AN ABSTRACT OF THE DISSERTATION OF

Trevor Kent Howard for the degree of Doctor of Philosophy in Nuclear Engineering presented on May 29, 2018.

Title: On Primary Vortex Shedding Regimes between Long Tandem Plates

Abstract approved:

Wade R. Marcum

Vortex shedding is a phenomenon relevant to any industry dealing with fluid flow. The shed vortices often produce oscillatory forces, which have been suspect in the catastrophic failure of airplanes and bridges alike. To prevent further engineering failures a better understanding of the underlying physics is needed. It has been well established that tandem plates exhibit different flow phenomena than cylinders, yet the study of the flow field around tandem plates is insufficient in providing a reasonable prediction of the Strouhal numbers for given geometry. This study fills the void in knowledge in three ways. First, it provides a review of the relevant literature related to vortex shedding for plates as well as that for cylinders, which have been well studied. Second, it develops the theory behind vortex shedding for plates through leveraging previous studies and applying a scaling analysis to the data. Third, it verifies the theory through a PIV analysis of various plate geometries. The results from the analysis are compared directly to the results of cylinders. The general plate has an l:b ratio of 20.

Reynolds numbers (Reb) range from 150 (below which vortex shedding does not occur) to 1100 (the “constant” Strouhal limit of cylinders). Gap spacing ratios are taken from a value of zero to infinity. The results of the study provide a means of more accurately determining the Strouhal number for both single and tandem long plates.

©Copyright by Trevor Kent Howard May 29, 2018 All Rights Reserved On Primary Vortex Shedding Regimes between Long Tandem Plates

by Trevor Kent Howard

A DISSERTATION

submitted to

Oregon State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Presented May 29, 2018 Commencement June 2019 Doctor of Philosophy dissertation of Trevor Kent Howard presented on May 29, 2018

APPROVED:

Major Professor, representing Nuclear Engineering

Head of the School of Nuclear Science and Engineering

Dean of the Graduate School

I understand that my dissertation will become part of the permanent collection of Oregon State University libraries. My signature below authorizes release of my dissertation to any reader upon request.

Trevor Kent Howard, Author ACKNOWLEDGEMENTS

First and foremost, I would like to sincerely thank my advisor, Dr. Marcum. He provided me with countless hours of support. He always went above and beyond what was required. Without his mentorship I would not have been nearly as successful as I was in this program.

I would like to thank the ARCS foundation, specifically my ARCS donors, Steve and Lynn Pratt who provided me additional funding to ensure my success as a Ph.D. student and beyond. It was such a wonderful pleasure being your ARCS scholar.

I would like to thank my friends, and specifically those in my research group who were extremely supportive of my research endeavors. In particular, I would like to thank Laura Oliveira, Tommy Holschuh, Aaron Weiss, Chad Nixon, Griffen Latimer all of whom at one point or another, despite bearing the brunt of my harebrained ideas, half- finished sentences, and severe algebraic mistakes, were willing to listen to me and help steer me in the right direction.

I would also like to express additional gratitude to Sam Goodrich who mentored me on the PIV system and Emory Brown who allowed me to use his 3d printer. Both of whom saved me countless hours that may have otherwise been wasted.

Finally, I cannot express enough thanks to my parents, Victoria and Tracy, who have always been there to support my pursuit of science throughout my life, and my brother Trenton who was always there and willing to provide support both emotionally and academically. I could not wish for more wonderful family.

A final thanks to all my friends, peers, professors, and coworkers who have made my stay at Oregon State University such a wonderful experience.

TABLE OF CONTENTS

Chapter Page

1 Introduction ...... 1

1.1 Vortex shedding ...... 1

1.2 Vortex shedding in the Context of Engineering ...... 3

1.3 Motivation ...... 7

1.4 Objectives and Outcomes ...... 8

1.5 Overview of Proceeding Sections ...... 9

2 Survey of Literature ...... 10

2.1 Vortex shedding Theory and a Universal Strouhal Number ...... 12

2.2 Vortex shedding for Cylinders ...... 21

2.2.1 Single Cylinders ...... 21

2.2.2 Tandem Cylinders ...... 29

2.3 Vortex shedding for Plates Parallel to the Flow ...... 35

2.3.1 Vortex shedding over Short Plates ...... 36

2.3.2 Vortex shedding over Long Plates ...... 39

2.3.3 Tandem Plates ...... 40

2.4 Additional Vortex Shedding Observations ...... 42

2.4.1 Experimental Studies ...... 43

2.4.2 Numerical Studies ...... 45

2.5 Study Justification ...... 47

3 Theory...... 48

3.1 Fluid Dynamics ...... 48 TABLE OF CONTENTS (CONTINUED)

Chapter Page

3.1.1 Inviscid Flow Equations ...... 48

3.1.2 Navier Stokes Equations ...... 50

3.1.3 Considerations for Turbulence ...... 53

3.2 Characteristics of Flow Over a Plate ...... 56

3.2.1 Interaction with the Leading Edge ...... 56

3.2.2 Boundary Layer Development along the Mid-Span ...... 57

3.2.3 Separation and Vortex Shedding at the Tailing Edge ...... 62

3.2.4 The Wake ...... 65

3.3 Derivations of the Scale of Vortex Shedding ...... 67

3.3.1 Scaling Analysis of Vortex Shedding from a Bluff Body ...... 68

3.3.2 Scaling Analysis of Vortex Shedding from a Plate ...... 72

3.3.3 Dimension of the Vortex Street ...... 73

3.3.4 Derivation of Vortex Shedding from Long Tandem Plates ...... 78

4 Methodology...... 81

4.1 Wind Tunnel ...... 81

4.1.1 Seed Region ...... 83

4.1.2 Flow Conditioner ...... 85

4.1.3 Inlet Section ...... 86

4.1.4 Working Test Section ...... 88

4.1.5 Outlet Section...... 90

4.2 Measurement Instrumentation ...... 92

4.2.1 Non-PIV measurements...... 93

4.2.2 Particle Image Velocimetry Equipment ...... 95 TABLE OF CONTENTS (CONTINUED)

Chapter Page

4.3 Conduct ...... 96

4.4 Test Matrix ...... 97

4.4.1 Length Scale Reference ...... 97

4.4.2 Reynolds Numbers ...... 98

4.4.3 Gap Spacing Ratios ...... 99

4.4.4 Plate Sizing ...... 100

5 Results ...... 103

5.1 Examination of Flow over a Single Plate ...... 103

5.1.1 Comparison of Boundary Layer...... 103

5.1.2 Comparison of the Average Wake ...... 106

5.1.3 Comparison of Vortex Shedding Frequency...... 111

5.2 Single Plate Regimes (Reynolds dependent phenomenon) ...... 115

5.2.1 Pulsating Regime ...... 117

5.2.2 Oscillatory Regime ...... 121

5.2.3 Vortex Shedding Regime ...... 127

5.2.4 Summary ...... 134

5.3 Tandem Plate Regimes (Gap Spacing Dependent Phenomenon) ...... 134

5.3.1 Trapped Regime ...... 136

5.3.2 Trapped Sub-regimes ...... 139

5.3.3 Incident Sub-regimes ...... 152

5.3.4 Summary ...... 153

5.4 Map of Regimes ...... 153

5.4.1 Description of Long Plate Regimes ...... 154 TABLE OF CONTENTS (CONTINUED)

Chapter Page

5.4.2 Comparison to Cylinders ...... 163

5.5 Summary ...... 166

5.5.1 Regimes...... 166

5.5.2 Effects on Strouhal number ...... 169

6 Conclusions ...... 172

6.1 Observations ...... 172

6.2 Relevance of Work ...... 172

6.3 Assumptions and Limitations ...... 173

6.3.1 Length to Thickness Ratio ...... 173

6.3.2 Smooth Plates and Turbulence...... 174

6.4 Future Work ...... 174

7 References ...... 176

8 Appendix: Incident Regime Plots ...... 188

9 Appendix: Vorticity & StreamLInes ...... 195

LIST OF FIGURES

Figure Page

Figure 1.1: Progression of vortex shedding...... 2

Figure 1.2: Vortex parameters for tandem plates ...... 3

Figure 1.3: Engineering process integrated with design tools ...... 5

Figure 2.1: von Karman potential vortex street streamlines...... 14

Figure 2.2: Igarashi’s flow map of tandem cylinders...... 32

Figure 2.3: Carmo’s flow map of tandem cylinders...... 34

Figure 2.4: Bull’s two primary regimes with hysteresis...... 41

Figure 3.1: Flow over the leading edge ...... 56

Figure 3.2: Boundary layer diagram...... 58

Figure 3.3: Boundary layer description over a plate...... 60

Figure 3.4: Blasius Solution for f and its three derivatives ...... 61

Figure 3.5: Difference of vortex dimensions at the tailing edge...... 63

Figure 3.6: Breakdown of vortex shedding in the wake...... 66

Figure 3.7: The potential vortex street...... 73

Figure 3.8: Vortex streamlines for an infinite street...... 75

Figure 3.9: Average velocity profile between vortices...... 76

Figure 4.1: Wind Tunnel Diagram ...... 82

Figure 4.2: Pictures of the seed region...... 84

Figure 4.3: Images of flow conditioner section...... 85

Figure 4.4: Inlet section...... 87

Figure 4.5: Test section and PIV equipment...... 88

Figure 4.6: Test section cross section...... 89 LIST OF FIGURES (CONTINUED)

Figure Page

Figure 4.7: Plate holders...... 90

Figure 4.8: Outlet section...... 91

Figure 4.9: Air blower and Variac controller...... 92

Figure 4.10: Instruments in the test section...... 94

Figure 4.11: Camera used for PIV...... 95

Figure 4.12: Laser system used for PIV...... 96

Figure 4.13: Blunt and rounded plate edges...... 101

Figure 5.1: Sample curve fit of boundary layer data (b = 8.4 mm, Re = 4380) ...... 104

Figure 5.2: Single plate boundary layer data...... 105

Figure 5.3: Comparison of theoretical boundary layer to measured boundary layer.106

Figure 5.4: Comparison of wake evolution across Reynolds numbers...... 108

Figure 5.5: Comparison of wake evolution across plate sizes...... 110

Figure 5.6: Comparison of single plate data via different scaling...... 113

Figure 5.7: Free pulsating regime: average values...... 118

Figure 5.8: Free pulsating regime: time resolved velocity...... 120

Figure 5.9: Free oscillatory regime: average values...... 122

Figure 5.10: Free oscillatory regime: time resolved velocity...... 124

Figure 5.11: Free oscillatory regime: time resolved velocity...... 125

Figure 5.12: Free vortex shedding regime: average values...... 128

Figure 5.13: Free vortex shedding regime: time resolved velocity...... 131

Figure 5.14: Free vortex shedding regime: time resolved velocity...... 132

Figure 5.15: Symmetric trapped regime: average values...... 136 LIST OF FIGURES (CONTINUED)

Figure Page

Figure 5.16: U-shaped trapped regime: average values...... 138

Figure 5.17: Trapped pulsating regime: average values...... 140

Figure 5.18: Pulsating, trapped: time resolved total, u, and y velocity ...... 143

Figure 5.19: Trapped oscillatory regime: average values...... 145

Figure 5.20: Trapped oscillatory regime: time resolved values...... 147

Figure 5.21 Trapped vortex shedding regime: average values...... 149

Figure 5.22: Trapped vortex shedding regime: time resolved velocity...... 151

Figure 5.23: Locations of plate regimes...... 154

Figure 5.24: Comparison of Bull’s region of hysteresis...... 158

Figure 5.25: Strouhal number for the various plate regimes...... 159

Figure 5.26: Relative expected frequency transition ...... 161

Figure 5.27: Strouhal maps for plates (top), and Xu’s cylinders (bottom)...... 163

Figure 5.28: Difference in Strouhal number, total (top), normalized (bottom)...... 165

Figure 8.1: Incident pulsing regime: average values...... 189

Figure 8.2: Incident pulsing regime: time resolved velocity...... 190

Figure 8.3: Incident oscillatory regime: average values...... 191

Figure 8.4: Incident oscillatory regime: time resolved velocity...... 192

Figure 8.5: Incident vortex shedding regime: average values...... 193

Figure 8.6: Incident vortex shedding regime: time resolved velocity...... 194

Figure 9.1: Free Pulsating Regime: images, streamlines, and vorticity...... 196

Figure 9.2: Incident pulsating: images, streamlines, and vorticity...... 197

Figure 9.3: Trapped pulsating: images, streamlines, and vorticity...... 198 LIST OF FIGURES (CONTINUED)

Figure Page

Figure 9.4: Free oscillating: images, streamlines, and vorticity...... 199

Figure 9.5: Free oscillating: images, streamlines, and vorticity...... 200

Figure 9.6: Incident oscillating: images, streamlines, and vorticity...... 201

Figure 9.7: Trapped oscillating: images, streamlines, and vorticity...... 202

Figure 9.8: Free vortex shedding: images, streamlines, and vorticity...... 203

Figure 9.9: Free vortex shedding: images, streamlines, and vorticity...... 204

Figure 9.10: Incident vortex shedding: images, streamlines, and vorticity...... 205

Figure 9.11: Trapped vortex shedding: images, streamlines, and vorticity...... 206

LIST OF TABLES

Table Page

Table 2.1: Survey of Literature Summary ...... 11

Table 2.2: A comparison of vortex shedding dimensions...... 16

Table 2.3: Comparison of various Strouhal numbers with various shapes ...... 17

Table 4.1: Target Reynolds numbers...... 99

Table 4.2: Target gap spacing ratios...... 100

Table 4.3: Plate sizes used...... 101

Table 5.1: Values for plate comparison...... 107

1

1 INTRODUCTION As the world population grows, inventions are created and improved to address society’s changing demands: new aircraft are designed with new materials to make travel faster and more economical, wind turbines and offshore platforms are built to address growing energy concerns, and skyscrapers are constructed ever larger. While it may seem a jet and skyscraper are extremely different, both undergo fluid-structure interactions (FSI).

Fluid-structure interactions create design challenges. Historically, FSI have resulted in the failure of many designs: passenger jets, bridges, and nuclear fuel plates to name a few. Within the realm of FSI is vortex induced vibrations (VIV). Vortex induced vibrations are the vibrations in a structure generated by the shedding of vortices at a frequency resonant to the natural frequency of the structure. The vortex shedding frequency is well known for circular cylinders. The cylinder frequency is so well known, it is often assumed to represent any bluff body. The assumption is valid in most cases; however, it is seen to be strikingly different in plates [1] and tandem cylinders [2]. Yet, no study to date synthesizes the complexity of plates in tandem; hence, the proposed study is to find a relationship between the plate spacing and the frequency of vortex shedding.

1.1 Vortex shedding

Any bluff body placed in a fluid flow will generate vortices. Small vortices begin to form as a fluid flows across the backside of the body. The vortices are symmetric, so long as the fluid velocity is low (Figure 1.1 A). As flow increases, the vortices begin to elongate (Figure 1.1 B). Elongation of the vortices results in them becoming unstable. The vortices oscillate between sizes producing an unsteady flow field downstream of the body. Further increases in the flow rate results in the instability becoming large enough vortices shed away from the body (Figure 1.1 C). 2

Figure 1.1: Progression of vortex shedding. The pattern of shedding vortices, referred to as a “vortex street” is characterized by the Strouhal number,

fb St  , (1) U based on the body thickness, b, freestream velocity, U, and the vortex shedding frequency, f. The Strouhal number is often presented as a relationship with the Reynolds number,

Ul Re  , (2)  which provides the relationship between the inertial forces to the viscous ones. The Reynolds number in addition to velocity and a length scale, l, is determined by the fluid properties of density, ρ, and viscosity, μ. For bluff bodies, it is typical to define the length scale,

lb , (3) as the thickness of the objects for bluff bodies. For plates, where there is not significant flow separation, it is of consequence that,

lb , (4) and the characteristic length is different from the characteristic thickness. For a plate, the geometric parameters are shown in Figure . For a bluff body, b is the dimensional height for the y-direction and l is dimensional length for the x-direction. 3

Figure 1.2: Vortex parameters for tandem plates

If the plate is arranged parallel to the flow as presented in Figure 1.2, b is equivalent the plate thickness; l is equivalent to the chord length of the plate. For the fluid parameters, a freestream velocity of U approaches the plate along the x-direction. The fluid flows over the plate and sheds vortices over the back end of the plate at a given frequency, f. The vortex moves downstream at a vortex velocity, υ, from the plate. Using the freestream as a frame of reference, the vortex propagation velocity, defined as

p U , (5) describes the difference between the freestream velocity and the vortex velocity*. The distance between two subsequent vortices shed from the same side of a body is defined as λ. The vertical distance between two vortices on alternate sides of a body is defined as β. The height of the wake region is given by η.

1.2 Vortex shedding in the Context of Engineering

Within the discipline of engineering, vortex shedding is of interest due to vortex- induced vibration (VIV) of structures and components. Vortex induced vibration becomes problematic when the frequency of vortex shedding is near the frequency modes of the object. As the vortex shedding frequency approaches the natural

* If the plate were moving and the water quiescent, this would be the vortex velocity. 4

frequency of the body, lock-in of the system occurs resulting in large amplitude vibrations [3]. The role of an engineer is to ensure the component will not fail due to vibrations. Failure is significantly less likely to occur when the vortex shedding frequency and modal frequency of the component are different enough to avoid lock- in.

Avoiding lock-in is a problem to be solved with the engineering process. The engineering process is the method an engineer uses to get from a defined problem to a manufactured solution. The process begins with research on the problem to provide a set of potential solutions. From the solution set, the solution is developed and tested until a final manufactured design is produced. Throughout the engineering process, various tools are used to produce a final design. The tools are used at different stages as appropriate. Early stage tools, such as empirical correlations and analytic solutions, are paramount to the development of a viable solution set. On the other hand, tools such as models and simulations work to develop a solution to be tested with a prototype. The relationship between the engineering process and design tools is shown in Figure 1.3. 5

Figure 1.3: Engineering process integrated with design tools

In Figure 1.3, the initial problem faced by the engineer is to design a component to avoid the adverse effects of lock-in. A solution to the problem is to ensure the vortex shedding frequency does not align with the frequency of the component. If one assumes the frequency of the component is known (often a bold assumption), then assessment of lock-in is performed by determining the vortex shedding frequency. 6

To determine vortex shedding frequency, a variety of tools can be implemented at various stages in the engineering process. A thorough review of available literature reveals analytic solutions, empirical correlations, and sets of data. In terms of vortex shedding, only a few studies provide analytic solutions [4]–[6] and empirical correlations [1], [7]–[9]. Both offer a potential solution to the problem, depending on the assumptions used. The biggest assumption often being the geometry of the potential solution is representative of the geometry of the problem.

If the assumptions employed are deemed inappropriate, models and simulations are used to develop a solution to the problem. Models exist for determining vortex shedding frequency [10]–[12]. The use of vortex shedding models is limited because the shedding frequency is determined after the input of experimentally determined parameters. Simulation tools like computational fluid dynamics (CFD) (which makes use of fluid models) determine vortex shedding frequency without the need of experimental data. Any simulation provides a solution, but the user must be aware of the limitations of the simulation’s ability to provide a realistic solution.

If simulations fail to provide a reasonable solution, the next step is to test a solution directly via experiment. The experiment typically involves a prototype or representative geometry. It is of economic significance to ensure the test is a success with as little user cost as possible, i.e. limiting the use of simulations and construction of prototypes. A desire to reduce cost provides the motivation for development of early stage tools such as models, correlations, and solutions.

Determination of the vortex shedding frequency in practice is done by assuming the structure is a cylinder, or simply the Strouhal number is 0.2, which corresponds to the nominal value of the cylinder. The solution is valid for a wide variety of shapes and Reynolds numbers, so the solution in Figure 1.3 is often reached without a need to test the solution. The problem occurs when the assumed value of the Strouhal number is significantly different from the actual value. For vortex shedding between tandem plates, there are no studies devoted to determination of the Strouhal number over a 7

range of Reynolds numbers. As mentioned earlier, only tandem cylinders or single plates have data to be used for comparison. Tandem cylinders capture two-body mechanics of vortex shedding, but the flat edges and length of plates invalidates the tandem cylinder solution. Single plate studies capture the edge effects of plates, but fail to account for the influence the trailing plate has on the vortex shedding frequency. No study has fully characterized the vortex shedding frequency for tandem plate geometries. Such a study would aid the engineer in developing a more complete solution set.

1.3 Motivation

A complete solution set is important for all engineering fields. In the field of nuclear engineering, fluid structure interactions have recently been given more attention. The increase in interest began with the creation of the Global Threat Reduction Initiative (GTRI).

The GTRI was established by the National Nuclear Security Administration (NNSA) in 2004 with the mission of protecting and reducing the vulnerability of civilian nuclear sites around the world. One of the highlights of the program is to convert research and test reactors at home and abroad from Highly Enriched Uranium (HEU) to Low Enriched Uranium (LEU) fuels. By switching directly from HEU to LEU, an LEU reactor produces fewer neutrons than an HEU reactor, and in some cases, results in the reactor being unable to achieve criticality. For a majority of research reactors, exemplified by the Oregon State TRIGA Reactor (OSTR), criticality is achieved and the original neutron fluxes are restored with LEU fuel by adding more fuel elements to the core. Not all reactors, exemplified by the Advanced Test Reactor (ATR) at the Idaho National Lab (INL), have the luxury of simply adding more fuel to the core. In such cases, entirely new designs must be created, tested and qualified for use as nuclear fuel. Qualification of new fuel designs requires the engineering process be followed and tests be conducted for all phenomena including FSI. For qualification of new ATR fuel, several tests have occurred, but one test emphasized the importance of completely 8

understanding the problems associated with vortex shedding. This test was AFIP-6 Mk II.

The AFIP-6 Mk II experiment experienced conditions that resulted in a trailing plate breaking off while in the core of the Advanced Test Reactor (ATR) at the Idaho National Lab (INL) [13]. Initial engineering analysis of the experiment was done taking into account the most recently available literature. Post failure, an additional computational analysis was performed which linked vortex induced vibrations to the failure of the plate within the ATR. The conclusion was several geometry factors altered both the frequency of vortex shedding and the excitation frequency of the plate. One of the primary factors was the tandem plate geometry.

To what extent the tandem plate geometry of AFIP-6 Mk II altered the frequency of vortex shedding is unknown. In literature, the only tools available for determining vortex shedding frequency are the solutions for tandem cylinders and single plates. An enhanced understanding of the vortex shedding phenomenon for tandem plates is paramount for safe designs in the future.

1.4 Objectives and Outcomes

Vortex shedding is a design concern and tools need to capture the phenomenon efficiently and accurately. While tools have been developed substantially for cylinders [14], [15], only a few studies address tandem plates [16]–[19]. The tandem plate studies provide some insight into the nature of the vortices, but do not completely address the complex nature of flow pattern or the vortex shedding frequency. Consequently, barring direct numerical simulation (DNS), there are significant limitations in the ability to calculate vortex shedding frequency between tandem plates.

The study adds to the body of knowledge by providing an improved understanding of the vortex shedding phenomena between tandem plates. Characterization of the vortex shedding frequency provides insights into vortex shedding and improves the ability to predict the vortex shedding frequency of tandem bluff bodies. The introduction of a 9

secondary plate downstream of the leading plate generates a pressure field reducing the vortex velocity, which subsequently reduces the vortex shedding frequency of the leading plate. This study shows the changing regimes of vortex shedding between plates. The lack of knowledge as to the nature in which the trailing plate influences the vortex shedding of the leading plate has been addressed in three ways:

1. Determination the vortex shedding frequency for tandem plates. 2. Development of a regime flow map for vortex shedding between tandem plates. 3. Provision a relation for the Strouhal number defined from the Reynolds number and gap spacing ratio (b/G).

These three outcomes have been addressed via empirical means through use of particle image velocimetry (PIV), the data has been synthesized into flow regime map, which provides both frequency of shedding and regimes. This dissertation details the basis for this work, the methods for experimental data collection, the approach for synthesizing and interpreting the experimental data and the results and conclusion of the work.

1.5 Overview of Proceeding Sections

The following sections paint a portrait of the study. Starting with what has been done for other geometries and directly relating this to what was be done in order to provide a flow map for tandem plates. Culminating in the synthesis and analysis of the results. In chapter 2, a literature review of work related to vortex shedding is presented with the primary focus being plates in tandem with the flow and tandem body interactions. Additionally, available tools used to assess vortex shedding are addressed. The chapter presents what has been done for heavily studied bodies (not to mention what could still be done) and relates the research to what has been done for parallel plates ultimately showing the novelty of the study. Chapter 3 provides a synthesis of current theory specific to vortex shedding between plates. As well as addressing the finer details not explicitly addressed in chapter 2. Chapter 4 provides the nature of the experiment and how the data was collected. Chapter 5 provides the experimental results while chapter 6 discusses the conclusion of the results. 10

2 SURVEY OF LITERATURE

Within the context of vortex shedding, the most well studied objects are circular cylinders. Cylinders are a simple shape that can be representative of generic bluff bodies, so cylinder solutions are applicable to most shapes. Because they are a simple representative shape, cylinders have received the greatest amount of research attention. The amount of research on plates pales in comparison to that of cylinders, despite it being a shape of relatively large importance.

Of the research that does exist and examines plate geometries, the majority of these studies are focused on varying angles of attack (primarily angles that produced large vortices from the leading and trailing edges that oscillate between each other). For fuel plates, however, the angle of attack is nominally zero degrees, which produces a symmetric vortex street similar to that of a cylinder.

If the available literature were not limited enough, there is even less literature investigating the effects of a second body in the wake of the first. The studies, which deal with multiple bodies, are often looking at multiple cylinders, or a splitter plate designed to suppress vortex shedding entirely. Table 2.1 presents a summary of the literature on vortex shedding most relevant to the study.

