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The Influence of End Conditions on Vortex Shedding from a
Circular Cylinder in Sub-Critical Flow
by
Eric Khoury
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 Eric Khoury 2012
ii
The Influence of End Conditions on Vortex Shedding from a Circular
Cylinder in Sub-Critical Flow
Eric Khoury
Master of Applied Science in Engineering
Institute of Aerospace Studies
University of Toronto
2012
Abstract
The effect of end boundary conditions on the three-dimensionality of the vortex shedding from a
circular cylinder in sub-critical flow has been studied experimentally, with a focus on the unsteady
nature of the vortex filaments. Analysis of the near-wake of the cylinder was undertaken to
determine the dependency of the spanwise uniformity of the vortex shedding on the end conditions.
Flow visualization was performed downstream of the cylinder, and the temporal variation of the
vortex filament angle was observed. Vortex dislocations were found to occur in this Reynolds
Number regime regardless of the end boundary conditions. Having a cylinder bounded by two
elliptical leading edge geometry endplates at an L/D value of five yielded parallel shedding with a
reduction in the time-based variation of the vortex filament angle, and was shown to be the ideal end
conditions for modeling an infinite cylinder in a free-surface water channel.
iii
Acknowledgments
I would like to thank my supervisor Dr. Ekmekci for her guidance at my time at UTIAS. I would
also like to thank my research committee, Dr. Zingg, Dr. Lavoie, Dr. Steeves and Dr. Ekmekci for
their very useful advice throughout the entire thesis process. I also wish to thank all the students in
the office for creating a great work environment. Specifically I am very grateful for all the hours of
experimental setup help and explanation my lab mates Tayfun Aydin and Antrix Joshi gave me
during my time at UTIAS. Finally I wish to thank my family for all the support they have given me
during this process, if it was not for them I would not be in the position I am in today.
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Table of Contents
Contents
Acknowledgments.......................................................................................................................... iii
Table of Contents ........................................................................................................................... iv
List of Tables ................................................................................................................................. vi
List of Figures ............................................................................................................................... vii
Introduction .................................................................................................................................1 1
1.1 Motivation and Background ................................................................................................1
1.2 Literature Review.................................................................................................................2
1.2.1 Flow Past an Infinite Cylinder .................................................................................2
1.2.2 End Effects on the Spanwise Uniformity of the Cylinder Near Wake ....................4
Experimental Setup and Analysis Techniques ............................................................................9 2
2.1 Experimental Setup ..............................................................................................................9
2.1.1 Water Channel .........................................................................................................9
2.1.2 Test models ..............................................................................................................9
2.1.3 Particle Image Velocimetry ...................................................................................11
2.1.4 Constant Temperature Anemometry ......................................................................14
2.2 Analysis Techniques ..........................................................................................................16
2.2.1 Time-Averaged Recirculation Region ...................................................................16
2.2.2 Space-Time Plots ...................................................................................................17
2.2.3 Continuous Wavelet Transformation .....................................................................18
2.2.4 Fast Fourier Transform and Short-Time Fourier Transform .................................19
2.2.5 Time Evolution of Phase-Angle Difference Between Two Probes .......................20
v
Results .......................................................................................................................................27 3
3.1 Time-Averaged Characteristics of Vortex Shedding .........................................................27
3.2 Unsteady Nature of Vortex Filaments ...............................................................................29
3.2.1 Visualization of Vortex Filaments .........................................................................29
3.2.2 Effect of End Configuration on the Vortex Filament Orientation .........................31
3.2.3 Vortex Splitting ......................................................................................................34
Conclusion and Future Work ....................................................................................................60 4
Appendix A ....................................................................................................................................63
References ......................................................................................................................................66
vi
List of Tables
Table 1: Acquisition details for PIV measurements ........................................................................... 12
Table 2: Probe Characteristics ............................................................................................................ 15
Table 3: Error in calculated recirculation region as a function of Reynolds number ......................... 17
vii
List of Figures
Figure 1: The various boundary conditions investigated. a) A cylinder bounded by the channel floor
and free-surface. b) A cylinder bounded by the channel floor and the channel cover. c) A cylinder
bounded by a sharp leading edge geometry endplate on the bottom and the free-surface on top. d) A
cylinder bounded an elliptical leading edge geometry endplate on the bottom and the free-surface on
top. e) A cylinder bounded by a sharp leading edge geometry endplate on both the top and bottom. f)
A cylinder bounded by an elliptical leading edge geometry endplate on both the top and bottom. ... 22
Figure 2: Schematic detailing the plane on which PIV experiments are being performed. ................ 23
Figure 3: Vortices shed off the shoulder of the cylinder induce positive and negative variations in the
free-stream velocity, allowing for visualization of the vortex filaments. The bottom image shows an
instantaneous streamwise velocity contour plot obtained via PIV on the side-plane. Positive sign
vortex filaments are seen as red, while negative sign filaments are seen as blue-green. .................... 23
Figure 4: Method to determine the length of the recirculation region along the span. ....................... 24
Figure 5: Generation of space-time plots of contours of the streamwise velocity in the z-t plane. The
z axis (spanwise direction) is normalized by D and time t is normalized by D/uo. The plot is
constructed from the time trace of streamwise velocity signals obtained in the y/D = 0.5 plane, at a
chosen streamwise location (designated as xo) along the entire z direction. The spanwise line along
which the streamwise velocity signals were extracted as a function of time is shown in the figure
with the line vector . .......................................................................................................................... 25
Figure 6: CWT was used to calculate the phase angle variation along the span based on the
instantaneous streamwise velocity field in the plane of y/D = 0.5. The calculated phase lag along the
span is converted to a streamwise distance (right figure), and the slope of the linear approximation to
the curve is converted to an approximation for the vortex filament angle (θ). ................................... 26
viii
Figure 7: Ensemble averaged results of the streamwise velocity yield a demarcation line between the
recirculation and positive velocity flow. Formation length as a function of spanwise location at
varying Re is shown for a cylinder bounded by the channel floor and the free-surface. As Re
increases free-surface effects are causing oblique shedding. .............................................................. 37
Figure 8: Formation length as a function of spanwise location at varying Re is shown for a cylinder
bounded by the channel floor and the channel top cover. As Re increases there is no major change to
shedding orientation. ........................................................................................................................... 37
Figure 9: Formation length as a function of spanwise location at varying Re is shown for a cylinder
bounded by sharp endplates at L/D of 2.5. As Re increases there is no major change to shedding
orientation. .......................................................................................................................................... 38
Figure 10: Formation length as a function of spanwise location at varying Re is shown for a cylinder
bounded by elliptical endplates at L/D of 2.5. As Re increases there is no major change to shedding
orientation. .......................................................................................................................................... 38
Figure 11: Free-surface effect causes a downward flow near the top, rear portion of the cylinder.
This leads to spanwise non-uniformities and introduces three-dimensionalities to the flow. The right
image shows the velocity vector map superimposed over the streamwise velocity contours. ........... 39
Figure 12: The velocity vector map is superimposed over the streamwise velocity contours for three
cases. A reduction in spanwise flow, and hence a decrease in three-dimensionalities is found if a top
cover or endplate is used. a) Cylinder is bounded on top by the channel cover. b) Cylinder is
bounded on top by a sharp leading edge geometry endplate. c) Cylinder is bounded on top by an
elliptical leading edge geometry endplate. .......................................................................................... 40
Figure 13: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by the channel floor and the free-surface. Vortex splitting is seen in the
right most images of this figure. ......................................................................................................... 41
ix
Figure 14: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for the top field of view of a cylinder bounded by the channel floor and channel cover. Minimal
variation in the vortex filament angles are found for this configuration. ........................................... 42
Figure 15: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for the bottom field of view of a cylinder bounded by the channel floor and channel cover.
Slightly more variation is found than in the previous image, but it is still minimal compared to most
cases. ................................................................................................................................................... 43
Figure 16: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by a sharp leading edge endplate at L/D = 1 and the free-surface. ....... 44
Figure 17: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by a sharp leading edge endplate at L/D = 2.5 and the free-surface. .... 45
Figure 18: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by a sharp leading edge endplate at L/D = 5 and the free-surface. ....... 46
Figure 19: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by an elliptical leading edge endplate at L/D = 1.0 and the free-surface.
............................................................................................................................................................. 47
Figure 20: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by an elliptical leading edge endplate at L/D = 2.5 and the free-surface.
The fourth set of data for this configuration was corrupted. ............................................................... 48
Figure 21: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by an elliptical leading edge endplate at L/D = 5.0 and the free-surface.
............................................................................................................................................................. 49
Figure 22: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by a sharp leading edge endplate at L/D = 2.5 on the top and bottom. 50
x
Figure 23: Space-time plots of the streamwise velocity and their corresponding vortex filament angle
plots for a cylinder bounded by a sharp leading edge endplate at L/D = 5 on the top and bottom. ... 51
Figure 24: Probability density function of the vortex filament angle for a cylinder bounded by a
sharp leading edge endplate and the free-surface for various L/D values. ......................................... 52
Figure 25: Probability density function of the vortex filament angle for a cylinder bounded by an
elliptical leading edge endplate and the free-surface for various L/D values. .................................... 53
Figure 26: Probability density function of the vortex filament angle for a cylinder bounded by the
channel floor and the free-surface. ...................................................................................................... 54
Figure 27: Probability density function of the vortex filament angle for a cylinder bounded by a
sharp leading edge endplate on top and bottom for two L/D values. ................................................. 55
Figure 28: Probability density function of the vortex filament angle for a cylinder bounded by an
elliptical leading edge endplate on top and bottom for two L/D values. ............................................ 56
Figure 29: Vortex Splitting was observed in the space-time plots for all boundary conditions
analyzed. ............................................................................................................................................. 57
Figure 30: Comparing the stream-wise velocity signals for a point in the flow where a split has
occurred and one that has undergone normal shedding. The vortex split causes attenuation of the
velocity signal. .................................................................................................................................... 58
Figure 31: The stream-wise velocity signal in a window where there is a reduction in the spectral
density of the Karman Strouhal number shows attenuation similar to that found when a vortex split is
present. ................................................................................................................................................ 59
Figure 32: Bar Graphs displaying the number of occurrences of each range of Vortex Filament
Angles prior to a vortex split for cases in which the end configuration does not promote parallel
shedding. Note that the most likely shedding orientation prior to a vorte .......................................... 64
xi
Figure 33: Bar Graphs displaying the number of occurrences of each range of Vortex Filament
Angles prior to a vortex split for cases in which the end configuration promotes parallel shedding.
