to the addition of vorticity from the upstream
separation, which is caused by using a sharp leading edge.
67
Figure 4.4: Time series evolution of vorticity in the upstream junction of a cylinder-sharp leading edge endplate λ=5. Each column of frames represents a
time series evolution of vorticity. The horizontal blue region represents the endplate, and the vertical blue region represents the cylinder. It can be seen that
there is one steady primary horseshoe vortex which does not diminish in size periodically due to the entrainment of upstream vorticity. Pockets of negative
vorticity can be seen surrounding the primary vortex, but do not significantly diminish the size over time, due to the addition of vorticity from the upstream
separation.
68
Figure 4.5: Time series evolution of vorticity magnitude for upstream junction of elliptical leading edge endplate: λ=1. Each column of frames represents a
time series evolution of vorticity. The horizontal blue region shows the endplate, and the vertical blue region represents the cylinder. A periodic movement,
and reduction in size of the primary vortex as it approaches the cylinder can be seen clearly. The secondary vortex amalgamates with the reduced primary
vortex when the vortex diminishes in size, and is close to the larger oncoming secondary vortex.
69
Figure 4.6: Time series evolution of vorticity magnitude for upstream junction of elliptical leading edge endplate: λ=2.5. Each column of frames represents a
time series evolution of vorticity. The horizontal blue region shows the endplate, and vertical blue region shows the cylinder. The primary vortex can be
seen approaching the cylinder and reducing in size and strength, when the secondary vortex reaches the primary vortex it amalgamates with the primary
vortex.
70
Figure 4.7: Time series evolution of vorticity magnitude for upstream junction of elliptical leading edge endplate: λ=5. Each column of frames represents a
time series evolution of vorticity magnitude. The horizontal blue region shows the endplate, and the vertical blue region shows the cylinder. The periodic
reduction of vorticity magnitude of the primary vortex can be seen as it approaches the cylinder. When the secondary vortex reaches the reduced primary
vortex it amalgamates with the primary vortex.
71
Figure 4.8: Plots of frequency spectra taken for all endplate configurations, and no end plate case. Spectra were sampled at the location of time averaged
maximum vorticity. Results of the spectral analysis demonstrate a dominant frequency of 0.36, when no endplate is used. An endplate with a sharp leading
edge generated significant upstream separation, which caused the primary vortex to retain its size and strength, and not periodically diminish. Therefore
the spectra appear broadband, and no dominant frequency is detected. The results presented for the elliptical endplate with no upstream separation are
shown in the bottom half of the figure. A dominant frequency can be seen for each case. An inversely proportional relationship is seen between the leading
edge distance and the dominant frequency, with larger leading edge distances resulting in smaller frequency values.
72
CHAPTER 5
Conclusions and Future Work
The goal of this research project was to undertake a comprehensive set of measurements relating
to the use of endplates in cylinder flow research in the sub-critical regime. The introduction
outlined the theoretical construct of the work and described the various experimental parameters
affecting the two-dimensionality in the near wake of a circular cylinder including; aspect ratio,
ReD, endplate size and shape, etc. This investigation focused on the effect of different end plate
configurations, and their impact on the two dimensionality of the flow in the near wake. In
addition to this research, experiments were done at the junction of the cylinder and endplate to
study the effect of the leading-edge shape of the endplate, and the cylinder position on the
horseshoe vortex system occurring in the junction region. Because this study varied many
parameters, it was decided to fix the Reynolds number and the cylinder diameter during the
course of the investigation.
The flow physics in the near wake was investigated using Particle Image Velocimetry along the
span of the cylinder for each endplate configuration, and the results were discussed in terms of
the demarcation line occurring along the span, which was defined as the line where the
streamwise velocity in the near wake region changes from negative to positive orientation. This
corresponds to the streamwise location of the boundary of the separation bubble, i.e., the distance
from the base of the cylinder to the demarcation line represents the length of the recirculation
bubble in the near wake. In a perfectly two dimensional flow, the demarcation line would be
parallel to the cylinder span. This was used to compare with as a quantitative measure of the
parallelism of the demarcation line along the span, and thus represents an excellent indicator of
the flow two-dimensionality.