11

Table 2.1: Survey of Literature Summary Geometry Data Collection Author(s) Year Cylinders Flat Plates Available Tool Other Experiment Numeric Single Tandem Single Tandem Von Karman [5], [6] 1911 X X Analytic X Heisenberg [4] 1922 X Analytic Relf & Simmons [20] 1925 X Flow Map X Dryden & Heald [21] 1926 X X Fage & Johansen [22], [23] 1927 X X X Hollingdale [24] 1940 X X Kovasznay [25] 1949 X X Roshko [7], [8] 1953 X X Correlation X Delany [26] 1956 X X X Taneda [1] 1958 X Correlation X Roshko [27] 1961 X X Abernathy & Kronauer [28] 1962 X X Eagleson et al [29] 1963 X X X Bloor [30] 1964 X X Thomas & Kraus [31] 1964 X X Goldburg et al [32] 1965 X X Bearman [33] 1965 X X Regime Map X Gerrard [34] 1966 X X Takami & Keller [35] 1969 X X Bearman [36] 1969 X X Clements [11] 1973 X Model Zdravkovich & Pridden [37] 1977 X X Gerrard [38] 1978 X X Igarashi [2], [39] 1981 X Regime Map X Okajima [40] 1982 X Flow Map X Schewe [41] 1983 X X Parker & Welsh [42] 1983 X Flow Map X Zdravkovich [43] 1987 X X X X Williamson [9] 1988 X Correlation X Stoneman et al [44] 1988 X X X Knisely [45] 1990 X X Oertel [46] 1990 X X X X X Nakamura et al [47] 1991 X X Ohya et al [48] 1992 X X Roshko [49] 1993 X X Regime Map Nakayama et al [50] 1993 X X Ljungkrona & Sunden [51] 1993 X X Nakamura et al [52] 1996 X X Bull et al [16] 1996 X Regime Map X Nakamura [53] 1996 X X Chen & Chou [54] 1997 X X X Edamoto & Kawahara [55] 1998 X X Bosch & Rodi [56] 1998 X X Luo et al [57] 1999 X Flow Map X Guillamue & LaRue [58] 2001 X X X Yao et al [59] 2001 X X Hourigan et al [60] 2001 X X Albhorn et al [10] 2002 X Model Noack et al [12] 2003 X Model Xu & Zhou [61] 2004 X Flow Map X Mills et al [62] 2005 X X Ryan et al [63] 2005 X X X Papaioannou et al [64] 2006 X X X Deng et al [65] 2006 X X Ozgoren [66] 2006 X X X Carmo & Meneghini [67] 2006 X X Thompson et al [68] 2006 X X X Blazewicz et al [17], [18] 2007 X X Regime Map X Yen et al [69] 2008 X X Leclercq & Doolan [19] 2009 X X X Sumner [70] 2010 X X X X X Singha [71] 2010 X Correlation X Carmo et al [72] 2010 X Regime Map X Shi et al [73] 2010 X X Naghib-Lahouti et al [74] 2014 X X Zhang & Liu [75] 2015 X X X 12

2.1 Vortex shedding Theory and a Universal Strouhal Number

Since one objective of the work is to develop a flow map, it is important to establish what has been done in literature to determine the vortex shedding frequency, which corresponds directly to vortex shedding regimes. In addition, another objective is to derive a relationship between the Strouhal number, Reynolds number and gap spacing ratio. In order to assess the feasibility of the goal, it is imperative to lay out previous attempts to develop correlations for Strouhal numbers.

Observations and investigation of the vortex shedding phenomenon were conducted as early as the 19th century by von Helmholtz, who investigated mathematically, the formation of a vortex [76], and Strouhal, who investigated the frequency of oscillations of wires in the wind [77]. Von Helmholtz’ mathematical work was further expanded upon by Kelvin [78]. Kelvin examined the drag and creation of “dead water” (wake) regions. These observations were the genesis for the first major investigation and publication to develop a solution for the vortex shedding phenomenon. A work presented by von Karman in 1911 [5]*.

Von Karman’s work specifically referenced shedding of vortex sheets behind a bluff body, but his work focused mainly on the calculation of the drag generated from the formation of vortices behind bluff bodies. Conveniently, the derivation presumes nothing about geometry, so it applies to plate geometries. The derivations for these were performed by assuming consistent spacing of the vortex filament and only one pair of vortices could move freely. In the aforementioned work, von Karman described the propagation velocity,

   812 , (6) p 

*Equations from this reference were either translated from, or are transcribed incorrectly in, the original work and thus have been updated to accurately reflect their values as mentioned later in [4]. 13

in terms of the strength of the vortex, Γ, and the separation distance between the same rotationally shed vortices, λ. The vortex strength, being a function of the freestream velocity and separation length can be replaced producing a propagation velocity

12  p  8 U , (7) in terms of the freestream velocity, U. The relationship between separation height, , and length, λ, was further defined by von Karman as,

 cosh 212. (8) 

Combining these terms, von Karman produced an equation for the drag imposed by vortex shedding where the drag

DU1812  , (9) could be described in terms of the separation length, density and freestream velocity. Despite the work being groundbreaking in terms of relating the vortex shedding to the drag of a body, the work required one to determine the separation length or height in order to calculate drag.

The following year, experiments performed by von Karman showed the empirically derived relationship between the propagation velocity and freestream velocity,

 p  0.20, (10) U to be substantially different than the derived values, but provided a relationship between the separation length and plate thickness,

  5.5, (11) b which was measured [6]. 14

Heisenberg in 1922 attempted to expand upon the work of von Karman by adding dimension to it [4]. Taking a similar approach to von Karman, but utilizing laws developed by Kirchhoff, Heisenberg derived a relationship between the unknown parameters and known values. Heisenberg took von Karman’s values for the vortex parameters and developed solutions to relate the size and scale of the ideal vortex street to a meaningful velocity and length. One such relation is the strength of the vortex is,

 U 2  , (12)  2 is directly related to the stream moment. This provides a value for the propagation velocity of

p  0.229U , (13) based on the freestream velocity. The second solution comes from an assumption based on observations of the streamlines of von Karman’s potential vortex street. The vortex street streamlines, without an applied velocity field is shown in Figure 2.1.

Figure 2.1: von Karman potential vortex street streamlines.

Heisenberg noted that the street pictured in Figure 2.1 could be divided into two regions: a vortex region (red) and a backwards flowing stream region (blue). Between 15

the two regions exists a streamline that extends towards infinity (dashed line). Heisenberg concluded that the integrated flow in the stream region was equal to the flow displaced by the plate. Assuming this provides a relationship between the flow around the plate,

 Ub  , (14)  and the flow between the vortices. The solution of which provides a vortex length to plate thickness,

  5.45, (15) b and tends to be very close to the values found by von Karman.

The work of von Karman and Heisenberg looked at drag, but surmised nothing about the frequency of shedding despite the fact the shedding frequency can be calculated from the given parametric ratios. Additionally, the works assumed vortex shedding from a plate positioned normal to the flow. Fage and Johansen acknowledged these shortcomings and addressed them with experimental data [22]. While Fage collected data for varying angles of attack, he found that the shedding of vortices was far from parallel, which had been an assumption in the analysis of both von Karman and Heisenberg. Fage also observed the frequency of vortex shedding was proportional to the incident velocity and that the Strouhal number of the present work matched that of von Karman’s experimental work [6]. Up to this point, no correlation between the Strouhal number and Reynolds number had been calculated, however, the analytic solutions assume a constant value of the Strouhal number. A comparison between the vortex dimensions of von Karman, Heisenberg, and Fage is presented in Table 2.2. An additional Strouhal number based on the vortex shedding distance is given as well. 16

Table 2.2: A comparison of vortex shedding dimensions. Fage & Johansen [22] at x/b Parameter von Karman [5] Heisenberg [4] = 10, 90  0.365 0.283 0.381*   0.592† 0.771 0.765 U

St 0.216 0.218 0.2915 St n/a 0.141 0.146

Fage and Johansen further investigated the nature of the vortex shedding wake and comparison of Strouhal numbers [23]. For this study, they examined the wake structure left behind by various objects. What they found was something quite interesting. Strouhal numbers for shape dimensions were very different, but if the length scale of the Strouhal number was changed from a body dimension to a more appropriate vortex- wake-height dimension,

f St  , (16)  U the results should be universal. Consequently, this proved to improve the comparison of Strouhal numbers for various geometries drastically. These geometries and Strouhal numbers are presented in Table 2.3, with an additional column added by the author for the Strouhal number based upon the longitudinal spacing of the vortices calculated from Fage’s data. The third column is presented based on the assumption,

 St  St , which is true for von Karman’s approach, and the wake size, in some cases may not be representative of the actual vortex spacing. Additionally, the author has supplied a

* This ratio was observed to change with x/b (x being the distance from the trailing edge of the body). † Heisenberg references von Karman’s formulae, which differ from the ones in this paper. 17

fourth row for the relative standard deviation to quantitatively, but subjectively, compare the universality of various Strouhal numbers.

Table 2.3: Comparison of various Strouhal numbers with various shapes fb f f  Shape St  St  St  U  U  UU Flat Plate (α = 90⁰) 0.146 0.270 0.919 Flat Plate (α = 40⁰) 0.231 0.277 0.771 Cylinder 0.187 0.271 0.799 Airfoil (α = 45⁰) 0.210 0.275 0.820 Airfoil (α = 10⁰) 0.735 0.418 0.767 Wedge 0.238 0.273 0.821 Ogive 0.271 0.298 0.856 Extend Ogive 0.254 0.332 0.846* Relative Standard 61.4% 18.2%† 5.6% Deviation (RSD)

Fage’s results empirically provided evidence of a universal Strouhal number – a Strouhal number independent of geometric shape of the vortex generating body. The concept of a universal Strouhal number was looked at again in a 1953 study by Roshko [7]. Roshko’s primary interest was not in the universality of Strouhal numbers, but the vortex shedding behind a body. The study greatly upon vortex shedding theory. Roshko noted three primary differences between the idealized vortex shedding street discussed by von Karman and a real vortex shedding street.

(1) The vortex street is finite. The street losses its identity as the vortices travel downstream, but the assumption is valid if the street maintains its identity for at least 10 diameters. (2) Vortex spacing is not constant along the street. More importantly, the vertical spacing increases downstream.

* Fage notes the value increases steadily with distance behind the model. † Even with the outlier of the airfoil at 10 degrees removed, RSD is greater than 10 percent. 18

(3) Vortex cores are finite. Vortices grow downstream and ultimately diffuse into one another and the turbulent freestream.

Roshko defined the vortex spacing using different parameters from previous studies. First, the frequency measured from the shed vortices,

 f  , (17)  must be the same as the Strouhal number shedding frequency,

U f  St . (18) b

When combined, equations (18) and (17) produce a useful relation,

1  , (19) bUSt for the vortex length in comparison to the dimensional height. Roshko then notes that Fage’s defined height for the Strouhal number would be approximately that of the difference between the vortex core centers with the introduction of a small error. In essence, they would be the same, so the relationship between Fage’s edge parameter and the vortex center is

 1   . (20) 

Roshko then combines equation (19) with (20) to provide,

 1   St , (21)   p 1  U a relationship between Fage’s Strouhal number, defined in equation (16), and the error term. From here Roshko notes the factor on the right hand side is approximately unity 19

and Fage’s Strouhal number is approximately 0.28, which in turn provides a value of 0.28 for β/λ, approximately the value for von Karman’s theoretical ratio. Finally, in looking at the wake width, Roshko estimated the spread of the vortex wake,

1/2 Ux   , (22)  b grows downstream at a rate of the 1/2 power. On the nature of vortex shedding frequency, Roshko left the following note:

“There is yet no adequate theory of the periodic vortex shedding, and it is not clear what is the principal mechanism which determines the frequency.”

Despite the detailed investigation into vortex shedding, there still was no clear cause as to the vortex shedding frequency and whether or not the spacing of the vortex street had any relationship to the frequency. Roshko decided to pursue a solution for this problem while investigating all bodies, not just cylinders. Roshko’s goal was a universal Strouhal number.

The aforementioned pursuit resulted in the collection of large amount of data published by Roshko in 1954 [8]. Expanding largely on his previous work, Roshko sought to find a universal Strouhal number correlation. First, he defined a wake Strouhal number in terms of vortex dimensions and velocity such that,

U  St*  St , (23) Ubs making it a function of the calculated Strouhal number and the relationship between the length dimensions of the vortices and the bluff body as well as the vortex velocity to the freestream velocity. The value Us is the separation velocity. Furthermore, he defined a wake Reynolds number, 20

U  Re*  Re s , (24) Ul in a similar fashion. The velocity ratio was calculated from the pressure difference in the wake of the bluff body as,

12 Us RCU1  ps   , (25) U with the vortex separation distance being calculated from theory. To this end, the wake Strouhal number can be calculated as,

* 0.281 St 0.164 . (26) RUU 

The above value will hold so long as

U s  , which is true for higher Reynolds number and all shapes, but for lower Reynolds number, the relationship fall apart, likely due to the nature of turbulence. Roshko then provided a methodology for calculating drag form the collected data using only the vortex shedding frequency or the frequency from the pressure in the wake of a body.

With advances in theory, Gerrard took a look at the mechanics of the formation of vortices [34]. In particular, two major studies were examined. Those studies were Roshko’s [8], previously mentioned, and Goldburg’s study the previous year which noted for supersonic flows, the Strouhal number,

f  St  2 (27) 2 U was a function of the momentum thickness [32]. Gerrard’s review answered some questions but ultimately raised more questions than it answered [34]. Of particular note was the importance of the wake region and fluid entrainment on the shedding 21

frequency, which Gerrard concluded was fundamental to determine a scale for the vortex shedding frequency. For circular cylinders, it was noted the free-stream turbulence affected the Strouhal number and shedding frequency.

2.2 Vortex shedding for Cylinders

While it may seem odd to have a section devoted to cylinders in a document about long plates, the reason for doing so is three-fold. First, the literature on cylinders is ubiquitous, whereas the literature on vortex shedding from plates parallel to the flow is limited. Second, the direct observation of phenomenon starting from a single body and extending it to a two-body problem is easily realized. Third, as both Fage and Roshko observed, there is a somewhat universal nature to vortex shedding between bodies of various shapes and sizes, so long as the characteristics of the vortex shedding is accounted for appropriately.

2.2.1 Single Cylinders

For the single body problem, the most recent review was presented by Williamson in 1996 [79] and multiple books have been published more recently [15], [70], [80]. The nature of vortex shedding in cylinders is described through changes in regimes. Cylinder regimes changes are dependent upon the Reynolds number. Regime transitions result in changes to the relationship between the Strouhal number and the Reynolds number. Despite cylinders being heavily studied, there is no concrete consensus on the flow regimes of a cylinder, but there is agreement in the basis for the varying transition points, albeit the regimes do not entirely agree upon the corresponding Reynolds number or number of regimes. For a single cylinder, the most comprehensive regime map includes nine major regimes further divided into 12 minor regimes. The regime map delineated in Sumer’s book with all regimes includes the following:

Creeping Flow 22

(Re < 5): Flow moves over the cylinder without any noticeable separation resulting in no vortices being shed. Since flow is continuous, no frequency component exists.

Symmetric Separation

(5 < Re < 40): Clear vortices are seen on the backside of the cylinder and do not shed, but act as a recirculation zone behind the cylinder. The vortices are symmetric. Since no vortices are shed, there is no major frequency component.

Laminar Street

(40 < Re <64): The symmetric vortices do not remain so and result in a vortex street being generated. In this regime, the vortex shedding is completely laminar. A very clean frequency signal exists.

(64 < Re < 200): Increased Reynolds numbers result in parallel shedding of the vortices is becoming instable. Oblique shedding of the vortices results in a drop in the frequency.

Transitional

(200 < Re < 250): The transitional regime is marked by the transition of the wake from laminar to turbulent near the cylinder, or rather motion in the out of plane direction begins to become significant. Erratic behavior in the signal exists, a direct consequence of the transition to turbulence. The frequency shifts between the laminar and turbulent shedding frequencies and experiences a significant drop.

(250 < Re < 300): A continuation of the transition regime, the stable state is at a higher frequency than the previous, but is still transitional.

Subcritical 23

(300 < Re < 1000): The wake is completely turbulent; however, the boundary layer separation is still laminar. Three-dimensional effects are still developing. The Strouhal number increases with formation length and base suction drops. The frequency signal is marred with turbulence, but a frequency component can easily be found.

(1000 < Re < 3x105): Same as above, however in this region is marked by a turnaround in measurement parameters, that is to say the Strouhal number and formation length drop while the base suction increases.

Critical

(3x105 < Re < 3.5x105): One side of the wake develops turbulent boundary layer separation while the other is laminar. The sides of the cylinder switch back and forth between laminar and turbulent. The critical region is mark by an immediate jump in the frequency.

Supercritical

(3.5x105 < Re < 1.5x106): The frequency jumps again out of the critical region as both sides develop turbulent boundary layer separation. A high burst peak can be seen, with a slightly lower peak present in the frequency signal, both of which collapse into one occasional burst peak.

Upper Transition

(1.5x106 < Re < 4x106): One side develops a turbulent boundary layer over the entire face. In the upper transition, no well-defined frequency exists; only an approximate peak in the regular Strouhal range exists. The occasional high frequency burst of the supercritical regime will appear.

Transcritical

(4x106 < Re): Turbulent boundary layer develops on both sides resulting in a

peak that becomes more defined as the Reynolds number increases. 24

The Regimes provided are based upon transitions to turbulence at various location in the flow field and are largely based upon the Regimes developed by Roshko [27] and Schewe [41].

The creeping flow problem was studied early on, most notably by Stokes, who found a solution for a sphere but not a two-dimensional cylinder. The creep regime is interesting because two dimensional flows around a body do not have a non-trivial steady state solution for Newtonian fluids – referred to as Stokes’ paradox – but do have a solution for pseudo-plastic fluids [81]. Creeping flow does not produce vortices, so its relevance to the study is limited.

The next transition is to the generation of symmetric vortices. While creeping flow is typically defined as flows having a Reynolds number significantly less than one, changes to the cylinder solution and the wake dynamics are not readily observed until the Reynolds number reaches about five. As the Reynolds number increases, so does the length of the recirculation region. A numeric solution for the symmetric separation and creep regimes was developed by Takami and Keller in 1969 [35]. While vortices are generated, the vortices do not shed resulting in a constant stream beyond the separation region.

The laminar street regime first appears as the symmetric separation regime becomes unstable and vortices begin to be shed. While the vortices being shed remain laminar, the Strouhal number surprisingly has a discontinuity around Re = 64 resulting in two different sub regimes. Various theories emerged for the causes of the discontinuity such as turbulence levels, a change in the mean flow through the test section, transition of the flow, and vibration of the cylinder. Williamson proved there was only one true discontinuity that was not caused by any of the aforementioned theories [9]. The cause was three-dimensional effects, or specifically, oblique shedding of the vortices. This discontinuity was studied in detailed and differences in the Strouhal number were corrected by Williamson. Williamson was able to collapse the laminar shedding regime data onto a single universal Strouhal curve, 25

St 3.3265 St   0.1816  1.6x 104 Re (28)  cos Re by dividing by the oblique shedding angle. The upper range of the laminar shedding regime was thoroughly investigated early on by Kovasznay [25]. Kovasznay noted, for Reynolds numbers below 150, vortex shedding was relative stable. The onset of the laminar street regime was initially hypothesized to be a laminar to turbulent transition. It was Roshko who observed the onset of shedding of vortices was not a laminar to turbulent transition, but instead one of two different regions of stable viscous flow [7]. The stability was shown because any disturbance inflicted upon the stable condition would be damped out. For the laminar street regime, Roshko created the empirical correlation,

21.2 St 0.212 1 , (29) Re to fit the data. As the Reynolds number increased beyond the laminar street regime, Roshko observed a transition range where the vortex shedding would occur in a stable manner, but would experience sharp burst and irregularities in the frequency making it difficult to determine the frequency.

It was this region where the transition from laminar to turbulent flow occurred. Roshko speculated that this transition region based on Reynolds number would be influenced by the freestream turbulence or cylinder roughness. In this transition regime, the wake becomes three-dimensional and transition to turbulence occurs very near to the cylinder. These observations have been observed by several researchers [7], [38], [82] with Bloor investigating the transitional Regime in detail and showing a Reynolds number of 400 produced an entirely turbulent wake [30].

The transition regime has been shown to end around a Reynolds number of 300. At this point, the vortex shedding is predominantly turbulent. Roshko noted the irregularities seen in the transition regime continued on into the subcritical regime, but the frequency 26

became easier to determine and was initially referred to as the irregular range [7]. Roshko fit another empirical curve to the subcritical data. For Reynolds number below 2000,

12.7 St 0.212 1 (30) Re was found to be representative of the data. If the empirical correlation were to be extrapolated, it is within 4% if applied up to a Reynolds number of 10,000. Further investigations by Roshko led to a split of the subcritical regime into two sub regimes [49]. For Reynolds numbers less than 1000, the Strouhal number continues to increase, base-suction decreases, and formation length increases. Above a Reynolds number of 1000, in what is referred to as the “Shiller-Linke” regime, the Strouhal number steadily drops, formation length decreases and base suction increases. Unlike the sharp discontinuity seen in the laminar street regime, the transient between the two subcritical sub-regimes is less of a discontinuity and more a continuous transition. Above Reynolds numbers of 10,000 another sharp transition occurs.

The transition to the critical regime is another transition to turbulence, this time in the flow over the cylinder. In the critical regime, the boundary layer separation is turbulent on one side and laminar on the other resulting in a very large lift coefficient on the cylinder. The super critical Reynolds number had yet to be addressed for cylinders, previously noted by several authors [20], [26], [27]. In 1969, the phenomena was understood significantly better by Bearman who observed an instant jump in the shedding frequency followed by significant broadening of the spectral density peaks [36]. This shift was proposed as the onset of the transition to critical shedding over a circular cylinder.

Information beyond the critical point of a cylinder was studied largely by Schewe [41]. It was noted that critical shedding is do the asymmetry of the turbulent shedding. When the shedding is turbulent on both sides of the cylinder, the cylinder enters the supercritical regime. In the supercritical regime, the vortex shedding frequency is much 27

more predictable. While the trip to turbulence may be asymmetric on either side of the cylinder, this does not influence the general trend of vortex shedding until the turbulence trip reaches the leading edge of the cylinder.

The turbulent boundary layer reaching the cylinder leading edge initially only occurs on one side as opposed to both. This bias creates the upper transition region. A bias in turbulent and laminar layers results in a variation of lift on the cylinder.

As the Reynolds number is increased even further, the boundary layer over the entire cylinder is turbulent. At this point, the cylinder has entered the transcritical regime. Flow over the cylinder no longer generates the variation of lift seen in the upper transition region and the Strouhal number of the cylinder is again predictable.

While the regimes for circular cylinders are extremely diverse, attempts have been made to unify them. Unification of the cylinder regimes for a single cylinder based upon a single reference parameter would yield a potential source universal Strouhal number correlation. One such attempt was made by Roshko [49]. Roshko reviewed general progress and assessment of bluff body dynamics that had been made over the years. Roshko defined six regimes for a cylinder (reference to the primary regimes in parenthesis).

Regime 1 (“Laminar Street”) (50 < Re < 180): Vortex shedding starts around a Reynolds number of 50 and the wake becomes unsteady. For this region, the wake is entirely two- dimensional. Regime 2 (“Transitional”) (180 < Re < 300): In this range, the vortex shedding begins to develop instability in three dimensions. Small changes to the drag are noticed, but in general, direction is maintained. Regime 3 (“Subcritical I”) 28

(300 < Re < 1000): In this range, vortex shedding’s influence on the drag is reduced comparted to the previous ranges, but it still plays a role. Additionally, base suction begins to decrease. Regime 4 (“Subcritical II”) (1000 < Re < 1x105): Base suction begins to rise. In this regime, vortex formation occurs closer to the base and the wake is shortened. Regime 5 (“Critical”) (1x105 < Re < 4x105): Critical transition occurs. In this range, the drag behind a bluff body decreases immensely. The reason for this is the separation point begins to move downstream for a cylinder. Regime 6 (“Supercritcal”) (4x105 < Re): The post critical regime exists. The separation of a cylinder reaches its maximum point downstream.

These regimes were determined quantitatively in terms of the drag coefficient. Coincidentally, these regimes relate directly to the observed Strouhal number of a cylinder. An observation made by Roshko was that the Kelvin-Helmholtz frequency,

U s fKH  0.017 , (31) 2 which is a function of a jet stream and the momentum thickness was very close to the shedding frequency. Alternatively, the Bloor frequency for cylinders,

0.5 ffB  0.11 Re , (32) correlates well to the shedding frequency. In 2002 Ahlborn ambitiously expanded on Roshko’s work to collapse the Strouhal number and Reynolds number into one empirical correlation [10]. The relationship was derived semi-empirically. Ahlborn provide a relation,

4N c 1 St2  St D , (33) 2Re 2 3/2  2  2 29

which relates the Strouhal number to the Reynolds number as a function of the drag coefficient and vortex parameters. While the correlation does provide a single equation between the Strouhal number and Reynolds number, the other coefficients require a complete measurement of the system resulting in a limited applicability of the equation.

2.2.2 Tandem Cylinders

Whereas single cylinders are studied in depth and have a full regime map, such is not the case for tandem cylinders. To the author’s knowledge, no book dedicated to flow over two or more cylinders exists. However, in the literature, a relatively recent review of two cylinders in cross-flow was published by Sumner in 2010 [14]. Because of the lack of any governing text, Regime maps and transitions of flow are on a per article basis.

A simplified flow map was proposed by Xu [61]. Xu collected copious amounts of data over a wide range of spacing ratios and Reynolds numbers. Xu’s data also observed a significant change in the Strouhal number around a G/b ratio of five suggesting the value as a basis of change for single cylinder and tandem cylinder flow regimes. A similar change had been first observed by Zdravkovich via the drag coefficient [37]. While single cylinders have eight different regimes that generate vortices, Xu’s data only focused on the subcritical region. Xu defined the regimes based upon the location where the vortex reached the leading cylinder. Xu defined three tandem cylinder regimes (4 sub regimes) for the subcritical cylinder:

Extended Body Regime

(G/b < 1): Vortices become trapped and the vortex shedding that does occur is directly from the trailing cylinder. In this regime, the cylinders essentially become a singular bluff body.

Reattachment Regime 30

(1 < G/b < 2): Vortices are generated by the leading vortex; however, they cannot be completely shed. The consequence is vortices begin to roll up along the tailing cylinder.

(2 < G/b < 4): Same as above, however vortices are reattaching at the leading face of the cylinder as opposed to the trailing face.

Co-shedding Regime

(G/b > 4): In this regime, the gap spacing is sufficient to allow complete vortex shedding from the leading cylinder. Vortices from the lead cylinder do reach the trailing cylinder; however, vortices are shed by both bodies.

Of course Xu’s regime definitions were based largely on the regimes put forward by Zdravkovich [43]. The regimes put forth by Zdravkovich were synthesized in his second review of the subject of flow past multiple cylinders. At the time, a lack of organization of the literature prompted Zdravkovich to write another review detailing the interference between two cylinders in various locations. Three primary regimes of importance were identified by Zdravkovich based on separation distance. The transition spacing ratios of the regimes were no so easily defined as the regimes exhibited a strong hysteresis. Zdravkovich noted that the regimes were dependent, to some extent, upon Reynolds numbers.

While Zdravkovich, made observations on tandem cylinder regimes with respect to Reynolds number, most of the pioneering work explicitly defining the flow regimes for tandem cylinders was performed in two articles by Igarashi [2], [39]. In which, Igarashi identified six distinct flow regimes as opposed to the three. It is important to note that, like Xu’s data, these regimes would be for a single cylinder entirely in the subcritical regime. Igarashi classified six different flow patterns for Reynolds numbers ranging from 1x104 to 1x105. The map of these patterns can be seen in Figure 2.2. The regimes were defined as follows (Xu’s corresponding regimes are parenthesis):

Regime A (“Extended Body”) 31

For very small gap spacing, the cylinders are close enough such that both cylinders act as a singular body and the separation region off the leading cylinder doesn’t reattached Regime B (“Extended Body”) If the Reynolds number is sufficiently high, synchronization of the vortex shedding occurs between the two cylinders. Vortex shedding frequency is indifferent to changes in velocity. Regime C (“Reattachment I”) As the gap spacing increases for regimes A and B, stationary vortices are formed between the two cylinders. Regime D (“Reattachment I”) Increases in the gap spacing results in the stationary vortices developing instabilities and vortex shedding is intermittent from the lead cylinder. Regime E (“Reattachment II”) The gap size becomes sufficiently large such that the shear layer separated from the leading cylinder rolls up onto the trailing cylinder. The region is a bi-stable regime between D and F, with E’ being a continuation where the roll up pattern remains for a long time. Regime F (“Co-shedding”) Both cylinders shed defined vortices. Regime G (“Extended Body”) Is a pseudo-regime, where A, B, and C are all unstable and each regime can exist. 32

Figure 2.2: Igarashi’s flow map of tandem cylinders.