Note that the vortex filament angle takes on nearly all values before a vortex split .......................... 65
1
Introduction 1
1.1 Motivation and Background
Due to their practical importance, flow dynamics related to vortex shedding behind a bluff body has
been the subject of intense research across a variety of engineering fields. The main model used to
represent bluff bodies is traditionally a cylinder. Early researchers characterized the flow around
cylinders for a range of upstream flow conditions. The alternate shedding of vortices on the cylinder
was shown to lead to large pressure imbalances and subsequent vortex-induced vibrations (VIVs) on
the body. VIVs have negative effects on the structure, and can lead to a loss of structural integrity,
threatening the fatigue life of the system. Therefore being able to understand and control the near-
wake of a cylinder, and hence suppress the VIVs, have been the goal of many researchers.
The majority of the bluff bodies in practical applications, such as riser tubes, cables, towers, bridges
and chimney stacks, have a length much greater than their diameter, and hence can be rendered as
infinitely long. For such bodies, the wakes and the resulting VIVs are not influenced by the end
boundaries. However, in a finite experimental setup, where the effects from the end boundaries of
the body cannot be underestimated, a method to minimize the end effects is deemed necessary for
the experimental work to ensure that any concepts developed would not be limited to the specific
laboratory arrangement, but would have universal effectiveness. Consequently the primary goal of
this thesis is to examine how different end conditions applied on a cylinder affect its near wake, and
to generate a method to properly model an infinite cylinder in a laboratory experiment.
The first chapter of the thesis will provide an introduction to the topic and a summary of the previous
work related to this field of study. Chapter two will discuss the experimental setup and the different
analysis techniques employed. Chapter three includes the results and findings on test models with
different end conditions. Finally, chapter four concludes the paper and presents recommendations for
future work on the subject.
2
1.2 Literature Review
1.2.1 Flow Past an Infinite Cylinder
The mechanism of vortex formation was qualitatively explained first by Gerrard [1]. He suggested
that as a vortex forms and increases its strength from one side of the cylinder, it draws the shear
layer from the opposite side, across the wake centerline. Eventually, this opposite shear layer cuts of
the supply of vorticity to the growing vortex. This process repeats alternately between the two shear
layers, leading to the alternate shedding of Karman vortices.
A few key definitions are needed before progressing in the literature review: The first definition is
the formation length, FL, which is the length of the mean recirculation region in the near-wake of a
cylinder. This is a bubble shaped region, which is symmetric with respect to near-wake centerline.
The formation length can be identified as the point downstream of the bluff body where the velocity
fluctuations make a maximum. The second key definition is the base suction coefficient, -Cpb, which
is defined as the negative value of the pressure coefficient at the back of the cylinder.
The characteristics of the flow past a cylinder are shown to greatly depend on the Reynolds number
of the flow. Following is a breakdown of the different regimes (based on the Reynolds number) [2].
For small Reynolds numbers (Re < ~49), the flow is said to be in the laminar steady regime. In this
regime, the flow is time independent, and the mean recirculation zone is two symmetrical fixed
eddies. As the Reynolds number increases (Re = ~49 to ~190), the flow becomes unsteady and the
flow enters the laminar vortex shedding regime. Due to instabilities in the near-wake, a von Karman
vortex street is developed in this regime. With increasing Reynolds number (Re ~190 to 260), the
flow enters the wake transition regime. Here, three-dimensional characteristics are introduced to the
wake. Also, two discontinuities in the wake formation are found. These discontinuities are due to
two different instabilities, called as mode A and mode B, which appear to have hysteretic
characteristic. Up to this point, as the Reynolds number increases, the base suction coefficient, and
the Strouhal frequency increases, while the formation length decreases. At a Reynolds number of
approximately 260, there is a peak in Reynolds stresses in the near wake and the trends mentioned
above all reverse with an increase in Reynolds number. As Reynolds number increases from 260
until about 1,000, three-dimensional fine scale streamwise vortex structures become increasingly
3
disordered in the near wake while the boundary layer stays laminar. The next regime, known as the
sub-critical regime (or shear-layer transition regime), encompasses the Reynolds number values of
Re = 1,000 to 200,000. The experiments in this thesis fall within this regime. The main
characteristics of this regime are that, with increasing Reynolds number, fluctuation increases, base
suction increases, Strouhal number decreases, and the formation length decreases. These trends are
caused by the increased unsteadiness of the shear layers, separating from the sides of the body. In
addition, in this regime, small-scale vortical structures in the separating shear layers due to the
Kelvin-Helmholtz instability (also known as shear-layer instability) develop and additionally
increase the base suction and the Reynolds stresses. As the Reynolds number increases in this sub-
critical regime, the turbulence transition point (i.e., the onset location of the shear-layer instability)
moves upstream toward the surface of the body. So the turbulent transition occurs somewhere within
the separating shear layers. In a narrow Reynolds number range past 200,000, the turbulent transition
point moves upstream such that it enters the boundary layer on the cylinder surface. The boundary
layer becomes turbulent at the separation point, but this occurs at only one side of the cylinder, while
the boundary-layer separation remains laminar on the other side. As a result, a separation-
reattachment bubble forms on the side where turbulent separation occurs, causing an asymmetric lift
vector that can have quite a large magnitude. Turbulent boundary-layer separation switches in this
regime from one side of the cylinder to the other, causing a change in the lift vector direction
occasionally. The flow with these characteristics is said to be in the critical (or lower transition) flow
regime. Both the base suction and the drag decrease drastically in this regime due to a phenomenon
known as drag crisis. Past the critical regime, the flow enters the symmetric reattachment regime,
also known as the supercritical regime. In this regime the flow is characterized by turbulent
boundary-layer separation on both sides of the cylinder and the flow is symmetric with two
symmetric separation-reattachment bubbles. In this regime, the transition point in the boundary layer
is somewhere between the stagnation point and the separation point, that is, the boundary layer is
partly laminar and partly turbulent. Finally, at higher Reynolds numbers, the entire boundary layer
on the surface of the cylinder becomes turbulent. This is known as the boundary-layer transition
regime, or the post-critical regime.
4
1.2.2 End Effects on the Spanwise Uniformity of the Cylinder Near Wake
Eisenlohr and Eckelmann [3] showed that in the laminar vortex shedding regime, the oblique angle
of vortex shedding can be as high as 30o
. As the oblique angle further increased, the vortex would
split, in a phenomenon known as vortex splitting. While attempting to cause vortex splitting by using
larger diameter cylinders at the end of the original cylinder, they discovered that the flow had
become more parallel. Eisenlohr and Eckelmann postulated that the original oblique shedding is due
to the end of the vortex axes being curved by the horseshoe vortex (HSV) formed by the boundary
layer of the wall. This curvature causes strain on the entire vortex axes and leads to an oblique angle
being formed. Upon placing larger diameter cylinders at the ends, the vortex splits before it can be
curved by the horseshoe vortex, preventing any strain on the initial axes.
Eisenlohr and Ecklemann were among the first to observe vortex dislocations. They showed that the
Strouhal number (St) is not always constant along the span of a cylinder. This can occur for a
multitude of reasons, but is most likely due to non-uniformities in the oncoming flow. If there is a
large difference in St along the span, oblique shedding is formed. This may cause a phenomenon
known as vortex dislocation to occur if the shedding at the interface of two cells of different
frequency is out of phase. Eisenlohr and Ecklemann placed cylinders of slightly larger diameters on
either end of the test cylinder to ensure that there would be a jump in the frequency of shedding.
They observed vortex dislocations occurring at the beat frequency between the cylinders of different
diameters. This led them to believe that as the phase angle between the two shedding cycles grows in
magnitude a dislocation of the vortex filament is found, and due to the conservation of circulation
the filament must either be short-circuited with its counterpart or divide up its circulation among
neighboring filaments of the same vorticity. The latter option is referred to as vortex splitting. At
lower Re, Re<300, vortex dislocations are found only when oblique shedding is present and there are
cells of different frequency along the span.
Also within the laminar vortex shedding regime, Williamson [4] was able to manipulate the end
conditions of the flow, by introducing slanted endplates, so as to create a quasi-two dimensional
flow. In the absence of endplates, the end cells of the cylinder have a higher base pressure than the
midspan, which enlarges the vortex formation region, in turn reducing the shedding frequency.
Parallel shedding can be induced by decreasing the base pressure at the end cells; increasing the
5
shedding frequency near the ends to match that of the midspan. This allowed for the development of
a continuous Strouhal-Reynolds curve, which was shown to be reproducible in other experimental
facilities.
Williamson [4] also observed interesting interactions between cells of different vortex shedding
frequencies. His experimental setup was such that near the ends of the cylinder there was a lower
frequency cell found, due to non-ideal endplate use. It was found that if the phase angle between
cells is low the vortices of the low frequency cells get induced downstream by its neighboring cell.
As the phase angle increases, a critical value is hit and the vortex filament is found to dislocate. He
showed that the number of shedding cycles between splitting can be found to equal the ratio of
frequency at the mid span to the beat frequency. At this low Re, dislocations are shown to only occur
near the interaction of these cells and therefore always occur near the end cells at a near constant
spanwise location. This is shown to be the case for both free and fixed ends.
Hammache and Gharib [5], [6] were able to properly model an infinite cylinder in a novel method by
influencing the base pressure at the ends of the test cylinder. They placed cylinders orthogonal to the
flow and main cylinder. The cylinders were placed such that their center wake would have a base
pressure equal to that of the midspan of the test cylinder, ensuring that there were no pressure
gradients along the span, further illustrating that within this Re regime uniform flow can be induced
by matching the flow conditions at the end cells to that of the midspan.
At higher Re flows, flow in the sub-critical or shear-layer transition regime, the effect of outward
angled endplates in inducing parallel shedding was also found. Prasad and Williamson [7] performed
signal analysis on a variety of flow conditions, ranging from large angles of shedding to near parallel
filaments. For flow conditions that had minimal angle of shedding they found that signal analysis
had a single dominant peak at the von Karman shedding frequency, while oblique shedding cases
had a much broader peak.
