The Influence of End Conditions on Vortex Shedding from a Circular Cylinder in Sub-Critical Flow

Transcription

1 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

2 ii The Influence of End Conditions on Vortex Shedding from a Circular Cylinder in Sub-Critical Flow Abstract Eric Khoury Master of Applied Science in Engineering Institute of Aerospace Studies University of Toronto 2012 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.

3 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.

4 iv Table of Contents Contents Acknowledgments... iii Table of Contents... iv List of Tables... vi List of Figures... vii 1 Introduction Motivation and Background Literature Review Flow Past an Infinite Cylinder End Effects on the Spanwise Uniformity of the Cylinder Near Wake Experimental Setup and Analysis Techniques Experimental Setup Water Channel Test models Particle Image Velocimetry Constant Temperature Anemometry Analysis Techniques Time-Averaged Recirculation Region Space-Time Plots Continuous Wavelet Transformation Fast Fourier Transform and Short-Time Fourier Transform Time Evolution of Phase-Angle Difference Between Two Probes...20

5 v 3 Results Time-Averaged Characteristics of Vortex Shedding Unsteady Nature of Vortex Filaments Visualization of Vortex Filaments Effect of End Configuration on the Vortex Filament Orientation Vortex Splitting Conclusion and Future Work...60 Appendix A...63 References...66

6 vi List of Tables Table 1: Acquisition details for PIV measurements Table 2: Probe Characteristics Table 3: Error in calculated recirculation region as a function of Reynolds number... 17

7 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 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 Figure 4: Method to determine the length of the recirculation region along the span 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/u o. 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 x o ) 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 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

8 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 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 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 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 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 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

9 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 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 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 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 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 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 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 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 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

10 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 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 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 Figure 26: Probability density function of the vortex filament angle for a cylinder bounded by the channel floor and the free-surface 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 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 Figure 29: Vortex Splitting was observed in the space-time plots for all boundary conditions analyzed 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 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 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

11 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

12 1 1 Introduction 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 nearwake 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.

13 2 1.2 Literature Review 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, F L, 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, -C pb, 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

14 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 subcritical 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 separationreattachment 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.

15 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 30 o. 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

16 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

17 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 twinpeaks 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 10 5 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 10 3 to 1.4 x 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

18 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 twodimensionality 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*10 3 <Re<10 4, and as Re increases to the range of 10 4 <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 10 5 ). 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

19 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.

20 9 2 Experimental Setup and Analysis Techniques 2.1 Experimental Setup 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 4x x10 3 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, u o, was adjusted accordingly 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.

21 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 freesurface at the top. As indicated above, the bevel angle of the sharp leading edge was 23.6 o. (S/D = 12.3) d) A cylinder bounded by the endplate with an elliptical leading edge at the bottom and the freesurface 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 23.6 o. (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)

22 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 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 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 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

23 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 10 3 Sampling Δt between Shedding Cycles Samples per Frequency pairs of images Observed shedding cycle (Hz) ( x10-3 seconds)

24 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 sideplane 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.

25 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 = 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 1.6 o when the probes were placed 3D apart.

26 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 endplate 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) Transverse location (y/d) Spanwise location (z/d) Sensor Resistance 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

27 16 Desired Sensor Temperature 40C 40C Overheat Ratio Analysis Techniques 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 (F L ) 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>/u o = 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

28 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 ( F L ), was therefore dependent on the free-stream velocity and hence the Reynolds number. F L was calculated by determining the possible streamwise location values of <u>/u o = 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 F L /F L 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 = x o ) 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 (x o ), 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 x o /D values used to produce such space-time plots were in the range of 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

29 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 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 x o 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 spacetime 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:

30 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 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, 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.

31 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 F s is the sampling frequency (48Hz), N is the window size (128), F k 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 F K 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.

32 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 -63 o 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 θ max. The θ max was found to be 44.5 o, which is constant along the span within a vortex filament such that θ max is the same for both 1D and 3D probe separations. Therefore, this same θ max 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; Φ max 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 Φ max 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.

33 22 Figure 1: The various boundary conditions investigated. a) A cylinder bounded by the channel floor and freesurface. 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.

34 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 freestream 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.

35 Figure 4: Method to determine the length of the recirculation region along the span. 24

36 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/u o. 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 x o ) 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.

37 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).

38 27 3 Results 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 10 4 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 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 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 = 10 4, 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

39 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 freesurface 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 = 36x10 3 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 10 4 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

40 29 focus on the Reynolds number of 10 4 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 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 10 4 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 and 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

41 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 spacetime 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 tu o /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

42 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 Effect of End Configuration on the Vortex Filament Orientation Dual-CTA experiments were performed at a Reynolds number of 10 4 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 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 timeevolution of phase angle difference, from which the vortex filament angle (θ) was calculated as a function of time (see section 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

43 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 -15 o, 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 =

44 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 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 10 4 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

45 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 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 5x10 3 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 spacetime plots of the streamwise velocity contours visually showed appearance of this phenomenon at some instant in time at a high Reynolds number value of 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.

46 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 = 10 4 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/u o 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/u o 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

47 36 feature of the flow regardless of the end condition used at a sub-critical Re = 10 4 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 10 4 (see Appendix A for plots and details).

48 37 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. 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.

49 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.

50 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.

51 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.

52 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.

53 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.

54 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.

55 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.

56 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.

57 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.

58 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.

59 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.

60 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.

61 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.

62 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.

63 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.

64 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.

65 54 Figure 26: Probability density function of the vortex filament angle for a cylinder bounded by the channel floor and the free-surface.

66 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.

67 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.

68 Figure 29: Vortex Splitting was observed in the space-time plots for all boundary conditions analyzed. 57

69 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.

70 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.