Figure 2.2 was adapted directly from Igarashi [2]. The data was adapted through plot digitization of the data presented in the article to capture the transitions lines as accurately as possible. While Igarashi and Xu both examined regimes for subcritical flow over tandem cylinders, lower Reynolds number regions were left out. To examine the low Reynolds number regimes, a numerical investigation and perturbation analysis was performed by Carmo [72]. Carmo’s approach being numerical looked at calculable parameters, in the middle of the wake as opposed to visual observations. To this end, Carmo found three primary regions based upon shedding in the gap region (corresponding regimes in parenthesis):

Steady Gap (SG) Regime (“Extended Body”)

Flow is steady in the gap, i.e. no vortex shedding or symmetric vortices are generated in the gap region.

Alternating Gap (AG) Regime (“Reattachment”)

Flow is alternating in the gap, but vortices are not shed. 33

Wake in the Gap (WG) Regime (“Co-shedding”):

Flow generates a wake in the gap; or rather, vortices are being shed.

Carmo further classified the flow as either having dominant 3-dimensional or 2- dimensional components. In addition to the five major regimes, there were other regions where two different regimes were both stable, and thus subject to hysteresis. The regimes presented by Carmo are shown in Figure 2.3. 34

Figure 2.3: Carmo’s flow map of tandem cylinders.Figure 2.3 was adapted directly from Carmo [72]. The figure was adapted by digitizing the data directly from the plot. The solid lines indicate a direct transition while the dashed lines indicate a region of bi-stability. The lighter dashed lines indicate where the separation between the dual states occurs between the hysteresis regions. While all the regimes can be linked to the three primary regions, there are still discrepancies between the various regions and the three main regions. Additionally, the maps provided by Carmo and Igarashi are not identical. One covers low Reynolds and the other high Reynolds numbers, but there are only two regimes covered at the highest Reynolds number and five regimes covered by Igarashi’s lowest Reynolds number. Even Xu’s three primary regions do not completely align with the various proposed regions. Based on the definitions provided, the two regions (three if the bi-stable region is included) for the large Reynolds numbers presented by Carmo could be expanded to the five regions in the lower bound of Igarashi’s work. Depending on the cut-off, the SG, or 2-d AG region would extend to become Regime A. The 3-d WG region would yield the E and F regimes. The bi-stable regime shown by Carmo would likely be Regime D if presented by Igarashi. Ultimately, the maps do not overlap without speculation. 35

While the regimes of tandem cylinders are complex, there is the possibility the Strouhal number, Reynolds number, and spacing ratio may be combined into a single equation. If one knows the gap spacing and Reynolds number, then the Strouhal number can be calculated. Singha provided a partial equation for the tandem cylinder problem [71]. Singha developed an empirical correlation, such that the Strouhal number could be plotted on a single line. The Strouhal number,

8/5 4/5 St 1.6x 104 Re G / b  5.5 x 10 3 Re 1/2  G / b  0.16 , (34) is described as a function of the gap ratio and the Reynolds number. While a valuable equation, the equation becomes invalid at higher Reynolds numbers, and thus is restricted in its applicability.

2.3 Vortex shedding for Plates Parallel to the Flow

The previous section developed a basis for different regimes of a cylinder and then expanded it for tandem cylinders. The following section does the same for plates, first examining the difference between a short plate and a long plate and expanding it to two plates in tandem. Furthermore, it examines the nature of vortex shedding from plates, highlighting the differences due to geometry.

There is an extensive amount of literature for vortex shedding over plates. The vast majority of the literature deals with plates at an angle and leaves only a sliver of research on plates, parallel to the flow. The first study on parallel plate geometry was conducted by Hollingdale in 1939 who provided the first look at vortex shedding from parallel plates [24]. Hollingdale made observations of the wake for Reynolds numbers below 1000, but did not provide much in lieu of frequency relations or flow regimes. Later researchers realized that even the length of the plate played a fundamental role in the vortex shedding phenomenon. 36

2.3.1 Vortex shedding over Short Plates

Over the proceeding decades, plates were treated very similarly to any generic bluff body, however, in 1982, Okajima presented a critical study of rectangular prisms of increasing ratios [40]. The results flew in the face of the idea of a universal Strouhal number. Square prisms compared very well to previous results, but the elongated rectangular prisms had some different results. Prisms with an l/b ratio of two experienced a discontinuity in the Strouhal numbers around a Reynolds number of 500. Plates with a ratio of 3:1 experienced a similar trend for low Reynolds numbers but instead of a single discontinuity, multiple Strouhal numbers were observed. Perhaps most interesting was the absence of any discontinuity for the plate with a ratio of 4:1.

These discontinuities were further confirmed by Parker who was investigating the effects of sound on the flow [42]. Parker was using plates with square leading and trailing edges. Parker observed a series of flow regimes at a Reynolds number of about 20,000, associated with the length of the plate compared to the thickness.

Bluff Body Regime (l/b < 3.2): The flow was observed to separate completely from the plate without any reattachment. Periodic Reattachment Regime (3.2 < l/b < 7.6): In this regime, reattachment would occur, but the reattachment occurred periodically and created a well-defined vortex street. Propagation Regime (7.6 < l/b < 16): Reattachment of the separated flow was constant but the vortices from the leading edge would propagate down the plate and interfere with vortex shedding from the trailing edge resulting in irregular shedding of vortices. Long Plate Regime (16 < l/b): The separated flow reattaches and diffuses into the boundary layer before reaching the trailing edge. 37

While Parker did distinguish the difference between these regimes, the data was very limited, and the Reynolds number was relatively constant.

Knisely replicated the results of Parker using a wider variety of Reynolds numbers [45]. Knisely found discrepancies in the Strouhal numbers when comparted to Parker’s data, in particular for very low and very high aspect ratios of the plates. The discrepancies were attributed to different causes. For the low aspect ratio plates, the cause was attributed to the Reynolds number’s influence on the vortex shedding phenomenon. For higher aspect ratios, geometric blockage of the flow was to blame.

Due to the observations made by Knisely, further exploration of vortex shedding for short plates was needed. The work was conducted by a team of scientists who ran a series of experiments to further investigate the influence of plate length on vortex shedding starting with Nakamura in 1991 [47]. Nakamura observed discrete values where the Strouhal number based on the chord length would remain constant and then shift to a higher number as opposed to remaining constant. Nakamura’s work was predominately in the periodic re-attachment regime. The result of Nakamura were later shown to be a direct consequence of the vortex shedding via numerical simulation performed by Ohya [48]. Ohya showed the “lock-in” type effects were geometry based and noted that the interactions between the leading edge and trailing edge of the plates. In 1993, Nakayama, performed a similar simulation and experiments [50]. For a Reynolds numbers of 400, the results of the previous studies were repeated; however, when the Reynolds number was decreased to 200, the results showed a linear increase in the Strouhal number based upon the b/l ratio. Nakamura again further investigated short plate vortex shedding to attempt to determine what Reynolds number the impinging shear layer instability occurred [52]. It was observed that transition began to occur at a Reynolds number of 250, and was present at a Reynolds number of 300 for the experiments, which agrees to the cylinder transition region. Shi further examined the short plate vortex shedding problem using PIV [73]. 38

Guillamue additionally provided a comparison for vortex shedding from a single plate over a series of ratios for a plate and array [58]. Guillamue showed the vortex shedding lock-in phenomenon was suppressed when the short plate was placed in an array of plate. The question as to what plate array spacing is needed to suppress the phenomenon remains unresolved.

Guillamue suppressed the lock-in by effectively reducing the channel thickness. Mills suppressed the lock-in by using plates with elliptical leading edges [62]. Mills observed no discontinuities in the Strouhal numbers once the rounded leading-edge plate achieved an l/b ratio greater than six. Mills’ observations meant a plate with a rounded edge had a much smaller regime change than a square leading-edge plate. In other words, with a rounded leading edge, a plate reaches the long plate regime at a b/l ratio of 6:1, rather than 16:1.

Hourigan later examined the vortex shedding for long plates via numerical simulations for both elliptical leading edges and square leading edges for low Reynolds numbers [60]. In the case of the elliptical leading edges, the change in shedding frequency was continuous, whereas in the case of the square leading edge, the shedding frequency would take on discrete values. Hourigan additionally took into account the feedback system from the trailing edge, which determined the vortex shedding frequency.

Blazewicz in 2007 provided additional descriptions of the flow regimes for a single plate presented by Parker [17]. In his study, Blazewicz confirmed the lock-in presented by previous authors. The study results were eventually expanded to tandem plates.

Insights into short plate geometries are useful, but the leading edge vortex shedding influences the vortex shedding from the trailing edge. If one is to look directly at the impact of a trailing plate on the vortex shedding from a leading plate, it is necessary to examine long plates where the leading edge plays an insignificant role on the vortex shedding form the trailing edge. 39

2.3.2 Vortex shedding over Long Plates

Unlike cylinders, no regime map exists for plates, let alone long plates. While some of the regimes for cylinders translate to plates, not all of them do, in particular the regimes beyond the subcritical cylinder. Long plates exhibit another problem making it difficult to translate regimes directly to a Reynolds number: plates develop large boundary layers.

It was Taneda who recognized this and one of the other fundamental differences between vortex shedding from flat plates and bluff bodies [1]. The second observation, which played a larger role in the preceding section, was separation of the boundary layer generates vortices in the case of a bluff body. In the case of a plate, separation occurs at the leading edge and the trailing edge of the plate. Taneda defined the characteristic thickness as the length of the plate. The result is a Strouhal number,

fl St  , (35) l U based upon the plate length not the thickness. Taneda’s results showed the following proportionality:

1/2 Stl  Re . (36) a case where the length scale is dependent upon the plate length not the width.

To empirically account for the boundary layer growth along the plate, Naghib-Lahouti performed a series of experiments [74]. The frequency of vortex shedding was accounted for by the Roshko number,

Ro ReSt (37) a combination of the Strouhal and Reynolds numbers. Additionally, the characteristic length was defined by a combination of the displacement thickness, δ1 of the boundary layer and the plate thickness. The resulting correlation, 40

0.289Re ,if Re 10,000  bb2211 Rob2   (38) 1 0.231Re ,if 10,000 Re  bb2211 provides a user with a means of determining the frequency if the boundary layer is calculated.

Beyond boundary layer growth and the influence of the leading edge, there is yet another small, but important difference between plates and shorter bodies. It was an assumption that plates underwent an identical transition to turbulence. This assumption was looked at by Ryan who examined the three-dimensional transition of the wake for long plates [63]. Ryan showed that plates did not undergo an identical transition to turbulence when compared to a circular cylinder or square cylinder. Ryan used numerical stability analysis to gather transition information on this transition. Ultimately, the long plate had a turbulent spanwise wavelength much greater than that of a cylinder.

A good portion of the early work dealing with vortex induced vibration of plates was performed by Eagleson who was looking at flow and vibration problems associated with plates [29]. Eagleson’s work examined the Strouhal numbers associated with plates if vibrations could occur and to what extent trailing edge geometry affected the solution. Eagleson’s plate frequencies provided near constant Strouhal numbers. The more important aspects of the work were the influence of plate vibrations on the shedding frequency The plates underwent a transition from matching the Strouhal number frequency to a lock-in phase where the velocity had little impact on the shedding frequency to a ramp up phase which returned the frequency to the expected Strouhal frequency of a stationary plate.

2.3.3 Tandem Plates

While no regimes have been explicitly identified for long plates, tandem plate regimes have been observed on a small scale. Bull in 1996 produced one of the first studies on vortex shedding from flat plates in tandem [16]. For the study, a rounded leading plate 41

was placed in front of a square trailing plate and the gap sized was varied. Bull identified the vortices as either trapped or free. In the trapped region, the plates would act as one body and no vortices would be shed from the leading plate. If freed, the vortices would shed in a fashion similar to a normal bluff body. Bull also observed hysteresis in the regimes while increasing and decreasing the flow rate. Figure 2.4 shows the two flow regimes identified by Bull [16].

Figure 2.4: Bull’s two primary regimes with hysteresis. Figure 2.4 is adapted directly from Bull [16]. The lines were obtained through plot digitization of the plot. The region of hysteresis is quite small, spanning at most a gap spacing ratio of 0.1. Later in 2007, Blazewicz provided a possible list of regimes for tandem plates of varying length [18]. To date, this proceeding is the closest study done to the proposed work. While the experimental results were limited, the study included description of proposed plate regimes. The proposed regimes were based around whether or not the vortices were trapped. The study essentially combined the regimes of Bull with the regimes of Parker. It did not capture the variety of changes that are similarly seen for cylinders. 42

Blazewicz and Bull provided a simplified regime schematic, and a few others have investigated the fundamental problem of vortex shedding between plates. For example, in 2009, Leclercq found a solution for the Strouhal number numerically [19]. The simulation was 2-dimensional and used low Reynolds numbers. The results under- predicted the Strouhal number but were comparable leading some credence to simplified numerical solutions as a predictor for vortex shedding.

From the experimental side, Chen examined the effects of vortex shedding on a flat blunt plate subject to vortices generated upstream of the flow [54]. Chen was primarily focused on finding the impact of regular shedding at different locations. The impact of shedding was monitored by keeping the Reynolds number and separation distance constant and altering the vertical offset distance of the leading body.

The most recent study looking at vortex shedding for tandem plates was conducted by Zhang in 2015 [75]. Like Chen’s study, a cylinder was used to generate the vortices that impacted the trailing plate. The study by Zhang directly compared plates with and without incident vortices being shed.

Ultimately, the studies for tandem plates are limited; the nature of influence of the trailing plate on the leading plate vortex shedding frequency is unknown. For this reason, development of a flow map for tandem plates based on regime changes is paramount to understanding the influence of the trailing plate on the leading plate vortex shedding. Determination of regimes and frequencies further allows a user to make a rational decision on the risk involved with vortex induced vibrations.

2.4 Additional Vortex Shedding Observations

While the previous sections dealt with cylinders, plates, or generic bodies, a few studies have been conducted which provided further insights into cylinders or the nature of vortex shedding. The remaining studies are divided into experimental studies and numerical studies. While experimental studies are reality, measurements are sometimes 43

difficult to obtain. Numerical studies on the other hand provide a wealth of information but may or may not accurately represent the true solution of the problem.

2.4.1 Experimental Studies

While Fage’s study was listed first, Fage was not however, the first to observe the universality of the Strouhal number. The first major work showing the Strouhal number was identical for similar geometries is the paper written by Relf [20]. Relf took measurements of cylinders as opposed to plates in a wind tunnel and found that the Strouhal number for cylinders of varying diameters were identical for the same Reynolds number and compared to data from earlier experiments. Additionally, it was noted that vortex shedding did not occur for Reynolds numbers below 100, the Strouhal number remained relatively constant after a Reynolds number of 1,000 but showed a sharp increase after a Reynolds number of above 100,000. A similar study was also conducted by Dryden and Heald in 1926 [21]. The study focused more on drag coefficients than an investigation into the relationship between the Strouhal number and Reynolds number.

The first work truly dedicated to the vortex shedding of tandem bodies is that of Biermann who studied the drag generated by two bodies in tandem of varying shapes and dimensions [83]. While the work was one of the first on flow around tandem bodies, it was dedicated to the calculation of drag forces.

It would not be until the 1960’s (after Roshko’s work) that multiple body problems would be heavily researched. One of the first major publications on tandem body interactions was the work of Thomas and Kraus [31]. Their worked look at two cylinders at one Reynolds number and the flow patterns behind them. It was found when the cylinder spacing was on odd interval of half of the vortex separation distance, the cancellation of the vortex street was at a maximum.

In 1965 Bearman conducted an interesting investigation of a splitter plate introduced into the wake of a plate [33]. He identified five different regimes under which a splitter plate influenced the vortex shedding. 44

Regime 1

(l2/b1 < 1.0): Vortex shedding remained relatively unaffected, frequency increased slightly and the vortex shedding location was pushed downstream the length of the splitter plate. Additionally, the base pressure coefficient was proportional to the splitter plate length. Regime 2

(1.0 < l2/b1 < 1.5): Both vortex shedding frequency and the base pressure coefficient increased. The shear layers began to break down beyond the edge of the splitter plate and vortices were formed just beyond the edge of the plate. Regime 3

(1.5 < l2/b1 < 2.0): Vortices began to develop further downstream once again and the frequency of shedding began to drop, similar to the first regime. Regime 4

(2.0 < l2/b1 < 3.0): Vortices no longer exist, but the flow has not reattached to the splitter plate, rather the system is behaving like a low Reynolds number flow. Regime 5

(3.0 < l2/b1): Vortex shedding is completely suppressed; reattachment of the flow has occurred.

In 1988 a study was performed by Stoneman [44]. The study examined the effects of sound induced lock-in between two tandem plates with rounded edges. Stoneman showed that lock-in could occur in a duct if near the resonance of chamber.

Over the next few years, many studies were performed on tandem cylinders but most of them looked at the effect of pressure fluctuations and drag on the bodies themselves as opposed to shedding frequencies. Ljungkrona presented results based upon separation ratios [51].

Oertel in 1990 reviewed the understanding of vortex shedding, in particular, the advances made within the last five years of the review [46]. Resonances can occur 45

when an object is placed in the flow field. This creates global pressure waves that create an acoustical wake tone and creates acoustic feedback. For a single bluff body, vorticity waves propagate up and downstream and drive the vortex shedding frequency.

In 1996 Nakamura examined vortex shedding from an assortment of bluff bodies, including two normal plates in tandem and examined how the Strouhal number was affected [53]. Nakamura argued that Roshko’s universal Strouhal number is applicable for single body geometries, however when multiple bodies are added the wake structure changed significantly.

In 1999 Luo investigated the flow dynamics downstream of two square cylinders in Tandem [57]. Luo observations indicated a lock-on region at low Re, followed by a constant Strouhal number when measured at a given L/D location.

Ozgoren looked at the wake downstream of a cylinder and compared that to diamonds and squares [66]. Ozgoren’s study provided a direct comparison between shedding from a rectangular edge and a circular separation.

While others looked at tandem cylinders, Yen observed tandem squares [69]. Yen essentially described the tandem square regimes in the same way tandem cylinders were observed, however, there were differences in the behavior of the Strouhal number.

2.4.2 Numerical Studies

Contrary to experimental work is the use of numerical simulations in order to investigate vortex shedding between plates.

Yao in 2001 used direct numerical simulations to analyze flow over a rectangular trailing edge [59]. Yao determined that a box of 20b:16b:6b was sufficient to model all 3-dimensional effects for the plate at a Reynolds number of 1000.

Eight years after Roshko’s observations, Abernathy ran a numerical simulation on a model for the growth of vortex streets [28]. In this work, two vortex streets were 46

initially assumed parallel, and the growth and alterations were tracked. For the first time the growth of vortex streets was modeled.

In 1998 Edamoto performed a finite element analysis of square cylinders in tandem [55]. Edamoto showed very detailed images of the transitions of flow over square cylinders, which had not been previously observed.

For years computational simulations had been maturing, and in 1998 Bosch and Rodi examined some of the models that had been used and compared the results for vortex shedding over a single square cylinder [56]. The compared models were all variants of the k-ε turbulence model; they found near-boundary conditions influenced the simulation but if conditions were appropriate, the simulations produced reasonable results.

It would be Thompson in 2006 who looked at the transition of the wake from 2D to 3D for axisymmetric bodies, i.e. an elongated ellipse and a cylinder [68].

Papaioannou looked at the three-dimensional effects of Tandem cylinders [64]. The paper in turn added insight as to why the flow regimes showed less of a dependence on Reynolds number and more of one on gap thickness as opposed to Reynolds number.

Similar to Papaioannou, Deng in 2006, also looked at the three-dimensionality of the wake of tandem cylinder numerically as well [65].

Carmo in 2006 modeled tandem cylinders numerically [72]. Carmo provided a very detailed comparison between the influence of separation distance and Reynolds number on the Strouhal number.

While numerical studies provide a non-invasive means of determining the vortex shedding frequency, to be an accurate (while not entirely true) representation of vortex shedding, direct numerical solutions (DNS) are need. At the current state of computational power, DNS is not a feasible option for desktop computing, as it requires a cluster to be performed in a reasonable period. 47

2.5 Study Justification

As a whole, vortex shedding for cylinders has been studied in depth and the mechanisms of shedding are well understood. When it comes to plates, there are some similarities, but the biggest differences being the development of the boundary layer and location of the separation region. The natural starting point for developing plate theory is acknowledging the differences between the two bodies. To present tandem plates, the theory for a single plate must be understood.

A single plate may be compared to a single cylinder, but a direct comparison between tandem plates and tandem cylinders is not so easily done. While single cylinders are understood, when a second body is introduced in tandem, the nature of study is less a full study and more of a characterization. Even Sumner in a recent 2010 review stated,

“For tandem cylinders, additional study of the gap fluid dynamics is needed to better characterize the shear layer reattachment and gap vortex shedding processes within the reattachment regime.” [14]

As the literature on tandem cylinders requires further study on the gap dynamics, the investigation of vortex shedding for tandem plates is extensively new to the field of research. 48

3 THEORY Vortex shedding from tandem plates is another fluid flow problem, and as such, it is governed by the same equations as all fluid mechanics problems (section 3.1). The governing equations (also known as the Navier-Stokes equations) are non-linear meaning the problem of vortex shedding will likely not have a neat analytical solution. Despite this, attempts at deriving a solution for the vortex shedding frequency have been made from both a scaling analysis and control volume approach (section 3.3). To understand the validity of the scaling approaches, it is useful to examine the assumptions employed and the general characteristics of flow along the plate. Flow along the plate is described in detail in section 3.2.

3.1 Fluid Dynamics

Fluid dynamics, as the name suggests, is the study of fluid flow. There are several equations useful in fluid mechanics, however only three groups of problems will be addressed herein: inviscid, laminar, and turbulent. First is inviscid flow, which is solved with the Euler equation and potential flow theory. Potential flow theory has been used to describe the von Karman vortex street [4], [5], [7], [8]. Second are laminar flow problems. These problems are solved with assumptions to the Navier-Stokes equations (often two-dimensionality). The Navier-Stokes equations are used to solve virtually all fluid flow problems. Finally, turbulent flow is discussed. Turbulent flow is often approximated through semi-empirical closure relations from the Navier-Stokes equations.

3.1.1 Inviscid Flow Equations

Inviscid flow means the flow is without viscosity. Viscosity is a fluid’s ability to resist deformation by stress. Since viscosity is an important aspect in the solution of flow problems, inviscid solutions are limited. Despite this, inviscid flow solutions are useful in determining approximations for viscous flow solutions. As with all governing equations, mass momentum and energy are conserved to derive a solution based on given conditions. Inviscid flow is solved using the Euler equation: 49

Du  P  g  h . (39) Dt

The Euler equation assumes that the material derivative of the velocity vector is dependent on the gradient of the pressure, P, and the gradient of the fluid height, h, with respect to gravity, g.

Potential flow theory is a variation of potential theory used to describe the fluid velocity field. Potential flow theory makes two primary assumptions about the velocity. First the divergence of the velocity field,

u  0 , (40) is equal to zero. This relation satisfies continuity. Second, the flow is irrotational which means the curl of the velocity,

u  0 , (41) is also equal to zero. From these two equations, the velocity,

u   , (42) may be calculated as a potential, φ. Combining (42) with (40) shows the Laplacian of the velocity potential,

2 0 , (43) is zero, and thus satisfies a Laplacian equation. From the velocity potential values, streamlines are created or, giving rise to a stream function,

  , (44) which is orthogonal to the velocity potential. 50

Vortex shedding is largely a 2-dimensional phenomenon. As such, it is useful to assume the flow is 2-dimensional when using potential flow theory. Assuming 2-dimensional flow for (44) and (42), the velocity potential and stream function,

       uv ,    , (45) x  y  y  x are directly related back to the fluid velocity in the x and y-directions. Conveniently, using 2-dimensions also allows for transforming the flow potential into complex number functions. Both the flow potential functions and co-ordinates are transferred to the imaginary plane creating a transform function,

f()() x iy   i   f    , (46) which is dependent on one variable. Several flows have known potential flow equations including that of uniform flow,

U , (47) and an irrotational vortex,

i    ln   , (48) 2 where Γ is the vortex strength and the sign indicates the direction of rotation. In section 3.3.3 (47) and (48) are combined to provide a potential flow equation for a vortex street. For now, it is important to acknowledge the usefulness of the equations, but also realize a vortex cannot form without viscosity, and thus a means of analyzing the flow with viscosity must be sought.

3.1.2 Navier Stokes Equations

As a fluid begins to move, it first does so in an organized manner. As the velocity of the fluid increases, the flow becomes more chaotic due to small perturbations in the flow. If there were no viscosity, energy would not dissipate, and flow would become 51

purely chaotic in motion. The viscosity of the fluid provides a resistance to shear which allows for two things. First, it allows for the dissipation of kinetic energy as heat. Second, it acts a resistance to the inertia of the fluid. If the viscous forces of the fluid are greater than the inertial forces of the fluid, the flow is laminar. If the opposite is true, then the flow is turbulent. Both laminar and turbulent flows are characterized by the Navier-Stokes equations. Mathematically, laminar flow solutions assume the flow solution is invariant to perturbations. Physically, laminar flow is characterized by the fluid organizing itself into lamina or streamlines.

The Navier-Stokes equations are valid for most fluid conditions. The Navier-Stokes equations assume the fluid is in a continuum and the fluid exhibits Newtonian behavior. For this study, both conditions are satisfied, and the Navier-Stokes equations are applicable. The Navier-Stokes equations are broken down into the conservation equations of mass, momentum and energy. First, conservation of mass is written as

  u  0. (49) t

Note that if equation (49) is assumed incompressible, it reduces to (40). The second equation is that of momentum. The momentum is broken up into x, y and z components. Written as a single equation, momentum,

u u  u   p      S , (50) t is a function of the density, pressure gradient, gradient of the shear (τ), and a source function, S. The source function includes any momentum source including gravity. The shear tensor,

du 2 i j,2i  p   u dxi 3   , (51) ij u u ij,  i j  xxji 52

contains terms that are dependent upon the viscosity. For the region of vortex shedding being analyzed the flow is assumed to be incompressible, the source terms acting on the flow are assumed to be negligible, and the viscosity constant. These assumptions reduce the momentum equation,

u u  u   p  2 u , (52) t to equations based only on viscosity, velocity, and pressure. Despite the simplifications, the equation is still non-linear. The final equation is that of the energy of the flow. The energy equation (which will only look at internal energy),

 e    eu   q   k     q (53) t  describes the relationship between heat transfer of the flow. The internal energy, e, is dependent upon heat generation, q, the density, the temperature, θ, and the conductivity, k. The heat generation has been split into two terms, a general source term and the viscous dissipation term designated with a subscript μ. For isothermal flow, this equation is not needed. If the flow is not isothermal, the energy equation may be decoupled from mass and momentum if the viscous dissipation term is negligible, or the heat generation from the viscous dissipation term has no effect on the momentum equation, i.e. the temperature change of the fluid has a negligible impact on viscosity and density.

Again, the equations are more useful if appropriate simplifications are made. Since vortex shedding over a plate is predominately 2-dimensional, the Navier-Stokes equations ((49) and (52)) are written in two dimensions: 53

uv continuity :0 xy u  u  u1  p 22 u  u x momentum:  u  v    22  . (54) t  x  y  x  x  y v  v  v1  p 22 v  v y momentum:  u  v    22  t  x  y  y  x  y

These equations are manipulated further to derive a sense of scale from the vortex shedding phenomena in section 3.3. Since the third dimension is removed, the equations are no longer used to calculate turbulent flow for vortex shedding, and thus are reserved for laminar flow. Any analytic solution derived from the problem is inherently laminar. One of the most famous solutions is the Blasius solution, which characterizes laminar flow over a plate. For now, however, it is important to consider why the 2-dimensional assumption negates turbulence.