As Re increases past the critical value of 5000, Prasad and Williamson [7] among others postulated
that vortex dislocations are inherent to the flow, regardless of end conditions. The vortex
dislocations were observed as attenuation to the stream-wise velocity signals, but due to the lack of
experimental technology, not with quantitative visualization. Experiments were performed under
6
near-ideal end conditions that allowed for a proper model of a uniform cylinder. They showed that at
a Re of approximately 5000, there was a discontinuous decrease of the St, an observation of twin-
peaks in the spectra of the frequency, and the inception of vortex dislocations. This appears to be due
to an inverse of the mode A to mode B transition that is found in the wake-transition regime, but is
only given as a theory.
Stansby [8] established the basic requirements for endplate use in the sub-critical regime by means
of base pressure measurements. The results showed that the endplates should be mounted outside of
the channel wall boundary layer, and should be positioned far enough upstream that the effects of
horseshoe vortices are reduced but not have such a large leading edge distance that the boundary
layer growth on the endplate itself will affect the flow. The recommended endplate position was
such that the distance from the leading edge of the endplate to the cylinder was between 2.5 and 3.5
diameters, and the distance from the trailing edge to the cylinder is 4.5 diameters.
Stager & Eckelmann [9] showed that cells of low frequency are found at the end sections of the
cylinder in the shear-layer transition regime as well. Similar to Eisenlohr and Eckelmann [3], these
low frequency cells are thought to be due to the interaction of Karman vortices with the HSV in the
boundary layer. As Re increases the size of these low frequency cells tends to decrease, and at a Re
of approximately 4800 this affected region is nearly negligible. Correspondingly, the ratio of
endplate size to cylinder diameter must increase as Re increases for any affected region to be
noticeable. Fox and West [10] had contradictory findings and stated that even at Re as high as 105
interference from the end effects can be found at a spanwise distance of 3.5D from the endplate.
Szepessy and Bearman [11] were able to show the influence of aspect ratio, the ratio of the length of
the cylinder to its diameter, on the effectiveness of endplates. It was shown that an increase in
fluctuating lift implies an enhanced spanwise correlation of the flow. They showed an increase in the
fluctuating lift for reduced aspect ratios in the Re range of 8 x 103 to 1.4 x 10
4. At a Re of 10
4 the
influence of aspect ratio between one and ten causes little or no effect on the midspan flow. They
also found that the free-surface introduces an element of three dimensionalities to the flow. This is
due to regular alternating shedding having higher fluctuating values then the mean flow around the
cylinder, and these higher Reynolds stresses lead to a higher base suction coefficient. The different
7
values of base suction coefficient lead to a span wise flow, and an increase in oblique shedding near
the free-surface
Szepessy [12] showed that any phase drift in the vortex shedding will cause instantaneous pressure
gradients along the span, disrupting the vortex shedding regularity. To generate ideal two-
dimensionality in shedding in the shear-layer transition regime a minimum leading edge distance
between the cylinder and endplate must be 1.5D while the trailing edge distance must be at least
3.5D. For different Re regimes, the trailing edge distance must always be longer then the vortex
formation region to ensure a uniform base pressure along the cylinder. Szepessy further shows that
for proper endplate use, there is a minimal pressure gradient two diameters downstream of the
cylinder in the spanwise direction.
Szepessy [13] used a range of pressure sensors along the span of the cylinder to determine if there
were any spanwise deviations. He was able to show that for proper endplate use in the subcritical
regime, the phase angle of the vortex filaments, hence the vortex filament angles, varies but the
distribution is centered along zero phase difference and has a Gaussian like distribution. This was
only tested for one endplate configuration, and examining the distributions gathered from a variety
of endplate usages would allow for the discovery of an optimal endplate configuration. Furthermore
he found that a disturbance in the flow causes attenuation of the velocity signals, and what appears to
be vortex splitting in the following shedding cycles.
Norberg [14] set out to determine the minimum aspect ratio requirements so as to ensure that the
midspan of the cylinder was not being influenced by the end conditions for multiple Re regimes. He
was able to give both Strouhal and base-pressure coefficient curves versus Reynolds number for the
case of an infinite cylinder. Norberg states that to properly represent an infinite cylinder at the
midspan, an aspect ratio of 60 is needed for 4*103<Re<10
4, and as Re increases to the range of
104<Re<4*10
4 an aspect ratio of only 25 is needed.
Norberg proposed that due to the discontinuity at Re of 5000, the subcritical regime should be
broken into two parts; the lower subcritical regime (260 < Re < 5000) and upper subcritical regime
(5000 < Re < 2 x 105). Norberg [14] showed that the discontinuity is present for a variety of aspect
ratios, showing that it is not dependent on end conditions. This supports the thought that the presence
8
of vortex dislocations is a fundamental feature of the flow in this Re regime. Along with Prasad and
Williamson[7], both Norberg and Szepessy [13] showed the presence of dislocations in this Re range
for near uniform shedding.
This thesis will further examine the unsteady nature of vortex shedding in the sub-critical regime
proposed by Szepessy [13], by means of visualization of the near-wake. Furthermore multiple end
configurations will be analyzed as opposed to just the single case studied by Szepessy, allowing for
insight into how different end conditions impact the flow past a cylinder. Finally the ability to
quantitatively visualize the flow in the near-wake will allow for the confirmation of the presence of
vortex splitting in the sub-critical regime. This will help confirm the findings of previous authors
[7], [13], [14], as well as further the study into the formation and characteristics of the phenomenon.
9
Experimental Setup and Analysis Techniques 2
2.1 Experimental Setup
2.1.1 Water Channel
Data acquisition was undertaken in the experimental fluids research laboratory at the Institute for
Aerospace Studies at the University of Toronto. The experimental models were placed in a state-of
the-art recirculating water channel. The main test section extends 5 m in in the horizontal direction,
and has a cross-section of 610 mm by 686 mm. The channel has a flow speed controller, a returning
plenum, a settling chamber composed of a honeycomb and a set of screens, and a 6:1 contraction
section. This channel can provide continuous flow in the horizontal direction either in free-surface
mode or through the placement of top covers in fully-covered mode as a tunnel. In free-surface
mode, free-surface turbulence intensity was less than 0.5% and the flow uniformity was better than
0.3%. When the top covers of the channel were in place, it achieved a turbulence intensity of less
than 0.4% and a flow uniformity of better than 0.1%. The range of Reynolds numbers examined was
4x103
- 36x103 based on the cylinder diameter. The water temperature was shown to vary from 18 °C
to 24 °C, depending on the outside temperature. To avoid fluctuations in the value of the Reynolds
number due to such temperature changes, the temperature of the water was measured frequently over
the course of experiments and the free-stream velocity, uo, was adjusted accordingly.
2.1.2 Test models
The cylinders were mounted in the vertical orientation inside the water channel, equidistant from
both side walls. Each cylinder was fixed to a traverse outside of the water channel, which ensured
that there was no cylinder vibration. The length of the cylinder in which shed vortices will be
investigated is denoted as S and D represents the diameter. The cylinders used in the experiments
have a diameter of D = 50.8 mm. This gave a maximum aspect ratio S/D of 13.5 for the water
channel being used, which is below the minimum aspect ratio to be able to neglect end effects [14].
A cylinder of diameter of 50.8 mm is selected to ensure that the flow dynamics of the wake can
properly be visualized with the vector resolution available from PIV.
10
A right handed Cartesian coordinate system will be used for all the experiments presented in the
thesis. The origin of the axis will be at the center of the bottom of a cylinder with no endplate, and
the x, y and z axis represent the stream-wise, transverse and spanwise directions respectively.
To explore the effect of various end conditions on the wake three-dimensionality, different end
boundaries were designed for cylinder models. For those involving an endplate boundary,
consideration was given to two different leading-edge shapes: One involved a sharp leading edge
with a bevel angle of 23.6° and the other had a super-elliptical nose shape with an axes ratio of 6.
The latter shape adopted with an axes ratio of 6 and above was reported to result in laminar
boundary layer along the plate by Narasimha and Prasad [15]. All end plates had a thickness of 12.7
mm, a total length of 7.5D (following the recommendations of Stansby [6]) and a width equal to the
entire width of the channel 12D. The different end configurations considered in the present
investigation are sketched in figure 1. They are as detailed below:
a) A cylinder bounded by the channel floor at the bottom end and the free-surface at the top.
(S/D = 13.5)
b) A cylinder bounded by the channel floor at the bottom and a top-cover at the top. This is also
referred to, in the text, as the wall-wall boundary condition. The top-cover was designed to
have an opening for the cylinder to pass through. (S/D = 13.5)
c) A cylinder bounded by the endplate with a sharp leading edge at the bottom and the free-
surface at the top. As indicated above, the bevel angle of the sharp leading edge was 23.6o.
(S/D = 12.3)
d) A cylinder bounded by the endplate with an elliptical leading edge at the bottom and the free-
surface at the top. As indicated, the axes ratio of the elliptical nose shape was 6. This
elliptical leading edge was found to prevent flow separation at the leading edge of the
endplate by Blackmore [16] (S/D = 12.3)
e) A cylinder bounded by endplates at both ends. This scenario involved the use of two
endplates with the sharp leading edge, bevel angle of which was 23.6o. (S/D = 10.8)
f) A cylinder bounded by endplates at both ends. Both endplates used in this case had the
elliptical leading edge. (S/D = 10.8)
11
The cylinder was mounted approximately 21D downstream from the entrance of the test section in
each experiment. At this location, the boundary layer forming along the channel floor, with no
cylinder present, was determined to be 0.25D at a Reynolds number value of 104. For those
configurations where an end plate was used at the bottom end, the endplate was lifted such that the
bottom end of the cylinder was 1.25D way from the channel floor to ensure that there would be no
direct influence from the channel floor’s boundary layer. As for the configurations in which a top
plate was used, the end plate was placed such that the end of the cylinder had a distance of 1.5D
from the free-surface, which corresponds to a location of 0.87 z/S. This is a sufficient distance from
the free-surface as Farivar [17] found that the maximum in fluctuating pressure due to free-surface
effects occur closer to the free-surface, at a z/S value of 0.95.
Let L designate the distance between the leading edge of an endplate and the center of the cylinder.
For the configurations where endplates were employed, multiple L/D values were tested to
determine the significance of the cylinder location on the end plate. For situations in which the
cylinder was bounded by one endplate at the bottom and the free-surface at the top, L/D values range
from 1 to 6. For the case of the cylinder being bounded by endplates on both the top and the bottom,
the range of L/D values examined was 2 to 6 due to restrictions on the ability to properly fix the
cylinder between the two endplates.