The measurements were undertaken using four endplate configurations, with the position of the
cylinder on the endplate varied in each configuration (i.e., when endplates were used). The
leading-edge distance of the plate from the cylinder centerline was normalized by the cylinder
diameter, and thus the cylinder position was represented by λ=L/D. The four experimental
arrangements tested were as follows:
73
Cylinder with no endplates bounded by the channel wall at the bottom and the free
surface at the top.
Cylinder bounded by a single endplate with sharp leading-edge geometry at the bottom
and the free surface at the top.
Cylinder bounded by a single endplate with elliptical leading-edge geometry at the
bottom and the free surface at the top.
Cylinder bounded by two endplates each with the sharp leading-edge geometry
The results indicate a definite trend in relation to the optimal leading edge distance and endplate
configuration. The demarcation line factor, which compares the RMS of the recirculation region
length to the mean recirculation region length, was calculated for each configuration and cylinder
position. The cylinder with no end plate had a demarcation line factor of approximately 9%. The
cylinder position that optimized the demarcation line factor was λ =2.5; this value was also
observed by Stansby (1974). The arrangement with two endplates generated the best results, and
the demarcation line factor was on the order of 2%. The use of a single endplate with elliptical
leading-edge geometry showed a qualitatively similar trend in cylinder position optimality, but
gave a better demarcation line factor result for all λ values tested compared to the use of a single
plate with sharp-leading edge geometry.
The findings of this research provide insight to experimentalists on a number of factors for the
design of endplates to promote two-dimensionality in the near wake of a cylinder. Results clearly
show an advantage of using two endplates with sharp leading edge rather than one, and further
indicate that the optimal λ value is approximately 2.5 for the subcritical ReD = 10,000. The work
also describes in detail how the two dimensionality in the near wake can be quantified using
Particle Image Velocimetry, which is in contrast to most of the works done in this research area.
The majority of these studies relied on hot wire or pressure measurements or flow visualization
rather than Particle Image Velocimetry.
Experiments were conducted to elucidate the unsteady horseshoe vortex system characteristics at
the junction between the cylinder wall and cylinder endplate(s). Temporal evolution of the
contour patterns of vorticity were evaluated for all of the experimental arrangements tested
74
herein and, in addition, point-wise autospectral density of velocity fluctuations were calculated to
detect if the horseshoe vortex dynamics are dominated by any frequencies. The cylinder mounted
flush to the channel wall involved a periodic horseshoe system, where the primary horseshoe
vortex was stationary and the secondary vortex approached the primary vortex. During this
process, the primary vortex began to diminish in size and strength until the growing secondary
vortex amalgamated the primary vortex and became the new primary vortex. Volumetric 3-
Component Velocimetry measurements showed that the decrease in the strength of the primary
horseshoe vortex upstream of the cylinder coincided with an increase in the vorticity magnitude
in the legs of this primary horseshoe vortex around the cylinder. Streamwise velocity spectra,
which were computed at the location of time-averaged maximum vorticity, showed the dominant
frequency of the system to be approximately St = 0.36.
Experiments performed with an endplate which had sharp leading-edge geometry displayed
significant flow separation at the leading edge. It also was shown that, in the junction flow
measurements made upstream of the cylinder-endplate junction, the upstream flow separation
had significant implications on the development of the horseshoe vortex system. Temporal
evolution of vorticity contour plots, evaluated for varying λ values, showed a horseshoe vortex
system consisting of one primary horseshoe vortex which never diminished in size and strength
due to the addition of vorticity from the separated upstream region. This result contrasts sharply
to the use of the elliptical leading-edge geometry endplate, as this case eliminated the upstream
separation. In the time series presented with these measurements, the horseshoe vortex system
was periodic and the same mechanism which was seen in the experiments with no endplates was
visualized, in which the primary horseshoe vortex periodically diminishes in size and strength
until it amalgamates with the oncoming secondary vortex. Streamwise velocity spectra
measurements indicated that there was a dominant frequency of the horseshoe vortex system, and
that this frequency was inversely proportional to the leading edge distance of the endplate i.e. λ.
