1 DRAG FORCES on PILE GROUPS by RAPHAEL CROWLEY a THESIS

DRAG FORCES ON PILE GROUPS

RAPHAEL CROWLEY

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2008

To my Dad.

ACKNOWLEDGMENTS

I would like to thank my committee (Dr. Alex Sheremet, Dr. Robert Thieke, and my advisor, Dr. D Max Sheppard). I thank Kornel Kerenyi for giving me the opportunity to run experiments at the Turner Fairbank Highway Research Center (TFHRC) in McLean, Va. Thanks also go to the TFHRC staff, especially Matthias Poehler, Thies Stange, Dan Brown, and Jesse

Coleman. This thesis would not have been possible without the lab and its staff. Thanks also go to all the professors I’ve had through the years--especially Richard Crago at Bucknell University for giving me my start in research.

TABLE OF CONTENTS

page

ACKNOWLEDGMENTS ...... 4

LIST OF FIGURES ...... 8

ABSTRACT...... 14

1 INTRODUCTION ...... 15

Motivation for Research ...... 15 The 2004 Hurricane Season...... 15 The 2005 Hurricane Season...... 16 Scope of Research...... 17 Methodology...... 18

2 LITERATURE REVIEW ...... 19

Schlichting’s Far Wake Boundary Layer Theory...... 19 Relationship for Wake Half-Width and Velocity Reduction Factor ...... 19 Substitution into Boundary Layer Equations ...... 22 Extension to Multiple Piles ...... 25 M.M. Zdravkovich...... 28 Two Cylinders ...... 28 Hori’s pressure fields around tandem cylinders...... 29 Igarashi’s pressure field around tandem cylinders...... 30 Drag coefficients at lower Reynolds Numbers for tandem cylinders ...... 31 Drag coefficients for tandem cylinders at higher Reynolds Numbers ...... 32 Williamson’s smoke visualizations past side-by-side cylinders ...... 33 Hot-wire tests of flow the field behind two side-by-side cylinders ...... 34 Drag on side-by-side cylinders...... 35 Origins of the bistable flow phenomenon ...... 36 Pile Groups ...... 37 Shedding patterns behind three in-line cylinders ...... 37 Pressure fields behind three in-line cylinders ...... 38 Four in-line cylinders ...... 40 Square cylinder clusters ...... 42 Flow in heat exchangers...... 43 Literature Review Summary...... 45

3 MATERIALS AND METHODS...... 47

Schlichting’s Far Wake Theory...... 47 Near Wake Region...... 49 Pile Configurations ...... 50 Particle Image Velocimetry ...... 51 Problems with Traditional Measuring Techniques...... 51 Particle Image Velocimetry...... 52 Turner Fairbanks Highway Research Center PIV Flume...... 54 Setup...... 54 PIV measurement ...... 57 Potential errors in PIV measurements...... 59 Verification with ADV Probe...... 60 PIV Data Analysis ...... 62 Force Measurements...... 64 TFHRC Force Balance Setup ...... 64 Measurements in the Force Balance Flume...... 68 Methods Summary...... 69

4 EXPERIMENTAL RESULTS...... 70

PIV Data ...... 70 Velocity Fields...... 70 Velocity Profiles from PIV data...... 70 Vorticity Data ...... 70 Strouhal Number Comparison...... 70 Measured Force Data...... 99 One Pile ...... 99 Aligned Piles ...... 100 Side-by-Side Piles ...... 101 Pile Group Configurations...... 102 Results Summary...... 104

5 DISCUSSION...... 105

PIV Data Analysis ...... 105 Average Velocity Field Measurements ...... 106 One pile arrangement ...... 106 Three pile arrangement...... 107 Nine pile arrangement ...... 108 Two Pile Arrangement ...... 109 Demorphing PIV Data...... 109 Velocity Profile Measurements ...... 112 Vorticity Measurements ...... 112 Force Balance Data Analysis...... 113 One Pile Arrangement ...... 113

Two Side by Side Pile Arrangement ...... 113 Three In-Line Pile Arrangement ...... 114 Force Decrease at High Reynolds Numbers in the Three-Pile Configuration ...... 115 Complex Pile Arrangements Including Nine Pile Arrangement ...... 116 Statistical Analysis of Force and PIV Data ...... 117 No-Pile PIV experiments...... 119 Single-Pile Experiments ...... 123 Re ~ 5x103 ...... 123 Re ~ 4x103 ...... 126 Three-Pile Experiments...... 129 Re ~ 5x103 ...... 129 Re ~ 4x103 ...... 131 Re ~ 3x103 ...... 133

6 FUTURE WORK...... 136

Additional Pressure Field Measurements ...... 136 Additional Force Balance Measurements...... 136 Other Drag Force Measurements...... 136 Inertial and Wave Force Measurements ...... 137 Large Scale Testing ...... 138 Future Work Summary ...... 139

LIST OF REFERENCES...... 140

BIOGRAPHICAL SKETCH ...... 143

LIST OF FIGURES

Figure page

2-1. Definition Sketch for Flow around a Single Pile...... 19

2-2. Example of Reduced Velocity Past a Circular Pile Based on Equation 2-30...... 24

2-3. Column and Row Definition Sketch...... 25

2-4. Olsson Definition Sketch...... 26

2-5. Plot of Equation 2-36...... 27

2-6. Mean Pressure Distribution Around Tandem Cylinders ...... 29

2-7. Mean Pressure Distribution for Re = 35,000...... 30

2-8. Compilation of Drag Coefficients on Tandem Cylinders...... 31

2-9. Drag Coefficient and Strouhal Number Variation at Higher Reynolds Numbers...... 32

2-10. Smoke Visualization Downstream from Two Side-by-Side Cylinders, Re = 200. Top Figure has s/d = 6 & Bottom Figure has s/d =4...... 33

2-11. Diagram Showing the scattered 108 Hz and 47 Hz frequencies past the cylinders ...... 35

2-12. Interference Coefficient...... 36

2-13. Flow Past Three In-Line Cylinders Showing Shedding Behind the Second Cylinder, Re = 2000...... 38

2-14. Pressure Coefficient on Three Cylinders for Different Values of s/d ...... 38

2-15. Smoke Visualization with Three In-Line Cylinders, Re = 13,000...... 39

2-16. Variation in Drag Coefficient for Three In-Line Cylinders, Re =13,000...... 39

2-17. Pressure Distributions Around Four In-Line Cylinders. (a) is mean and (b) is fluctuating ...... 40

2-18. Drag Coefficient Variation for Four In-Line Cylinders. Aiba’s data is on the left and Igarashi’s data is on the right...... 41

2-19. Smoke Visualization past Four Cylinders for Four Different Values of s/d ...... 41

2-20. Drag Coefficient for a 2x2 Matrix...... 43

2-21. Pressure Coefficient on a 5x9 Cylinder Matrix (a.) Free Stream Turbulence = 0.5%. (b.) Free stream Turbulence = 20%. S/D = 2.0 for both plots ...... 44