2.1.3 Particle Image Velocimetry
Quantitative visualization of the flow in the wake region will be obtained via a cinema technique of
Particle Image Velocimetry (PIV). There are four main components of a PIV setup: a CCD camera, a
double-pulsed Nd:YAG laser system, a data acquisition computer equipped with the appropriate
frame grabber, and a synchronizer. Tracer particles that are made up of small glass beads with
neutral buoyancy and diameter of about 14 microns are mixed into the water channel. A set of lenses
(cylindrical and spherical) are used to transform the laser beam to a laser sheet to illuminate the
tracer particles as they pass through the visualization plane of interest. The thickness of the laser
sheet was kept constant for all experimental configurations. The camera is setup perpendicular to the
laser sheet. Both the digital camera and the laser system are connected to the synchronizer and the
computer. The camera provided a magnification factor of 4.7 pixels per millimeter and a spatial grid
resolution of 0.67D was used for all experiments. Particle images are captured in pairs, and the
12
cross-correlation of the images in each pair through software named INSIGHT provides the velocity
vector field. A recursive Nyquist grid algorithm compared the two images that were captured at a
specified time difference to develop a flow field in pixels per second for each pair. The Δt between
pairs was chosen carefully to optimize the output of the algorithm by ensuring there was the
appropriate average displacement of particles for each set of frames. After processing the images
with INSIGHT, a second program named CleanVec was used to remove any spurious vectors
produced by the algorithm. Finally, a post-processor was used to convert the units of the velocity
vectors to mm/s as well as positions to mm as opposed to pixels.
As outlined by Raffel et. al. [18] to compensate for the out-of-plane loss of particle pairs the laser
light sheet was arranged to have a thickness of 1 mm. The PIV system employed can provide only a
coarse frequency resolution as opposed to the constant temperature anemometers, which are
described in the following section. This bottleneck of the PIV system is due to both the limitation of
the capturing time over a sequence due to the available RAM memory of the computer, and the low
sampling rate. That is, the PIV system used has a maximum sampling frequency of 14.5 Hz, and is
limited to acquiring 200 sets of images per experiment. Hence, the frequency resolution of the data
acquired by this system is 0.07 Hz. Despite the disadvantage of coarse frequency resolution, PIV
measurements provide a non-intrusive method for quantitative visualization of the global flow
features. With the knowledge of the average shedding frequency at different Re, given by [14], an
optimal sampling frequency of data acquisition for each experimental Re was calculated as shown in
the table 1. The sampling frequency for a given Re was chosen such that enough data points were
sampled per shedding cycle while also ensuring that a large enough number of shedding cycles was
observed.
Table 1: Acquisition details for PIV measurements
Re x 103
Sampling
Frequency
(Hz)
Δt between
pairs of images
( x10-3 seconds)
Shedding Cycles
Observed
Samples per
shedding cycle
13
The orientation of the camera was such that the observable plane of interest in the wake had a
normal in the transverse (y) direction, that is, its field of view covered a spanwise-streamwise (z-x)
plane. For each experimental configuration, quantitative visualization of this flow field was
performed at two camera elevations: one of them covered the top half of the cylinder wake and the
other covered the bottom half. This allowed for increased vector resolution as opposed to observing
the entire span at once. The field of views (FOV) was setup so that there was a slight overlap region
and had a size of 6.75D by 4.5D in the spanwise and stream-wise directions respectively, as seen in
figure 1. The amount of overlap depended on the specific experimental configuration, and varied
slightly to ensure that the vector resolution was kept nearly constant regardless of the experimental
setup.
Experiments were performed in two spanwise-streamwise planes: namely, the mid-plane and side-
plane of the cylinder as illustrated in the sketch of figure 2. In mid-plane (y/D = 0), the illuminated
plane in the wake was aligned to be coincident with the spanwise plane of symmetry of the cylinder
wake, whilst in side-plane (y/D = 0.5), the illuminated plane was half a cylinder diameter offset in
the transverse direction from that in the mid-plane experiments. PIV measurements in the mid-plane
provided the time-averaged recirculation region, giving a value for formation length, along the entire
span. The streamwise velocity contours, obtained from the side-plane PIV experiments, incorporated
the signatures of the vortices, as illustrated in figure 3. That is, the vortex filaments induce negative
and positive streamwise velocity components in the side-plane, allowing the identification of vortex
filaments forming in the wake.
10 14.5 2.2 10.2 19.6
20 14.5 1.1 19.4 10.8
30 14.5 0.80 29.4 6.3
36 14.5 0.74 34.5 5.8
14
2.1.4 Constant Temperature Anemometry
Constant Temperature Anemometers (CTA) were also used to acquire stream-wise velocity signals.
CTA can achieve higher sampling rates than PIV and allows for longer sampling time as the memory
requirements are much smaller than PIV. On the other hand, each probe used was limited to
acquiring the streamwise velocity signals at only one point in the flow as opposed to the global
information obtained from PIV and the probe causes a slight disturbance due to the fact that it is an
intrusive measuring technique.
A constant temperature anemometer aims to keep the temperature in the probe’s wire constant by
altering the voltage across the wire based on the flow speed of the water. The practice is based on the
convective heat transfer of the wire in the moving fluid. Determining the voltage needed to maintain
the constant temperature at known speeds gives a method of creating a calibration curve between
voltages and velocities. This curve allows for a continuous one to one conversion for voltage
obtained from the anemometer to a velocity value.
Experiments using two CTA probes were performed simultaneously to determine how the
streamwise velocity signal varied with spanwise location. Both of the probes were placed at a
streamwise location 3.5D downstream of the cylinder center axis (x/D = 3.5). The spanwise
locations of the two probes were chosen to be 3D apart, 1.5D above and below the mid-point of the
span (z/D = 8.25 and z/D = 5.25 respectively). The two probes were placed off of the same shoulder
of the cylinder, at transverse locations of y/D = 2.25 and y/D = 2.75. A summary of the locations of
the two probes can be seen in Table 2. These probes were placed away from the vortex formation
region, allowing for the effects of the vortex shedding to be observed without introducing too much
noise to the signal. Ideally the two probes would have the exact same transverse distance but it is
unfeasible in design. The offset in transverse location will cause a difference in the amplitude of the
velocity signals, but will not factor into frequency analysis with respect to dominant frequencies.
Experiments were performed with the two probes at the same spanwise location to determine the
offset in phase angle due to the difference in transverse probe location. The phase angle difference
between the two probes when they are at the same spanwise height was calculated to equal 0.015π
which correlates to a vortex filament angle error of 1.6o when the probes were placed 3D apart.
15
The CTA systems used were from Dantec Dynamics. The probes were connected to a MiniCTA
system, which in turn was connected to the computer via a shielded connector block from national
instruments (NI BNC 2110). This allowed for acquisition of stream-wise velocity signals from
multiple probes. The characteristics of each probe are given in Table 2. The probes were placed in
probe holders supplied by Dantec Dynamics (part 55H22) that had a 90° bend to allow for
acquisition of streamwise velocity signals while being mounted to the same traverse that the cylinder
is mounted to. CTA data was not possible with the top channel covers in place because there was no
opening to mount the probe holders. Likewise, for end boundary configurations where a top end-
plate had a small leading-edge distance (L/D < 3), CTA measurements were not possible due to the
long trailing distance of the endplate preventing the probe from reaching the desired downstream
location.
Table 2: Probe Characteristics
Probe 1 Probe 2
Streamwise location (x/D) 3.5 3.5
Transverse location (y/D) 2.25 2.75
Spanwise location (z/D) 8.25 5.25
Sensor Resistance 7.23 6.06
Sensor Lead Resistance 0.5Ω 0.5Ω
Support Resistance 0.42Ω 0.42Ω
Cable Resistance 0.15Ω 0.15Ω
Sensor TCR 0.38%/K 0.38%/K
16
Desired Sensor Temperature 40C 40C
Overheat Ratio 0.08 0.08
2.2 Analysis Techniques
2.2.1 Time-Averaged Recirculation Region
Velocity vector fields were acquired in y/D = 0 plane, i.e., the mid-plane. The span of the cylinder
was broken into two planes (upper and lower mid-planes) so as to increase the vector resolution. 800
sets of data, corresponding to roughly 40 shedding cycles, were ensemble averaged for each field of
view, and then merged to obtain an average velocity vector field along the entire span. The border
between the negative streamwise velocity vectors, i.e., the recirculation zone, and the positive
streamwise velocity vectors, defined a spanwise demarcation line over the field, as illustrated in
figure 4. The demarcation line quantifies how the formation length (FL) varies along the span; an
indicator of how two-dimensional the flow is. For ideal shedding from an infinite uniform cylinder,
there would be no end dependence, and the average recirculation region length would be near
constant along the span, yielding a demarcation line that would be parallel with the cylinder.
Correspondingly, an oblique shedding would show a demarcation line that is angled at the average
shedding angle of the filament. As the demarcation line is determined from the average streamwise
velocity vectors, it hides any changes in the angle of shedding with time. Therefore this analysis
technique is especially useful for analyzing cases in which non-parallel shedding is being observed,
but observing an averaged parallel demarcation line does not necessary indicate that the shedding is
parallel at every instantaneous moment.
Previous analysis of the experimental flow conditions showed that the velocity vectors obtained by
PIV measurements are accurate up to 2% of the free-stream velocity. To calculate the streamwise
location of the zero streamwise velocity in the time-averaged field along the span, that is, <u>/uo =
0, an interpolation was used between the negative streamwise velocity values closest to zero, to the
smallest positive streamwise velocity values. Both points were chosen such that their absolute value
17
was greater than the PIV measurement error in velocity. This ensured that the interpolation was
always performed between negative and positive streamwise velocity values. The error in the
determination of demarcation line, or in other words the error in formation length (∆FL), was
therefore dependent on the free-stream velocity and hence the Reynolds number. ∆FL was calculated
by determining the possible streamwise location values of <u>/uo = 0 based on the errors in
streamwise velocity. The table below shows the error obtained in formation length as a function of
Reynolds number.