3.1.3 Considerations for Turbulence

Turbulent flows are characterized by irregularity, diffusivity, large Reynolds numbers, 3-dimensional vortex fluctuations, and dissipation. Turbulent flows are a continuum, and turbulence is property of flow not the fluid [84]. Each of these may be put into context of vortex shedding. The major transition regimes for cylinders are dependent upon where the flow transitions to turbulence along the surface of the cylinder. The cylinder transitions are general described as two-dimensional (with a notable exception in the laminar region). As such, it is tempting to consider vortex shedding a two- dimensional turbulent phenomenon.

It is interesting to note that while 3-dimensional vortex fluctuations are typical of turbulent flows, they are not necessary. Consequently, (54) is not inherently laminar. The equations may be discretized in two dimensions and produce turbulent results. Turbulence in two dimensions is not a purely mathematical construct, however the physics requires a fluid depth scale several orders of magnitude smaller than the planar scale, e.g. the earth’s atmosphere (10 km to 103 km) [85]. While using a 2-dimensional 54

solution to the phenomena of turbulent vortex shedding is possible, it would result in the dissipation of energy cascading to larger structures as opposed to smaller ones. This is the opposite of the dissipation path of 3-dimensional turbulence where energy is dissipated in the smaller scales.

Dissipation is one of the more important features of turbulence. While describing a turbulent vortex as 2-dimensional is not appropriate, there is a question to whether the vortex itself may be considered turbulent. Unless the flow is inherently turbulent, the vortices produced by vortex shedding persist far past the point of shedding, which makes the vortex street, ergo; vortex shedding is not necessarily a turbulent feature. While a shed vortex may be affected by turbulence, it is not inherently born from turbulence. This is a key factor when discussing how the regimes may differ between a cylinder and a plate at varying Reynolds numbers.

A useful exercise is examining the effect of the non-linearity with a perturbed system. The turbulent system is first broken down such that any scalar, represented here by the x-velocity,

u()() t u u t (55) may be broken up into a time averaged and fluctuating component. It is also useful to show that the left-hand side of the momentum equation (52),

u  u  u u  u   u u  u  u   u u , (56) t  t  t may be re-written assuming that the flow is incompressible. Substituting (56), (55), (52), and (49), gives the Navier-Stokes equations 55

uu continuity :0ii xxii uu u u u u u  u u  u  momentum :ii i j  i j  i j  i j  ..., (57) t  t  xjjjj  x  x  x 1 pp 22uu ...   ii  22  xi  x i  x j  x j in terms of their average and fluctuating components. Considering that the time average of the velocity

TT1 u dt0  u  u dt  u , (58) 00T is zero, a temporally averaged flow relation for (57) may be written as,

u continuity :0i  xi . (59) 2  ui  u i1 p  u i uuij momentum:  u j    2  t  xj  x i  x j  x j

The velocity fluctuations are typically referred to as apparent stresses. They form the basis for the calculation of Reynolds-Averaged Navier-Stokes (RANS) equations. RANS equations approximate the fluctuations with a closure relation that allows the flow field to be solved assuming steady state. If they are insignificant, the flow will fit a laminar profile. These stresses are small if the Reynolds number is small. For turbulent flows, the point of transition to turbulence is defined by a Reynolds number.

There is a stark difference between turbulent transitions for a plate and for a cylinder. Transition to turbulence on the cylinder surface occurs at a width Reynolds number around 300. For a pipe, this occurs at a Reynolds number of 2300. For flow over a plate, this transition occurs at a Reynolds number of 100,000 based on the length, not the thickness. The increased Reynolds number means that a plate may experience a transition well beyond that of the cylinder; however, this ignores the implication of 56

sharp edge of the plate. The discontinuity (as opposed to a smooth round surface) will provide a more consistent Strouhal number as function of all Reynolds numbers. Fundamentally, turbulence has a much smaller impact on the Strouhal number for plates than cylinders. It is of merit to discuss why this is.

3.2 Characteristics of Flow Over a Plate

Fluid flowing over a plate interacts with both leading and tailing edges resulting in unique interactions at various locations along the plate. Further complicating things, the wake behind a bluff body is inherently unstable and develops its own vortices at a frequency with little relationship to the vortex shedding generated directly behind the body. Flow over a plate can be divided into four general regions: the leading edge, the mid-span, the tailing edge and the far wake.

3.2.1 Interaction with the Leading Edge

For low Reynolds numbers, the interaction with the leading edge of a plate the flow creeps around edges of the plate. At higher Reynolds numbers, recirculation zones appear resulting in adjustments to the height of the boundary layer. As the Reynolds number increases, the flow will separate and has the potential to shed its own vortices. The flow lines are diagramed in Figure 3.1.

Figure 3.1: Flow over the leading edge

In Figure 3.1 two primary leading-edge flow conditions are examined. On the left, the low Reynolds number flows produce a recirculation zone at separation. On the right, 57

the separation results in vortices being shed. A concern for developing a flow regime map is the impact of the leading end on the plate. Several studies have examined interactions with the leading plate edge [17], [18], [42], [47], [48], [50]. The consensus between these studies is leading edge effects will dominate the vortex shedding at the tailing edge if the plate is short. If the plate is long, vortices are either shed beyond the flow area of the plate or absorbed back into the boundary layer.

Interactions between the leading and tailing edge have been described but are not well characterized. Since this study is focused on the leading to tailing plate interactions, minimal impact from the leading edge is desired. Parker suggests there are no effects if the length to width ratio is greater than 16 [42]. An alternative to removing leading edge effect is shaping the leading edge so separation is reduced. This approach was used by Taneda as well as Bull and Pickles [1], [16]. The plates studied herein are referred to as “long” plates. Long here refers to the fact that leading edge effects are assumed to have a negligible impact on vortex shedding form the tailing edge. This assumption means a boundary layer develops along the mid-span of the plate between the leading and tailing edge.

3.2.2 Boundary Layer Development along the Mid-Span

As flow moves past the leading edge of the plate, shear forces between the plate and the fluid result in the fluid at the surface stopping. The slowdown of the fluid at the plate results in the fluid being pushed upward. This results in what is known as the boundary layer. The boundary layer is defined as being the region where the velocity of the fluid is less than that of the freestream. Typically, the boundary layer is defined as the region where the velocity is less than 99% of the freestream velocity. A diagram of the boundary layer is presented in Figure 3.2. 58

Figure 3.2: Boundary layer diagram.

Figure 3.2. shows that a fluid column, H, entering at a constant velocity will reach a certain point where the fluid column makes up the entirety of the boundary layer height, δ. The fluid column height is determined from the boundary layer thickness,

H 1 Udy  uydy()()  H   uydy , (60) 0 0U 0 through conservation of mass. Since the choice for the boundary layer thickness value is arbitrary (in theory, the value would be infinite). Defining the fluid column is not meaningful. What is more meaningful is the difference between the boundary layer height and the height of the fluid column. The difference between the two will approach an asymptotic limit. This value,

1   u  H  dy  u dy 1  dy , (61) 1    0UU 0 0  is known as the displacement thickness. A physical interpretation of the value is the amount the plate would have to move upwards if the flow were inviscid to account for the boundary layer growth.

Vortex shedding, an inertial phenomenon, may be better represented by a thickness based on momentum as opposed to mass. The thickness needs to be based off the change in height for the same flow rate. The entrance momentum height is therefore, 59

H 2 22 u()() y u y U dy  u( y ) dy  H  1 2  1  dy , (62)    UU 1 00 and is defined as the difference between the entrance height and the displacement height. This gives an asymptotic limit for what is referred to as the momentum thickness,

 u2  u   u  u    H    dy  11  dy   dy ; (63) 21   2      0 0UUUU  0   a function of the velocity profile. Both the momentum thickness and the boundary layer thickness have assumed nothing about whether the flow is turbulent or laminar. Thus, these relations hold for all flows. The devil is in the details for the velocity profile needs to be first defined. The velocity profile carries the burden of assuming whether or not the flow is turbulent.

The boundary layer of a plate, like most flows, undergoes a turbulent transition if the Reynolds number is high enough. For a constant velocity, and fluid properties the Reynolds number,

Ux Re (x )  (64) x  is a function of position. Consequently, the flow transitions from laminar to turbulent along the plate. This transition is presented in Figure 3.3. 60

Figure 3.3: Boundary layer description over a plate.

In Figure 3.3. flow comes over a plate and the boundary layer developing is laminar (assuming the leading edge is smooth). This flow will remain laminar up until a critical Reynolds number. Various values for this number are given, but typically, values given are on the order of 105. At this point, the flow enters a region where it transitions to turbulence. The turbulent region is divided into three regions: a turbulent layer, a buffer layer, and a viscous sub layer. The viscous sublayer maintains a laminar flow profile. The buffer layer acts as a transition to turbulence from the sublayer. The turbulent layer maintains the full characteristics of turbulent flow.

An analytic solution for the laminar boundary layer exists. This solution takes the Navier-Stokes equations and transforms them into two variables,

Uu yf,   , (65)  xU which then reduce the equations to a single equation,

ff20 f  , (66) referred to as the Blasius solution. The Blasius solution requires three* boundary conditions,

* Four boundary conditions are listed, but the first two are not unique. 61

f0  f 0  0, f  0  0, f    1, (67) which are provided by maximum shear at the wall, no-slip at the wall in the x and y- directions, and the freestream condition of the fluid at an infinite range. Calculating the solution numerically with an infinite boundary of η = 10, produces the results shown in Figure 3.4.

Figure 3.4: Blasius Solution for f and its three derivatives

Figure 3.4 provides the value for the various derivatives of the velocity. From the plot, the two most important derivatives are f’ and f”. The first derivative, f’ is a nondimensional version of the velocity profile. The second derivative, being the derivative of the former, is a nondimensional version of shear. The other two derivatives have some meaning, but in general, their use here beyond a requirement to solve the similarity solution is limited. With a solution for the velocity profile, the values for the displacement thickness (61) based on plate length,

 1 =1.72Re1/2 , (68) l and the momentum thickness (63) with respect to plate length,

 2 =0.664Re1/2 , (69) l 62

are derived. The two thickness values are only applicable to laminar flow. For turbulent flows, there is no analytic solution for the size of the momentum or displacement thicknesses. Such solutions are semi-empirical in nature and rely on correlation of experimental data. Using the form of Blasius similarity and applying a Reynolds stress correlation, the values for the displacement thickness,

 1  0.046Re1/5 , (70) l and the momentum thickness,

 2  0.036Re1/5 , (71) l are obtained [86]. As opposed to being theoretically exact, the values of the momentum and displacement thicknesses are obtained through approximation of the shear stress in the Blasius solution. Given that the values express the boundary layer in terms of a length scale, they are important when considering the impacts to the vortex shedding at the end of the plate.

3.2.3 Separation and Vortex Shedding at the Tailing Edge

When the flow reaches the tailing edge of the plate, it encounters a sharp adverse pressure gradient created by the tailing edge of the plate. This adverse pressure gradient creates the conditions necessary for vortex shedding to occur. At the point of shedding there are some important considerations to be made that will affect the vortex shedding. First, a streamline may be drawn that forms the boundary between the flow generating the vortex wake and the flow that remains with the freestream. This streamline is moving at velocity that may be different from that of the freestream. This velocity is referred to as the separation velocity.

The separation velocity, previously mentioned, is the velocity at which the flow separates from the body at the location of the vortex formation. The separation velocity is further described through the separation angle, α, which is the angle between the 63

primary flow direction and the separation velocity. As the flow separates, it carries with it a specific vortex strength. The vortex strength entering the flow,

in   out   , (72) is split along the streamline where part of the vortex strength leaves the flow and the rest remains in the vortex. Finally, the wake develops and the distance between the streamlines not involved in vortex shedding is given as η. A diagram of these parameters is presented in Figure 3.5.

Figure 3.5: Difference of vortex dimensions at the tailing edge.

In Figure 3.5 two different diagrams are presented to highlight the differences between a long plate on the right and the bluff body of same characteristic thickness on the left. The plate normal to the flow was chosen to represent a bluff body as it has the same leading and tailing edge of a long plate, and is the plate used in theoretical derivations of vortex shedding [4], [5], [7], [8].

The velocity at the point of separation is different for both plates. For the bluff body case, the separation velocity is higher than the freestream velocity due to the acceleration of the flow around the body. For the long plate the separation velocity, is around that of the freestream velocity. The notable exception to this is if the separation occurs within the boundary layer. If such were the case, then the separation velocity would be less than the freestream velocity. 64

For both plates, an angle at separation exists. For the plate, the y velocity is negligible compared to the local x velocity. This means the separation angle is negligible as well, even in the boundary layer of the plate. For the bluff body this is not the case as the flow attempts to move around the plate but separates as it moves vertically away from the plate. The separation angle plays a role in the overall strength of the vortex.

The flow, as it approaches the point of separation, carries with it an amount of vorticity. The vorticity,

  u , (73) is the curl of the velocity. If this vorticity is integrated over the surface of the vortex, the value for the strength of the vortex,

  ndAˆ , (74) A also referred to as the circulation is obtained. The rate of circulation generated from instant flow separation was originally thought of in the context of an instantaneous vortex formation at the point of separation. In this regard, the circulation generated for a bluff body,

U 2  f   (75) in in 2 may be thought of as the magnitude of an imaginary vortex that is generated at the frequency of the boundary. Here the generation is only considered for one side of the body as opposed to both. For a plate, all the vorticity being generated is carried through the boundary layer of the plate. The flow rate of the circulation, may then be thought of as

 u dy (76) in z 0 using the Blasius solution variables, equation (76) is transformed by, 65

f  u 2 f   y u U,,   U  y   , (77)  yy  2    and rewritten as,

  U2 f f   (78) in  0

Conveniently, this equation means the solution is independent of the boundary condition. The solution after integration,

U 2  , (79) in 2 is identical to the instantaneous shear assumption. The circulation available for each individual vortex is not necessarily the circulation carried in by the system. The circulation flow will be split between the vortex and the remainder of the freestream. Unfortunately, there is no analytic solution to this for the bluff body. Roshko and Fage both noted that the fraction of vorticity lost to the freestream was always greater than 0.5 and dependent on the body [8], [23]. For a long plate, the boundary layer profile is known, so it would be possible to calculate the vorticity entering the vortex if the location of the streamline is known. Such a location may be made from the size of the wake, η.

The wake size is also different between the two bodies. Early studies showed that the size of the wake is tied to the vortex geometry as well as the frequency of vortex shedding [7], [8], [20], [22], [23]. Therefore, the wake may be considered an integral component in the calculation of the vortex shedding frequency.

3.2.4 The Wake

The wake region is the region beyond the initial formation of the vortices. The wake is somewhat arbitrarily divided into the near wake and far wake regions. In the near wake, if vortices are shed, they form a uniform street. In the far wake, the vortex street breaks 66

down due to the secondary instability of the boundary layer flow. In the far wake, vortices may form due to the shear in the wake which results in a second instability. This secondary instability can also generate vortices that can reach the tailing plate. While the secondary wake is of interest, observations of it are within the context of the leading plate wake as the focus of the study is on primary vortex shedding on the lead plate.

As presented in section 2.3, Bull et al. proposed two primary regimes: a trapped and free vortex shedding regime [16]. These regimes are determined by the Reynolds number and the plate location in the presence of the wake. It is of more use to first look at the physical interactions between the wake and the secondary plate. Such interactions are presented in Figure 3.6.

Figure 3.6: Breakdown of vortex shedding in the wake.

Without assuming anything about the flow conditions, other than that they are sufficient to generate vortices, there exists three potential regions where a force may influence vortex shedding as shown in Figure 3.6. A formation region, an interaction region and an acoustic feedback region. These regions are shown along with the interaction with the tailing plate.

Closest to the leading plate is the formation region. The formation region is the area physically required to form alternating vortices. If a second plate is placed in this region, two counter rotating vortices are formed. This region would fall entirely within 67

the “trapped vortex” regime. The size of the region is on the order of half the separation length, λ.

Farthest form the plate (but close enough to have an impact) is the acoustic feedback region. The acoustic feedback region exhibits only an acoustic effect on the plate. The region falls entirely within the “free vortex” regime. The vortices in this region are free of the domain of the plate. Only the general pressure field produced affects the vortices being shed. The implication is the acoustic feedback region is greater than the separation distance of the vortices.

The middle region is the interaction region. In the interaction region, the vortices are forming along the leading edge of the tailing plate. The vortices may be considered “trapped” or “free”, but the important distinction is that the vortices are in direct contact with the leading edge of the tailing plate.

It is important to note the regions are based on the scale of the vortex length for the free plate; however, the vortex separation distance changes with the presence of a second plate. As the plate moves closer, the vortices are slowed. The slowing of the vortices reduces the separation distance, thus changing the separation length of the vortices and with it the location for the regions. To this end, it is very important to develop a sense of scale for vortex shedding from a plate.

3.3 Derivations of the Scale of Vortex Shedding

There have been a few attempts to derive a sense of scale from vortex shedding, and all of them do so for a plate normal to the flow [4], [5], [8]. First von Karman derived a relative scale to a vortex street based on a street [5]. This analysis, while derived with a plate in mind, made no assumptions about the body generating vortices. Heisenberg took von Karman’s result and added closure relations estimated the scale of the vortices being shed [4]. Finally Roshko expanded on Heisenberg’s approach and made adjustments to account for the real flow around a plate [8]. 68

This section considers these approaches, along with other known approaches to derive a sense of scale for vortex shedding. This is achieved first through scaling of the Navier- Stokes equations, and then examines the methods to derive the scale of vortex shedding from first principles.

3.3.1 Scaling Analysis of Vortex Shedding from a Bluff Body

The scale of the vortex shedding may be obtained from the Navier-Stokes equations. The scaling performed is a nondimensional analysis that seeks to scale the relationship between the terms in the equations by assuming nominal parameters. To the author’s knowledge, no such scaling has been performed. The first step in the nondimensional analysis is choosing terms such that the magnitude of the variable terms is of the order one. Since vortex shedding is largely a planar phenomenon, the 2-dimensional Navier- Stokes equations are used. In these equations, the nondimensional forms of the parameters,

u v x y t pˆ uˆ,,,,, v ˆ  x ˆ y ˆ  tˆ  p  (80) UVXYTP are denoted with a hat, and are the values divided by the expected magnitude, which is the variable capitalized with a tilde.

Substituting (80) back into the Navier-Stokes equations (54),

U uˆˆ V v continuity :0 X xˆˆ Y y U uˆ U2  u ˆ UV  u ˆ x momentum: uˆˆ  v  ... T tˆ X  xˆˆ Y  y P pˆ11 22 u ˆ  u ˆ ... U 2 2  2 2 . (81)  X xˆ X  x ˆ Y y ˆ V vˆ VU  v ˆ V2  v ˆ y momentum: uˆˆ  v  ... T tˆ X  xˆˆ Y  y P pˆ1122 v ˆ v ˆ ... V 2 2 2 2 Y yˆ X  x ˆ Y y ˆ 69

yields a nondimensional relation. First, an assumption can be made for the far field of the scale, if a pure street is formed. This assumption is the time and distance scales are linked such that,

1   , (82)  tx

The velocity scales and time scales are completely linked to one another. Consequently, this results in a collapse from three to two variables: time and y-direction. Before assuming anything from the scale, it is useful to reduce the number of equations. From continuity, it can be shown that,

11 UY uˆ  v ˆ  UY  u ˆ   v ˆ  UY VVV         , (83) X xˆ y ˆ  T  tˆ  y ˆ  T the y velocity scales to the dimension of the x velocity and the x and y scales. With continuity used, equation (83) and equation (83) used to change the momentum equations to nondimensional approximations of

UUP2 11 x momentum:   U 2 2  2 TTTTY   . (84) UY U2 Y P UY 11 y momentum : 2 2 2   2 2  2 TTYTTY    

Note the advection terms became one scaled parameter. The basis for the choice was that the values for the u and v components and their derivatives are orthogonal such that the combined scale is of order 1,

The next task is determining the scale such that any variable (represented by the velocity),

uˆ 1 (85) U is of order one. The time scale is simple, 70

Tf 1 , (86) as it is just the period of the system. This leaves the velocity scale, the y-distance scale and the pressure scale. The pressure scale, is the next easiest to determine as the fluid moves from a full velocity to a stagnation velocity at the plate. For this reason, the pressure scaling is assumed the full dynamic pressure of the fluid velocity,

1 PU  2 2 , (87) since it stagnates at the plate. The next scaling factor is the choice for the velocity. The velocity has three primary choices: the freestream velocity, the vortex velocity, or the propagation velocity. These three terms are relatively close, with the propagation velocity being the smallest of the three. The argument being made here is the appropriate choice to make is vortex velocity,

U  , (88) as the vortices never truly stop, merely slowdown from the freestream. Regardless, any value chosen is on the same order. The last scaling, parameter, and hardest to justify is the length scale. The plate thickness may be chosen, but such a choice, while logical, arbitrarily assumes the scale is based purely upon the plate thickness. Instead, the length scale is based on the boundary layer “thickness” of the vortex street. The assumptions involved are the momentum thickness from (63) for the length scale is half the plate thickness,

b bU 2 1 YY   , (89) 22UU  p and the new scaling parameters are based on relationship between the momentum displaced by the plate at the freestream velocity, and the momentum returned to the 71

vortex velocity of the wake. Substituting (89),(88),(87), and (86) into (84) produces the following relations

fb11 fb U2  f 2 b 2 422 x momentum : p 2 2 4 U42 U Ub U . (90) fb 2 1   b22 f 4  2  2 y momentum : p p p 3 2 2 U fb U2 Ub U

If the Reynolds number based on the thickness is sufficiently large, then the last term in both equations disappears. From the x momentum equation, it is shown,

U 2 , (91)  the relationship between the vortex velocity and the freestream velocity is constant on the order of 2, which coincides with the studies presented in section 2.1. From the y momentum equation, the Strouhal number,

fb  p , (92) UU2 scales with the relationship between the propagation velocity and the vortex velocity to the freestream velocity. From (91), the Strouhal number is approximately constant,

St 0.25 , (93)

With a value on the order of 0.25. Using the values given from Heisenberg [4], for the values of the propagation velocity and the vortex velocity the value for the Strouhal number, is about 0.18, which, along with the scaling value in (93) agree with the values observed by cylinders in the subcritical regime. This also provides a strong case for a universal Strouhal number. 72

3.3.2 Scaling Analysis of Vortex Shedding from a Plate

Despite, the scaling evidence to the contrary, values for the Strouhal number are very dependent upon the geometry. The most egregious violator of the scaling is that of the plate which was shown to have a Strouhal number that was not constant, but scaled with length [1], [47], [74].

To account for this, we return to (89) and redefine the momentum displaced by the plate as

b   b  U 2 22 1  YY      , (94) 22UU    p a function of the plate combined with the momentum thickness. Conveniently, the values for the scaled thickness are the only values that have changed. Thus, equation (91) does not change and equation (92) combined with (94) and (92) yields,

1 2 f  St  2 , (95) 4 U a relationship between the Strouhal number, and the momentum thickness. The next step is replacing the boundary layer value parameter with an equation based on the dimensions of the plate. For both laminar (69) and turbulent (71) flows, the momentum thickness (as well as the displacement thickness) fits an equation

1/n 2  Cl Re , (96) where the constants C and n are determined from the nature of the flow be it laminar or turbulent. Substitution back into (95)

l 1/n 1 St 1 2C Re , (97) b 4 produces a scaled correction for the Strouhal number. If the plate length is short, the second term disappears, and the Strouhal number returns to its constant value as seen 73

in (93). If the length of the plate is sufficiently long, then the Strouhal number scales as

1 b St Re1/n , (98) 8Cl which is no longer a constant and a function of the Reynolds number. This scaled value is better written as a Strouhal number,

1 St Re1/n , (99) l 8C based on the length of the plate as opposed to the thickness of the plate. The scaling analysis provides a sense of the scale of the vortex shedding but does not explicitly calculate any of the values associated with the dimensions of vortex shedding.

3.3.3 Dimension of the Vortex Street

The first derivation of the size of the vortex street was performed by von Karman [5]. The values calculated were in excellent agreement with those observed from experiment. The derivation itself assumed nothing about the body generating the vortices, only that they would form a stable street. To this end, von Karman used potential flow equations to describe the flow of a vortex street. A diagram of the flow field is presented in Figure 3.7.

Figure 3.7: The potential vortex street. 74

The vortex street in Figure 3.7 is made up of two rows of vortices that are offset from each other. These rows continue infinitely along the x-direction. In addition to the vortex street, there is an applied velocity. This velocity is the velocity at which the vortices move is relative to that of the flow and as such, it is the propagation velocity. Each vortex carries the circulation in the center. The separation distance between the vortices is assumed the same as that of the real vortex street. The biggest assumption is that the street itself is parallel. This assumption is consistent with observations made of the vortex street a few vortices from the plate.

The vortex street being a two-dimensional flow potential can be described as a complex potential .The complex potential equations for an individual vortex and that of the freestream were presented in (47) and (48). These equations are combined to provide the complex potential for the vortex street,

i sin   i  2   ln  . (100) 2  i  p sin    22

The vortex potential is used to calculate the velocity field of the system. The vortex parameters are undefined. These parameters can be related by two conditions. The first condition comes from the observation that the vortex center will have a velocity at the center if the vortices move along with the bulk fluid, thus the vortices will move relative to the bulk flow at the propagation velocity. The solution to this condition,

  p  tanh , (101) 2 provides the propagation velocity with respect to vortex strength and the vortex dimensions. The second relation comes from von Karman’s perturbation analysis of the system [5]. The analysis shows the stability condition is such that the relationship between the vortex spacing in the x and y-directions, 75

 1  cosh1 2 1/2  , (102)  is a constant. Inserting (102) into (101) gives the propagation velocity,

   , (103) p 81/2 in terms of the vortex strength and length scale. The corresponding streamlines for a nondimensional system are plotted on Figure 3.8.

Figure 3.8: Vortex streamlines for an infinite street.

The top row of vortices rotates clockwise while the bottom row rotates counter clockwise. The streamlines are different from those featured earlier in Figure 2.1. This is due to the applied velocity condition. The condition creates three separate stream line regions, the locked vortex cores (red), the streamlines flowing between the cores (dark blue) and the streamlines flowing outside the cores (light blue). These three regions are separated by the dashed lines. Flow from the streamlines is predominantly left to right. This flow pattern negates the back-flow region observed by Heisenberg; however, based on superposition, the back-flow region still exists. 76

For the potential flow solution, the average x velocity of the flow everywhere is equal to that of the freestream. However, if one assumes repetition for the infinite street, and takes the average flow over the vortex separation length, this value is something different. As seen in Figure 3.9.