2-22. Average Drag and Lift Coefficients for Circular Tubes in a 7x7 Matrix...... 45

3-1. Definition sketch for Schlichting’s 1917 theory for flow downstream from a circular pile (Top View)...... 47

3-2. Definition sketch for scenario of offset piles (Top View)...... 48

3-3. Definition Sketch of Side-by-Side Piles (Top View)...... 49

3-4. Pile arrangements used in this study...... 50

3-5. Typical PIV setup schematic drawing (Oshkai 2007)...... 52

3-6. Typical PIV image pair...... 53

3-7. TFHRC PIV Flume in McLean, VA ...... 54

3-8. Photograph of “trumpet” used to ensure uniform flow ...... 54

3-9. SoloPIV laser...... 55

3-10. MegaPlus Camera used in the Experiments ...... 55

3-11. Photo of camera-mirror setup...... 56

3-12. PIV rig setup used during experiments...... 57

3-13. PIV with laser on piles...... 57

3-14. PIV output with the correct time delay. The large arrows represent “errors.” Time delay adjustments are completed until the number of errors is small...... 58

3-15. ADV Probe Measurements...... 60

3-16. Average Velocity Image in PIV at First Velocity ...... 61

3-17. Average Velocity Image in PIV at Second Velocity...... 61

3-18. Average Velocity Image in PIV at Third Velocity...... 62

3-19. Masked PIV Image (9 Piles)...... 62

3-20. Example of a Correlation Image (no piles) ...... 63

3-21. Average Velocity Image Example...... 64

3-22. TFHRC Force Balance Setup ...... 65

3-23. Flapgate Setup in Force Balance Flume...... 66

3-24. Trumpet Setup in Force Balance Flume...... 66

3-25. SonTek MICRO-ADV Robot...... 67

3-26. View of force balance, piles, and flume looking downstream...... 68

4-1. Re = 5.13x103, 1 Pile Average Velocity Image...... 71

4-2. Re = 3.85x103, 1 Pile Average Velocity...... 72

4-3. Re = 2.57x103, 1 Pile Average Velocity...... 72

4-4. Re = 5.13x103, 2 Piles Average Velocity...... 73

4-5. Re = 3.85x103, 2 Piles Average Velocity...... 73

4-6. Re = 2.57x103, 2 Piles Average Velocity...... 74

4-7. Re = 5.13x103, 3 Piles Average Velocity...... 74

4-8. Re = 3.85x103, 3 Piles Average Velocity...... 75

4-9. Re = 2.57x103, 3 Piles Average Velocity...... 75

4-10. Re = 5.13x103, 9 Piles Average Velocity...... 76

4-11. Re = 3.85x103, 9 Piles Average Velocity...... 76

4-12. Re = 2.57x103, 9 Piles Average Velocity...... 77

4-13. Velocity Profiles through the Center of the First Pile for Re = 5.13x103 ...... 77

4-14. Velocity Profiles 20mm from the Center of the First Piles for Re = 5.13x103 ...... 78

4-15. Velocity Profiles 40mm from the Center of the First Pile for Re = 5.13x103 ...... 78

4-16. Velocity Profiles 60mm from the Center of the First Pile for Re = 5.13x103 ...... 79

4-17. Velocity Profiles 80mm from the Center of the First Pile for Re = 5.13x103 ...... 79

4-18. Velocity Profiles 100mm from the Center of the First Pile for Re = 5.13x103 ...... 80

4-19. Velocity Profiles 120mm from the Center of the First Pile for Re = 5.13x103 ...... 80

4-20. Velocity Profiles 140mm from the Center of the First Pile for Re = 5.13x103 ...... 81

4-21. Velocity Profiles 160mm from the Center of the First Pile for Re = 5.13x103 ...... 81

4-22. Velocity Profiles 180mm from the Center of the First Pile for Re = 5.13x103 ...... 82

4-23. Velocity Profiles through the Center of the First Pile for Re = 3.85x103 ...... 82

4-24. Velocity Profiles 20mm from the Center of the First Pile for Re = 3.85x103 ...... 83

4-25. Velocity Profiles 40mm from the Center of the First Pile for Re = 3.85x103 ...... 83

4-26. Velocity Profiles 60mm from the Center of the First Pile for Re = 3.85x103 ...... 84

4-27. Velocity Profiles 80mm from the Center of the First Pile for Re = 3.85x103 ...... 84

4-28. Velocity Profiles 100mm from the Center of the First Pile for Re = 3.85x103 ...... 85

4-29. Velocity Profiles 120mm from the Center of the First Pile for Re = 3.85x103 ...... 85

4-30. Velocity Profiles 140mm from the Center of the First Pile for Re = 3.85x103 ...... 86

4-31. Velocity Profiles 160mm from the Center of the First Pile for Re = 3.85x103 ...... 86

4-32. Velocity Profiles 180mm from the Center of the First Pile for Re = 3.85x103 ...... 87

4-33. Velocity Profiles through the Center of the First Pile for Re = 2.57x103 ...... 87

4-34. Velocity Profiles 20mm from the Center of the First Pile for Re = 2.57x103 ...... 88

4-35. Velocity Profiles 40mm from the Center of the First Pile for Re = 2.57x103 ...... 88

4-36. Velocity Profiles 60mm from the Center of the First Pile for Re = 2.57x103 ...... 89

4-37. Velocity Profiles 80mm from the Center of the First Pile for Re = 2.57x103 ...... 89

4-38. Velocity Profiles 100mm from the Center of the First Pile for Re = 2.57x103 ...... 90

4-39. Velocity Profiles 120mm from the Center of the First Pile for Re = 2.57x103 ...... 90

4-40. Velocity Profiles 140mm from the Center of the First Pile for Re = 2.57x103 ...... 91

4-41. Velocity Profiles 160mm from the Center of the First Pile for Re = 2.57x103 ...... 91

4-42. Velocity Profiles 180mm from the Center of the First Pile for Re = 2.57x103 ...... 92

4-43. Re = 5.13x103 Average Vorticity for One Pile...... 92

4-44. Re = 3.85x103 Average Vorticity for One Pile...... 93

4-45. Re = 2.57x103 Average Vorticity for One Pile...... 93

4-46. Re = 5.13x103 Average Vorticity for Two Piles ...... 94

4-47. Re = 3.85x103 Average Vorticity for Two Piles ...... 94

4-48. Re = 2.57x103 Average Vorticity for Two Piles ...... 95

4-49. Re = 5.13x103 Average Vorticity for Three Piles ...... 95

4-50. Re = 3.85x103 Average Vorticity for Three Piles ...... 96

4-51. Re = 2.57x103 Average Vorticity for Three Piles ...... 96

4-52. Re = 5.13x103 Average Vorticity for Nine Piles...... 97

4-53. Re = 3.85x103 Average Vorticity for Nine Piles...... 97

4-54. Re = 2.57x103 Average Vorticity for Nine Piles...... 98

4-55. Published Strouhal Number Data (Sarpkaya 1981)...... 98

4-56. Strouhal Number Data from PIV dataset...... 99

4-57. Drag Coefficient for 1 Pile ...... 100

4-58. Results for One Row of Piles ...... 101

4-59. Results for Two Side-by-Side Piles...... 101

4-60a. First Two Complex Configurations Run in the Force Balance ...... 102

4-60b. Second Two Complex Configurations Run in the Force Balance Flume...... 103