Table 3: Error in calculated recirculation region as a function of Reynolds number
Re 10000 20000 30000 36000
∆FL /FL 0.082 0.089 0.094 0.097
2.2.2 Space-Time Plots
Space-time plots of contours of streamwise velocity, where the horizontal axis is the time axis and
the vertical axis is the spanwise direction, were created from the PIV data acquired on the side-plane
of the cylinder (y/D = 0.5 plane). In other words, these plots showed the contours of streamwise
velocity in the z-t plane. To construct these plots, an appropriate streamwise location (x = xo) was
chosen at the y/D = 0.5 plane (side-plane), and the streamwise velocity (u) vectors along the entire
span (z direction) were extracted for every instant in the PIV data sequence. The streamwise
coordinate (xo), was chosen for each end boundary configuration studied in the present work such
that key features of the flow would be illustrated by watching compilations of all the instantaneous
streamwise velocity contours in movie mode. The xo/D values used to produce such space-time plots
were in the range of 1.8 - 2.5. Iso-contours of streamwise velocity were then plotted over the grid
region formed by spanwise direction and time as shown in figure 5. These space-time plots of
streamwise velocity contours are limited to a single field of view and cannot be merged to obtain an
entire spanwise view because the data was not acquired simultaneously for both the top and bottom
half of the span. The space-time plots are very useful in observing the dynamics of the vortex
filaments, and specifically the angle of the vortex filaments being shed from that shoulder of the
cylinder. For clarity purposes only the vortex filaments shed from the shoulder of the cylinder
18
closest to the plane of data acquisition are shown in the present work. Furthermore, the space-time
plots of streamwise velocity contours clearly illustrated where vortex splitting was present. This was
very helpful in creating detection algorithms for vortex splitting.
2.2.3 Continuous Wavelet Transformation
Continuous Wavelet Transformation (CWT) is a useful method for performing frequency analysis on
a signal that may not be consistent in time, either in dominant frequency or in amplitude. CWT aims
to match a specific wavelet to the signal in different windows in such a way that a very good
temporal and frequency resolution is obtained [19]. The wavelet is chosen such that it matches the
overall signal as much as possible. For analysis of the velocity signal, a complex Gaussian wavelet
of order three was used. Using a complex wavelet was beneficial in that it allowed for obtaining
phase angle values along the span of the cylinder in time.
CWT was performed on the streamwise velocity signals obtained via PIV on the side plane (i.e., y/D
= 0.5) in the present study. The phase angle variation (∆ϕ) of all the grid points were determined
along the span at the same xo coordinate where the space-time plots were constructed. This analysis
was done on instantaneous plots of the streamwise velocity for each filament observed in the space-
time plots. The phase angle difference (∆Φ) between velocity signals gives a value for how much
one signal is lagging or leading the other. This phase lag or lead can be converted to give a value for
streamwise distance between the peaks in the velocity signals, and hence the streamwise
displacement of the vortex filament between the two probes. Based on figure 5 and figure 6, the
convection speed of the vortex filaments was calculated to be nearly 0.8u. This allowed for the
distance between vortex filaments of the same sign to be calculated as 80% of the free-stream
velocity multiplied by the period of shedding. Therefore the streamwise displacement of the vortex
filament based on the phase difference is given by:
19
A line of best fit on the streamwise displacement of the vortex filament yields the average linear
shape of the filament, giving the average angle (θ) of the vortex filament (see the right side of figure
6). This gives a linear approximation for the angle of the filament, which may hide some of the
features of the filament orientation but gives a good indication of its shape. θ is defined as positive
for counter-clockwise rotations away from the vertical axis. The vortex filament angle was
calculated separately for each vortex filament in the near-wake, and as such the results are presented
for each filament observed.
2.2.4 Fast Fourier Transform and Short-Time Fourier Transform
Velocity signals, obtained from the two constant temperature anemometers in the wake, as explained
in the previous section, had a nearly sinusoidal shape due to the induced velocities from the alternate
shedding of vortices. Fast Fourier Transformation (FFT) of the signals converted this signal from the
time domain to the frequency domain. This was useful in illustrating which frequencies were
dominant in the shedding, as well as their respective amplitudes. The frequency resolution is given
by the ratio of the sampling frequency to the number of samples obtained. Due to the ability to
sample for extended time, 45000 data points were acquired at a sampling frequency of 50Hz, which
allowed for a frequency resolution of 1.1 x 10-3
in FFT analysis.
To observe if the dominant frequency was changing in time, Short-Time Fourier Transformations
(STFT) were applied to the stream-wise velocity signals. STFT breaks the signal into smaller
sections of signal, or windows, before performing the FFT analysis. This allows for observation of
how the frequencies and phases of the vortex shedding changes in time. The size of the window in
STFT, that is how many data points within the window are present, must be chosen to obtain
appropriate temporal and frequency resolution. To obtain high temporal resolution in STFT, a small
window size must be chosen. However, having a small window size reduces the frequency resolution
due to the fact that resolved frequencies are discrete values. Therefore, it is important to ensure that
the window size is chosen such that the dominant Karman frequency is resolved. Otherwise, the
frequency data will have noise that is contributed from the analysis technique.
20
2.2.5 Time Evolution of Phase-Angle Difference Between Two Probes
As explained in section 2.1.4, dual-CTA experiments were performed to acquire simultaneous
velocity signals from two different spanwise locations in the flow. By calculating the phase
difference between these points, it is possible to quantify how oblique the shedding is relative to the
span of the cylinder. The phase angle between the two signals was calculated as a function of time
by means of STFT. The window size for the STFT analysis was chosen such that two shedding
cycles were observed in a single window. The sampling frequency was determined such that the
Karman frequency would be resolved within the window, that is:
Where Fs is the sampling frequency (48Hz), N is the window size (128), Fk is the Karman shedding
frequency (0.75) and the factor 2 ensures that two shedding cycles are present within the window.
The phase angle difference (∆Φ) corresponding to FK between the signals within each window was
calculated and recorded. Due to the fact that ∆Φ was shown to change rapidly in time, each
successive window was chosen to overlap with the previous window. This overlap region covered
7/8 of the window size to yield more continuous phase angle variation in time.
The phase angle difference between the two streamwise velocity signals gives a value for how much
one signal is lagging or leading the other. This can be converted to a streamwise displacement value
as seen in section 2.2.3, and the linear approximation of the vortex filament angle, for the case of
two hot-wire probes that are 3D apart is therefore given by:
(
)
Time evolution of the vortex filament angle (θ) was used to generate the probability density function
of how likely each vortex filament angle was. This distribution was nearly a Gaussian curve for all
boundary configurations investigated, and as such, the middle portion of the curve would indicate
the θ values that dominate in time.
21
All ∆Φ values obtained by STFT analysis were between -π to π. The true value for ∆Φ can be 2π
plus or minus the ∆Φ calculated by the algorithm due to the fact that it is not possible to determine
which direction the phase lag/lead is occurring. To resolve this issue, one experiment setting the two
probes a small spanwise distance (1D) apart was conducted to see what the maximum vortex
filament angle in time is at this Re. For this small separation, it is known that ∆Φ will always be
between -π and π because this corresponds to a vortex filament angle of -63o to 63
o and such large
oblique angles are not possible behind a cylinder. The maximum ∆Φ measured between the probes
that are 1D apart was converted to a maximum vortex filament angle θmax. The θmax was found to be
44.5o, which is constant along the span within a vortex filament such that θmax is the same for both
1D and 3D probe separations. Therefore, this same θmax value can be used to determine the
maximum magnitude of ∆Φ for experiments in which the probes are separated in the spanwise
direction by 3D; ∆Φmax was found to be 1.53π. From there, the correction in the ∆Φ value was done
such that the ∆Φ was allowed to increase or decrease by 2π if the difference in ∆Φ from the previous
value in time would decrease, so long as the absolute value of ∆Φ was less than ∆Φmax calculated
above.
The topic of vortex splitting will be discussed in more detail in section (3.2.3), but for illustrative
purpose, the top right image of figure 13 shows a space time plot of the streamwise velocity that
displays the phenomenon of vortex splitting. The vortex split is shown to initiate at a normalized
time of approximately 25, and the merging of filaments, or hence forth known as branching, is
observed in the subsequent filaments. After the vortex split, a different amount of filaments are
found above and below the spanwise location of the dislocation. The differing amount of vortex
filaments means that during a vortex split, and the corresponding branching, the vortex filament
angle cannot always be calculated with certainty. This is due to the fact that a filament above the
dislocation is connected to two different filaments below the split. Therefore during a split two
different vortex filament angles could be calculated, one with a positive oblique angle and the other
being negative. This uncertainty in θ during regions of splitting leads to variability within the sign of
oblique angles of shedding, and affects the tail regions of the PDFs generated for the different
boundary conditions.
22
Figure 1: The various boundary conditions investigated. a) A cylinder bounded by the channel floor and free-
surface. b) A cylinder bounded by the channel floor and the channel cover. c) A cylinder bounded by a sharp
leading edge geometry endplate on the bottom and the free-surface on top. d) A cylinder bounded an elliptical
leading edge geometry endplate on the bottom and the free-surface on top. e) A cylinder bounded by a sharp
leading edge geometry endplate on both the top and bottom. f) A cylinder bounded by an elliptical leading
edge geometry endplate on both the top and bottom.
Note that FOV in the figure designates the field of visualization and is marked with dotted rectangular
regions, S shows the spanwise length of the cylinder, D shows the diameter of the cylinder and L shows the
distance of the cylinder center from the leading edge of the plate.
23
Figure 2: Schematic detailing the plane on which PIV experiments are being
performed.
Figure 3: Vortices shed off the shoulder of the cylinder induce positive and negative variations in the free-
stream velocity, allowing for visualization of the vortex filaments. The bottom image shows an
instantaneous streamwise velocity contour plot obtained via PIV on the side-plane. Positive sign vortex
filaments are seen as red, while negative sign filaments are seen as blue-green.
25
Figure 5: Generation of space-time plots of contours of the streamwise velocity in the z-t plane. The z axis (spanwise direction) is
normalized by D and time t is normalized by D/uo. The plot is constructed from the time trace of streamwise velocity signals
obtained in the y/D = 0.5 plane, at a chosen streamwise location (designated as xo) along the entire z direction. The spanwise line
along which the streamwise velocity signals were extracted as a function of time is shown in the figure with the line vector .
26
Figure 6: CWT was used to calculate the phase angle variation along the span based on the instantaneous streamwise
velocity field in the plane of y/D = 0.5. The calculated phase lag along the span is converted to a streamwise distance
(right figure), and the slope of the linear approximation to the curve is converted to an approximation for the vortex
filament angle (θ).