The implications of the endplate design on the behavior of the horseshoe vortex system are clear
from the measurements discussed in this research, and it has been shown that the condition of the
flow on the tip of the endplate must be carefully investigated prior to running experiments. The
assumption of an unaffected, newly growing boundary layer at the leading edge of the endplate is
not always correct, and must be validated.
75
This research project represents a comprehensive set of measurements describing the use of
endplates on cylinder flows, which has been an area of research dominated by either intrusive hot
wire and pressure measurements, or qualitative flow visualization. Particle Image Velocimetry is
a non-intrusive whole flow field measurement technique, which was used to expand the research,
and improve the understanding of endplate design to ensure spanwise uniformity of the flow in
the near wake. The measurements of horseshoe vortex dynamics demonstrated the need for a
properly designed leading edge if the assumption of a newly growing laminar boundary layer is
assumed in the measurements. This will be important to researchers in both cylinder flows and
experimentalists studying junction flows who are designing experimental setups where endplate
like designs are often used to control the thickness of the boundary layer.
Recommendations for future work will be briefly discussed in what follows.
Firstly is the need to test greater λ ranges with two sharp leading edge geometry endplates in
order to fully represent the effect of the cylinder position on spanwise wake uniformity. Current
experiments have only tested three λ values. These were in the intermediate range, and in the
short and large range. Experiments on the use of two elliptical-leading-edge-geometry endplates
should also be conducted to determine if the difference in leading edge design which showed a
marked improvement for a single endplate will occur with two endplates.
In summary, the goals of this research project have been achieved, the first of which was to
comprehensively test various endplate arrangements, and their efficacy on the promotion of
parallel shedding in the near wake. The results showed quantifiable differences in endplate
efficacy and, in addition, a method for evaluating the two dimensionality in the near wake based
on the demarcation line has been used which will provide a basis for experimentalists using
Particle Image Velocimetry. The second goal was to investigate the dynamics of the horseshoe
vortex systems, resulting from various endplate experimental arrangements. The results of the
junction flow measurements highlighted the need for a properly designed leading edge by
demonstrating the significant effect on the horseshoe vortex dynamics of upstream separation.
76
References
Baker CJ. The laminar horseshoe vortex, Journal of Fluid Mechanics, 1979, Volume 95
Chong MS, Perry AE, Cantwell BJ, A general classification of three-dimensional flow fields,
Physics of Fluids A: Fluid Dynamics, 1990, Volume 2.
Chou J.H., Chao S.Y. Branching of a horseshoe vortex around surface mounted rectangular
cylinders, Experiments in Fluids, 2000, Volume 28.
Dantec. How to Measure Turbulence with Hot-Wire Anemometers - a Practical Guide. 2002.
Gerich D, Eckelmann. H. Influence of end plates and free ends on the shedding frequency of
circular cylinders, Journal of Fluid Mechanics, 1982, Volume 122.
Gerrard JH. The wakes of cylindrical bluff bodies at low Reynolds number, Phil. Trans. of the
Royal Society of London, Series A, Mathematical and Physical Sciences, 1978, Volume 288.
Gupta A.K. Hydrodynamic modification of the horseshoe vortex at a vertical pier with junction
ground. Physics of Fluids, 1986, Volume 30.
Hammache, Gharib, A novel method to promote parallel vortex shedding in the wake of circular
cylinders. Physics of Fluids Letters, 1989
Kang KJ, Kim T, Song SJ. Strengths of Horseshoe Vortices around a Circular Cylinder with an
Upstream Cavity, Journal Of Mechanical Science And Technology, 2009, Volume 23.