4-61. Results for Complex Pile Arrangements ...... 103

5-1. Schematic Drawing of Light Bouncing Off Mirror...... 110

5-2. Average Velocity Image in PIV at First Velocity ...... 110

5-3. Average Velocity Image in PIV at Second Velocity...... 111

5-4. Average Velocity Image in PIV at Third Velocity...... 111

5-5. Strouhal Number vs. Reynolds Number from Sarpkaya and Issacsson (1981)...... 113

5-6. Deduced Drag Forces Based on Measurements on a Three Inline Pile Arrangement ...... 115

5-7. Labeling Scheme for PIV Spectral Analysis...... 118

5-8. 0-Pile Time-Series in Middle of PIV Window, Re = 5.13x103 ...... 119

5-9. 0-Pile Spectrum in Middle of PIV Window, Re = 5.13x103 ...... 120

5-10. 0-Pile Time Series in Middle of PIV Window, Re = 3.85x103 ...... 120

5-12. 0-Pile Time Series in Middle of PIV Window, Re = 2.57x103 ...... 121

5-13. 0-Pile Time Series in Middle of PIV Window, Re = 2.57x103 ...... 122

5-14. De-Meaned Velocity vs. Time for 1 Pile at Point A1, Re = 5.13x103 ...... 124

5-15. Velocity Spectrum for 1 Pile at Point A1, Re = 5.13x103 ...... 124

5-16. De-Meaned Force vs. Time for 1 Pile, Re = 4.72x103 ...... 125

5-17. Force Spectrum for 1 Pile, Re = 4.72x103...... 125

5-19. De-Meaned Velocity vs. Time for 1 Pile at Point A1, Re = 3.85x103 ...... 127

5-20. Velocity Spectrum for 1 Pile, Re = 3.85x103 ...... 127

5-21. De-Meaned Force vs. Time for 1 Pile, Re = 3.83x103 ...... 128

5-22. Force Spectrum for 1 Pile, Re = 3.83x103...... 128

5-23. Force Spectrum for 3 Piles, Re = 4.76x103 ...... 129

5-24. U-Velocity Spectrum for 3 Piles, Re = 5.13x103 ...... 130

5-25. V-Velocity Spectrum for 3 Piles, Re =5.13x103 ...... 130

5-26. Force Spectrum for 3 Piles, Re = 3.85x103 ...... 131

5-27. U-Velocity Spectrum for 3 Piles, Re = 3.85x103 ...... 132

5-28. V-Velocity Spectrum for Re = 3.85x103 ...... 132

5-29. Force Spectrum for 3 Piles, Re = 2.86x103 ...... 133

5-30. U-Velocity Spectrum for 3 Piles, Re = 2.57x103 ...... 134

5-31. V-Velocity Spectrum for 3 Piles, Re = 2.57x103 ...... 134

Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science

DRAG FORCES ON PILE GROUPS

Raphael Crowley

May 2008

Chair: D. Max Sheppard Major: Coastal and Oceanographic Engineering

Particle Image Velocimetry (PIV) was used to measure the flow field in the vicinity of groups of circular piles of various configurations. A force balance was used to measure the drag forces on these pile groups. Both data sets showed good agreement with existing data. With the force balance, drag coefficients for a single pile seemed to level off at 1.0 indicating that the correct force value is being measured.

When three piles are aligned, PIV data proves that the second pile in-line induces changes in the first pile’s wake. A significant zero velocity zone exists in the wake of the first pile in- line, which encompasses the second pile. The approach velocity in front of the third pile in alignment is significantly larger than the velocity approaching the second pile. In fact, PIV data shows that a negative drag coefficient should be expected for the second pile in the alignment, and existing pressure field data supports this hypothesis. Ultimately, this shows that a velocity reduction method is not a viable option for predicting the drag force on a group of piles. Instead, a more complex method is needed to accurately predict these forces.

CHAPTER 1 INTRODUCTION

Motivation for Research

Wave loading on coastal structures has become a particular area of concern in recent years because of the recent increase in hurricane activity and intensity. The North Atlantic Hurricane

Seasons of 2004 and 2005 were the most active and most costly hurricane seasons on record.

During these storms, coastal structures were devastated causing unprecedented losses of life and property. Particularly, several bridges were destroyed during these storms. The goal of this research is to investigate a possible mode of bridge failure so that bridges in the future can be built to withstand stronger-intensity storms.

The 2004 Hurricane Season

In 2004, there were fifteen tropical storms, nine hurricanes, and six major hurricanes - a hurricane category three or greater – in the North Atlantic Ocean. Five hurricanes impacted the

United States that year. Florida received the worst of the 2004 season as four hurricanes made landfall on the peninsula – Charley, Frances, Ivan, and Jeanne. According to The National

Oceanic and Atmospheric Administration (NOAA), the 2004 hurricane season cost the United

States an estimated forty-two billion dollars. At the time, this was the most expensive hurricane season on record. The second-most expensive season was in 1992 when Hurricane Andrew struck the Miami-Dade area, and cost taxpayers an estimated $35 billion (NOAA 2007).

Hurricane Ivan, which made landfall just west of Pensacola, Florida is of particular interest. Ivan made landfall as a category three storm on September 16, 2004. As the storm ravaged the coast, the Interstate I-10 Bridge that spans Escambia Bay collapsed into the bay.

The storm surge and local wind setup elevated the water level to a height that was just below the

bridge girder elevation for most of the bridge spans. Waves, superimposed on the storm water level, battered the bridge deck. The surge and wave loading exceeded the weight and tie-down strength of many of the spans, and they were either shifted or completely removed from the substructure. By the time Ivan had passed, most of the I-10 Escambia Bay Bridge spans were completely destroyed.

The 2005 Hurricane Season

The 2005 Hurricane Season is the most active, most destructive, and most costly North

Atlantic Hurricane Season on record. Although the other storms of the 2005 season are often overshadowed by the unprecedented devastation caused by Katrina, the lesser-publicized storms also played their role in ravaging the Gulf Coast.

In 2005, twenty-five named storms developed over the North Atlantic Ocean, breaking the old record of twenty-one named storms set in 1933. Of these twenty-five named storms, fourteen of them developed into hurricanes. The previous record for number of hurricanes in a season, which was set in 1969, was twelve. During 2005, five category five hurricanes formed; previously, the record for number of category five hurricanes was two.