27
Results 3
3.1 Time-Averaged Characteristics of Vortex Shedding
To assess how different end boundary conditions affect the spanwise uniformity of the near wake of
a cylinder in time-averaged sense, velocity vector fields obtained via PIV on the mid-plane of the
cylinder (y/D =0 plane) were analyzed over the range of Reynolds numbers from 104 to 3.6x10
4. As
explained in the preceding chapter, the entire spanwise field of visualization was divided into two
separate regions with a slight overlap to obtain increased vector resolution (see section 2.1.3 for
details), and 4 sets of 200 image pairs were acquired and ensemble averaged for each region in order
to compute the time-averaged streamwise velocity distribution, from which the demarcation line
between the recirculation flow and the downstream flow was calculated along the length of the span
(see section 2.2.1 for details).
The demarcation lines at four different Reynolds numbers are given in figure 7 for the case of a
cylinder bounded by the channel floor at the bottom and the free-surface at the top. Inspection of
these lines show that, except for Re = 104, the demarcation lines at all Reynolds numbers depict
significant spanwise non-uniformity, degree of which increases as the flow Reynolds number
increases. The change in the time-averaged recirculation region becomes more dramatic toward the
free surface. It can, therefore, be concluded that the free-surface boundary condition influences the
spanwise uniformity of the flow greatly at higher Reynolds numbers. Taken as a whole, figure 7
suggests that the flow under the presence of free surface becomes more and more three-dimensional
as the Reynolds number increases.
For cases in which the top end of the cylinder is bounded by either a channel wall or an endplate, the
spanwise uniformity even at higher Reynolds numbers is greatly improved. This inference can
clearly be seen from an inspection of figures 8 to 10, where the time-averaged recirculation length
along the span are given at four different Reynolds numbers for the following end conditions: (i) a
cylinder bounded by the channel floor at the bottom and the channel cover at the top (figure 8), (ii) a
cylinder bounded at both ends by endplates having the sharp leading edge, where the distance
between the leading edge of the plate and the cylinder axis is L = 2.5D (figure 9), and (iii) a cylinder
bounded at both ends by endplates having the elliptical leading edge; for which again the distance
28
between the leading edge and the cylinder is kept at L = 2.5D (figure 10). These plots do not
demonstrate a significant increase in formation length near the top of the cylinder, unlike the free-
surface end condition.
What we have seen so far is that the free-surface boundary condition disturbs flow past a cylinder
significantly at high Reynolds numbers. In figure 11, for a cylinder at Re = 36x103 with free-surface
boundary condition, the time-averaged velocity vectors are superimposed over the time-averaged
streamwise-velocity contours near the free surface. Also, in figure 12 (a) to (c), corresponding plots
are shown for the cases where, at the top boundary of the cylinder, the channel wall, the endplate
with sharp the leading edge, and the endplate with the elliptical leading edge are employed
respectively. Note that for the cases where the top boundary is an endplate, a similar endplate was
also placed on the other side of the cylinder to keep symmetry in boundary condition, and the
cylinder axis is kept at a distance L = 2.5D from the leading edge of both the top and the bottom
endplates. For the case of a cylinder bounded by the free surface on its top (figure 11), there exists a
large downward flow from the water-air interface, while for the configurations where a top wall or a
top endplate is present (figure 12 (a) to (c)), there is no appreciable flow in the spanwise direction.
The downward flow observed in the case of the free-surface type boundary can be attributed to the
large pressure gradient between the suction region at the base of the cylinder and the ambient air. As
Re increases, the base suction at the rear of the cylinder increases, and causes a large dip in the water
height near the rear-top of the cylinder. This decrease in water height and the downward flow from
the free-surface were clearly observed even by bare eyes during the data acquisition process. It is this
downward flow that influences the time-averaged recirculation length near the free surface (i.e., the
demarcation line) and introduces a great spanwise non-uniformity in the near-wake, leading to a
condition that does not properly model an infinite cylinder. Therefore, at higher Reynolds numbers,
it is clear that either a top wall or a top endplate is needed to prevent free-surface effects.
It was observed that at a Reynolds number of 104 all experimental configurations studied in the
present work, even those with a free-surface condition, show a demarcation line that is nearly
parallel to the spanwise axis of the cylinder, implying that the time-averaged vortex shedding at this
Reynolds number is quasi-two dimensional. Although this may be the case, averaging the velocity
vectors in the near-wake greatly hides many of the key details of the flow. The next sections will
29
focus on the Reynolds number of 104 and show that the orientations of the vortex filaments at this
sub-critical Reynolds number have an unsteady nature, and vary in time from being near parallel to
largely oblique for the same boundary conditions. Acquisition of 4 sets of 200 image pairs via PIV
provides a total of 40 shedding cycles at this Reynolds number. As we will see in what follows, 40
shedding cycles do not cover all possible vortex-filament alignments, and as such the time-averaged
PIV results discussed in this section are not fully converged. Furthermore, even if a much larger
sample of PIV data were to be time-averaged, a nearly parallel demarcation line might result in if the
oblique angles in one direction cancel the angles in the opposite direction. Attention is, therefore,
directed in the following section towards the unsteady features of the vortex filaments in order to
investigate the variations of their alignment in time and how often the shedding is oblique compared
to being parallel.
3.2 Unsteady Nature of Vortex Filaments
3.2.1 Visualization of Vortex Filaments
To observe how the orientation of vortex filaments in the near wake changes in time under different
end boundary conditions, velocity vector fields were obtained by PIV on the side-plane of the
cylinder (y/D =0.5 plane) at a Reynolds numbers of 104
for a variety of end configurations. In a
similar manner to the mid-plane PIV experiments presented in the preceding section, the entire
spanwise near-wake field was divided into two separate visualization regions with a slight overlap to
obtain increased vector resolution, and 4 sets of 200 image pairs were acquired for each field of
view. The temporal evolution of the velocity vector fields on the side plane were then used to create
space-time plots of the streamwise velocity component and to estimate the variation of the vortex
filament angle (θ) for each end boundary configuration (see section 2.2.2 and 2.2.3 for details on the
construction of these plots). In figures 13 to 23, the space-time plots of streamwise velocity contours
for four separate sets of PIV data are provided on the top row, and the average vortex filament
angles (θ) corresponding to each filament are given on the bottom row. The space time plots in these
figures display a spanwise field of view only in the top half of the cylinder near wake. The spanwise
field of visualization in the bottom half of the near wake were also studied and were found to show
analogous characteristics. Hence, in order to avoid repetition, only the patterns from the top half of
30
the visualization field are reported in the present work. Nevertheless, the general characteristics
observed were pertinent over the entire spanwise near-wake region behind the cylinder.
Overall examination of figures 13 to 23 show that for all boundary conditions examined, the
orientation of the vortex filaments in the near wake is time dependent. For a given end condition, the
vortex filament is parallel to the cylinder span at one instant in time, while it becomes oblique at
another instant. Furthermore, the direction of obliqueness also largely differ in time, that is, θ values
plotted in the bottom row show a change in time from being positive to negative.
Interestingly, the trends observed for the vortex filament angle in figure 14 is different from the rest:
vortex filament angles are near θ = 0° and show little variability with time for the particular case
where the cylinder is bounded by the channel floor on its bottom end and the channel cover on its
top. To examine if this lack of variability in vortex filament angle is due to the low sample size of
PIV measurements, figure 15 shows the space-time plots of streamwise velocity and the average
angle θ of each filament as a function of time for the bottom half of the cylinder near wake. The θ
values show slightly larger variability in this visualization field, but not to the same level of
variations as shown in other end configurations. Due to the inability to place CTA probes through
the top channel cover however this boundary configuration was limited to only PIV measurements.
Therefore it was not possible to tell whether this lack of variability in vortex filament angle is due
purely to the low sample size obtained by PIV compared to the CTA results to be discussed in the
next section, or is a feature of the flow for this end configuration.
The space-time plots of streamwise velocity contours allowed the quantitative visualization of the
phenomenon called vortex dislocation, also known as vortex splitting, in the near-wake of the
cylinder. The vortex dislocations will be examined in more detail in section 3.2.3, but before closing
this section, we would like to point out how vortex dislocations can be distinguished on the space-
time plots of the streamwise velocity and the corresponding vortex filament θ angle versus
normalized time plots. For example, see the rightmost space-time plot in figure 13: nearly around the
normalized time of tuo/D = 20 the streamwise velocity contours that are indicative of vortex
filaments in the space-time plot show bifurcation between successive vortex filaments of same sign
and different number of vortex filaments above and below the bifurcation (vortex splitting) location.
The corresponding plot showing the variation of the vortex filament angle (θ) with time also shows
31
large angular changes between the two subsequent vortex filaments near the time a split initiates.
These jumps in θ value are due to the bifurcation of the filaments, and that for the same filament
before the branching two different θ values could be calculated, one for each branch. It should be
emphasized that vortex dislocation phenomenon has been observed before via the smoke or dye
visualization of the flow for smaller Reynolds numbers by previous researchers [3], [4]. However,
there have been no studies visually showing the existence of this low-Reynolds number phenomenon
at much higher (subcritical) Reynolds number values due partly to the inability to visualize flow
using such qualitative techniques at high Reynolds numbers. The presence of dislocations at high
Reynolds numbers was suggested from the modulations in pressure/velocity measurements along the
span [7], [13]. Hence, cinema technique of PIV and construction of space-time plots enable the first
visualization of this event at high Reynolds numbers.
The space-time plots of the streamwise velocity presented in the present section showed that
examination of 4 sets of 200 velocity vector fields, in other words roughly 40 shedding cycles, are
not sufficient to properly characterize if and how a given end condition affects the orientations in
which the vortex filaments are shed behind the cylinder or to generalize which shedding angles are
observed the most for different end boundary conditions. The need for acquisition of longer samples
of data necessitated the dual-CTA measurements that will be presented now.
3.2.2 Effect of End Configuration on the Vortex Filament Orientation
Dual-CTA experiments were performed at a Reynolds number of 104 to properly analyze the time
dependency of the vortex filament angles shed in the near-wake of the cylinder for all the end
conditions studied in the present investigation. The CTA probe locations and characteristics are
given in section 2.1.4. Short time Fourier Transformation (STFT) of the streamwise velocity signals,
measured simultaneously at two locations that are 3D apart in the spanwise direction, gave the time-
evolution of phase angle difference, from which the vortex filament angle (θ) was calculated as a
function of time (see section 2.2.5 for details). As the vortex filament angle was time dependent, the
probability density function (PDF) of θ was generated to evaluate the relative likelihood of shedding
orientations for a given end condition.