Kelso R,M, Smits A.J. Horseshoe vortex systems resulting from the interaction between a
laminar boundary layer and a transverse jet. Physics of Fluids, 1995, Volume 7.
Lin C, Chiu , Shieh. Characteristics of horseshoe vortex system near a vertical plate – base plate
juncture. Experimental Thermal and Fluid Science. 2002, Volume 27
Luo S.C, Tan. Induced parallel vortex shedding from a circular cylinder at Re of O(104) by using
the cylinder end suction technique, Experimental Thermal and Fluid Science. 2009, Volume 33.
Narasimha R., Prasad N., Leading edge shape for flat plate boundary layer studies. Experiments
in Fluids, 1994, Volume 17
Norberg C. An experimental investigation of the flow around a circular cylinder: influence of
aspect ratio, Journal of Fluid Mechanics. 1994, Volume 258.
Okamoto K., Nishio S, Saga T, Kobayasi T. Standard images for particle-image velocometry,
Measurement Science and Technology, 2000, Volume 11.
77
l men S , Simpson R . Influence of passive flow-control devices on the pressure fluctuations
at wing-body junction flows, Journal of Fluids Engineering. 2007, Volume 129.
Ozturk A, Akkoca A, Sahin B. PIV measurements of flow past a confined cylinder, Experiments
in Fluids, 2008, Volume 44.
Paik J, Escauriaza C, Sotiropoulos F. On the bimodal dynamics of the turbulent horseshoe vortex
system in a wing-body junction, Physics of Fluids. 2007, Volume 19.
Pereira F, Gharib M. Defocusing digital particle image velocimetry and the three-dimensional
characterization of two-phase flows, Measurement Science and Technology. 2002, Volume 13.
Pereira F, Gharib M, Dabiri M. Defocusing DPIV a 3C3D DPIV measurement technique -
application to bubbly Flows, Experiments in Fluids, 2000, Volume 29
Pereira F, Stuer H, Graff E, Gharib M, Two frame 3D particle tracking, Measurement Science
and Technology, 2006, Volume 17.
Prasad A, Williamson CHK. Three-dimensional effects in turbulent bluff-body wakes, Journal of
Fluid Mechanics, 1997, Volume 343.
Raffel M., WillerT C., Wereley S., Kompenhans J., Particle image Velocimetry: A practical
guide, 2nd
Edition, 2007, Springer Press
Roshko A. On the development of turbulent wakes from vortex streets. NACA Report 1191.
1954.
Roshko A. Perspectives on bluff body aerodynamics. Journal of Wind Engineering and
Industrial Aerodynamics, 1993, Volume 49.
Sahin B, Ozturk NA. Horseshoe vortex system in the vicinity of the vertical cylinder mounted on
a flat plate. Flow Measurement and Instrumentation, 2007, Volume 18.
Seal CV, Smith CR, Akin O, Rockwell D. Quantitative characteristics of a laminar, unsteady
necklace vortex system at a rectangular block-flat plate juncture. Journal of Fluid Mechanics,
1995, Volume 286.
Simpson R.L. Junction flows. Annual Reviews of Fluid Mechanics, 2001, Volume 33.
Simpson R.L. Turbulent boundary layer separation, Annual Review of Fluid Mechanics, 1989,
Volume 21.
Stager, Eckellman, Effect of endplates on the shedding of circular cylinders in the irregular
range. Physics of Fluids, 1991, Volume 3.
Stansby. Effects of end plates on the base pressure coefficient of a circular cylinder.
Aeronautical Journal, 1974, Volume 87.
78
Sumner, Heseltine, Dansereau, Wake Structure of a finite circular cylinder of small aspect ratio,
Experiments in Fluids, 2004, Volume 37.