The United States was hit by seven named storms during 2005 – Arlene, Cindy, Dennis,

Katrina, Rita, Tammy, and Wilma. As usual, Florida received more than its fair-share of destruction; four of these seven storms struck the Sunshine State. The 2005 season was by far the most costly on record. According to NOAA, estimated losses due to hurricanes and tropical storms in 2005 are in the neighborhood of one hundred billion dollars (NOAA 2007).

As in 2004, a significant portion of this cost was due to the collapse of coastal bridges. In -

2005, Hurricane Katrina was the culprit. During Katrina, The I-10 Lake Ponchetrain Causeway

Bridge in New Orleans, Louisiana, The US-90 Biloxi-Ocean Springs Bridge in Biloxi,

Mississippi, and The US-90 Bay St. Louis Bridge in Bay St. Louis, Mississippi, all suffered failures similar to The I-10 Escambia Bay Bridge failure during Hurricane Ivan in 2004.

Scope of Research

The University of Florida has been working on problems associated with bridge failure during hurricanes since the Escambia Bay Bridge collapse during 2004. D Max Sheppard and his students have led efforts to determine the horizontal and vertical forces and associated moments on bridge decks. At present, significant progress is being made in understanding the wave forces on bridge decks.

There are aspects of hydrodynamic forces on bridge substructures that are not understood.

Many of the older bridge substructures were relatively simple in shape. Most new pier designs are complex in shape and consist of pile groups, pile caps, and columns. While much work has been done on current and/or wave loading on vertical structures such as pile caps, columns and single piles, there is less known about hydrodynamic forces on groups of piles. The goal of this project is to investigate steady current induced drag forces on pile groups. This work provides the basis for one of the components of wave induced forces on these types of structures. On single, smaller structures, where diffraction forces are small, wave forces can be computed using the Morison Equation:

F T = F I + F D 1 F = C ρ AU | U | D 2 D 1 ∂ U F = C ρ A I 2 I ∂ t

Where CD is the drag coefficient, ρ the mass density of water, A the projected area, U the fluid velocity just upstream of the structure, and CI the inertial coefficient. As stated above, this work focuses on drag forces in steady flows.

Storm surge and wind induced currents can be very large, thus producing large drag forces.

A better understanding of these forces also provides a basis for understanding one of the components of wave induced forces. A first step in understanding and predicting these forces is to measure the flow field in the vicinity of a pile group and to measure the forces for a range of flow velocities. The work reported on in this thesis addresses these issues for a limited range of conditions.

Methodology

This study approaches the problem from the standpoint of 1) quantifying the complex flow field in the vicinity of the pile group and 2) measurement of the forces on different pile arrangements.

Particle Image Velocitemetry (PIV) is used to measure the instantaneous and average

velocity flow field within a group of circular piles for a range of Reynolds Numbers. The

intent is to use this information to better understand they flow hydrodynamics in the

vicinity of pile groups.

A force balance is used to measure drag forces directly on the pile group. The force

balance arrangement can only measure the force on the entire pile group, so a variety of

pile groups were studied at different Reynolds Numbers to obtain a robust dataset of

forcing on pile groups.

CHAPTER 2 LITERATURE REVIEW

Schlichting’s Far Wake Boundary Layer Theory

H. Schlichting (1979) provides a means for estimating the time mean flow velocities in the far-field wake region for steady flow around a single circular pile. Schlichting’s theory is summarized below:

Relationship for Wake Half-Width and Velocity Reduction Factor

Approximate Extent of b Wake y U∞

x u

Figure 2-1. Definition Sketch for Flow around a Single Pile

In Figure 2-1, U∞ is the free stream velocity, d is the pile’s diameter, and b is the wake half-width. Applying the momentum equation to a control surface which encloses the pile of height, h gives

∞ ∞ F = hρ u(U − u)dy = hρ u U − u dy (2-1) D ∫∫∞ 1()∞ 1 yy=−∞ =−∞

2 2 Because u1 is small, O(u1 ) (terms on the order of u1 ) [i.e.] can be neglected, and the result is

∞ F = hρU u dy (2-2) D ∞ ∫ 1 y=−∞

The drag force is typically expressed as

1 F = C ρhdU (2-3) D 2 D ∞

Equating the expressions in 2-2 and 2-3 yields

1 ∞ F = C ρhdU = hρU u dy (2-4) D D ∞ ∞ ∫ 1 2 −∞

Therefore

1 ∞ C dU 2 = u dy (2-5) D ∞ ∫ 1 2 −∞

According to Schlichting, there is a direct relationship between change in linear momentum and the drag on a pile,

F = ρ u U − u dA . (2-6) D ∫ ()∞

He assumes that the control surface has been placed far enough behind the pile that the static pressure will equal the static pressure in an undisturbed stream. Far enough downstream, u1 is small compared with U∞, so, the following simplification can be made:

u()()U∞ − u = U∞ − u1 u1 ≈ U∞u1 (2-7)

Substituting this into the momentum integral:

F = ρU u dA (2-8) D ∞ ∫ 1

For a wake behind a pile, dA is simply hb where h is the pile’s height and b is the half- width of the wake. This can be rewritten as follows

FD ~ ρU∞u1hb (2-9)

2 Using the expression for drag force, FD = 1/2CDρ U∞ hd, and equating according to the expression above, one obtains

u C d 1 ~ D (2-10) U∞ 2b

Schlichting (1979) cites Prandtl’s work, and Prandlt said that “the following rule has withstood the test of time”

Db ~ v' (2-11) Dt

Where v’ is the transverse velocity and D/Dt represents the total derivative. Schlichting

(1979) says the following about the origins of the transverse velocity component:

“Consider two lumps of fluid meeting in a lamina at a distance y1, the slower one from

(y1-l) preceding the faster one from (y1+l). In these circumstances, the lumps will collide

with a velocity 2u’ and will diverge sideways. This is equivalent to the existence of a

transverse velocity component in both directions with respect to the layer at y1…This

argument implies that the transverse component v’ is of the same order of magnitude as

u’….”

⎛ du ⎞ ⎜ ⎟ | v'|= const | u'|= const⎜l ⎟ (2-12) ⎝ dy ⎠

Another way of writing this is that v’ ~ l du/dy. Therefore,

Db ∂u ~ l (2-13) Dt ∂y

At the wake boundary, we have

Db db = U (2-14) Dt ∞ dx

and if it is assumed that the mean value of du/dy taken over the half-width of the wake is proportional to u1/b, the following expression is also true:

Db 1 = const * u = const * βu (2-15) Dt b 1 1

where, again, u1 = U∞ – u. Equating these two expressions yields

db 1 U ~ u = βu (2-16) ∞ dx b 1 1

Where β is a constant. Rewritten this gives

db u ~ β 1 (2-17) dx U∞

From the expression obtained from the momentum equation

u C d 1 ~ D (2-18) U∞ 2b

and substituting, the following expression is obtained:

db 2b ~ βC d (2-19) dx D

or,

1 2 b ~ ()β xCD d (2-20)

Inserting this expression for the wake half-width into the expression from the momentum equation gives the velocity reduction factor downstream of the pile.