Figures 24 to 28 show the probability density functions of the vortex filament angle for various end
conditions. Ideal shedding would have a Gaussian like probability density function (PDF) of θ that is
32
centered on θ = 0° and have a distribution over a narrow band of θ values. The former would imply
that the most frequently occurring orientation of the vortex filament is parallel to the cylinder axis,
and the later would entail minimal variability in θ with time. Therefore, end conditions that promote,
quasi-two-dimensional shedding, can systematically be sorted out by examining the distribution of
the probability density function (PDF) of θ.
Figure 24 shows the PDFs of θ for the case of a cylinder bounded by the endplate having sharp
leading edge and the free-surface. Therein, three different leading edge distances (L/D = 1.5, 3 and
5) are compared. The largest leading edge distance considered (L/D = 5) results in a sharp peak at a
θ value of nearly -15o, with almost all the shedding being largely oblique. This clearly demonstrates
a boundary condition that should be avoided if quasi-parallel shedding is to be promoted. On the
other hand, shorter leading edge distances of L/D = 1.5 and 3 achieve a much better situation with
the shedding orientation mostly centered around a smaller θ value of about -5°. Nevertheless, the
shedding at these smaller L/D values is still slightly slanted most of the time.
Figure 25 shows the PDFs for a cylinder bounded by the endplate having elliptical nose on the
bottom and the free-surface on the top for three leading edge distances of L/D = 1.5, 3 and 5. For all
the L/D values considered, the PDF distributions look alike with the most probable θ values slightly
shifted toward negative oblique values.
Figure 26 shows the PDF of θ for the cylinder bounded by the channel floor on the bottom and the
free-surface on the top. The PDF is centered on a θ value of about -5° for this case. A comparison
with figures 24 and 25 shows that even the use of an endplate at the bottom end of the cylinder does
not improve the shedding orientation when the free-surface bounds the cylinder at the top. That is,
PDF distributions are all centered around -5° even when an endplate is used at the bottom at the L/D
value that achieves the most improved filament angle distribution in presence of the free-surface.
The observations so far suggest that to negate the free-surface affect, use of a top endplate might be
necessary. We, therefore, concentrate in the next figures on configurations where both ends of the
cylinder are bounded by endplates.
The probability density functions (PDFs) of θ corresponding to a cylinder bounded at both of its
ends by endplates having a sharp nose are shown in figure 27 for the leading edge distances of L/D =
33
3 and 5. For this case, the distribution of PDF strongly depends on which value of the leading edge
distance L/D is chosen. For L/D = 3, the PDF of θ shows a symmetric distribution about θ = 0°,
while for L/D = 5, the mean of the PDF is non-zero with a value centered on θ = -10°. Hence, in this
case, a leading edge distance of L/D = 3 achieves quasi-parallel shedding conditions and is favored
over L/D = 5.
Figure 28 shows PDFs of θ for a cylinder bounded at both ends by endplates having an elliptical
nose shape. Again, the effect of two leading edge distances are investigated (L/D = 3 and 5). It can
be seen that both L/D values achieve a probability density function that is roughly symmetric about a
mean value of θ = 0°, and hence show that the most probable shedding orientation is parallel.
However, the probably density function depicts a much tighter bound when the leading edge
distance is chosen to be L/D = 5, indicating lesser temporal variability in θ.
The key finding out of the results in the present section is the presence of free-surface influence,
even at a Re of 104. For all cases where the top of the cylinder is bounded by the water-air boundary,
there is a shift in the PDFs away from zero, that is the peak of the PDF is not at θ = 0, and the
distribution is not symmetric about parallel shedding conditions. This implies that it is most likely
that the filaments are being shed at a negative oblique angle. This is similar to the negative angle of
the demarcation line shown at higher Re for flow past a cylinder bounded by the free-surface.
Therefore at a Re of 104 or greater there is a need for either a top endplate or top wall cover to negate
the effect of the free-surface. Overall comparison of figures 24 to 28 shows that the use of endplates
at both the bottom and top ends of the cylinder at certain L/D values achieve a scenario where the
majority of the vortex filaments are shed parallel to the cylinder axis. Furthermore, the leading edge
distance L/D is found to have a much more critical influence on the orientation of vortex filaments
when an endplate with sharp leading edge is being employed (figures 24 and 27) as opposed to an
endplate with elliptical leading edge shape (figures 25 and 28). The use of an endplate having a
sharp leading-edge shape at high L/D values considerably worsened the shedding orientation
compared to the case with even no endplate at all (figure 26). As indicated in section 2.1.2,
Blackmore [16] showed that the nose shape of the endplate controls whether flow separation occurs
at the leading edge of the endplate or not. With an endplate having sharp leading edge, flow at the
leading edge of the endplate shows separation and then a reattachment further downstream, while the
34
elliptical leading edge shape prevents such a flow separation [16]. The location of the cylinder
relative to the separation at the leading edge of the endplates with a sharp nose therefore greatly
affects the two-dimensionality of the flow, while for the elliptical leading edge there is no such
phenomenon.
3.2.3 Vortex Splitting
Previous research shows that a vortex dislocation, or also known as a vortex split, occurs when a
vortex filament branches and merges with the neighboring filaments of same sign. This phenomenon
is thought to be an inherent feature of the flow at Reynolds numbers greater than 5x103 regardless of
end conditions [7]. Visualization of the phenomenon at high Reynolds numbers has not been
possible before due to the inability to use dye/smoke visualization studies at high flow speeds.
For all the different end configurations considered in the present work, examination of the space-
time plots of the streamwise velocity contours visually showed appearance of this phenomenon at
some instant in time at a high Reynolds number value of 104. A compilation of these plots is
presented for different end configurations in Figure 29. Occasional appearance of vortex splitting
phenomenon can be distinguished in all the space-time plots clearly, i.e., there are time intervals
when the vortex filament bifurcates and connects to the neighboring vortex filament of same sign. It
can also be seen in Figure 29 that the location along the span where the filament splits changes in
time for the cases. During the time when the vortex splitting persists a different number of vortex
filaments can be counted in space-time plots above and below the split. For example, let's consider
the left most plot in the top row of figure 29. The location of the split changes in time and it can be
seen that the filaments on top are splitting and merging to two of its neighboring filaments below.
Correspondingly, there are 11 filaments shown at locations below a z/D value of 8, and only 10 at a
z/D value of greater than 10. A further characteristic is that due to the bifurcation of the vortex
filaments, the vortex filament angle θ tends to vary suddenly near the instants when the vortex
splitting arises. Large phase angle variations along the span and consequent jumps in vortex filament
angle θ in time were observed to be key features of vortex splitting. Overall consideration of Figure
29 shows that splitting occurs in some cases when the vortex filament is being shed at a largely
oblique angle, while in other cases it occurs right after a filament shed parallel to the cylinder span.
35
Another common characteristic related to the vortex splitting phenomenon is represented in Figure
30, where time-traces of the streamwise velocity signals obtained via the PIV measurements on the
side-plane (y/D = 0.5 plane) are presented at two points in the flow. These points are selected from
the space-time plot of the streamwise velocity contours such that one of the points is located near the
split location while the other is far away from it. The velocity signals near the split location are
significantly attenuated and somewhat disturbed during the time when the vortex splitting
phenomenon occurs, while the signals away from the split show no significant sign of change in
amplitude during that time. From here, it can be concluded that highly distorted and attenuated
velocity signals distinguish points near the split location.
In order to study the distribution of spectral density of frequencies in the streamwise velocity signals,
time-frequency analysis using Short-Time Fourier transformation (STFT) was performed on
streamwise velocity signals acquired through CTA measurements. This provided the temporal
dependence of the frequencies in velocity signals. In figure 31, a representative time-frequency
spectrogram of a CTA signal for the streamwise velocity outside of the near-wake (as detailed in
section 2.1.4) is provided. A generic characteristic, observed for all the end configurations studied at
Re = 104 is that there exist periods with significant attenuation of the peak spectral amplitudes in the
time-frequency spectrograms (as can be seen in figure 31). Furthermore, in figure 31, the streamwise
velocity u/uo signal over a duration when the peak spectral amplitude is high is compared with the
signal over a duration when the peak spectral amplitude is attenuated. The streamwise velocity u/uo
signal during when the peak spectral amplitude attains high levels in figure 31 resembles the
streamwise velocity signal observed at a point away from the vortex splitting location given in figure
30, whilst the signal corresponding to the attenuation of peak spectral amplitude in figure 31 looks
similar to that by the split location shown in figure 30.
Taken as a whole, figures 29 to 31 imply that a sudden large change in vortex filament angle, a
disturbance in the velocity signal and a reduction in the peak spectral amplitude in time-frequency
spectrogram are all distinctive characteristics of vortex splits. These features can be used to identify
the presence of vortex splits in a flow without the visualization of the flow, such as performing CTA
measurements at two points separated a distance in the spanwise direction. Performing such dual-
CTA measurements, observations from figure 29 were confirmed. That is, splitting is inherent
36
feature of the flow regardless of the end condition used at a sub-critical Re = 104 and it occurs in
general whether the filament previously shed is parallel or oblique. However, evaluation of the
probability density functions of vortex filament angle θ for end configurations where the cylinder
was bounded by a free-surface on top showed higher likelihood of splitting when the filament angle
θ became greater than approximately a value of 25° in the negative direction. These splits were
preceded by a gradual decrease in θ until this largely negative oblique angle was found, after which a
sharp change in θ was observed. For end configurations that promoted parallel shedding there was
no distinct trend of θ value at which the vortex splitting occurred, although if shedding is to become
largely oblique than once again splitting is likely. This once again shows that in a water channel
there are clear free-surface effects even at a Re as low as 104 (see Appendix A for plots and details).
37
Figure 8: Formation length as a function of spanwise location at varying Re
is shown for a cylinder bounded by the channel floor and the channel top
cover. As Re increases there is no major change to shedding orientation.
Figure 7: Ensemble averaged results of the streamwise velocity yield a
demarcation line between the recirculation and positive velocity flow.
Formation length as a function of spanwise location at varying Re is
shown for a cylinder bounded by the channel floor and the free-surface.
As Re increases free-surface effects are causing oblique shedding.
38
Figure 9: Formation length as a function of spanwise location at varying Re
is shown for a cylinder bounded by sharp endplates at L/D of 2.5. As Re
increases there is no major change to shedding orientation.
Figure 10: Formation length as a function of spanwise location at varying
Re is shown for a cylinder bounded by elliptical endplates at L/D of 2.5. As
Re increases there is no major change to shedding orientation.