Szepessy, S. End wall interaction and aspect ratio effect on vortex shedding and fluctuating
forces on a circular cylinder. Research Thesis, Chalmers University of Technology. 1988
Szepessy S, Bearman PW. Aspect ratio and end plate effects on vortex shedding from a circular
cylinder. Journal of Fluid Mechanics, 1992, Volume 234.
Szepessy S, On the control of circular cylinder flow by endplates, European. Journal
of.Mechanics B/Fluids, 1993 Volume 12.
SzepessyS, On the spanwise correlation of vortex shedding from a circular cylinder at high sub-
critical Reynolds number. Physics of Fluids, 1994, Volume 6.
Taneda S. Experimental investigation of the wakes behind cylinders and plates at low Reynolds
numbers. Journal of the Physical Society of Japan, 1956, Volume 11.
Thomas ASW. The unsteady characteristics of laminar juncture flow. Physics of Fluids: Letters,
1986, Volume 30.
Tropea, Yarin, Foss, Springer Handbook of Experimental Fluid Mechanics, 2007, Springer Press
Unal M.F., Rockwell D. On vortex shedding from a cylinder. Part 1: The initial instability.
Journal of Fluid Mechanics, 1988, Volume 190.
Visbal M.R. Structure of laminar juncture flows. AIAA Journal, 1991, Volume 29.
Wang H.F, Zhou Y, Chan C.K, Lam KS. Effect of initial conditions on interaction between a
boundary layer and a wall-mounted finite-length-cylinder wake. Physics of Fluids, 2006, Volume
18.
Wang J.M, Bi W.T, Wei Q.D. Effects of an upstream inclined rod on the circular cylinder–flat
plate junction flow. Experiments in Fluids, 2009, Volume 46.
Wei Q, Wang J.M., Chen G., Lu Z.B., Bi W.T., Modification of junction flows by altering the
section shapes of the cylinders, Journal of Visualization, 2008, Volume 11.
Willert CE, Gharib M. Digital particle image velocimetry. Experiments in Fluids, 1991, Volume
10.
Williamson C.H.K., Oblique and parallel modes of vortex shedding in the wake of a cylinder at
low Reynolds number, Journal of Fluid Mechanics, 1989, Volume 206.
Williamson CHK. Vortex Dynamics in the Cylinder Wake. Annual Review of Fluid Mechanics.
1996, Volume 28.
79
Appendix A – Uncertainty Analysis
A.1 Estimation of Measurement Uncertainty
The propagation of errors through an experiment must be carefully examined if one wishes to
discuss the results with certainty. This appendix will detail the uncertainty analysis done on the
results presented in this thesis.
If a result can be described as a function of many variables, then the uncertainty associated with
each variable in the function can be expressed as the square root of the sum of the squares of the
error in each variable multiplied by a sensitivity coefficient. The sensitivity coefficient is the
partial derivative of the function with respect to the variable.
),...,( nxxxfr 21 (1)
22
2
2
2
1
1
n
n
XXrx
r
x
r
x
r ...
(2)
Where r is the result and x is a particular variable in which the result is a function of. The error in
each variable is , commonly available as a manufacturer specification. Thus, if the error term in
each variable making up the result is known a priori then it is a simple matter to calculate the
total error in the measured quantity.
In PIV tracer particles are used to measure the velocity of the fluid by measuring the
displacement of the particles in the flow over time. Therefore the result of a PIV measurement is
velocity which can be expressed as
t
Xu
(3)
The measurement uncertainty is therefore a function of the detected particle displacement and
time duration between successive laser pulses. Expressed in the form seen in Equation (2) it is
represented as
80
22
tSU
t
U
X
U
(4)
Where s is the maximum displacement allowed by the PIV experiment, which is usually ¼ of
the interrogation area, as seen in Raffel et al (2007). The uncertainty measurement is thus due to
the uncertainty in the detection of the particle displacement and uncertainty in the time duration
of the laser pulse.