Figure 3.9: Average velocity profile between vortices. Figure 3.9 shows the average velocity profile surrounding the two center vortices over one separation length (λ). The profile is show for several finite numbers of vortices and the infinite street condition. For the finite vortex street, the velocity peaks above the freestream velocity and decays away. As the number of vortices increases, this value asymptotically approaches the propagation velocity. The inner region asymptotically approaches some backflow velocity,

  U  , (104) b  which may be calculated as a function of the vortex geometric parameters. If it were assumed that the backflow is equivalent to the displacement flow of the plate, then the value for the relationship of the vortex street would end up being Heisenberg’s relation from (14). However, it is important to note the fluid flow everywhere continues to move forward, which means the region of backflow relating to a displacement flow of the 77

plate does not necessarily agree. Instead the assumption is that the momentum deficit of the flow,

U()()() U bb   U    b   U  , (105) is equal to that of the plate. The left-hand side divides the center drag and the edge drag. This provides the following relationships for the flow of a vortex

b 2  , (106) U which is twice the value derived by Heisenberg. Combining this value with the relationship derived in equation (79) gives the final relationship

fb b St   0.28 (107) UU for the Strouhal number. Since relationship of the bluff body thickness is in a form similar to that of the propagation velocity, (106) is combined with (103) and then (107) to give a value for the Strouhal number,

   St 321/2 p , (108) UU dependent only on the velocity of the vortex street since the relationship between vortex separation height and length is constant. The value from (107) is about twice that of the plate normal to the flow. However, it is higher than that for the flat plate, which has a value of about 0.14, and a cylinder, which is approximately 0.21. The difference between these values is attributed to the loss of shear of the flow around the plate which was discussed by both Roshko and Fage [8], [23]. Part of the lost shear is made up by the fact the velocity flowing around the body is higher than that of the bulk flow. This factor is determined by adding constants to equation (12) (the relationship between generated circulation and the circulation flowing with the vortex street) yielding, 78

U 2  (109) 2  which shows the flow rate is dependent upon the fraction of vorticity (ε) reaching the vortex street. The value may be greater than unity if the shear velocity,

UUs  , (110)

is greater than the flow velocity.

Following Rosko’s procedure [8], both the propagation velocity and the vortex velocity fit the solution

1/2 1  11   . (111) 1/2 U 22

This relationship means the Strouhal number from (107) is equivalent to

 St   , (112)  where, so long as the boundary is consistent, the Strouhal number is dependent only upon the fraction of the vorticity that enters the flow. Using Roshko’s values for the shear velocity around a cylinder of 1.3 and a fraction of vorticity of 0.45, the Strouhal number for a cylinder according to (112) would be about 0.213, which is in excellent agreement with the experimental values.

3.3.4 Derivation of Vortex Shedding from Long Tandem Plates

In the case of the plate, the boundary layer is going to have two effects. First it will add an additional momentum buffer from which the plates. Second, if the layer is sufficiently large, it will reduce the perceived velocity of the free-stream. The flow into the wake region is determined by integrating the flow over the boundary layer,

 u dy  . (113) 0 79

The boundary layer cannot be arbitrarily chosen as the wake region is driven by the size of the vortex street, which is in turn determined from the size of the plate. For this reason, the first step is determination of the relative size of the wake region. The wake region has an average flow of the propagation velocity,

2Uh b  , (114) and is related directly the back-flow region and the fluid velocity region,

2h . (115)

Solving the equations gives a value for the wake height,

 p 81/2  , (116)  as a function of only the vortex separation distance. Relating the separation distance to the value of the wake height means the value may be used to determine the amount of the boundary layer needed to generate the vortices. The influx of the circulation,

()y  U2 f f   , (117) in  0 can be determined through the integration of a relationship such as the Blasius equation. Similarly, the flow velocity

1 ()y U Uf   , (118)  0 may also be determined through integration. The last variable needed to determine the relationship between the Strouhal number and geometric dimensions is that of the true thickness. The true thickness may be related as the thickness of the plate plus the momentum thickness deficit of the flow generated by the surface of the plate, i.e. the momentum thickness. This means the 80

b  2 2  , (119) U

The momentum thickness for the flow is determined as

()y ff1   , (120) 2    0 if the boundary layer of the vortex shedding does not exceed that of the flow. In general, plates experiencing vortex shedding from the tailing edge as opposed to a secondary wake instability will entrain flow outside of the boundary layer. In this case, a plate with a sufficiently small boundary layer, the average velocity and propagation velocity are unaffected and the Strouhal number,

fb b St 0.28 , (121) Ub 22 will be a constant and a ratio between the momentum thickness and plate thickness. This relationship follows the scaling analysis in the previous section.

Since the tailing plate prevents the vortex street from forming, it is less likely that the values for tandem plates can be related directly back to the values of the vortex street. The best approximation for the value is through the observation of equation (112). As the tailing plate approaches the leading plate, the vortex velocity, and therefore the amount of circulation entering the flow, will decrease until the circulation no longer flows and becomes completely trapped between the two plates.

81

4 METHODOLOGY To meet the goals outlined in 1.4 several plates were created for testing. The plate sizes were chosen to cover a variety of different gap spacing ratios and Reynolds numbers following the theory outlined in chapter 3. The basis for plate dimensions and testing target values are provided in 4.4. Key values such as velocity (and corresponding derivatives) and frequency are obtained using particle image velocimetry (PIV). PIV uses a camera, light source, and seed particles to provide the velocity field at a given location. This equipment is described in section 4.2. The equipment is used to get before and after pictures that are then used to calculate the velocity vectors in the flow field as part of the PIV process. Doing this requires appropriate selection of algorithms combined with a procedure to ensure the equipment settings are appropriate (further elaborated in 4.3). A wind tunnel and PIV equipment existed for this purpose, however, the wind tunnel was deemed insufficient to achieve the desired flow conditions. Therefore, a new wind tunnel was designed and constructed. Design of the wind tunnel is presented in the following section (4.1).

4.1 Wind Tunnel

Boundary conditions play a key role in FSI. Variation in turbulent intensity or the impinging velocity profile drives the physical phenomena experienced by the structure. Control and characterization of the boundary conditions is important to perform any study. To achieve the desired boundary conditions a wind tunnel was constructed.

The wind tunnel was built out of 1.5-inch t-slot aluminum framing. The aluminum framing allowed for attachment of necessary instrumentation and control as well as providing a sturdy structure for the wind tunnel. The use of t-slot framing also allows for modular adjustments to the design. The wind tunnel is divided into five main regions based on function: seeding, flow conditioning, inlet, test section and outlet. A diagram of the wind tunnel is shown in Figure 4.1. 82

Figure 4.1: Wind Tunnel Diagram 83

In Figure 4.1 the fog is generated by the fog machine. Once the fog leaves the machine, it enters the seed region. The fog generated from the fog machine as well as additional air is drawn into the test section from the blower at the opposite end of the wind tunnel. As the fog is distributed in the seed region, it mixes in with the air and reaches the entrance of the flow-conditioning region. In the flow-conditioning region, excess turbulence is removed through the aid of a honeycomb grid and a series of screens. After the last screen, the air enters the inlet section. In the inlet, the cross section of the flow area is reduced and reshaped to enter the test section. In the test section, the laser generates a light sheet to illuminate the fog seed particles as they flow over the tailing edge of the leading plate. When the fog passes through the test section, it enters the outlet. In the outlet region, the flow is expanded, excess fog is filtered out and the blower pushes the air to the ventilation. The next sections describe each region in further detail.

4.1.1 Seed Region

The seed region is the area at the entrance of the wind tunnel used to seed the flow evenly with fog. The seed region consists of two components, a shroud and a fog machine. The fog machine is a Rosco fog machine using either Rosco fog fluid or Rosco light fog fluid for particle seeding. The shroud is a box nominally 17x17x36 inches. The shroud includes an entrance for the flow as well as a small cutout for the fog machine to inject fog. The seed region is pictured in Figure 4.2. 84

Figure 4.2: Pictures of the seed region.

The fog machine depicted in Figure 4.2 (highlighted in lower right) is in an unorthodox position for general seeding. General wind tunnel seeding typically occurs after the flow conditioner and before the inlet of the test section. The amount of fog produced by the fog machine is significantly large compared to the velocities and size of the wind tunnel. Even at higher velocities, the fog production was only raised slightly above the minimum amount produced.

The initial design involved not using a shroud and having the fog injected directly into the flow conditioner. Unfortunately, at low velocities, the jet of fog produced created an instability beyond the flow conditioner and resulted in significant turbulence inside the test section. To avoid this the fog machine was removed further back from the inlet. Doing this removed the turbulence issue; however, the fog seeding became highly susceptible to fluctuations to airflow in the room. At this point the shroud was created the fog machine was placed perpendicular to the flow. Since the air and fog was being drawn in at the beginning of the flow over a significant length, the fog was spread evenly and allowed for a good seeding of the air as it entered the flow conditioner. The entrance to the flow conditioner is shown at center bottom with the top of the shroud removed. 85

4.1.2 Flow Conditioner

The inlet of the wind tunnel consists of the flow conditioner. The flow conditioner consists of a flow straightener (also referred to as a honeycomb spacer), a holding screen and three additional screens to condition the flow. The purpose of the flow conditioner is to reduce the overall amount of turbulence in the flow prior to entering the test section. A picture of the flow conditioning section is shown in Figure 4.3.

Figure 4.3: Images of flow conditioner section.

The flow conditioner in Figure 4.3 is split into a series of boxes. At each box interface, a screen was mounted. The entrance section is slightly larger and contains a series of straws placed in a honeycomb pattern. The straw entrance is highlighted with a green double line. Approximately half of the straw length extends into the box. The straws were placed into an aluminum honeycomb grid and held in place by friction. The straws are nominally 100 mm in length with a diameter of 6 mm. The straw area is 292 mm in the x-direction and 279 mm in the z-direction. A black piece of card stock paper restricts 86

flow around the wind tunnel. To prevent flow around the straws, the spare space was sealed with caulk.

Behind the straws is a holding screen to keep the straws from being pulled into the wind tunnel as well as to remove the jet instability generated by the straw honeycomb. The holding screen was larger than the other conditioning screens further on in the wind tunnel. The holding screen was placed directly against the straw ends. The steel holding screen is highlighted with a blue dot-dash line in Figure 4.3. The screen grid spacing was about 2.5 mm. After the holding screen, there were three more screens in the flow conditioner. All of these screens were identical. The screen mesh was less rigid and comprised of aluminum strands that were spaced 1.8 mm apart from one another. At the exit of the flow conditioner, the amber colored inlet nozzle of the inlet section can be seen.

4.1.3 Inlet Section

After the flow conditioner is the inlet section, the inlet section reduces the area from the flow conditioner to the test section. The reduction in area serves two purposes. First, it reduces the pressure drop through the flow conditioner. The pressure drop,

1 p U2 k , (122) 2 is proportional to the velocity squared and the loss coefficient of the screens and flow straightener. Since the loss coefficient and the air density remain nearly constant, the velocity of the air drives the overall pressure drop. Reduction in the pressure drop allows for higher velocities in the test section. The reduction in the area is achieved with a clear amber acrylic nozzle. The nozzle reduces the diameter down from about twice that of the test section. Past the nozzle is a black PLA diffuser that allows the flow to enter smoothly into the test section. The inlet section is show in Figure 4.4. 87

Figure 4.4: Inlet section.

The amber nozzle in Figure 4.4 is held tightly in place through pressure from the flow conditioner and the test section. To remove air leakage into the test section, caulking was used to seal the space between the nozzle and the aluminum framing of the test section structure. Remnants of this are seen on the top left picture. Originally, the inlet nozzle led directly into the test section; however, due to the irregular test section shape (seen clearly in the bottom right photograph) an additional diffuser was included to smooth the airflow as it entered the test section. The diffuser was placed abut to the inlet nozzle.

The black inlet diffuser (as seen in the upper right photograph of Figure 4.4) contains an additional two screens. The screens function as an additional flow conditioner and allow the flow to expand smoothly after the nozzle contraction. The first mesh comprised of steel is highlighted with a green double line. The mesh is approximately one-third down the length of the diffuser. The mesh is about a 1 mm grid. The second mesh (highlighted in with a blue dot-dash line) is comprised of brass and has a grid of about 0.8 mm. The diffuser was printed in four separate parts and was bonded together 88

after printing. To ensure appropriate spacing, a series of gray cross section skeletons (1 mm thick) were printed to maintain the shape and hold the diffuser screens in place. The specialized diffuser allowed for flow to smoothly transition into the test section.

4.1.4 Working Test Section

Following the inlet is the working test section, the region in which experiments may be placed. The test section was framed with standard 1.5-inch aluminum t-slot profiles. Between the framing, clear acrylic panels were placed as the walls to allow for flow visualization from any side. The test section is shown in Figure 4.5.

Figure 4.5: Test section and PIV equipment.

The PIV equipment is attached to the outer frame of the wind tunnel using the t-slot aluminum framing and is shown in Figure 4.5. The camera is mounted vertically over the top of the test section. The laser is positioned a distance away from the test section and aimed at the mid-plane. Both instruments can be moved the full length of the test section. The amber nozzles mark the bounds of the primary test section frame. 89

The inlet and outlet to the test section are 152.4 mm by 152.4 mm squares. However, the framing with the acrylic panels created a slightly irregular cross section as seen in Figure 4.6.

Figure 4.6: Test section cross section.

The frame cross section, as seen in Figure 4.6, was created by separating the aluminum t-slot extrusions by 152.4 mm (6 inches). The windows to the test section are placed in the aluminum frame and held in place with gaskets. This approach leaves a 15 mm gap between the window and the nominally square section. The bottom of the frame has an 90

additional piece running down the center to hold the instrumentation and the test plates. The test plates were positioned vertically and reached most of the distance to the top of the test section. The plates were held in place by plate holders. The plate holders are shown in Figure 4.7

Figure 4.7: Plate holders.

Figure 4.7 shows the two different plate holders: standing alone on the left and as they fit in the test section on the right. Two different plate holder designs were used. For the larger plates, a smaller plate holder (featured in orange on the right) was fixed to the leading and tailing ends of the plate. The smaller plate holders were designed to twist and lock directly into the center channel. For the smaller plates, only one plate holder was used (featured in gray on the right). These plate holders sit on the outside of the aluminum frame. The slots in the holder are designed to fit the entire length of the plate. The design allows the plate to be held strait in-line with the channel. Past the plates, the air flows into the outlet section.

4.1.5 Outlet Section

The outlet section consists of four components: a nozzle, a diffuser, a junction box and a blower. The outlet region exists to provide a smooth transition from the test section to the atmosphere. The outlet region is featured in Figure 4.8. 91

Figure 4.8: Outlet section.

At the left end of the bottom photograph of Figure 4.8, the outlet nozzle is shown. The nozzle was 3D-printed in parts to carefully match the irregular shape of the test section as well as allow a pass through for instrumentation wiring. The nozzle provided a smooth transition from the test section to the square inlet of the outlet diffuser. The outlet diffuser is identical in size to the inlet nozzle. At the end of the outlet diffuser is a junction box, which allows the outlet diffuser to be connected to the air blower. On the inside of the junction box is an air filter. The filter location is highlighted with a double green line and shown in Figure 4.8. The filter served two purposes. First, it reduces the amount of fog entering the blower fan. Second, it creates a large pressure drop between the test section and the blower. The pressure drop decreased the amount of air attempting to bypass the flow conditioner as well as stabilized the airflow at the outlet of the test section by removing turbulence that was created by flow conditions of the blower.

At the outlet of the test section, a blower was attached to the junction box. The blower is an Air Foxx AM4000A blower with a 3-speed drive. The blower has two inlets to 92

draw in air, however only one was connected to the wind tunnel. Air circulated from the blower outlet to a duct, which blows the air out near the room ventilation to reduce the accumulation of fog. Speed from the blower is controlled by three different means. The speed switch dictated the maximum speed of the blower. If a further reduction in speed, or more fine-tuning was needed the blower was moved further from the outlet of the wind tunnel. Fine control of the velocity was obtained by using a Variac TDGC2- 2 autotransformer that had voltage marks from 0-130V every 5V. The blower and autotransformer are shown in

Figure 4.9: Air blower and Variac controller.

The blower was chosen to be placed at the outlet, thus drawing the air out of the test section as opposed to blowing it through the test section. Placing the blower at the outlet reduces the turbulence intensity. The primary disadvantage of this arrangement is maximum flow through the test section is reduced. Given that the maximum flow exceeded the capabilities of the PIV system, this was of little concern.

4.2 Measurement Instrumentation

For all experiments, there are five primary measurements to be taken: velocity, pressure, temperature, time, and distance. All other parameters for the experiments can be derived from those values. The most complex, and the one with the most significant uncertainty is that of the velocity. The non-PIV equipment is relatively simple in nature and readout from the instrumentation was achieved using an NI-cDAQ-9185 sending 93

the digital signal to the LabVIEW software. The PIV instrumentation is a two-step process. Imaging of the flow conditions and post processing of the data.

4.2.1 Non-PIV measurements.

The non-PIV measurements can be further divided into two different categories, geometric measurements and flow condition measurements. All of the geometric measurements were performed with Mitutoyo calipers that provide an accuracy of 0.025 mm. The calipers were used to measure all relevant dimensions of the plate as well as the gap spacing between the plates of each test.

For the flow condition measurements, three different instruments were used. Outside of the test section, a barometric pressure meter was used to get the pressure to an accuracy of 10 Pa. Inside the test section, an RTD was used to get temperature and a differential pressure meter was used to get the pressure difference between the static pressure in the test section and the pressure atmosphere. These instruments are shown in Figure 4.10. 94

Pressure Tube

Outlet

RTD

Flow

Figure 4.10: Instruments in the test section.

In Figure 4.10 the RTD and the pressure tube are shown lying flat inside the grove of the t-slot aluminum frame. The instruments are located at the end of the test section. The pressure differential between the atmosphere and the test section was on the order of the uncertainty of the barometer due to the low density and velocity of the air at the highest velocities. Since the only parameter that required the pressure reading was the kinematic viscosity calculation, and this value was more dependent on temperature, the value of the in-channel pressure measurement was limited. The Omega PT 100 RTD measured the temperature to an accuracy of 0.15 C.

Over the course of all the tests deviation of the temperature and pressure resulted in a change of kinematic viscosity of less than 2% from the extremes. The drop in pressure within the test section was insignificant compared to the general fluctuations in the barometric pressure. In general, the largest portion of the uncertainty associated with any measurement is linked directly back to the velocity of the system and thus the PIV system. 95

4.2.2 Particle Image Velocimetry Equipment

The PIV system consists of three major components. The fog machine to generate the seeding particles. The laser to illuminate a plain of the seed particles and the camera to capture images from one time step to another.

The vortex shedding was visualized using a Rosco 1700 fog machine with variable control to produce the seeding (glycerol-water solution) for the flow field. Air and fog is pushed through the blower into the wind tunnel duct. In the duct, two parallel acrylic plates are placed at varying distances to generate the vortex shedding.

For determining flow field data, the system includes a Dantec Dynamics PIV system that can be used to acquire time resolved flow fields. The camera is a two Mega pixel, 8-bit greyscale camera capable at capturing data at 1000 HZ at full resolution using a 60 mm Nikon lens with a 980 µs exposure time. The camera is shown in Figure 4.11.

Figure 4.11: Camera used for PIV.

The camera used is mounted vertically over the test section as seen in Figure 4.11. To accurately measure the distance downstream when multiple pictures are taken, the beam is marked for various positions to ensure accuracy of the location of the camera position. These marks can be seen in both pictures along the upper side of the bar. The 96

PIV camera is used in conjunction with a Dantec RayPower 532 nm laser. The laser is presented in Figure 4.12

Figure 4.12: Laser system used for PIV.

Figure 4.12 shows the laser (left) and the power system (right). The laser is cantilevered such that the laser plane crosses at the middle of the plate. The laser pulse timing is all controlled via the Dantec Dynamic Studio 2016a software. The software also processes the PIV images.

4.3 Conduct

For each test, the first step was warming up both the laser and the fog machine. During this time, the blower was turned on and set to the estimated Reynolds number value. All laser settings assumed a Strouhal number of 0.2 for the purposes of timing. Since the value of the Strouhal number is expected to be less than this for all cases, the value is inherently conservative. The laser duration was set to capture a total of 10 vortices based on the Strouhal number and 121 image pairs. The laser pulse was set to 350 microseconds or as large as the system would allow (whichever value was smaller). The individual laser pulse timing was set to resolve down to a 16-pixel window based upon the target velocity. Once the system was configured, fog was injected into the system. The fog seeding was controlled until an appropriate seeding was reached. When the seeding was appropriate, the PIV images were taken. After the images were 97

taken, the fog was turned off and an average correlation was used to examine the velocity of the flow field. If the velocity was reasonably close (dependent upon the Reynolds number criterion), the value was considered good. If not, the voltage was adjusted, and the test repeated until a valid velocity was achieved. If the velocity was valid, the vortex shedding frequency was examined. If less than 5 cycles were obtained, the image sample rate was cut in half and the process repeated.

The two major concerns with the general process were missing longer transient periods and, consequently any potential hysteresis of the system. To avoid hysteresis, a few minutes were allowed between individual tests. If a major transition was observed in the flow field, the velocity was ramped up and ramped down to the Reynolds number and the previous Reynolds number to see if the regimes were identical. If larger time scales, or general unsteadiness was observed, the images were also collected at one- tenth the frequency of the original collection.

The final step was loading and processing the PIV data. The PIV data was processed using the LabVIEW software. Specific recipes are discussed as they pertain to the results but the general PIV processing involves using the adaptive correlation resolving down to a 16-pixel window in three steps with additional testing for rotation motion. This process was chosen for its accuracy in solving the flow field.

4.4 Test Matrix

To ensure the varying domains of vortex shedding are captured, three different values must be looked at: the potential Reynolds numbers, the potential gap spacing ratios, and the nominal dimensions of the plate. The general theme of the developing the test matrix is the determination of how the vortex phenomenon scales.

4.4.1 Length Scale Reference

The biggest factor in determining this parameter relates back to the development of the boundary layer. Furthermore, the length scale could potential be different for all three parameters: the gap spacing ratio, the Strouhal number, and the Reynolds number. 98

Since the boundary layer size is arbitrary, there are three other more significant scales to examine. First, the scaling is proportional to the size of the plate,

* bb , (123) this scaling generally assumes the nondimensional quantity is nearly independent of the boundary layer. This scaling provides the minimum value for the length scale. Theory indicates this is not the case for Strouhal number but has the potential to be the case for the gap spacing ratio, which tended to have a more consistent value for the vortex shedding transitions. The second scaling worth examining,

* bb21 , (124) is dependent upon the displacement thickness. As mentioned earlier, the displacement thickness is proportional to the boundary layer thickness and provides the upper limit for the length scale. Alternatively, the gap spacing ratio may be more influenced by the boundary layer since most of the gap spacing ratios. Theory indicates that this parameter is unlikely to influence the Strouhal number which leads to the final scaling,

* bb22 , (125) which is dependent upon the momentum thickness. Theory indicates the momentum thickness is the most appropriate scale for the Strouhal number, but there is not enough information to determine if it is truly an appropriate scale for the gap spacing ratio or the Reynolds number as seen by the shedding of the vortex.

4.4.2 Reynolds Numbers

The Reynolds numbers of importance were determined from the theory of vortex shedding over cylinders. Sumer’s regimes were used to determine which Reynolds numbers were applicable as this regime list had the largest number of vortex shedding regimes. From this list, target Reynolds numbers were determined based on the plate thickness. The target values were the middle of the regime. The next step was determining the target numbers based on the momentum thickness values. If the 99

previously defined Reynolds numbers based on plate thickness were sufficiently in the regime based on momentum thickness, a new target Reynolds number was not created. The value was determined to be sufficiently close if the value was in the middle half of the regime. Once targets for the momentum thickness scaling were established, targets for the displacement thickness scaling were established in the same fashion. After all target values were determined, additional Reynolds numbers were added to fill in the large changes of the Reynolds number. The target Reynolds numbers are shown in Table 4.1.

Table 4.1: Target Reynolds numbers. Target Thickness Primary Purpose based Reynolds Number 8 Lower Laminar Regime for Displacement Thickness 23 Lower Laminar Regime for Momentum Thickness 52 Lower Laminar Regime 82 Lower Transition Regime for Displacement Thickness 112 Upper Transition Regime for Displacement Thickness 132 Upper Laminar Regime 152 Lower Transition for Momentum Thickness 193 Upper Transition for Momentum Thickness 225 Lower Transition Regime 275 Upper Transition Regime 450 Lower Subcritical for Momentum Thickness 650 Lower Subcritical Regime 875 Intermediate Value 1100 Upper Subcritical Regime

In total, there are 14 Reynolds numbers in Table 4.1. These values cover the six potential regimes of vortex shedding as observed for cylinders for three different length scales.

4.4.3 Gap Spacing Ratios

The basis of the gap spacing ratios were also chosen for the cylinders based on the length scales. Gap spacing ratios were chosen based on the best estimations of the various regimes provided by all of the tandem cylinder studies for low and high 100

Reynolds numbers. Overall, five primary target gap spacing ratios were chosen as 0.5, 1.5, 2.375, 3.125, and 4. For the gap spacing ratios, the same methodology was applied as the Reynolds numbers for different length scales. Target values were rounded to the nearest eighth. The various target gap spacing ratios are presented in Table 4.2.

Table 4.2: Target gap spacing ratios. Target Gap Spacing Ratio Reynolds Number 1 2 3 4 5 6 7 8 9 10 8 0.5 1.5 2.375 3.125 4 7 10 15 20 25 23 0.5 1.5 2.375 3.125 4 5.375 7 10 13.125 16.875 52 0.5 1.5 2.375 3.125 4 5.75 7.5 9.75 12.5 N/A 82 0.5 1.5 2.375 3.125 4 5.125 6.375 8.375 10.75 N/A 112 0.5 1.5 2.375 3.125 4 4.875 5.875 7.625 9.875 N/A 132 0.5 1.5 2.375 3.125 3.625 4 4.75 5.5 7.25 9.375 152 0.5 1.5 2.375 3.125 3.5 4 4.625 5.25 7 9 193 0.5 1.5 2.375 3.125 4 4.5 5.25 6.625 8.375 N/A 225 0.5 1.5 2.375 3.125 4 4.375 4.75 6 8.125 N/A 275 0.5 1.5 2.375 3.125 4 4.75 6 7.75 N/A N/A 450 0.5 1.5 1.875 2.375 3.125 4 5.375 6.875 N/A N/A 650 0.5 1.5 1.875 2.375 3.125 4 5 6 N/A N/A 875 0.5 1.5 1.75 2.375 2.875 3.125 3.75 4 4.75 6 1100 0.5 1.5 2.375 2.75 3.125 3.625 4 4.75 6 N/A

Table 4.2 shows a wide variety of tests (well over 100). During actual testing these numbers were consolidated based on experimental trials not yielding vortex shedding results. In some cases the gap spacing ratios, were changed to match nearby Reynolds numbers if they were relatively close to one another. These changes are discussed further in the results. With the gap spacing ratios and Reynolds numbers chosen, the final values to determine are the plate sizes.

4.4.4 Plate Sizing

For the purposes of testing different Reynolds numbers, there was a variety of plate sizes available for use. The purpose of the work is to establish a viable basis for the regime transitions for long plates. As such, the plates needed to have a length to 101

thickness ratio greater than 16:1 to avoid leading edge influences on the trailing plates. As a precaution against leading edge effects, the leading plate leading edge was rounded as seen in Figure 4.13.

Figure 4.13: Blunt and rounded plate edges.