1 2 u1 ⎛ CD d ⎞ ~ ⎜ ⎟ (2-21) U ∞ ⎝ β x ⎠

1/2 -1/2 In summary, b ~ x and u1 ~ x

Substitution into Boundary Layer Equations

In two-dimensional incompressible flow, the governing equations are

∂u ∂u ∂u 1 ∂τ + u + v = ∂t ∂x ∂y ρ ∂y (2-22) ∂u ∂v + = 0 ∂x ∂y

Where τ is the turbulent shear stress. All pressure terms have been dropped because we are assuming that pressure remains constant. Prandtl’s mixing length theory can be used in the above equations:

∂u ∂u τ = ρl 2 (2-23) ∂y ∂y

Since the y-velocity component gradient is small, after substituting into the governing equations, the following is obtained:

∂u ∂u ∂2u −U 1 = 2l 2 1 1 (2-24) ∞ ∂x ∂y ∂y2

Schlichting assumes that the mixing length is constant over the wake half-width and proportional to it so that l = βb(x). The ratio η = y/b is introduced as the independent variable that represents the similarity of the velocity profiles. In agreement with the proportions obtained from the momentum equations, the following are assumed to be true:

1 b = B()CDdx 2 1 − (2-25) ⎛ x ⎞ 2 ⎜ ⎟ u1 = U∞ ⎜ ⎟ f ()η ⎝ CDd ⎠

Inserting into the governing equation, a differential equation is obtained for f(η).

1 2β 2 ()f +ηf ' = f ' f '' (2-26) 2 B

At the free surface, the velocity reduction should equal zero and the y-gradient of the reduction factor should also equal zero. In other words, the boundary conditions are defined as at η = 1, f = f’ = 0. Integrating twice and applying these boundary conditions yields

2 1 B ⎛ 3 ⎞ f = ⎜1−η 2 ⎟ (2-27) 2 ⎜ ⎟ 9 2β ⎝ ⎠

From the momentum integral, the integration constant, B can be determined.

B = 10β (2-28)

With Schlichting assuming that

2 1 ⎛ 3 ⎞ 9 ⎜ 2 ⎟ ∫⎜1−η ⎟ dη = (2-29) −1⎝ ⎠ 10

The final solution to flow past a single pile then becomes

1 b = 10β ()xCDd 2 1 2 − ⎡ 3 ⎤ (2-30) u 10 ⎛ x ⎞ 2 ⎛ y ⎞ 2 1 = ⎜ ⎟ ⎢1− ⎜ ⎟ ⎥ ⎜ ⎟ ⎢ ⎥ U∞ 18β ⎝ CDd ⎠ ⎝ b ⎠ ⎣ ⎦

A plot of results from Equation 2-30 are shown below. This plot shows u1/UInf:

u /U 1 Inf 30 0.8

0.7 20

0.6

10 0.5

0 0.4

0.3 -10

Lateral Distance From Pile (cm) 0.2

-20 0.1

-30 0 5 10 15 20 25 30 35 40 45 50 Distance Downstream From Pile (cm)

Figure 2-2. Example of Reduced Velocity Past a Circular Pile Based on Equation 2-30

Parameters for computation of this velocity field are as follows: d = 1.905cm, CD = 1.0, and b = 30cm. Schlichting determined the constant β from measured values. According to measurements by Schlichting and H. Reichardt (Schlichting 1979), β = 0.18. The pile’s locus in

Figure 2-2 is at coordinate point (0,0).

Extension to Multiple Piles

When looking at more complex pile configurations, it is useful to define “rows” and

“columns” of piles. Pile “rows” are defined as piles that are one behind the other relative to the flow velocity and pile “columns” are defined as piles that are offset from one another in the y- direction.

Flow Direction

Column 1

Column 2

Column 3

Row 1 Row 2 Row 3

Figure 2-3. Column and Row Definition Sketch

R Gran Olsson studied flow past an array of multiple in-line piles both experimentally and theoretically. Although Olsson’s analysis is for an infinite row of bars, it will be used to represent a finite number of piles. Given a row with the following configuration

s s

u1 U ∞

Figure 2-4. Olsson Definition Sketch where s is the spacing between rows, U∞ is the velocity if there were no bars, and u1 is the reduced velocity, Olsson assumes that in a fully developed flow, the velocity distribution is expected to be a periodic function in y:

−1 ⎛ x ⎞ ⎛ y ⎞ u1 = U ∞ A⎜ ⎟ cos⎜2π ⎟ (2-31) ⎝ s ⎠ ⎝ s ⎠

with A being a free constant to be determined from experimental data.

Schlichting cites an extension of Prandtl’s mixing length theory

2 ∂u ∂u ⎛ ∂2u ⎞ τ = ρl 2 + l 2 ⎜ ⎟ (2-32) t 1 ⎜ 2 ⎟ ∂y ∂y ⎝ ∂y ⎠

and states that it seems reasonable to assume that l1 = s/2π. Taking the derivative and dividing both sides by ρ gives

−2 3 1 ∂τ 2 ⎛ x ⎞ 2 2 ⎛ 2π ⎞ ⎛ y ⎞ = l ⎜ ⎟ U ∞ A ⎜ ⎟ cos⎜2π ⎟ (2-33) ρ ∂y ⎝ s ⎠ ⎝ s ⎠ ⎝ s ⎠

Next, Olsson observes that at a certain distance from the row of piles, the width of the wake equals the spacing between the bars. At this distance, the velocity difference, u1 is small compared to U∞, so the two dimensional boundary layer equation (equation 3-20) reduces to

∂u 1 ∂τ −U 1 = (2-34) ∞ ∂x ρ ∂y

Substituting equation 3.31 into equation 3.32, and assuming the mixing length, l is constant yields

()s / l 2 A = (2-35) 8π 3 and the solution for velocity reduction becomes

2 U ∞ ⎛ s ⎞ s ⎛ y ⎞ u1 = ⎜ ⎟ cos⎜2π ⎟ (2-36) 8π 3 ⎝ l ⎠ x ⎝ s ⎠

An example of a plot of Equation 2-36 is presented below. Parameters for computation were as follows: s/d = 3.0, d = 2.0cm, and the mixing length, l, was assumed to be constant where l = 0.4cm. In this plot, piles are located at (0,0) and (0,+/-6).

u /U 1 Inf 10 10

8 8

6 6

4 4

2 2

0 0

-2 -2 -4 -4

Lateral Distance From Pile (cm) -6 -6 -8 -8 -10 -10 1 2 3 4 5 6 Distance Downstream From Pile (cm)

Figure 2-5. Plot of Equation 2-36

Olsson verified that this equation (equation 2-36) is valid for x/s > 4. Of course, most pile groups are spaced at around an x/s ratio less than 4. In fact, all of the pile groups studied for this thesis had values of x/s = 1. The question is whether or not the Schlichting Equations and the

Olsson Equations can be extrapolated to regions with a much smaller x/s ratio.