)(CCks (5)
Where k is the magnification factor in mm/pixel and CC is the particle displacement given by the
cross correlation analysis done by the PIV analysis program.
The uncertainty associated with the detection of the particle displacement is due to many factors,
and this is outlined comprehensively in Raffel et al (2007). Some of these are the detection of
sub-pixel displacement, seeding density, variation in particle size, cross interrogation window
velocity gradients, variation of image quantization levels, background noise etc. The error term
is thus a function of many experiment specific sources of error which are very difficult to
determine individually. In addition to the errors mentioned above, which are associated with the
displacement vector, the calibration performed to deduce the magnification factor also induces an
error. Thus the total error due to the displacement vector can be shown below.
22
kccs
k
s
CC
s (6)
The calibration is done with a reference plane such as a ruler where a known distance can be
imaged on the camera and related to the number of pixels it occupies in the image plane of the
CCD array:
c
c
n
lk
(7)
81
where cl is the reference distance used in the calibration and cn is the number of pixels the
reference distance occupies.
And the error in the calibration can be expressed as
22
cc n
c
l
c
kn
k
l
k
(8)
The table below lists the sensitivity factors based on the partial derivatives of equations 2-8
.
Table A.1: Sensitivity Factors in Partial Derivative Form and Experimental Parameter Form for 2C-PIV
Measurements
The vorticity calculations were done using a program called PostProcess.exe which used a
circulation based vorticity calculation for interior grid regions and forward/backward finite
Partial Derivative (Sensitivity
Factor)
Sensitivity Factor
(Experimental Terms)
X
U
t
1
t
U
2t
s
CC
s
k
k
s
s
cl
k
cn
1
cn
k
2
c
c
n
l
82
difference schemes for grid positions located near boundaries. The error associated with these
methods was presented in Raffel et al (2007) and can be seen in the table below. The circulation
method was used for the interior points due to the increased error accompanying estimates of
derivatives for discrete data points.
Vorticity Calculation Scheme Truncation Uncertainty
Forward Differencing
X
U
411.
Backward Differencing
X
U
411.
Circulation Method
X
U
630.
Table A.2: Truncation Errors from Raffel et al (2008) for various Derivation Schemes
In addition to the truncation error associated with a derivation scheme there is the normal
propagation of errors of the term xu / used in the calculation of vorticity. Humble (2008)
summarizes the error associated with the propagation of uncertainties related to each term in the
derivative. This term combined with the truncation error can be seen below in Equation (9)
222 41122
2 ).()()(/xxx
u
x
uxu
xu
(9)
The error in the vorticity is then
22 )( / xu (10)
where Xx , are the grid spacing and error in grid spacing respectively.
kpixelpacingFinalGridSX *))((
These will be used to determine the error in vorticity associated with the measurements in the
upstream junction region.
83
A.2 Summary of Results
The results for the uncertainty analysis in both the PIV measurements in the near wake along the
span of the cylinder and the upstream junction region of the flow can be seen below in Table A.3
Near Wake Measurements Upstream Junction Measurements
cl = 2” = 50.8 mm cl =0.5” = 12.7 mm
cl = 0.5 mm
cl = 0.5 mm
cn = 244 Pixels cn =347 Pixels
cn = 0.5 Pixels cn = 0.5 Pixels
k = 0.221 mm/Pixel k = 0.032 mm/Pixel
k =0.002183 mm/Pixel k =0.00132 mm/Pixel
CC = 0.1 Pixels CC = 0.1 Pixels
x =3.55 mm x =.55 mm
X = 0.0352 mm X = 0.021 mm
xu / = 3.4 1/s xu / = 16.6 1/s
s = 1.736 mm s = 0.256 mm
S = 0.032 mm S = 0.011 mm
84
U = 10.16 mm/s U = 4.012 mm/s
oUU/ = 1.46 % oU
U/ = 1.42 %
= 8.56 1/s = 18.45 1/s
Table A.3: Summary of Uncertainty Analysis
top related