The plates in Figure 4.13 also exhibit additional markings. The plates are named after their nominal size and split into A plates, which were used as the leading plates and B plates, which were used as tailing plates. The four corners of each plate were labeled as A, B, C, and D. Plate thicknesses were measured at each corner. The A and C corners made up the leading edge and the B and D corners made up the tailing edge, and consequently were used to determine the plate thickness. The A and B corners were faced up when the plate was placed in the test section. The length of the plate was taken as the average between A-B and C-D. The plate dimensions are listed in Table 4.3.

Table 4.3: Plate sizes used. Plate Thickness (mm) Plate Length (mm) Nominal Size l/b A B C D b A-B C-D l 1/2 A 12.01 12.48 12.06 12.52 12.5 246 245.7 245.85 19.7 1/2 B 12.13 12.17 12.34 12.18 12.175 258.6 259.7 259.15 21.3 7/16 A 11.56 11.44 11.45 11.7 11.57 248.2 247.6 247.9 21.4 7/16 B 11.34 11.2 11.37 11.43 11.315 246.5 247.1 246.8 21.8 5/16 B 8.28 8.31 8.58 8.51 8.41 185.2 184.6 184.9 21.0 5/16 A 8.13 8.18 8.48 8.56 8.37 184.3 184.4 184.35 22.0 1/4 A 6.13 6.16 5.75 5.92 6.04 128.8 129.52 129.16 21.4 1/4 B 5.73 5.73 6.17 6.11 5.92 129.14 127.68 128.41 21.7 102

Looking at Table 4.3 shows that there is some variability in the overall plate thickness for the given nominal size. The larger plates are typically less consistent because they were comprised of two separate pieces of acrylic whereas the smaller plates were cut from the same piece. The length to thickness ratios are about 21:1 for all plates. Keeping this ratio allows for consistency between the length based and thickness based Reynolds numbers. This is important because only a few plates can reach the higher Reynolds number values.

Optimal use of the PIV system and the wind tunnel falls in the 0.5-1.5 m/s range within the wind tunnel. Below this value, seeding is difficult to prevent from clouding the chamber. Above this value, the laser power is diminished due to the short pulse time required from the laser. To compensate for the range, the Reynolds numbers of the flow were driven by both the velocity and the plate thickness. The complete list of plate lengths and Reynolds numbers is presented in the next chapter. 103

5 RESULTS The original test matrix sought out to test all of the Reynolds numbers over the important corresponding ranges of a cylinder. It was discovered that below a Reynolds number of 125, there were no significant changes or observations in the local wake of the plate. As such, Reynolds numbers below this value were not investigated. As a result, the smallest plate was 6.04 mm thick and the largest was 12.5 mm thick.

The results are divided into five different sections. Section 5.1 examines flow over a single plate. This is done primarily to compare the results to other experiments and theory to add validity to the results. This is accomplished by comparing experimental results to well-established theory, testing for self-consistency and repeatability, and comparing to experimental results. The next two sections, 5.2 and 5.3, cover the main purpose of the work: developing and establishing the different regimes for vortex shedding over plate in the lower Reynolds number regimes. Section 5.4 discusses the overall regime map and compares it to that of a cylinder. The last section (5.5) summarizes the different regimes and relates them back to a cylinder.

5.1 Examination of Flow over a Single Plate

Prior to looking at the various regimes, it is important to look at the general flow over the flat plate. Examining the flow along and in the wake of the plate allows one to determine the validity of the results. Three primary areas are examined: the boundary layer, the wake, and the vortex shedding frequency.

5.1.1 Comparison of Boundary Layer

The first and most objective comparison is that of the boundary layer to the Blasius solution discussed earlier. This was performed by taking the average velocity profile a distance half of the plate thickness away from the plate as to avoid edge effects. A typical result is shown in Figure 5.1 104

Figure 5.1: Sample curve fit of boundary layer data (b = 8.4 mm, Re = 4380)

As seen in Figure 5.1 there are three data sets, the theoretical solution denoted with a dashed line, the experimental PIV results as squares, and a curve fit. The freestream velocity was calculated by using the freestream values at a distance 0.5b from the tailing edge of the leading plate. An sixth order polynomial was used to fit the data from the plate wall to where the experimental velocity was that of the average flow. The curve fit was used to calculate the momentum and displacement thicknesses of the plate. The curve was forced to yield a zero velocity at the wall. Doing this as opposed to only using the calculated experimental values provides a more reasonable result due to an increased uncertainty for points near the plate wall. The boundary layer was examined for all single plate flow rates. The culmination of the experimental boundary layer results is shown in Figure 5.2. 105

Figure 5.2: Single plate boundary layer data.

For all of the plates and Reynolds numbers there was good agreement with the Blasius solution profile. Figure 5.2 shows the spread of experimental data compared to theory. The various shapes indicate the plate size. Darker colors indicate lower Reynolds numbers and lighter, higher ones. The experimental values are plotted along the nondimensional Blasius parameters in order to collapse the various profiles on to one line. The spread is driven more by single plate tests than the Reynolds number as the largest and smallest plate (in thickness, length, and ratio) lay above the line, while the middle sizes are below it. There does however, appear to be a slight trend for data forming a shallower profile for higher Reynolds numbers and a steeper one for the lower profiles. For the collection of data, the momentum thickness, the displacement thickness, and the ratio of the two are plotted in Figure 5.3. 106

Figure 5.3: Comparison of theoretical boundary layer to measured boundary layer.

As seen in Figure 5.3, all three nondimensional thickness values fall along the expected value for theory. There is a bit more spread for the smaller plates as opposed to the larger ones. This is due to the relative size of the boundary layer compared to the imaging window of the camera. The larger plates have a relatively larger boundary layer in physical size and therefore have more data points with which to resolve the boundary layer. The ratio between momentum and displacement thickness is slightly higher than theory for all cases, while still in good agreement. This means the momentum thickness is proportionally larger than expected. The ratio is still well below the turbulence ratio and within 5% of the expected ratio. Beyond providing validity to the later results, exploring the general shape of the boundary layer provides insights into some of the assumptions associated with the theory. The good agreement with flat plate theory, especially considering the close proximity to the tailing edge means the general assumption of a Blasius solution for laminar flow over a plate is valid.

5.1.2 Comparison of the Average Wake

To verify that the characteristics of the wake are valid, a comparison of the largest plate and the smallest plate are made over three different Reynolds numbers that cover a 107

large portion of the range of Reynolds numbers tested. While the Reynolds numbers would ideally be identical, there are still deviations due to differences in the plate length to thickness ratio and the difficulty in fine-tuning the velocities. The values used and the differences in the calculated Reynolds numbers are presented in Table 5.1.

Table 5.1: Values for plate comparison. Re Re Plate Thickness Re Re Deviation Re l b b b l Deviation Target [mm] Measured [%] Measured [%] 12.50 190 3783 200 -5.4 -13.7 6.04 201 4300 12.50 423 8323 400 0.91 -7.5 6.04 419 8967 12.50 836 16450 800 -5.1 -13.4 6.04 880 18818

In Table 5.1 the differences between the two Reynolds numbers are largest at the highest and lowest Reynolds numbers used. This is due to the difficulty of hitting exact velocities at higher and lower values. The length-based Reynolds numbers have a larger error in the length due to the difference in the length to thickness ratios. The length- based error only contributes to the boundary layer error – which is to the order of one- half the Reynolds number. Consequently, the length-based error is half of the Reynolds number deviation. As the physics are driven by the plate thickness, boundary layer or some combination thereof, the errors due to duplicate results are bounded by the two errors. These errors are true errors and do not account for instrumentation uncertainty. First, it is important to make sure the wake profiles are consistent with themselves. A comparison of the two plates over the Reynolds numbers is presented in Figure 5.4. 108

Figure 5.4: Comparison of wake evolution across Reynolds numbers.

In Figure 5.4 the average wake behind the two plates are presented on two plots: 12.5 mm on top in blue and 6.0 mm in yellow on bottom. In each plot, the wakes for the three Reynolds numbers and seven different locations prior to and in the wake are shown. To compare the different sized plates, the ordinate is nondimensionalized to the plate thickness. To compare velocity and spacing the abscissa is the nondimensional length added to the nondimensional velocity. The nondimensional lengths used range from -0.5 to 5.5. Each value used is separated by one. Both plate cases have the same general trends. The first profile is attached to the plate that the velocity is zero at the 109

plate boundary (and nonexistent between ±0.5). The higher Reynolds numbers provide a steeper transition to the freestream and reach the freestream value sooner. Up to a distance of about two plate lengths away this trend continues, and the highest Reynolds number has a higher negative velocity at the plate centerline. Beyond this the higher the Reynolds number, the closer the velocity is to the freestream. This trend is more typical of a freestream wake. There is still substantial difference from wake theory as the wake is too close to the plate (usually one plate length, ~20b, is required) and the boundary layer is not significantly larger than the plate thickness.

It is important to consider the implications of the average velocity profile and its influence on various tandem plate regimes apart from a single plate. Near the plate, the velocity is negative or nearly quiescent at the centerline. This suggests that the lowest spacing regime exists up to about 1.5b. At which point higher Reynolds number will change regimes, whereas lower Reynolds number will maintain this regime for larger gaps. This trend is expected for most of the regimes based on the transition of the wake. The validity of this is dependent upon the two different plate sizes showing agreement with the velocity profiles at matching Reynolds numbers. A comparison of the plate profiles is shown in Figure 5.5. 110

Figure 5.5: Comparison of wake evolution across plate sizes. 111

Figure 5.5 presents the comparison of the two different plates for the three different Reynolds number. In all cases, the 12.5 mm plate has twice the resolution than the 6.04 mm plate due to the fact the boundary layer is twice as thick. This leads to a slightly rougher profile for the smaller plate for all cases. For all Reynolds numbers the first three profiles are nearly identical. Based on the previous comparison for each plate at given Reynolds numbers this is expected, as the profiles were quite similar in this region. Beyond the first three profiles, the profiles agree quite well for all Reynolds numbers and distances from the plate. There tends to be a small discrepancy a little farther from the plate. This results in the larger plate having a slightly steeper profile. The likely cause of this change is due to the blockage of the larger plate, which is just under 8%. While the effect is observable in the wake, previous experiments have demonstrated the effect is negligible on the frequency and amplitude of vibration for true fluid-structure interactions up to 10%.

Up until this point, only the average velocity profiles have been considered and compared for the various plate thicknesses. These comparisons showed that the flow characteristics over the plate are in agreement with theory, and the various plates sizes are in agreement with each other. These comparisons were made for the time average values; however, it is also important to compare the time dependent characteristics, e.g. the frequency of the plate, to both theory and experiment.

5.1.3 Comparison of Vortex Shedding Frequency

Unfortunately, there have been very few studies examining the vortex shedding frequency of long plates. Vortex shedding studies whose geometries are comparable to long plates usually fall into one of two categories: short plates and airfoils at various angles of attack. In the case of the former, the leading-edge effects play a substantial role in the overall vortex shedding frequency. In the second case, the geometry is asymmetric. While the results may be indicative of the solution for long plates, defining the characteristic thickness is not easy as airfoils are typically asymmetric, and the boundary layer thickness is often not detailed, nor can be reasonably calculated. 112

As mentioned in chapter 2, there are at least three studies detailing vortex shedding over plates: those of Taneda [1], Bull [16] and Naghib-Lahouti et al. [74].Their results, as well as the results of this study are presented in Figure 5.6. It is important to note that some of the data presentation techniques made it difficult to discern the true values for the Strouhal numbers based on other scaling lengths. In general, the spread of the data becomes worse when the data is changed to a different length scale, especially if values for the boundary layer and momentum thickness are not explicitly known. If not stated, general equations for the boundary layer and momentum thickness were used to determine the values based on whether it was turbulent or laminar. 113

A B C

Figure 5.6: Comparison of single plate data via different scaling.

For the composition of Figure 5.6, the work of four different authors were combined with the single plate results. The work includes that of Bull [16], Howard (this study), Naghib-Lahouti [74], Taneda [1], and Roshko [7].The data was adapted from the respective articles using a plot digitizer. In all cases, some degree of assumption about the displacement thickness and momentum thickness was needed to be able to plot against various scaling methods. If no other information was provided it was assumed the momentum thickness and displacement thickness followed equations (68) and (69) for a laminar flow over a plate and equations (70) and (71) for turbulent flow over a 114

plate. The primary plot shows all the results combined into a typical Reynolds vs. Strouhal plot. The lower three plots show the data using the three different scaling ways of presenting the data as it was presented in each of the studies. The color indicates whether the plate boundary layer was turbulent (red) or laminar (blue).

Two major observations can be made from the primary plot. First, despite a wide variety of plate sizes, the plot is relatively continuous without making any considerations for a scale based on the boundary layer thickness. The relative continuity of it is an indication that the boundary layer effect is minimal on both axes. Second, prior to very high Reynolds numbers, there is large disagreement between the vortex shedding between plates and cylinders as represented by Roshko’s equations. Additionally, the plates are much less clumped for Bull’s turbulent regimes indicating a dominant boundary layer effect for this regime.

The bottom three plots present the data using the suggested scaling of Bull (subfigure A), Taneda (subfigure B), and the momentum thickness scaling from chapter 3 (subfigure C). Bull’s scaling was explicitly referenced in Naghib-Lahouti et al. The first scaling in A is the most contiguous for all studies. However, it uses the Roshko number as opposed to the Strouhal number on the y-axis – the Roshko number being a product of the Reynolds number and Strouhal number. Since the Strouhal number approximately 0.2, and the x-axis is the Reynolds number, the result is, by definition, linear. Setting the x-axis scaling to a linear scaling makes any transition in the lower Reynolds number range unobservable. Unfortunately, this is the most dynamic range for the Strouhal number as a function of the Reynolds number. Bull’s results did extend to lower Reynolds numbers and would be useful to compare, however, the spread of the data on any of the other plots was far too large to warrant its inclusion. The length scale for A used twice the displacement thickness added to the plate thickness for both the Reynolds and Roshko number. The impact of this scaling in not fully realizable due to the use of the Roshko number. 115

The scaling in B uses a Strouhal number and Reynolds number based on the plate length. This effectively collapsed Taneda’s data to a single line, however it is clear the scaling is not valid as the other presented studies are scattered around the plot. Since the scaling is only using plate length, it ignores the effect of the plate thickness on the overall shedding and only looks at the influence of the boundary layer.

The final scaling is the one driven by theory, that is, the scale of vortex shedding is dependent upon the momentum thickness combined with the thickness of the plate. The scaling provides a continuous line that connects the variety of plates over a variety of Reynolds numbers. The most notable exception to this is the turbulent region of Bull’s data. However, this region is also the one with the largest uncertainty since the values for the momentum thickness were assumed from theory. It does align with Naghib- Lahouti’s turbulent values, which provided the ratio of the momentum thickness to the displacement thickness. In addition, the transitions in the general slope coincide with different forms of vortex shedding and the regimes.

5.2 Single Plate Regimes (Reynolds dependent phenomenon)

For single plate regimes, the basis for dividing the regimes was based around the structure formed near the base of the plate. These regimes are fundamentally driven by changes in the Reynolds number. Four principle regimes were observed:

Steady Regime

(Reb < 130) – Viscous forces dominate flow behind the plate. The flow is temporally independent and symmetric.

Pulsating Regime

(130 < Reb < 230) – Inertial forces begin to affect the wake formed behind the plate, but the viscous forces are larger than the inertial forces. The flow is axially symmetric but fluctuates temporally.

Oscillatory Regime 116

(230 < Reb < 350) – Inertial forces exceed that of the viscous forces near the base of the plate. The flow oscillates and is no longer symmetric.

Vortex Shedding Regime

(Reb > 350) – Inertial forces dominate the viscous forces and the characteristic vortex oscillations are observed.

The first two regimes produced no coherent fluctuations and were relatively steady. The distinguishing feature of the pulsating regime was the fluctuation of the vortices at the base of the plate. In the steady configuration, there were no observed changes in the structure of the vortex. On average, the wakes between the two were identical. For the two later regimes, there were visible asymmetric fluctuations in the wake. The following sections detail the difference between the average and temporal nature of each regime.

In this section and in section 5.3, average observations are presented, followed by time dependent observations. For the average observations, four different variables were examined. The first was the velocity in the x-direction. In chapter 3, it was shown that the frequency of vortex shedding is dependent on the wake velocity relative to the freestream. In addition to hinting at the vortex frequency, the wake velocity shows where the velocity is significantly lower than the free stream velocity. The results indicate where a second plate will strongly influence the behavior of the regime. Lines are presented to show where the local velocity is 1/20 of the free stream velocity. The total velocity was also considered; however, since the velocity is dominated by the x velocity, it provided little additional information.

The second average plot is the y-direction velocity. The y-direction velocity shows the regions where fluid from outside the wake is entering the flow. In addition, the y- direction velocity shows general boundary layer growth, and locations for counter rotating vortex pairs. Relative magnitude of the average y velocity provides insight into the relative magnitude of the oscillations. 117

To present an average sense of variability, the other two plots are the relative standard deviation and the correlation coefficient. The relative standard deviation is taken with respect to the free stream velocities as opposed to the average velocity. The results are important especially in oscillatory regions where the average flow may be small, but the fluctuations are large. The normalized standard deviation defined as:

1/2 N 2 uu  1 i1 i   . (126) UUN1 

Physically, the standard deviation is related to the local turbulent intensity in the x-y plane. Compare this to the correlation coefficient,

N u u v v  C  i1 ii (127) NN1/2 1/2 u u22 v v ii11 ii      which shows how tightly correlated the x and y velocities are. Mathematically the correlation coefficient ranges from +1 to -1, indicating a positive or negative correlation between the velocities. The value 0 indicates a location where there is no correlation, or more likely, the wake is invariant. Physically it highlights the locations were the velocity fluctuations are most strongly related. Effectively, this highlights the wake region behind the plate where vortices are traveling.

For the time resolved plots, only images and the velocity components are presented in the main body of the work. Both velocities are the most helpful in highlighting the differences in the regimes. The time resolved plots covered five different steps, enclosing roughly 1/5 of a complete period where applicable. Additional plots showing the streamlines and vorticity of these plots are presented in chapter 9.

5.2.1 Pulsating Regime

At the low Reynolds numbers, the wake is completely stable. However, as the Reynolds number increases, the wake begins to fluctuate and begins to pulse. The pulsating 118

regime was so called because the wake structure remained relatively symmetric. The wake did fluctuate with time as opposed to being relatively steady. Due to the fluctuation, the wake was considered a different regime than the steady regime. The average characteristics of the regime are presented in Figure 5.7. The average velocity characteristics were identical to that of a steady wake.

Figure 5.7: Free pulsating regime: average values.

Figure 5.7 provides the x velocity scaled to the free stream on the top left and the y velocity scaled to the free stream velocity on top right with the physical plate. For the average x velocity, a modified scale was used to emphasize the back-flow region. A red dashed line indicates the wake region where the velocity is less than 1/20 the free stream velocity. The center of the wake contains a negative velocity region that corresponds the center of the pair of counter rotating vortices at the base of the plate. The wake slowly changes slightly past the edge of the plate. The wake forms a general triangular shape and extends about 3b past the edge of the plate. The negative velocity 119

region is relatively small, and the larger values are positioned closer to the tailing edge of the plate.

In the top right plot, y velocity is in the negative direction of the y position for nearly the entire flow field with a couple exceptions. A gray null velocity region follows the centerline of the plate. Near the end of the plate, the largest average velocities are seen. The results were expected since the flow is converging after being separated by the plate. A unique feature for the average y velocity plot is the two spots at the edge of the plate. These spots are in the opposite direction of the main flow and indicate the left edge of the vortex pair generated behind the plate.

In three of the images (not including the correlation coefficient plot), some artifacts are seen. The streak observed past the plate is present due to the shadow generated by the plate edge. The spots in the far right are consequences of dimming in the laser near the edge of the image. These features are clearly present in the lower left plot, which presents the average wake variability in terms of the standard deviation and correlation coefficient of the velocity.

The standard deviation plot in the lower left is an indication of the variability or rather the steadiness of the pulsating regime. The top plot is the standard deviation of the total velocity as scaled to the free stream velocity. The same artifacts are more clearly seen in the standard deviation plot in the lower left. There are two notable features in this plot. First, there is an increased variability further from the end of the plate. This variability is indicative of the instability of the wake itself which, given sufficient distance, will produce vortices regardless of the plate base conditions. The second feature is the small triangular region at the centerline 3-4b away from the base of the plate. The triangular region is located where the changing wake size is most impacted.

The final plot on the lower right provides the correlation coefficient of the y and x velocities. Since the wake is relatively steady, there are no coherent regions. The lone observable feature (albeit very slight) is the more noticeable negative correlation coefficient difference on top and the more noticeable positive correlation coefficient 120

difference on bottom. The lack of features defines the average nature of the presented regime. The time resolved images of the velocity show a slight difference. The more general pulsing nature of the presented regime is shown in Figure 5.8.

Figure 5.8: Free pulsating regime: time resolved velocity. 121

Figure 5.8 shows the time resolved velocities for a single pulse. The far-left column demonstrates the corresponding image of the flow, the center column shows the x velocity and the right column displays the y velocity. For the x and y velocity, wake lines are drawn where the velocity is equal to 1/20 of the free stream velocity. From examination of the last two columns, the dominant component of the velocity proved to be in the x-direction for the pulsating regime. The center wake region - which has a very low velocity - is virtually zero throughout the pulse. In the center column, the wake is seen to grow and shrink with the pulse. A very slight negative region can be observed in the first image due to the reduced size of the wake forcing more backflow. The third column adds a little bit of insight into the overall nature of the pulsating regime, as there is a more visible fluctuation of the velocities. Throughout the transient, the y velocity on top remains negative and the y velocity on bottom remains positive. At the start of the pulse images, these velocities are strongest about 1.5b. As the pulse continues, the velocities move further away, reaching a maximum of about 3b. Ultimately, these regions appear to diminish in strength as they grow and eventually return to a location closer to the plate.

The pulse region is characterized by a pair of vortices whose wake was irregularly unstable. FFT analysis of the region did not produce any significant dominant frequencies. The implication of this is the pulse region is more or less a transition from a completely steady wake to one that is oscillatory in nature.

5.2.2 Oscillatory Regime

As the Reynolds number increased, the pulsing behavior gave way to an oscillation of the wake. As a result, the next identified regime was the oscillatory regime. Oscillations had been observed further downstream of the plate while in the pulsating regime. The oscillation being further downstream resulted in the general wake instability, not the instability from directly behind the plate. The oscillatory regime was characterized by the presence of a uniform dominant frequency. The wake itself was oscillatory near the base of the plate. Despite having observable frequencies, the characteristic pattern of 122

vortex shedding was not present. The regime does share some similar characteristics with the vortex shedding regime and is more similar to the vortex shedding regime than the pulsating regime. Regarding average velocity, the regime is similar to the pulsating regime. The average velocity for the oscillatory regime is seen in Figure 5.9.

Figure 5.9: Free oscillatory regime: average values.

The four plots in Figure 5.9 once again show the average values, specifically for the oscillatory regime. The two velocity profiles are seen on top with the left side being the x-direction velocity and the right being the y-direction velocity. It is interesting to note the presented average velocities are nearly identical to those of the pulsating regime, indicating a strong relative continuity between the two regimes. The rapid changes to the general flow are not occurring.

The x-direction velocity has an extended wake, expanding from 3b to 4b. The center of the region is largely negative. The biggest difference between the presented region and 123

the pulsating regime regarding average velocity is the stronger relative back flow region in the center of the wake.

The average profile of the y velocity is similar to the pulsating regime. Flow is largely negative above the centerline and positive below. The relative scale of the y velocity is similar to the pulsating regime. At the base of the plate, there is a region of counter rotating flow, producing velocities in the opposite direction, which was expected. For the average velocity, there is little discernable difference between the regime and the pulsating regime. A significant difference was observed in the variability of the velocity due to the standard deviation value.

The lower left plot shows the standard deviation of the velocity normalized by the freestream velocity. The bottom of the plot shows the correlation coefficient for the velocity. The standard deviation of the flow is relatively low up until 2b away from the tailing edge of the plate. Before this point, a dark region corresponding to the low velocity wake region is visible. Beyond this point, the variability in the wake increases and expands outward. This expansion coincides with the oscillations of the wake. The wake is oscillating heavily in the region, indicating a larger fluctuation in the velocity relative to the free stream.

The correlation coefficient plot shows a significant difference between the pulsating regime and the oscillatory regime. The most notable difference is two very clear regions: in the wake and outside the wake. Inside the low velocity region of the wake, the correlation coefficient is relatively uniform. Above the centerline, it is positive above and below it is negative. Outside of the low velocity region, the correlation coefficient takes on the opposite direction. An additional comparison can be made by examining the velocity field for this region. The velocities are shown in Figure 5.10 near the plate and Figure 5.11 for further downstream. 124

Figure 5.10: Free oscillatory regime: time resolved velocity.

125

Figure 5.11: Free oscillatory regime: time resolved velocity.

In Figure 5.10 and Figure 5.11 images of the flow are shown in the left column. While the average velocity indicated a symmetric wake nearly identical to the pulsating regime, the time dependent wake is not. The wake itself rises and falls as the vortices 126

are shed. Near the base of the plate, a low velocity region exists for 3b to 4b away from the plate. Near the base, the region is significantly consistent; however, around 2b, it begins to fluctuate. These fluctuations caused the pattern observed earlier in the standard deviation plot of Figure 5.9. Further downstream from the plate, the oscillations become larger including the total velocity, as it approaches the freestream velocity.

The same general patterns are observed in the center column, which displays the x- direction velocity. The wake region is highlighted by red and blue lines in which indicate where the velocity is 1/20 of the free stream velocity. The low velocity region extends to 3b to 4b away from the plate. The flow inside the wake region flows backwards slightly toward the plate. In general, the x velocity is virtually identical as the y-direction components are on the order of 1/10 the x velocity.

The y component of the velocity is presented in the right column. Within 1b of the plate, little to no action is observed, and the velocity is in the direction of the centerline of the plate. The largest velocities were observed being close to the same height as the plate edge. Between 1b and 2b, these velocity regions expand and alternate in magnitude. At 2b at the time when the velocity region has become the largest, it reaches the centerline of the plate and continues to expand. The larger velocity region continues well beyond and by about 3b the velocity region has grown to expand to reach the other side of the plate. At about 3b the limit of the expansion of the region away from the side it was formed; however, it continues to expand on the side it was formed. By about 4b the region has extended to a width of about 3b. From these plots, the change the oscillating region has brought is very apparent. Instead of a relatively steady convergence, large regions where the y velocity oscillates are observed in its place.

The oscillatory regime is characterized by a wake that coherently oscillates back and forth. The oscillation is due to the instability directly behind the plate; however, it is heavily influenced by the boundary layer. The implication of this is that vortex formation continues further downstream of the plate. Near the plate, only slight 127

oscillations are observed. Contrast this to the next regime, the vortex shedding regime, where formation of the vortex is very close to the base of the plate.

5.2.3 Vortex Shedding Regime

The final regime is that of the vortex shedding regime. In this regime, vortices are being shed completely. Significant changes to the characteristics of the flow field are observed. These changes are evident when the average values are examined, as presented in Figure 5.12. 128

Figure 5.12: Free vortex shedding regime: average values. 129

Figure 5.12 shows the average values for the free vortex shedding regime. The first row shows the average x velocity and the second shows the average y velocity. In the x velocity plot contour lines are drawn to show the location where the velocity is 1/20 of the freestream velocity. The x velocity in this plot is scaled to the freestream velocity. The bottom plot shows the y velocity of the region. The scale in this plot is limited to 1/10 the free stream velocity.