M.M. Zdravkovich

Work on flow beyond circular cylinders is extensive; over the years, there have been thousands of papers regarding various facets of fluid dynamics in the vicinity of a cylinder.

M.M. Zdravkovich summarized his research in the field of flow around circular cylinders into two volumes: Flow Around Circular Cylinders: Volume I: Fundamentals and Flow Around

Circular Cylinders Volume II: Applications. Each volume comprises over one thousand pages of material concerning several nuances regarding the flow around a circular cylinder. Zdravkovich acknowledges that even these extensive volumes are by no means the “complete collections” of work regarding flow past a circular cylinder. However, Zdravkovich’s volumes are the best and most extensive collection of material concerning flow around circular cylinders found to date.

Summary of previous work in this thesis is limited to studies that could be applied to pile groups. For a more detailed analysis of previous work concerning the broad topic of flow past circular cylinders, refer to Zdravkovich’s volumes.

Two Cylinders

According to Zdravkovich, the motivation for the study of two cylinders spaced closely together is not limited to marine applications. Previous study has been motivated by aeronautical engineering (struts on a biplane), space engineering (twin booster rockets), civil engineers (twin chimney stacks) electrical engineering (transmission lines), and even chemical engineering (pipe racks). Zdravkovich (2003) divides previous work into three classification – experiments

performed on tandem cylinders (one cylinder behind the other), side-by-side arrangements, and staggered arrangements (not covered in this thesis).

Hori’s pressure fields around tandem cylinders

In 1959, Hori measured the mean pressure distribution around two tandem circular cylinders at a Reynolds Number of 8,000 for s/d ratios of 1.2, 2, and 3, where S is the centerline spacing between cylinders and D is the cylinders’ diameters. His results are presented in Figure

2-6.

Figure 2-6. Mean Pressure Distribution Around Tandem Cylinders

Hori’s figure is backwards from what one would normally think. On the first cylinder, the negative pressures are plotted on the axis behind the cylinder while the positive pressures are presented in front of and within the first cylinder. On the second cylinder, the negative pressures are plotted in front of it and the positive pressures are plotted behind it.

Interestingly, Hori’s results show that the pressure distribution around each of the two cylinders is significantly different from one another. Most notably, pressure around the first cylinder is positive in the stagnation region. On the downstream cylinder, the gap pressure is

lower than the base pressure – thus inducing a negative drag coefficient on the second cylinder.

Based on this pressure distribution, one should expect significantly different forcing and drag coefficients on the first and second cylinders. That is, the second cylinder has a significant effect on the forces on the upstream cylinder.

Igarashi’s pressure field around tandem cylinders

In the early 1980’s, Igarashi (1981) conducted extensive pressure field measurements

p0 − p around tandem cylinders. His results are non-dimensionalized such thatC = where p0 p 0.5ρV 2 is the pressure in the free stream, p is the new pressure, ρ is the density of water, and V is the free stream velocity.

Igarashi’s results are presented in Figure 2-7:

Upstream Pile Downstream Pile Figure 2-7. Mean Pressure Distribution for Re = 35,000

Of note in this study: the pressure on the front cylinder in the two-cylinder tandem is different than it would have been with a single cylinder. In other words, the downstream

cylinder induces significant changes on the flow patterns on the upstream cylinder – thus changing the pressure coefficient for various values of s/d.

The variation of pressure coefficient does not seem to follow a regular pattern as s/d is varied. One cannot make generalizations such as “as spacing increases, pressure coefficient on the first cylinder increases” or “as spacing decreases, pressure coefficient on the second cylinder obeys a certain pattern.” Instead, the fluctuations of pressure coefficient behave uniquely for the different spacings studied in his experiments for the given Reynolds Number.

Drag coefficients at lower Reynolds Numbers for tandem cylinders

In 1977, Zdravkovich compiled decades of drag coefficient data from various authors including Pannel et al. (1915), Biermann and Herrnstein (1933), Hori (1959), Counihan (1963),

Wardlaw et al. (1974), Suzuki et al. (1971), Wardlaw and Cooper (1973), Taneda et al. (1973), and Cooper (1974). The single plot is shown in Figure 2-8. Closed symbols represent drag coefficients on the first cylinder, and open symbols represent drag coefficients on the second cylinder.

Figure 2-8. Compilation of Drag Coefficients on Tandem Cylinders

Zdravkovich says that from this plot, a few trends can be determined. First, there is negligible Reynolds Number effect on drag coefficients for the first cylinder in the line.

Secondly, there is a strong Reynolds Number effect on drag coefficients for the second cylinder.

At higher spacing ratios, the drag coefficient on the second cylinder becomes positive.

Drag coefficients for tandem cylinders at higher Reynolds Numbers

In 1977 and 1979, Okajima varied Reynolds Number from 40,000 to 630,000 and measured the variation in drag coefficient on each of the two tandem cylinders. He used S/D =

3.0 & 5.0. Figure 2-9 shows his results. St is the Strouhal Number which is defined as St = fvD/U, where fv is the vortex shedding frequency, D is the pile’s diameter, and U is the free- stream velocity. His results are presented in Figure 2-9.

Figure 2-9. Drag Coefficient and Strouhal Number Variation at Higher Reynolds Numbers

Williamson’s smoke visualizations past side-by-side cylinders

In 1985, Williamson carried out dye visualizations for flow past two side-by-side cylinders at Re=200 and s/d of 1.85. Some of his results are presented in Figure 2-10. As seen, two vortex streets transform into a large-scale eddy street, where the eddies are comprised of one or three separate eddies. Williamson said that the vortex shedding takes place at harmonic modes.

Figure 2-10. Smoke Visualization Downstream from Two Side-by-Side Cylinders, Re = 200. Top Figure has s/d = 6 & Bottom Figure has s/d =4

Hot-wire tests of flow the field behind two side-by-side cylinders

In 1946, Spivack used hot-wires to explore the flow field past two side-by-side cylinders in the range of Reynolds Numbers from 5,000 to 93,000. He used twenty-one different spacings, with s/d varying from 1 to 6.

Spivack found a single vortex-shedding frequency everywhere in the flow field when

Reynolds Number was 28,000 and s/d was between 1.00 and 1.09. From Re=5,000 – 93,000, the

Strouhal Number for flow past a single circular cylinder should level off around 0.2 (Sarpkaya &

Isaacson 1981). Spivack found that if the diameter in the Strouhal Equation was replaced with

2d, the frequency he was observing from 1.00 to 1.09 would level off at 0.2. Furthermore,

Spivack found that a single shedding frequency existed from s/d ranging from 2.0 to 6.0, which also led to a constant Strouhal Number of 0.2.

