The Use of End Plates for a Cylinder in the Sub …...ii The Use of End Plates for a Cylinder in Sub-critical Flow Regime Adam Douglas Blackmore Master of Applied Science in Engineering

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.

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.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.

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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

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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 /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

/Uo are represented by dashed and solid lines respectively ................................................. 41

Figure 3.1: Contours of time-averaged streamwise velocity /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.

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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

/Uo are represented by dashed and solid lines respectively. ................................................ 42

Figure 3.2: Time-averaged contours of streamwise velocity /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 /Uo are represented

by dashed and solid lines respectively. ...................................................................................... 43

Figure 3.3: Time-averaged contours of normalized streamwise velocity component /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

/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 /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


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


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


0.196 & 0.214. .......................................................................................................................... 49


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


0.196 & 0.214. .......................................................................................................................... 50


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


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

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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


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


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


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 /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 /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 /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 /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 /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 /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 /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 /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), /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 /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 /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 /Uo are represented by dashed and solid lines

respectively

42

Figure 3.1: Contours of time-averaged streamwise velocity /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 /Uo are represented by dashed and solid lines respectively.

43

Figure 3.2: Time-averaged contours of streamwise velocity /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 /Uo are represented by dashed and solid lines respectively.

44

Figure 3.3: Time-averaged contours of normalized streamwise velocity component /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 /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 /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


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


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


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


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

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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

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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

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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

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The Use of End Plates for a Cylinder in the Sub …...ii The Use of End Plates for a Cylinder in Sub-critical Flow Regime Adam Douglas Blackmore Master of Applied Science in Engineering