In the previous two regimes, the average velocity in the x-direction formed a very long region where the velocity extended upwards of 4b past the edge of the plate. The results were coupled with a negative region where the largest negative velocity was closer towards the base of the plate. For the vortex shedding regime, the wake is greatly reduced in size, being about 1.75b away from the plate. The region of maximum negative velocity sits closer to the edge of the wake than to the plate base at about 1.25b. The spread of the wake occurs a little further downstream and is visible at about 3b. Downstream the wake spreads much faster than the two previous regimes. This is a consequence of the regime occurring at a higher Reynolds number.

Examining the y velocity plot there are even more stark differences between this regime and the oscillatory regime. Near the plate, two counter rotating regions push fluid towards the outer edge. These regions were found in the previous regimes, however here they are more pronounced and not nearly as symmetric. From 0.5b to 3b, the average velocity reaches its strongest value outside of the counter rotating area and pushes toward the centerline of the plate. At 1.5b, the y velocities are all towards the centerline of the plate. This continues to about 4b where the direction of the velocity switches due to the expanding wake. This switch was not observed within 8b for the previous regimes. Even more differences are seen in the standard deviation and correlation coefficient as presented in the third and fourth rows.

The third and fourth rows show the standard deviation and the correlation coefficient respectively, for the vortex shedding regime. The former showing near the base of the plate and the latter showing further downstream. Here the plots, while different, are 130

much similar to the oscillatory regime, however, very in magnitude. The top plot shows the standard deviation of the velocity normalized to the free stream velocity. This plot gives a general sense of the relative scale of the velocity fluctuations. The bottom plot shows the correlation coefficient, which indicates regions where the x velocity is dependent upon the y velocity or vis versa.

Little to no fluctuation of the standard deviation was observed in the flow until about 0.25b away from the plate. Beyond this distance, the regions of high fluctuation increase and spread out. The increase along the centerline occurs up to the maximum fluctuation about 3b away from the plate. Afterwards, the maximum value of the standard deviation of the flow decreases along the centerline and continues to spread outward, similar to the oscillatory regime. The maximum value occurs closer to the plate and the relative fluctuations are much larger.

For the correlation coefficient, it possesses some similarity with pattern of the y velocity, with one major difference: no positive to negative switch further than the 1.5b location is observed. The correlation coefficient provides a general sense of the expanding wake and the formation of the vortices as they propagate downstream from the base of the plate. By 6b, the correlation region has exceeded 2b away from the centerline of the plate. At about 6.5b, the wake development settles with less of a correlation between the changes of the y and x velocities. Overall, the regime shares many similarities to the oscillatory regime but with a higher degree of correlation between the velocities.

The standard deviation and the correlation coefficient provide a sense of the fluctuations, but do not explicitly time resolve the vortex shedding domain. To this end, Figure 9.8 Figure 9.9 present images, streamlines and vorticity to highlight this transition. In addition to previous images, the time resolved velocity is also presented in Figure 5.13 and Figure 5.14 for both the total and components of the velocity. 131

Figure 5.13: Free vortex shedding regime: time resolved velocity. 132

Figure 5.14: Free vortex shedding regime: time resolved velocity.

Figure 5.13 and Figure 5.14 show the velocity is relatively similar to the oscillatory regime. The left column presents the images used for processing the data, the center 133

column presents the x component of the velocity and the right column presents y component of the velocity. In the x and y velocity plots, locations where the velocity is 1/20 of the freestream value are banded by red and blue for the positive and negative values respectively.

The x-direction velocity looks similar to the oscillatory regime. The low velocity wake in this region extends outward but fluctuates significantly in both size and direction. Contrast this to the oscillatory regime, where the wake size was relative consistent. The change size of the wake is more apparent in the plots of the x-direction velocity. The x- direction velocity maintains a much more erratic wake region. In this region, larger negative velocities are seen in the wake near the leading edge of the wake region. The results contrast to the oscillatory regime, where the larger values of negative velocity were seen closer to the base of the plate. Further from the plate, the wake oscillates more but quickly approaches the velocity of the freestream. The biggest differences for the velocity are seen in the y component.

In the oscillatory regime, the y components were largely evident on one side only and expanded across the centerline. In the vortex shedding regime, the y component of the velocity is present on both sides. The regions are initially separated by a near zero velocity at the centerline. The y-components of the velocity initially become visible about 1b away from the plate. The larger region is on the opposite side of the plate, e.g. the positive value for v is initially larger on the bottom. The larger region is the formation of the vortex on the side. The smaller region across the wake corresponds to the rolling of the previous vortex. These velocity regions ultimately expand until they converge together, which occurs at about 2.5b. In the far wake, the vortices begin to breakup and the velocities are diminished.

In general, the biggest change from the oscillatory regime to the vortex shedding regime is the roll-up of the vortices near the base of the plate. This roll-up coincides with a reduction of the length of the wake region. Since the vortices are developing 134

closest to the plate and the relative scale of the velocities are significantly larger, this regime affects the leading plate the most.

5.2.4 Summary

Three different primary regimes were discussed. The regimes were presented as the pulsating, oscillatory, and vortex shedding regimes. These regimes are characterized by a change in Reynolds number. Starting from a low Reynolds number, the flow creeps over the plate. This regime is the steady regime, which is usually not referenced for vortex shedding, as there are no dynamic characteristics of the wake. As the Reynolds number increases, the vortices at the tail of the plate begin to become unstable. At the onset of instability, the pulsating regime has been reached. The regime releases vortices nearly symmetrically and has no coherent observable fluctuation. The pulses occur randomly. Further increases in the Reynolds number result in the wake oscillating near the base of the plate. At the point where the oscillation becomes detectable, the oscillatory regime begins. A further increase in the Reynolds number causes the wake to begin to roll-up. This roll-up at the production of the vortex marks the vortex shedding regime. The vortex shedding regime is marked by a sharper increase in the Strouhal number (discussed in more detail in 5.4), a smaller and shrinking average wake, larger fluctuations, and an average backflow region that is pushed towards the downstream edge of the wake.

These three regimes are further referred to as the free regimes, as there is no tailing plate inhibiting the flow. The regimes discussed later have similar visual characteristics but experience changes that are significant. More importantly, changes that influence the Strouhal number and magnitude of the velocities.

5.3 Tandem Plate Regimes (Gap Spacing Dependent Phenomenon)

The previous section examined the effect of the Reynolds numbers on vortex shedding over plates. Three regimes were identified providing an estimated Reynolds number for transition. For the regimes that are dependent upon gap spacing, the lines are not 135

easily defined. The gap spacing regimes are dependent upon both the Reynolds number and the gap spacing. However, a couple trends are observed as gap spacing is increased.

First, only one true regime was added due to adding the second plate. This regime was referred to as the trapped regime. The trapped regime occurs at very small values for the gap spacing ratio. The trapped regime is identified by a pair of counter rotating vortices that occupied the space between the plates. The trapped regime was further subdivided into two regimes, the symmetric regime, and the u-shape regime. The u- shape regime appeared at higher Reynolds numbers and larger values for the gap spacing ratios. The primary difference between the two was the appearance of the vortices. In the symmetric regime, the vortices visually occupied the whole space. In the u-shape regime, the vortices formed similar to the vortices seen in the vortex shedding regime, however they did not shed. Instead, they were larger and shifted forward from the plate.

Second, after the trapped regime, the regime rarely transitioned to the free regime at the same Reynolds number. It usually transitioned to a form of the lower Reynolds number regime and then up to the free regime at the same Reynolds number. There is a notable exception to this trend, which is discussed in 5.4.

Third, two additional sub regimes were noted for each of the three originally defined regimes. These two regimes were defined as the trapped and incident sub regime. In each of these regimes, the flow took on characteristics of the free regimes, but the general characteristics of the regime were different from its free regime counterpart. Most importantly, there was a noticeable change in the Strouhal number. The incident regime was a location where there was any change, and the trapped regime was defined as a significant change. The explicit meaning is discussed later.

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5.3.1 Trapped Regime

With the introduction of a second plate, a new regime is introduced which cannot be described in terms of free regimes. The new regime is referred to as the trapped regime. The most important characteristic of it is formed vortices occupy the space between the two plates. Within the trapped regime, two sub regimes were identified: symmetric and u-shaped regimes. The average values of the former are shown in Figure 5.15.

Figure 5.15: Symmetric trapped regime: average values.

The average velocities for the trapped symmetric regime are shown in Figure 5.15. The top left plot shows the velocity in the x-direction and the top right plot presents the velocity in the y-direction. Dashed lines mark the regions where the average x velocity is at 1/20 of the free stream value. The center region between the plates has a large amount of backflow. This region is created by the pair for vortices that occupy the space between the plates. Outside the wake, the flow is relative consistent. Indicating the boundary layer is minimally affected by the presence of the vortices. 137

The y velocities tell a similar story. The y velocity sign is aligned with the sing of y- axis value. This means the boundary layer is still growing although it is slightly impacted by the vortices. At the edges of the plates, small regions are seen where the vortices reach the peak y velocity values recycle flow from the center region to the outside edge. As indicated by the small scale of the velocities in the region, the variability scaled to the free stream was small. The variability and correlation regions are presented on the bottom right and left respectively.

The standard deviation confirms the information seen in the velocity plots. There is not much happening between the plates. In the center towards the tailing plate, a small region of increased variability is observed where the largest back flow region exists. This indicates the vortices are not completely steady at this stage and do fluctuate slightly in size. As for the correlation coefficient, there is a slight correlation at the leading edge of the tailing plate. The region exists because of the occasional fluid that is ejected from the vortices. Other than that region, there is not much of a correlation, as would be the case for a steady regime.

It was originally anticipated the symmetric vortices would give way to vortex shedding as the gap between the plates increased. Observations indicated a significant change in the regime prior to shedding. In this regime, the vortices began to separate from the base region. The vortices never shed, but instead remained relatively constant in size apart from the occasional disruption in the flow field. Since the structure was visibly different and did not produce any transient behavior, it was referred to as a sub-regime of the trapped regime. The general vortex and wake structure formed a u shape as opposed to a symmetric shape, thus called the u-shaped trapped regime. The velocities for this regime are presented in Figure 5.16. 138

Figure 5.16: U-shaped trapped regime: average values.

Figure 5.16 shows the average x velocity and y velocity (top and bottom respectively) for the u-shaped trapped regime. The x velocities on top are marked with dashed lines showing general wake contours marking the locations where the velocities reach 1/20 of the free stream value with blue indicating negative and red indicating positive. The average velocity profile in both plots has a notable asymmetry along the y-direction. The asymmetry is due to the difference in the sizes of the plates at this location.

From x-velocity plot, the free stream velocity changes little from the end of one plate to the beginning of the next. As will be seen later, the areas along the second plate that indicated a larger change were a result of high uncertainty due to reflections off the plate. The largest difference between this regime and the symmetric regime is that the negative velocity region appears much closer to the tailing plate than it does the leading plate. In addition to being much more forward in position, the region is significantly stronger than the symmetric trapped regime. 139

Examination of the y velocity shows there is more similarity than difference between the regimes. Since the regime is still trapped, there is virtually no y component of velocity above or below the plates except for the growing boundary layer. The asymmetry of the vortices at leading edge of the tailing plate is visible here as well. The large upwards flow near the plate edge marks the left edge of the lower vortex. Similarly, the downward flow closer to the center of the gap marks the left edge of the lower vortex. When it comes to the y velocity, the presence of the reflow into the center being significantly stronger than circulation at the tailing edge of the leading plate defines this sub-regime.

In the lower left plot, the characteristic u-shape of the u-shaped sub-regime is seen in the standard deviation of the velocity. The relative fluctuations of the velocity are small at the base of the leading plate and on the outer edges. Near the tailing plate, the fluctuations become larger due to changing structure of the vortices. The largest fluctuation is at the boundary between the vortices at the right side of the gap. The fluctuations in this region are much higher and closer to the base of the plate. At the right end, there is a significant deviation, but this is a result of the values being influenced by the edge of the laser plane and the corresponding shadow of the tailing plate. For the correlation coefficient, there is again, little relationship between the flows due to the general steady nature of the regime.

If the gap spacing is increased from the trapped regime, the regime changes to one of the following regimes: the pulsating, oscillatory, or vortex shedding regime. At this point, the regime will have characteristics of the regime, but will often take on different average and dynamic values. Most importantly, the Strouhal number is impacted by the different sub-regimes. The first transition from the trapped regime as the gap spacing is increased it to the trapped sub-regimes.

5.3.2 Trapped Sub-regimes

As the gap spacing between the vortices is increased, the net result is a significant amount of backflow towards the plate extending form the leading edge of the tailing 140

plate. This means the characteristic triangular wake region in observed in the regimes that were free is not visible, as the average wake extends from one plate to the other. Despite the increased wake size, the transient characteristics of the wake are similar to the free sub-regime counterparts. The trapped sub-regime is relatively small, but it coincides with a rapid drop in Strouhal number compared to the more gradual changes to the Strouhal number when a plate was placed directly behind it further downstream. The trapped sub-regime is named because, on average, the velocity indicates a portion of the fluid is being trapped by the second plate.

Of the trapped sub-regimes, the pulsating regime is seen at lower Reynolds numbers. The average velocities are shown in Figure 5.17.

Figure 5.17: Trapped pulsating regime: average values.

The average velocities presented in Figure 5.17 show that the trapped pulsating regime maintains many of the characteristics of the pulsating regime, and even some associated with that of the trapped regime. Since both the trapped regimes and pulsating regime 141

are relatively steady, the average boundary layer changes very little, even in the gap between the plates. The top image provides the x-direction velocity while the bottom image provides the y-direction velocities. In the top images a red dashed line indicates the region where the velocity reaches 1/20 the value of the free stream.

The u-velocity plot shows the defining characteristic of the trapped-sub regimes, which is that the region of backflow extends from the leading plate to the tailing plate. Effectively, some amount is contained by the plates. The wake lines indicate that the region is trying to converge at the end, however the proximity of the tailing plate allows it to expand again, thus trapping the pulsating regime. Outside of the region between the plates, the flow looks like a developing boundary layer, similar to that of the trapped regimes. This is expected, as the pulsating regime is relative steady.

The average y velocity shows how this regime is different from the trapped regimes. At the edge of the leading plate, the y velocity shares the same sign as the y position. This is consistent with every regime, and is consistent with the single plate results, which show the boundary layer is nearly identical to theory up to the plate edge. At the base of the leading plate, the small circular regions of negative and positive velocity are consistent with both the pulsating and trapped regimes. The difference appears further into the flow. While the trapped regimes continued to grow the boundary layer, the boundary layer is shrinking in the trapped pulsating regime, i.e. flow switches from moving away from the centerline to moving towards the centerline. This trend continues all the way up to the second plate. At this point, another big difference between this regime and the trapped regime is observed. Since the flow is heading toward the centerline in the gap region, the flow is forced back out of the gap due to the presence of the second plate. This creates two regions of significantly higher velocity as the leading edge of the tailing plate. Further differences are seen in the fluctuations of the velocity around the plate as seen in the stand deviation plot in the lower left. 142

Since the pulsating regime is relatively steady, the correlation coefficient provides little information as to the nature of the flow. The information contained within the standard deviation plots shows a little bit more information about the flow in the gap region. Outside the gap, the fluctuations in the flow are relative low. About the 2b from the edge of the plate, the fluctuations begin to increase as the stability of the pulse wake decreases and pulses are observed. The highest fluctuations are observed past the leading edge of the tailing plate, where the flow is pushed out of the gap region. While the average values are useful, the transient values in Figure 5.18 provide a different view of what is occurring in the regime.

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Figure 5.18: Pulsating, trapped: time resolved total, u, and y velocity

Figure 5.18 shows the time resolved images, x velocities, and y velocities. The images being on the left column, the x velocities in the center and the y velocities on the right. As expected the y velocities and x velocities look similar to that of the symmetric 144

trapped regimes. This is expected because the pulsating regime was relatively steady already. The biggest differences are in the changing of the wake shape in conjunction with the pulse. The wake lines follow the general pattern of the images. With the primary wake collapsing in the center and expanding again outward over the edge of the tailing plate.

While the wake does indicate a slight transient, the wake is relatively steady, which is similar to the free pulsating regime. This is primarily because the velocity of the pulse is much lower than that of the freestream. The characteristic pulsing motion is best observed in the images, which retain information about where the flow came from. Outside the gap, the flow is well seeded. Inside the gap, steak lines from an early period are visible. These lines indicate the wake first contracts in the center. At the outer edges of the wake, the wake is pushed forward. As it is pushed forward, it grows and entrains flow from outside the gap. To compensate, the excess wake is pushed out over the tailing plate’s leading edge.

While this pulsing structure is unique in the trapped pulsating regime, the slow, general steady velocities make it less of an issue for fluid structure interactions. The same cannot be said of the trapped oscillatory regime. The average values for this regime are presented in Figure 5.19.

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Figure 5.19: Trapped oscillatory regime: average values.

Figure 5.19 shows the average values for the trapped oscillatory regime. There are several key differences between this regime and the free oscillatory regime. As with all trapped sub-regimes, the characteristic backflow exists from the leading edge of the tailing plate back to the tailing edge of the leading plate. The backflow region is seen in the upper right plot detailing the average x velocity. This results in no triangular wake forming, but instead a large negative velocity region in the right half of the gap. Furthermore, the oscillations result in the x velocity maintaining its profile. The x velocity looks more similar to the trapped u-shaped regime than the oscillatory free regime. Examining the other plots shows that the regime is relatively similar as well. This is also seen in the y velocity plot.

The y velocity is shown in the upper right corner. The y velocity shares several similar characteristics to the u-shaped trapped regime as well. In the center of the wake, the velocity profile is nearly identical to that of the u-shaped trapped regime. Additionally, 146

there is virtually no flow entering the center wake region. Examination of the average velocity shows no major differences between this regime and the trapped u-shaped regime.

The standard deviation seemingly indicates an even greater similarity to the u-shaped regime as the fluctuations are nearly inverted from the free oscillatory regime. The larger fluctuations are present in the center, again similar to the u-shaped trapped regime as opposed to along the outside, as was the case for the free oscillatory regime. Despite this, the variation increases and spreads the further downstream from the plate. This feature is more similar to the oscillatory regime than the trapped regimes.

The general steadiness of the flow despite oscillations means the correlation coefficient (lower right plot) is relatively the same everywhere in the flow. The only region that does have some consistent correlation is the leading edge of the tailing plate. This region of correlation means the flow is not steady and is oscillating around the plate.

The general shape of this regime indicates that it exists at a point just beyond the trapped regime. While it shares several of the same steady characteristics, the characteristics at the leading edge of the tailing plate indicate that this regime is not a true trapped regime, but instead a trapped sub-regime that maintains the characteristics of the oscillatory regime. These features are confirmed by examining the time resolved velocities in Figure 5.20. 147

Figure 5.20: Trapped oscillatory regime: time resolved values.

While the average appeared similar to the trapped regimes, Figure 5.20 shows the regime is very different from the u-shaped trapped regime in a temporal sense. In the left column, the images of the trapped oscillatory regime are seen. These images show 148

a lot more volatility in the wake region, especially compared to either the trapped or the oscillatory regimes. In some ways, this is much more characteristic of the vortex shedding regime. Examining the vortex structures, roll-up is not visible as would be the case for vortex shedding. The real information is gleaned from the center and right columns which show the x and y velocities respectively.

The x velocity shows that a constant wake exists between the two plates. The right edge tends to oscillate up and down whereas the left edge is more constant. This behavior is consistent with the delay in the formation of the vortices starting to be formed and then pushed further downstream with the oscillating wake. The region in between the plates while experiencing back flow is still small compared to that of the freestream. The time resolved x velocity does not strongly indicate that the regime is oscillatory.

According to the data, the y velocities clearly indicated the regime is oscillatory. The y velocities transitioned between up and down at regular intervals. The strength of the y velocity grows along with the vortex until is destroyed against the leading edge of the tailing plate. Some residual velocity in this region remains and slips past onto the tailing plate. The gray region of low velocity between the two regions of higher positive or negative velocity also shows the region is oscillatory as opposed to a vortex shedding regime. The trapped vortex shedding regime average values are shown in Figure 5.21. 149

Figure 5.21 Trapped vortex shedding regime: average values.

Figure 5.21 shows the average values of the trapped vortex shedding regime. These values share the same characteristics as that of the free vortex shedding regime. The most notable difference is the x velocity. The x velocity presented in the upper right has a region of backflow that extends from the tailing edge of the leading plate to the leading edge of the tailing plate. Contrast this to the free vortex shedding regime, which formed a shortened triangular wake behind the lead plate. The boundary layer of the flow is seen to expand at about 0.5b up to the leading edge of the tailing plate. The largest negative velocity still occurs at about 1.25b. While the minimum velocity location was not pushed back towards the plate, other plots indicate the formation of the vortices has been pushed significantly closer to base of the plate. This is exemplified in the y velocity plot in the upper right corner.

The y velocities in the regime switch the average direction 1b away from the base of the leading plate. This switch occurred further downstream for the free vortex shedding 150

regime. This indicates that the presence of the second plate is forcing the formation of the vortices closer to the base of the plate. This behavior is further seen in the standard deviation plot and correlation coefficient (lower right and lower left).

The correlation coefficient shows a nearly identical pattern to that of the free vortex shedding regime. The biggest difference is the pattern is influenced by the presence of the tailing plate. As was seen in the y velocity plot, the sign changes much sooner at a distance 1b away from the tailing edge of the plate.

The final plot featured is the standard deviation. The plot takes on a pattern similar to the free vortex shedding regime. A notable feature is that the relative strength of the fluctuations is smaller compared to that of the free vortex shedding regime. This is likely due to the presence of the second plate restricting the wakes ability to shed vortices. It is also notable that the fluctuations are significantly closer to the tailing edge of the leading plate. This again coincides with the location of vortex development being pushed closer and closer to the base of the plate. The fluctuations are better visualized through the time resolved images presented in Figure 5.22. 151

Figure 5.22: Trapped vortex shedding regime: time resolved velocity.

Figure 5.22 provides the time resolved images and velocity of the trapped vortex shedding regime. Looking at the images in the right column, the classic vortex structure 152

is seen as the vortices roll-up immediately upon formation. The images provide further evidence that the location of shedding has been pushed closer to the plate. As the vortices finishing forming they immediately impact the tailing plate and a new vortex is formed off the leading edge.

In the center column the time resolved x velocity is shown. The wake region is much more sporadic than that of the free vortex shedding regime as the wake both grows and shrinks significantly with large portions of backflow laying inside the wake. The erratic behavior of the wake is further indication of the influence of the tailing plate.

The final column shows the y velocity for the trapped vortex shedding regime. The behavior is similar to that of the free vortex shedding regime. Again, the larger difference is that the velocity fluctuations are pushed much closer to the tail edge of the plate as opposed to occurring further downstream.

The trapped sub-regime highlights a region where the characteristics of the free regime are most impacted by the presence of a plate downstream of the flow while still maintaining the major regime characteristics. The trapped sub-regime is relatively small and characterized by a steeper drop in the frequency of oscillations. Due to backflow in this regime, the shedding characteristics are suppressed. Further reduction of the plate spacing forces the trapped regime. An increase in the spacing changes the regime to the incident sub-regime.

5.3.3 Incident Sub-regimes

In between the trapped and free sub-regimes lie the incident sub-regimes. The incident sub-regimes are the transition to a free plate. As such, they share part of the characteristics of the free sub-regimes and that of the trapped regimes. For the most part the trapped regimes shared several similarities with the free counterparts. As such, the incident regime is merely the transition from the trapped sub-regime to the free sub- regime. Any average or time dependent plot will be a hybrid of the characteristics already discussed and will be similar to either the trapped or the free sub-regimes depending on which regime it is closer. For this reason, the individual plots are not 153

presented in this section, instead a sample of the average and time resolved incident sub-regimes are presented in chapter 8. The incident sub-regime begins when a wake can form, and flow trapped at the base of the lead plate does not reach the leading edge of the tailing plate. In the incident sub regime, the wake behind the plate begins to take on the more characteristic triangular shape seen in the free regime. Whereas the trapped sub-regime is identified by a steeper drop in the frequency as it approaches the trapped regime, the incident sub-regime has a much more gradual slope deviating only slightly.

5.3.4 Summary

By adding another plate, one new regime (trapped) with two sub-regimes (symmetric and u-shaped) was added as well as two additional sub-regimes (trapped and incident) for each of the previously defined. If a plate is introduced far enough downstream, it will have no influence on the on the behavior at the leading plate. As the trailing plate is moved closer, it enters the incident regime. Here the behavior of the flow changes slightly. The second plate changes the flow by forcing the wake to take on a different profile. This shift in profile, general means the frequency of oscillation is reduced as the wake velocity is also reduced. As the plate moves closer it eventually reaches a point where it directly impacts the formation of the flow structure at the base of the plate due to directly preventing flow from continuing downstream. The net effect is a stronger drop in the vortex shedding frequency. Further decreasing the gap spacing results in a change in regime to the trapped regime.

5.4 Map of Regimes

Section 5.2 and 5.3 described the various regimes observed the transitions as they generally occurred based on Reynolds numbers and gap spacing ratios. While it is convenient to think of them as direct transition, e.g. the vortex shedding regime transfers through all of its sub-regimes before reaching the trapped regime, the results shows locations where this is not the case. In the following sections, the transitions of regimes, and the impact of Reynolds and gap spacing ratio on the Strouhal number are 154

investigated. These dependencies are then compared to the transitions experienced by cylinders.

5.4.1 Description of Long Plate Regimes

From the previous section, four primary regimes were discussed in detail: the trapped, pulsating, oscillatory, and vortex shedding regimes. With the exception of the trapped regime, these regimes are split into three different sub-regimes, the free, incident and trapped sub-regimes. The trapped regime is divided into the symmetric and u-shaped sub-regimes. General trends as far as Reynolds number and gap spacing ratio were noted, however not explicitly detailed. These regimes, as well as their sub-regimes are plotted in Figure 5.23

Figure 5.23: Locations of plate regimes.

In Figure 5.23 all of the data collected for determining the Strouhal number has been presented. It is important to note that the free plates are plotted at a G/b proportional to 10 to present all of the data on a single plot. Each of the points is labeled with an initial 155

indicating the regime and a subscript indicating the sub regime. The letters of each correspond with the first letter of the regime, e.g. the incident vortex shedding regime is labeled with a Vi centered on the location where the data was collected. The graph is furthered divided by lines and colors highlighting various features of the regimes. Red text is reserved for both the oscillatory and vortex shedding regimes as these regimes displayed clear oscillatory patterns. The blue text indicates the trapped and pulsating regimes as these regimes did not display any clear oscillatory behavior. Between the blue and the red text runs a solid black line. This line approximates the location where the transition occurs between the oscillatory wake characteristics and the non- oscillatory ones. The two other lines indicate the other regime transitions. The dashed line indicates an approximate location for the change from the trapped regime to the pulsating regime. The dotted line indicates the change from the oscillatory regime to a vortex shedding regime.

In this plot, a series of trends become present. First, the trapped regime is highly dependent upon the gap spacing. The trapped regime was not observed for gaps larger than 2.5b nor was any other regime observed below this value. The lack of Reynolds influence on this regime indicates a relationship between the size of the vortex pair at the base of the plate and its overall stability. Within this regime, there are two primary sub-regimes: the symmetric and u-shaped vortex shedding regimes. These sub-regimes are divided by both Reynolds number and gap spacing. All gaps that were 1.5b or smaller produced the symmetric vortex shedding regime. The production of the u- shaped regime was only seen at Reynolds numbers above 300. The location of formation of this regime suggests it is directly related with the single plate’s ability to shed vortices.