Spivack discovered that additional frequencies were found outside the wakes, but these frequencies occurred at different positions in the field of flow. Sometimes, these different frequencies would be found simultaneously in the same place. An example is given in Figure 2-

11, which shows unrelated vortex frequencies of 47 Hz and 108 Hz scattered in the flow field.

Zdravkovich says that these two frequencies may be an indication of two different eddy streets, but that their simultaneous occurrence could not be explained. Spivack’s explanation was that there may be two modes of vortex formation at the outside of the cylinders and in the gap between them. Zdravkovich (2003) says that although this explanation is plausible, it is incorrect.

Zdravkovich says that this phenomenon is the embodiment of two seemingly “absurd paradoxes” that are actually taking place. First, the idea that flow around symmetrically arranged cylinders should also be symmetric is not correct. In a certain range of Reynolds

Numbers, the flow was actually asymmetric with narrow and wide wakes separated by the

“biased gap flow.” Secondly, the notion that only stable flow would be possible when fluid is flowing past two symmetric bodies is also incorrect. Two quasi-stable flows can exist, which produces the bistable biased flow, which in turn intermittently switches back and forth in this regime of Reynolds Numbers. This phenomenon was resolved by Ishigai in 1972 and Bearman and Wadcock in 1973 (Zdravkovich 2003).

Flow Direction

Figure 2-11. Diagram Showing the scattered 108 Hz and 47 Hz frequencies past the cylinders

Drag on side-by-side cylinders

On a single cylinder, drag force can be related to the width of the near-wake. Because of the existence of the bistable region between two cylinders, different drag forces are observed for the side-by-side cylinder configurations. Biermann and Herrnstein (1934) measured the drag force on two side-by-side cylinders with a spacing of s/d ranging from 1.0 to 5.0. Reynolds

Numbers ranged from 65,000 from 163,000. Biermann and Herrnstein defined a new term called an interference drag coefficient to account for this. The interference drag coefficient is defined

as the drag coefficient expected for a single cylinder minus the observed drag coefficient when two side-by-side cylinders are involved (Figure 2-12).

Figure 2-12. Interference Coefficient

Biermann and Herrnstein observed that the type of flow downstream from the cylinders changes rapidly based on cylinder spacing and it may even change when spacing is held constant. This was the first clue that there was a bistable flow pattern involved.

Origins of the bistable flow phenomenon

Recall the bistable gap flow phenomenon where two strange paradoxes form. First, an entirely symmetrical oncoming flow into an entirely symmetrical configuration leads to asymmetric narrow and wide wakes behind to identical side-by-side cylinders. Second, uniform and stable flow induces a non-uniform and random bistable flow. The origins of this phenomenon have been explored, but remain unresolved.

In 1972, Ishigai suggested that the Coanda effect is the culprit. The Coanda effect is when a jet attached to a curved surface gets deflected when following the surface. Bearman and

Wadcock (1973) performed a series of experiments to verify the Coanda hypothesis for bistable flow past two side by side cylinders, by using side-by-side flat plates. The biased, bistable flow was still found despite the absence of the curved surface, so the Coanda effect could not be the problem. Bearman and Wadcock suggest that the flow phenomenon could be due to wake interaction instead.

In 1977, Zdravkovich noticed that stable narrow and wide wakes were common for upstream and downstream cylinders in staggered arrangements. As the amount of staggering between cylinders approached zero (so that the cylinders became “side-by-side”), in terms of the wakes, one cylinder remained “upstream.” In other words, one cylinder’s wake remained larger than the other. When the cylinders were completely side-by-side, the asymmetric nature of preserved, but became bistable because neither cylinder was upstream or downstream of the other. According to Zdravkovich, the flow structure consisting of two identical wakes appears to be “intrinsically unstable, and therefore impossible” (Zdravkovich 2003).

Pile Groups

Previous work on pile groups in-line with the fluid flow is more limited than work on two- pile arrangements. However, there have been some studies completed.

Shedding patterns behind three in-line cylinders

For flow behind two in-line cylinders, there are two flow regimes. If s/d is less than a critical s/d value, the shedding behind the upstream cylinder is suppressed by the presence of the downstream cylinder. If s/d is greater than the critical s/d value, both cylinders shed eddies. The critical value for s/d strongly depends on free-stream turbulence (Zdravkovich 2003). A third cylinder placed in-line with the other two cylinders is subject to greater turbulence because of the presence of additional turbulence generated by the second cylinder. For the third cylinder then, the critical value for s/d is expected to be less than the critical s/d for the second in-line cylinder.

An example of this is given in Figure 2-13, which shows that shedding does not take place behind the first cylinder, but it does occur behind the second and third cylinder.

Figure 2-13. Flow Past Three In-Line Cylinders Showing Shedding Behind the Second Cylinder, Re = 2000

Pressure fields behind three in-line cylinders

In 1984, Igarashi and Suzuki studied flow past three in-line cylinders. They observed two types of average pressure coefficient distribution. The first type of distribution, which cylinder 1 always experiences in the spacing configurations studied, involves the stagnation point at zero degrees. The second type involves the reattachment peak between sixty and eighty degrees. The variation of pressure coefficient is presented in Figure 2-14. The dotted line represents what would have happened with two cylinders in tandem, without adding the third cylinder.

Figure 2-14. Pressure Coefficient on Three Cylinders for Different Values of s/d

Igarashi and Suzuki provided smoke visualization to supplement their pressure distribution measurements (Figure 2-15). They also computed the drag coefficients on each of the three cylinders based on pressure distribution (Figure 2-16).

s/d = 1.91 s/d = 2.06

s/d = 3.24 s/d = 3.53

Figure 2-15. Smoke Visualization with Three In-Line Cylinders, Re = 13,000

Figure 2-16. Variation in Drag Coefficient for Three In-Line Cylinders, Re =13,000

Four in-line cylinders

In the early to mid 1980’s, Aiba (1981) and Igarashi (1986) carried out experiments for four cylinders of equal diameter arranged in-line with the fluid flow. Pressure and drag coefficients were measured for each cylinder within the flow. s/d ranged from one to five, and

Reynolds Number ranged from 8,700 to 35,000. Zdravkovich (2003) compiled data from these two studies to show drag and pressure coefficients. Figure 2-17 shows each study’s pressure distributions, Figure 2-18 shows each study’s drag coefficients, and Figure 2-19 shows Igarashi’s accompanying smoke visualization.