39
Figure 11: Free-surface effect causes a downward flow near the top, rear portion of the cylinder. This leads to spanwise non-uniformities
and introduces three-dimensionalities to the flow. The right image shows the velocity vector map superimposed over the streamwise velocity
contours.
40
Figure 12: The velocity vector map is superimposed over the streamwise velocity contours for three cases. A reduction in spanwise flow, and
hence a decrease in three-dimensionalities is found if a top cover or endplate is used. a) Cylinder is bounded on top by the channel cover. b)
Cylinder is bounded on top by a sharp leading edge geometry endplate. c) Cylinder is bounded on top by an elliptical leading edge geometry
endplate.
41
Figure 13: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for a cylinder bounded by the channel
floor and the free-surface. Vortex splitting is seen in the right most images of this figure.
42
Figure 14: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for the top field of view of a cylinder
bounded by the channel floor and channel cover. Minimal variation in the vortex filament angles are found for this configuration.
43
Figure 15: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for the bottom field of view of a
cylinder bounded by the channel floor and channel cover. Slightly more variation is found than in the previous image, but it is still minimal
compared to most cases.
44
Figure 16: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for a cylinder bounded by a sharp
leading edge endplate at L/D = 1 and the free-surface.
45
Figure 17: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for a cylinder bounded by a sharp
leading edge endplate at L/D = 2.5 and the free-surface.
46
Figure 18: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for a cylinder bounded by a sharp
leading edge endplate at L/D = 5 and the free-surface.
47
Figure 19: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for a cylinder bounded by an
elliptical leading edge endplate at L/D = 1.0 and the free-surface.
48
Figure 20: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots
for a cylinder bounded by an elliptical leading edge endplate at L/D = 2.5 and the free-surface. The fourth
set of data for this configuration was corrupted.
49
Figure 21: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for a cylinder bounded by an
elliptical leading edge endplate at L/D = 5.0 and the free-surface.
50
Figure 22: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for a cylinder bounded by a sharp
leading edge endplate at L/D = 2.5 on the top and bottom.
51
Figure 23: Space-time plots of the streamwise velocity and their corresponding vortex filament angle plots for a cylinder bounded by a sharp
leading edge endplate at L/D = 5 on the top and bottom.
52
Figure 24: Probability density function of the vortex filament angle for a cylinder
bounded by a sharp leading edge endplate and the free-surface for various L/D values.
53
Figure 25: Probability density function of the vortex filament angle for a cylinder bounded
by an elliptical leading edge endplate and the free-surface for various L/D values.
54
Figure 26: Probability density function of the vortex filament angle for a cylinder
bounded by the channel floor and the free-surface.
55
Figure 27: Probability density function of the vortex filament angle for a cylinder
bounded by a sharp leading edge endplate on top and bottom for two L/D values.
56
Figure 28: Probability density function of the vortex filament angle for a cylinder
bounded by an elliptical leading edge endplate on top and bottom for two L/D
values.
57
Figure 29: Vortex Splitting was observed in the space-time plots for all boundary conditions analyzed.
58
Figure 30: Comparing the stream-wise velocity signals for a point in the flow where a split has
occurred and one that has undergone normal shedding. The vortex split causes attenuation of
the velocity signal.
59
Figure 31: The stream-wise velocity signal in a window where there is a
reduction in the spectral density of the Karman Strouhal number shows
attenuation similar to that found when a vortex split is present.
60
Conclusion and Future Work 4
The purpose of this thesis was to develop a method for modeling an infinite cylinder in a finite
experimental setup. This was done by manipulating the boundary conditions of the cylinder to
remove any end effects, which in turn would get rid of any spanwise flow or non-uniformities along
the span. Having quasi-two dimensional flow would allow for researchers to develop near-wake flow
control techniques that would be effective in multiple practical applications, and not in specific
laboratory configurations. Experiments were performed in a free-surface, recirculation water channel
at Reynolds numbers in the sub-critical flow regime, also known as shear-layer transition regime.
Multiple boundary conditions were examined and their effectiveness in creating parallel shedding
was determined. The results obtained by the various experiments undertaken will be summarized in
this section.
Time-averaged analysis of the near-wake of the cylinder showed that as the Re increases, free-
surface effects become more pronounced. For cylinders that were not bounded on top by an endplate
or channel wall, an average negatively oblique angle of shedding was found for flows with a Re
greater than 104. Using a top boundary condition in the form of an endplate or channel wall created
time-averaged shedding that had increased spanwise uniformity, and displayed shedding that was
nearly parallel to the cylinder axis. This is attributed to a reduction in spanwise flow at the top, rear
most portion of the cylinder. When the cylinder is exposed to the free-surface there is a large
mismatch in pressure between the ambient air, and the base pressure of the cylinder, causing a large
spanwise flow behind the cylinder near the free surface. This spanwise flow influences the
formation length at the top of the cylinder and introduces three-dimensionalities in the near-wake.
Analysis of the unsteady nature of the vortex filament angle showed that a cylinder bounded on top
by a free-surface was more likely to have vortex filaments shed at oblique angles even at a Reynolds
number value of 104. Experimental configurations that had the cylinder bounded on top by an
endplate with an appropriate leading edge distance led to shedding orientations that had an average
vortex filament angle θ that was parallel to the cylinder axis, with probability density function of
shedding angles distributed symmetrically about θ = 0°. These results made it clear that free-surface
effects are present for Re greater or equal to 104
and a top endplate or channel cover is needed to
reduce the three-dimensionalities caused by the water-air boundary.
61
Quantitative visualization of the near-wake of the cylinder, with the use of PIV, showed that the
orientation of the vortex filaments varies in time for all the end configurations investigated herein. It
was shown however that using different end boundary configurations for a cylinder leads to different
probability density function (PDF) distributions of the vortex filament angle. Therefore, it was
important to not only determine the average shedding orientation for each boundary condition
examined, but also the variability of the vortex filament angles in time.
Analyzing the probability density functions of the vortex filament angle for different experimental
setups demonstrated how the boundary conditions influence the flow near the mid-span of the
cylinder. For cases in which an endplate with sharp leading edge geometry is used, the streamwise
distance between the leading edge of the endplate and the cylinder axis was shown to be a much
more influential factor than configurations where an elliptical leading edge endplate is employed.
The nose shape was shown to control the flow along the plate previously. According to the
investigation of Blackmore [16] a plate with the sharp nose causes separation at the leading edge,
whilst a plate with the elliptical leading edge removes flow separation. Due to the separation at the
leading edge, the flow at the base of the cylinder for cases in which a sharp endplate is being used is
considerably more dependent on the location of the cylinder along the endplate as opposed to the
endplate with the elliptical leading edge, as the flow over the endplate is less dependent on the
distance from the nose.
One of the most common end configurations used in literature is that of a cylinder bounded by
endplates on top and bottom, at a leading edge distance of three diameters; a configuration
developed by Stansby [8]. Regardless of the shape of the nose, this experimental setup was shown to
induce near parallel shedding, and had a probability distribution that was greatly improved from the
case of a cylinder bounded by the channel floor and free-surface. As the leading edge distance
increased to 5, the use of two elliptical endplates showed markedly improvements in creating
parallel flow while the same configuration with a sharp nose led to oblique shedding due to
separation at the nose of the endplates as mentioned above. It is important to note that all cases in
which the top of the cylinder is bounded by the free-surface show a mean shedding that is negatively
oblique, and the use of an endplate on the bottom does not show discernible improvement regardless
of leading edge distance.
62
Flow visualization showed that there are vortex splits for all boundary conditions examined at a Re
of 104. Vortex splitting is shown to occur regardless of whether the vortex filaments directly before
the split are parallel or oblique to the cylinder axis. This implies that vortex splitting is an inherent
feature of the flow in the sub-critical regime. Although this phenomenon can occur during quasi-two
dimensionality, splitting was found to happen most often when the shedding was largely oblique,
which is similar to what was found at lower Re flows by previous investigators [3], [4].
Future work on the topic can involve determination of the variation of the vortex filament angle for a
cylinder with a large aspect ratio. This would help in understanding how the vortex shedding behind
a near infinite cylinder would vary in time, assisting in creating a model for practical applications.
Furthermore having data for a near infinite cylinder would allow for the observation of vortex
splitting regardless of the aspect ratio.
Further future work can include taking a larger range of leading edge distances for the use of
elliptical leading-edge endplates. This would allow for an optimization of endplate position relative
to the cylinder in generating parallel flow.
Finally, varying the Re of the flow and repeating the experiments that provide the variation in vortex
filament angle would allow for greater insight into the flow dynamics. This would give a more exact
range of values for where the free-surface effects on the mid-span flow are found, as well as helping
to determine in what flow regimes vortex splitting is present.
63
Appendix A
One of the goals of performing the Dual-CTA experiments was to determine if vortex splitting is an
inherent feature of the flow in the sub-critical Reynolds number regime, or if it is due to disturbances
caused by the experimental setup. Section 3.2.3 shows that vortex splitting is found to occur for all
boundary conditions investigated, and also is found regardless of the previous orientation of the
vortex filaments, implying it is an intrinsic feature of this regime. Further analysis will be done in
this appendix to determine the relationship between the end configurations and the orientation of the
vortex filaments prior to shedding.
The PDFs shown in figures 24-28 display the likelihood of the vortex filament angle (θ) for each
boundary condition investigated. All but three of the cases had a PDF that was centered on a non-
zero negative value. Figure 32 shows the value of θ directly preceding a vortex split for all cases in
which the most likely shedding orientation is oblique. Although the data may not be fully converged,
due to a lack of splitting observations in the sample size, there is a clear trend for most cases of θ
being a large negative value before the vortex splits. Figure 33 shows the same results but for the
three cases in which the most likely shedding orientation was parallel. Once again the sample size is
low, but here there is no discernible trend of θ value before a split is found. Furthermore the number
of splits is shown to decrease for cases that promote quasi-two dimensional shedding. This shows
that vortex splitting can be created in the sub-critical regime by causing oblique shedding, but cannot
be removed entirely even if uniform shedding is induced.
64
Figure 32: Bar Graphs displaying the number of occurrences of each range of Vortex Filament Angles prior to a vortex split for cases in which
the end configuration does not promote parallel shedding. Note that the most likely shedding orientation prior to a vorte
65
Figure 33: Bar Graphs displaying the number of occurrences of each range of Vortex Filament Angles
prior to a vortex split for cases in which the end configuration promotes parallel shedding. Note that
the vortex filament angle takes on nearly all values before a vortex split
66
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