Of the other regimes, the pulsating regime is the most limited by the Reynolds number as no pulsating regime was observed above a Reynolds number of 500. A majority of the pulsating sub-regimes were found to be trapped as opposed to incident as was the primary case for the oscillating and vortex shedding regimes. There are two major pulsating regions: the primarily free and incident region at low Reynolds number and 156

large gaps and the trapped region that exists from low to mid Reynolds numbers and smaller gap spacing ratios. Most of the pulsating regime observations were in the latter region. The large presence of trapped pulsating regimes is generally unsurprising because the pulsating regime is quasi-steady. Vortices do form at the base of the plate, but the wake is not stable. When the second plate is introduced at lower gap spacing ratios it is easier for these vortices to grow and connect to the other plate. The vortices are not stable due to their increased length; this instability leads to pulsing behavior but still allows backflow from the tailing plate.

The next regime most limited by Reynolds number is the oscillatory regime. For the oscillatory regime, two major locations exist. The first region includes the incident and free regimes that lie at smaller Reynolds numbers and larger gap spacing ratios. The second region is a small band that extends to high Reynolds numbers. This band contains only the trapped oscillatory regime. The band exists because the proximity of the second plate increases the resistance to the formation of vortices. The increase in resistance results in vortex shedding is no longer sustainable. At this point, the wake is still unstable but cannot completely shed vortices. Based on images and values provided by Bull [16], as well as a comparison to the regime map in Figure 5.23, this band was found to correspond to the region of hysteresis observed by Bull. It is important to note that the trapped oscillatory regime did have characteristics that were similar to that of vortex shedding.

The final regime is that of vortex shedding. For the most part, the vortex shedding regime exists for high Reynolds numbers with gaps larger than 3b. The vortex shedding regime was very rarely found to be trapped. The presence of backflow is what defines the trapped sub-regime. As vortex shedding requires roll-up of the vortices, backflow is difficult to achieve without the suppression of vortices. As such, the trapped vortex shedding regime is very rare and exists only near the edge of the trapped oscillatory regime. The lack of the trapped vortex shedding regime also helps explain why the trapped oscillatory regime is seen at higher Reynolds numbers. 157

In the previous sections, it was stated that regimes transitioned from pulsating to oscillatory and from oscillatory to vortex shedding regimes as the Reynolds number increases for a free plate. This observation holds true for all of the gap spacing observed in this study. For transitions from higher to lower gap spacing, the general trend assumed is that from the free-incident-trapped transition through the regime. While this is certainly a common trend, another possible transition is from the incident sub-regime to a different regime, usually one that is considered a lower Reynolds regime, e.g. from the incident vortex shedding regime to the incident oscillatory regime. In some ways, the reduction of the gap spacing is similar to a reduction in the Reynolds number. There is however, a region of particular interest, where oscillations and vortex shedding were effectively induced by the presence of the second plate. This region shows that general trends in the Reynolds number do not relate directly back to trends in the gap spacing. In this region, the pulsating regime leads to the oscillatory regime, which leads to the vortex shedding regime as the gap is reduced. In other words, vortex shedding is induced by the presence of the second plate. The region existed for relatively small plate Reynolds numbers from about 150 to 300.

While identifying the regimes is important, it is equally important to compare them to those of Bull et al. presented earlier in Figure 2.4 [16]. This comparison is shown in Figure 5.24 158

.

Figure 5.24: Comparison of Bull’s region of hysteresis.

Figure 5.24 provides a comparison of the displayed flow regimes and data of this study to the flow regime line and hysteresis provided by Bull et al. [16]. In general, Bull’s data and regime line matches well with the oscillatory-vortex shedding line. The biggest difference being at lower Reynolds numbers. The basis for this difference is likely due to the smaller amounts of data used by Bull to create the transition at these low Reynolds numbers. At the higher end, the hysteresis region is seen, and is quite small. Given its location and size, as well as the general downward sloping trend, it is unlikely to be a true hysteresis. The location occurs between the trapped oscillatory and trapped vortex shedding regimes. These regimes share several similar characteristics, and do not have a discrete shift in frequency as is seen with the hysteresis observed in cylinders. As such it is more likely the hysteresis observed by Bull was inherent to the experiment as opposed to a phenomenon of the fluid. 159

While it is of use to see where the different regimes exist, it is of greater importance to examine how the Strouhal number changes with both the gap spacing ratio and the Reynolds number. A map of the Strouhal number versus these parameters is shown in Figure 5.24.

Figure 5.25: Strouhal number for the various plate regimes.

The Strouhal map in Figure 5.24 is colored in blue to yellow with dark blue indicating regions where oscillations did not occur and yellow indicating the location of the upper limit of values observed for the Strouhal number (0.205).

The best way to describe the map shape is that of a plateau. The majority of the area exhibiting a non-zero Strouhal number exists has a value greater than 0.1. Around a Strouhal number of about 0.08, the Strouhal number drops off rapidly and no 160

oscillations are seen. The general outline of this plateau is very similar to the solid line drawn previously in Figure 5.24. This is expected as the line separates the oscillatory regimes from the non-oscillatory ones. Some general trends are seen along this plateau.

As was seen for single plates, the Strouhal number increases with Reynolds number. The highest Strouhal numbers exist at the highest Reynolds numbers and largest gap spacing rations. It is important to note the Strouhal number appears to change significantly less with changes in the gap spacing up until it reaches the lower edge of the plateau this is consistent with observations made by Bull [16], who noted the change in the Strouhal numbers with gap spacing decreased slightly, dipped, and then dropped off quickly.

Showing the Strouhal numbers are of interested, but a global map tends not to emphasize the regions that may be significant. Especially along the plateau where differences in frequency are not so easily observed. To this end, changes in the Strouhal number along gap spacing ratios and Reynolds number are better shown in Figure 5.25 161

Figure 5.26: Relative expected frequency transition

Figure 5.25 shows to different plots that both highlight regions of significance based on the transition of the Reynolds number and gap spacing ratio. The image is the change based on the gap spacing and the bottom image is the change based on the Reynolds number. The change is calculated by taking the difference between the expected Strouhal number and the Strouhal number and dividing by the maximum possible Strouhal number. 162

For the top plot, the expected Strouhal number is that of the free plate for the given Reynolds number and the maximum Strouhal number is the maximum Strouhal number that exists for the given Reynolds number. This way, regions where vortex shedding is heavily influenced by gap spacing are shown. On the left-hand side of the plot, a large gray region exists. This area is where oscillations never occurred for the free plate and did not occur for any of the gap spacing ratios. This region is relatively neutral because vortex shedding did not occur, nor was it expected. In the lower right quadrant is the blue region where oscillations have been completely suppressed. The region is ideal from an engineering standpoint as vortices that would otherwise occur are now being suppressed. In the upper right quadrant lies the bulk of the plateau. In this region, the vortex shedding frequency is shown to change very little, generally decreasing with the exception of the region near the larger red zone. The relative small change across the plateau is relevant and important, but not excessively, so as the fluctuations in Strouhal number is at most 10%. The most significant region is the large red zone towards the upper right. This region indicates where vortex shedding is being induced on a plate that would not normally experience vortex shedding.

For the bottom plot, the expected Strouhal number is that of the Strouhal number for the given gap spacing ratio at a Reynolds number of 2000 (the right end of the plot). The maximum Strouhal number is the Strouhal number is the maximum Strouhal number for the given gap spacing ratio. The normalized difference highlights regions that are more dependent on Reynolds number. The most notable feature is the lower gray region where the trapped regime exists. In this region, (gap spacing less than 2b) vortices do not ever shed, and the flow never oscillates. The Strouhal number for this range of Reynolds and gap spacing ratios is heavily dependent on the Reynolds number. As such, the remainder of the figure follows the general Strouhal number trend. Although here the plateau fluctuates little with gap spacing since the reference is from the higher end as opposed to the steady end, which would just return the Strouhal number pattern. 163

5.4.2 Comparison to Cylinders

A primary goal of the work was to examine the regimes of plates and the differences when compared to cylinders.

A Strouhal number map similar to the one presented in Figure 5.24 is presented alongside one for cylinders in Figure 5.26.

Figure 5.27: Strouhal maps for plates (top), and Xu’s cylinders (bottom). 164

In order to compare values accurately, the spacing ratio as opposed to the gap spacing ratio is used for the y-axis in Figure 5.26. The top plot is the map for plates and the bottom map is that for cylinders presented by Xu [61]. The data for Xu was obtained by digitizing the presented 2d plots and converting them into a 2d contour map so it could be compared directly with the results previously presented. In both maps, an identical color scheme is used. The color scheme is centered on a Strouhal number of 0.2. This value is the typically cited value for the Strouhal number of a cylinder. In both images, a red box shows the regions where the cylinder map and the plate map are being compared.

In the top plot, the same recognizable features can be seen as in the original Strouhal map. Since the spacing ratio is used as opposed to the gap spacing ratio, the plateau of Strouhal numbers is now on the bottom as opposed to the top. The other major difference is the plateau is has been condensed since the gap spacing ratios greater than four occupy the lower fifth of the graph as opposed to the majority of it.

The lower plot is for tandem cylinders. The data in this plot was collected by Xu. For tandem cylinders, the frequency is generally low for lower Reynolds numbers and increases and well into the subcritical regime. In the case of cylinders, vortex shedding is never completely suppressed as it is with plates. This is because the cylinders will switch to a single-body shedding mode. This feature is demonstrated in lower plot by the drop that exists in the lower middle of the lower plot. Because cylinders are both independently shedding, there is much more of an interference between the cylinders a both cylinders attempt to shed at their own frequency, the tailing one being driven by the leading one, all the while influencing the leading cylinder.

At first glance, the biggest difference between the cylinders and plates is the fact that the plates immediately drop off in Strouhal number while the cylinders often return to a higher number as they converge. In addition to that, the frequency of the cylinders appears less stable than the plates as there exists certain regions where the cylinders 165

drop in frequency much quicker than plates. A direct comparison between the red dashed line zones is presented in Figure 5.27.

Figure 5.28: Difference in Strouhal number, total (top), normalized (bottom).

Two different plots are shown in Figure 5.27. The top shows the total difference in the Strouhal number for plates and cylinders and the bottom showing the difference normalized to the maximum value of the two. The difference is with respect to the plates so red regions indicate a higher Strouhal number for cylinders and blue regions indicate a higher Strouhal number for plates.

At first glance, the most obvious feature is the large red region at the top of both plots. This region is the location where vortex shedding between plates has been completely 166

suppressed. In the bottom plot, the suppressed region is unity and in the top plot, it is the Strouhal number for the tandem cylinders. The lower region is then where the plateau for the tandem plates exists. Here is where the difference between shedding from cylinders and plates can be seen.

Along the lower edge, where the spacing ratio is zero, the cylinders are always higher up until the highest Reynolds number. This was shown previously for single plate. For most of plateau, the Strouhal numbers for the cylinders are higher than that of the plates. At the outer edge of the plateau, the Strouhal number for plates is higher than that for the cylinders. This is likely due to interference occurring between the cylinders being the strongest at this location. Whereas the Strouhal number for plates is relatively stable, the cylinders Strouhal number takes a larger dip because of the interference between the two different shedding frequencies. At a Reynolds number of about 800 and a Spacing ratio of about 0.1, the Strouhal number for plates is again higher. At this Reynolds number, cylinders are just entering the subcritical regime. The presence of the second cylinder likely has a stronger impact at this region as it likely influencing the transition on the lead cylinder. The plates on the other hand have a relatively stable Strouhal number that drops slowly resulting in temporarily higher Strouhal number.

5.5 Summary

5.5.1 Regimes

Observations of the wake behind long tandem plates, as well as measurement of both the velocity and frequency have provided insights into the transition of cylinders

Trapped Regime

The trapped regime was found to exist only for small gap spacing ratios. It was found at all Reynolds numbers investigated. The trapped regime only exists because of the presence of the second plate. The second plate forces the formation of vortices. The regime as a whole is insignificant to general FSI because the wake 167

between the plates allows fluid to streamline past the gap. The trapped regime is further divided into two sub-regimes:

Symmetric Sub Regime

In the symmetric sub regime, the flow around the plate occurs with little disruption. This regime parallels Igarashi’s regime C.

U-shaped Sub Regime

In the u-shaped sub regime, the general flow between the plates. The vortex structure is less stable and flow occasionally slips in and out of the gap. Two smaller vortices are formed in the downstream end of gap. The defining characteristic is that there is a large portion of backflow in the downstream half of the gap compared to the upstream half. This regime corresponds to Igarashi’s regime D.

Pulsating Regime

The pulsating regime was a characteristic low Reynolds number regime. The vortices formed at the base of the plate were slightly unstable and pushed flow downstream, usually in synchronization. This regime was divided into three other regimes. In general, the regime is insignificant from an FSI perspective because the wake is generally steady and the pulses were not coherent.

Free Sub-regime: The free sub-regime is the regime where the pulses were unaffected by the presence of the plate. Any pulsing characteristics that were seen did not exist by the time it reached the tailing plate.

Incident Sub-regime: The incident sub-regime was relatively rare. When the pulses were subjected to a tailing plate, either the wake became trapped by growing, or the regime switched to being the incident oscillatory regime.

Trapped Sub-regime: The trapped pulsating regime shared several similarities with the symmetric trapped regime. In some ways, this regime was more of an 168

extension of the trapped regime than the pulsating regime. The only major difference between the two was the stability of the wake resulted in pulses in the wake as opposed to being completely steady.

Oscillatory regime

The oscillatory regime marks the onset of regimes that are important from an FSI perspective. The oscillatory regime has coherent oscillations but would only have vortex roll-up further away from the plate if at all. Generally, this regime was observed at lower Reynolds numbers. The oscillatory regime has the same characteristic sub-remiges as the pulsating regime:

Free Sub-regime: The free sub-regime for the oscillatory regime is defined as the point where the oscillating frequency is no longer significantly different from that of the free regime. A proposed value is 0.99% of the oscillating frequency defines the lower bound of the free sub-regime.

Incident Sub-regime: The incident sub-regime was unique in that it often arose from the free pulsating regime. Consequently, this regime was often induced as opposed to resulting from a lower oscillation frequency of the free sub-regime.

Trapped Sub-regime: The trapped sub-regime is significantly different from the free sub-regime. The trapped sub-regime was only located at higher Reynolds numbers and was only found when transitioning down from the incident vortex. Additionally, the wake structure of this regime was more characteristic of the vortex shedding regime. What makes this sub-regime of particular importance is that it persisted up through the highest Reynolds numbers tested. This regime parallels Igarashi’s E regime.

Vortex Shedding Regime

The vortex shedding regime is the most important regime to consider when determining whether the oscillations of the wake will result in vibration of the plate. 169

The vortices are formed very close to the plate indicating a stronger alternating force on the lead plate. The vortex shedding regime also has three sub-regimes:

Free Sub-regime: The free sub-regime is the regime where there is no-effect on the vortex shedding behavior. It is defined as the point at which the frequency of shedding is 99% of the single plate value. In other words, at this location there is no effect from the tailing plate.

Incident Sub-regime: In the incident sub-regime, vortex shedding still occurs, but at a lower frequency. The formation of vortices is not significantly impacted but This regime relates to Igarashi’s F regime. The vortices are incident on the plate, but only after separating from the wake. Acoustic feedback from the tailing plate that reduces the shedding frequency.

Trapped Sub-regime: In the trapped sub-regime vortex shedding looks the same but is pushed incredible close to the plate. The frequency of shedding begins to drop significantly, and ultimately, be suppressed if the gap is reduced. The trapped sub-regime is defined as the point where vortex shedding occurs, but there is backflow towards the leading plate. This regime resembles Igarashi’s E’ regime. There is a direct interaction between the forming vortex and the tailing plate.

5.5.2 Effects on Strouhal number

The Strouhal number was shown to range significantly between tandem long plates and cylinders. The biggest difference is that the Strouhal number for plates jumped quickly and increased smoothly as the gap between plates was increased. For the Reynolds number, no major transition was observed in the wake. From comparison between cylinders and long plates, there are two major differences driving the difference in Strouhal number.

First, the plate geometry allows flow to travel smoothly along the surface. The flow over the plate results a significantly sized boundary layer, but also forces flow nearly 170

parallel to the surface of the plate. Cylinders on the other hand disrupt the flow resulting in a wake region that is separated from the cylinder. The short length of a cylinder limits any form of boundary layer. The geometry of long plates simultaneously lowers and raises the Strouhal number when compared to a cylinder. Since flow is moving parallel to the plate, more vorticity is incorporated into the generation of the vortex street. This raises the Strouhal number and explains why both Naghib-Lahouti and Bull reported Strouhal numbers much greater than that of a cylinder. In contrast, the boundary layer substantially lowers the Strouhal number and acts as a momentum sink for the formation of the vortices. This is why Taneda’s values for the Strouhal number were grossly smaller than even the smallest values for a cylinder. The data collected in this study successfully bridge the gap between the Reynolds numbers that produced both lower and higher Strouhal numbers.

The second biggest difference is vortex shedding is that vortex shedding is driven by one body for tandem plates and two for cylinders. Both plates and cylinders shed vortices from the tailing edge. By definition, shedding from a long plate is not dynamically influenced by the leading edge. Conversely, vortex shedding from the tailing edge of the tailing plate has no influence on the flow around the leading edge of the tailing plate. The opposite is true for cylinders. Cylinders are short which means there is a dynamic response between vortex shedding from one to the other. The smooth shape of cylinders allows for a change in location of the separation from the cylinder surface. This further increases the dynamic response between cylinders that are positioned close to one another.

The geometry of long tandem plates is both helpful and harmful in determining the vortex shedding frequency. The Strouhal numbers from long tandem plates are much more consistent than cylinders at a given Reynolds number. Long plates also experience a much smoother transition through Reynolds numbers due to the sharp discontinuity at the tailing edge of the plate. Unfortunately, long plates cover a much larger range of Strouhal numbers, and are dependent upon both the thickness and length of the plate. Fortunately, the wide range of the Strouhal numbers is due to the boundary 171

layer thickness and correction of the Strouhal number based on this value provides a much shorter range of Strouhal numbers for consideration.

172

6 CONCLUSIONS A PIV investigation of the vortex shedding phenomenon over tandem plates has been performed. This investigation has provided the regime changes for tandem plates as both a function of the Reynolds number and the spacing distance between the plates. Although investigations of long plates and tandem geometries exist, this is the first one that examines both.

6.1 Observations

An extensive literature review of the vortex shedding phenomenon for tandem plates and cylinders was performed. For the purposes of determining the Strouhal number theoretically for plates, two different analysis were performed, a scaling analysis and a calculation for the Strouhal number based on previous established theory. Both of these values were found to be in agreement with experiment values. Additionally, a comprehensive investigation was carried out to examine the behavior of vortex shedding over a range of Reynolds numbers and gap spacing ratios. The values for the Reynolds numbers and gap spacing ratios were chosen to coincide with key values that were observed for cylinders and other plate studies. The observations further established the bounds where vortex shedding would occur and additionally where it was most relevant.

For the data collected, the Strouhal number was recorded and compared to that of the cylinders. It was shown that vortex shedding is effectively delayed by the presence of a boundary layer. It was also observed the plate still undergoes its own transitions that are not linearly scalable to the boundary layer thickness.

6.2 Relevance of Work

The vortex shedding investigation herein provides a range of data and observations for a geometry that has been often overlooked. Additionally, it provides confidence for any assumptions that may be made about the Strouhal number for similar geometries. The investigation also provided insights into the nature of vortex shedding of tandem plates. 173

Determination of the changing regimes and Strouhal number is useful for future engineering analysis in which the behavior of vortex shedding may not be explicitly known. The observations for long plates establish a basis for further investigations and observations for vortex shedding frequencies.

With the determination of the flow regimes, further insight can be gleaned from the effect of the second plate on AFIP-6 mk. II. The separation distance between the plates was greater than ten times the thickness of the plates themselves in the AFIP experiment. This means the flow behind the leading plate was well within the incident vortex shedding regime, if not the free vortex shedding regime. At the given spacing, the vortex shedding was not significantly influenced by the presence of the second plate. A determination that could not be made prior to this study. To this end, it was other factors that influenced the Any influence would be insufficient to drop the shedding frequency below the Strouhal number below the standard 0.2 value.

6.3 Assumptions and Limitations

The study has provided valuable insights into the nature and behavior of vortex shedding between long tandem plates. The observations allow one to make sounds judgments as to the true Strouhal number, ergo vortex shedding frequency of the plate. There are however some limitations.

6.3.1 Length to Thickness Ratio

For all the plates used, the ratio of the length to the thickness was kept constant to allow the regime map to be comparable for the different plate sizes. The investigation did not examine the relationship between the length ratio and the transition of various regimes. While the vortex shedding frequency is clearly dependent upon the boundary layer thickness, the amount of effect the boundary layer had on the vortex shedding frequency and regime transitions was not determined, insofar at the scale of the momentum thickness and boundary layer thickness were both within the uncertainty of the appropriate scaled shedding frequency. It was shown, however, that even with a 174

wide variety of plate lengths, there is still some agreement even when the boundary layer is not accounted for when comparing single plates. Despite, the general agreement, there is still uncertainty associated with the effect of the boundary layer on the tandem plate regimes as this adds another relationship, especially since the gap spacing may not scale identically to the Strouhal number length scale.

6.3.2 Smooth Plates and Turbulence

The investigation herein used smooth. Observations made by Bull [16], show plates with initially turbulent boundary layers experienced a reduction in frequency that could not be directly attributed to a change in boundary layer size. As such, the regimes only apply to smooth plates where the boundary layer will undergo a normal development. This precludes plates with square leading edges. While there is a drop in the vortex shedding frequency, the general trends and observations for smooth plates are anticipated to be more valid than that of cylinders for rough plates.

6.4 Future Work

To expand upon the understanding of vortex shedding between long plates in tandem, a larger range of gap spacing ratios and Reynolds numbers would need to be covered. There are three areas of particular interest:

1. Observation of the direct transition from no shedding to shedding based regimes, especially at higher Reynolds numbers were the trapped vortex shedding regime was pushed closer to the base of the plate and ultimately suppressed. 2. An extension of Reynolds numbers for the upper vortex shedding regime through turbulent transition along the plate for and the impact of turbulence on the various tandem plate regimes. 3. Investigation into the mechanism of the induced oscillation regions where the presence of a second plate resulted in vortex shedding as opposed to suppressing it. 175

In addition to long plate geometries, vortex shedding between tandem plates of varying length is of importance, as very rarely are two long plates placed in tandem geometries. The geometry of AFIP-6 mk. II used different sized plates as opposed to two longer plates. A complete investigation of these regimes would allow a more sound engineering judgment while investigating FSI based phenomena.

176

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185

TABLE OF NOMENCLATURE

Symbol Definitions Dimensions b thickness l B channel height l l length l ε arbitrary error variable f frequency of vortex shedding t-1 f function and Blasius function n/a cD drag coefficient n/a D drag force mlt-2 δ boundary layer thickness l δ1 displacement thickness l δ2 momentum thickness l G gap size l g gravitational constant lt-2 h height l k Thermal conductivity Mlt-3θ-1 m mass m p pressure ml-1t-2 Ro Roshko Number n/a Re Reynolds Number n/a St Strouhal Number n/a t time t T period of vortex shedding t θ Temperature θ u x component of velocity lt-1 υ vortex velocity lt-1 U freestream velocity lt-1 -1 υp vortex propagation velocity lt -1 Us shear layer separation velocity lt v y component of velocity lt-1 w z component of velocity lt-1 x x-direction l y y-direction l z z-direction l α angle off x-axis n/a φ velocity potential t-1 ψ stream function t-1 ϕ oblique shedding angle n/a ξ complex coordinate system l Φ complex flow potential t-1 186

Symbol Definitions Dimensions S source term τ shear stress ml-1t-2 β vortex separation height l λ vortex separation length l η wake height l η Blasius variable n/a μ dynamic viscosity ml-1t-1 q heat generation per mass ν kinematic viscosity l2t-1 ρ density ml-3 υ vortex velocity lt-1 ζ vorticity t-1 Γ circulation lt2 ≡ is defined as n/a = is equal to n/a ≃ is asymptotically equal to n/a ≅ is approximately equal to n/a ≈ is estimated as n/a ∼ is on the order of n/a

187

TABLE OF ABBREVIATIONS

Abbreviation Definition AFIP Advanced test reactor Full size In flux trap Position ATR Advanced Test Reactor CFD Computational Fluid Dynamics DNS Direct Numerical Simulation FIV Flow Induced Vibration FSI Fluid-Structure Interaction GTRI Global Threat Reduction Initiative HEU Highly Enriched Uranium INL Idaho National Lab LES Large Eddie Simulation LEU Low Enriched Uranium LIFT Laser Imaging of Fluid Thermal NNSA National Nuclear Security Administration OSTR Oregon State TRIGA Reactor PIV Particle Image Velocimetry RANS Reynolds Averaged Navier-Stokes TRIGA Training Research Isotopes General Atomics URANS Unsteady Reynolds Average Navier-Stokes VIV Vortex Induced Vibration

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8 APPENDIX: INCIDENT REGIME PLOTS The following appendix presents the incident regime plots. The incident regime plots were left out of the main body of work because of their similarity to the other regimes.

The first figure of plots includes a two by two plot with the x velocity in the upper left corner, the y velocity in the upper right corner, the standard deviation normalized to the freestream velocity in the lower left and the correlation coefficient in the lower right. The velocities presented are the average velocities.

Following the average plots are time resolved plots. These plots contain images on the left column, x velocity in the center column, and y velocity on the left column. The three columns are split into five rows. Each row representing about 1/5 of an oscillation.

189

Figure 8.1: Incident pulsing regime: average values.

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Figure 8.2: Incident pulsing regime: time resolved velocity.

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Figure 8.3: Incident oscillatory regime: average values. 192

Figure 8.4: Incident oscillatory regime: time resolved velocity.

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Figure 8.5: Incident vortex shedding regime: average values. 194

Figure 8.6: Incident vortex shedding regime: time resolved velocity.

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9 APPENDIX: VORTICITY & STREAMLINES The following images provided are those for the streamline and the vorticity for all of the results presented in chapter 5. The left column provides the image being processed, the center column provides the streamlines and the right column provides the vorticity on a scale normalized to the freestream velocity divided by the thickness of the plate. The figure captions indicate the regimes, and the five rows cover roughly one period of motion or oscillation. 196

Figure 9.1: Free Pulsating Regime: images, streamlines, and vorticity. 197

Figure 9.2: Incident pulsating: images, streamlines, and vorticity.

198

Figure 9.3: Trapped pulsating: images, streamlines, and vorticity. 199

Figure 9.4: Free oscillating: images, streamlines, and vorticity.

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Figure 9.5: Free oscillating: images, streamlines, and vorticity.

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Figure 9.6: Incident oscillating: images, streamlines, and vorticity.

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Figure 9.7: Trapped oscillating: images, streamlines, and vorticity.

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Figure 9.8: Free vortex shedding: images, streamlines, and vorticity.

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Figure 9.9: Free vortex shedding: images, streamlines, and vorticity. 205

Figure 9.10: Incident vortex shedding: images, streamlines, and vorticity. 206

Figure 9.11: Trapped vortex shedding: images, streamlines, and vorticity.