Figure 2-17. Pressure Distributions Around Four In-Line Cylinders. (a) is mean and (b) is fluctuating

Figure 2-18. Drag Coefficient Variation for Four In-Line Cylinders. Aiba’s data is on the left and Igarashi’s data is on the right

Figure 2-19. Smoke Visualization past Four Cylinders for Four Different Values of s/d

Square cylinder clusters

In 1982, Pearcey et al. performed experiments on 2x2, 3x3 and 4x4 cylinder matrix arrangements. Reynolds Numbers varied from 40,000 to 80,000 and s/d was fixed at 5.0. Local drag coefficients were evaluated from the local pressure distributions at various angles for the

2x2 matrix. Results show that at a zero skew angle, drag coefficients for the first two cylinders in the matrix are about 0.61. Results for the second cylinders in the line of fluid motion are about the same as well; the cylinder on the left had a drag coefficient of 0.43 and the cylinder on the right had a drag coefficient of 0.44.

For his 3x3 and 4x4 arrangements, Pearcey’s Reynolds Number was fixed at 80,000 and his spacing ratio was 5.0. This study found that the maximum drag coefficient for flow without a skew angle occurred in the first row of cylinders, and the minimum drag coefficient occurred in the third row.

In 1995, Lam and Fang measured the pressure coefficient on cylinders in a square cluster for spacings ranging from 1.26 to 5.8 (where they call the spacing, P) at various skew angles, and a Reynolds Number of 12,800. Local drag coefficients were computed from the measured pressure distribution. To compute forcing on each pile, the local drag coefficient must be combined with the average upstream velocity. Results are presented in Figure 2-20.

Other experiments have been conducted on complex cylinder arrays (Ball and Hall 1980,

Wardlaw 1974), but their spacing ratios were much greater than the spacing ratios typically seen in pile groups. Wardlaw’s spacing ratio was fixed at 10 and Ball and Hall’s ratio was fixed at 8.

Most spacing ratios in pile groups for bridge foundations are between 3 and 5.

Figure 2-20. Drag Coefficient for a 2x2 Matrix

Flow in heat exchangers

Extensive studies have been conducted involving multi-tube arrays of cylinders. A multi- tube array is defined as a series of cylinders that are confined between walls. These multi-tube arrays are often used in heat exchangers.

There are two major differences between flow around heat exchanger tubes and flow around bridge foundation pile groups. First, flow around heat exchanger tubes is confined between two walls while pile group flows have a free surface. Secondly, the spacing of multi- tube arrays is smaller than the spacing seen for most pile groups. Maximum spacing (s/d) between tubes in multi-tube arrays is usually ~ 1.5, whereas the minimum spacing between piles within a pile group is about 3.0. Furthermore, many multi-tube arrays are spaced asymmetrically in the x and y directions. In other words, the longitudinal spacing ratio may be on the order of

1.2 whereas the horizontal spacing ratio may be on the order of 2.0. The pile groups studied in this thesis focused on cylinder arrays with symmetric spacing in the x and y directions.

Only a few experiments with multi-tube arrays could be found where the spacing was close to the spacing found in bridge foundation pile groups. In 1973, Batham measured the pressure field around a 5x9 matrix. His results are presented in Figure 2-21.

Column

Figure 2-21. Pressure Coefficient on a 5x9 Cylinder Matrix (a.) Free Stream Turbulence = 0.5%. (b.) Free stream Turbulence = 20%. S/D = 2.0 for both plots

Batham, like most authors of research involving multi-tube heat exchangers, is mostly interested in the effects of free-stream turbulence on the pressure distribution. The bottom plot is with free stream turbulence of 20% and the top graph is with a free-stream turbulence of 0.5%.

If U = u + u’, where U is the total velocity, u is the steady velocity component and u’ is the fluctuating velocity component, then the free-stream turbulence is defined as the ratio between u’ and u times 100%.

In 1987, Chen and Jendrezejczyk measured the average drag forces on different rows of tubes in a 7x7 matrix. They took the entire column of tubes, and used them to measure the average drag coefficient on that row. “Tube 1” means that these are the average values for the

piles in the first row, “Tube 2” means that these are the average values for piles in the second column, etc. Their results are presented in Figure 2-22.

Figure 2-22. Average Drag and Lift Coefficients for Circular Tubes in a 7x7 Matrix

As evidenced by these plots from Chen and Jendrzejczyk, the drag coefficients are very small at these Reynolds Numbers. For one cylinder, the drag coefficient decreases dramatically when Reynolds Number is greater than 104. It therefore is not unreasonable to assume that the drag coefficient may decrease as well at higher Reynolds Numbers when dealing with multi-tube arrays.

Literature Review Summary

First, work on complex pile arrays is severely limited. There have been very few experiments that have looked at the flow past a cylinder problem from a coastal engineering perspective. Most of the previous work has been completed in other contexts.

Secondly, existing literature has shown that the interference patterns between cylinders create a complex problem that is difficult to solve analytically. Existing work provides a good starting point for better understanding drag forces on piles within a pile group, but no one has yet

used this data to formulate predictive equations that will yield forces on generic pile groups of any configuration. Zdravkovich is the first person to combine the multitude of data for flow past circular cylinders, but even he has not used his anthology to formulate a method that will solve for drag forces on a pile group of arbitrary geometry.

CHAPTER 3 MATERIALS AND METHODS

Schlichting’s Far Wake Theory

Analytic methods were first tried to predict the flow field past a circular pile. The hope was that experiments with pile groups would produce similar results. H. Schlichting (1917) proposed a method to estimate the time average horizontal velocity profile downstream from a pile in a steady flow (see the definition sketch in Figure 3-1).

b y U Approximate ∞ x Extent of Wake u

Figure 3-1. Definition sketch for Schlichting’s 1917 theory for flow downstream from a circular pile (Top View).

The horizontal velocity, u, downstream of the pile should be a function of the distance downstream from the pile, x, the y-position normal to the flow, the original upstream velocity,

U∞, the pile’s drag coefficient, CD, and the wake half-width, b. Equation 3-1 was developed for this flow field (derivation of this formula is found in Chapter 2, Literature Review).

⎛ −1 / 2 3 / 2 2 ⎞ ⎜ u 10 ⎛ x ⎞ ⎡ ⎛ y ⎞ ⎤ ⎟ = ⎜ ⎟ ⎢1− ⎜ ⎟ ⎥ , (3-1) ⎜U 18β C d ⎝ b ⎠ ⎟ ⎝ ∞ ⎝ D ⎠ ⎣⎢ ⎦⎥ ⎠ where u is the new velocity beyond the pile, U∞ is the free stream velocity, x is the distance downstream from the pile, CD is the drag coefficient on the pile, y is the horizontal offset from the

centerline of the pile, b is the wake half-width, d is the pile diameter, and β is a constant (value given in Chapter 2).

The above equation assumes that only one pile is involved. If piles are aligned, it should be possible to extrapolate this theory to multiple piles downstream from one another. Suppose a line of piles is spaced along the x-axis in the above definition sketch. If the assumption is made that the downstream piles do not affect the flow at the upstream piles then Schlichting’s equation could be used to estimate the approach velocity at the second pile. This could be applied to the second pile to obtain the approach velocity for the third pile, etc.

In fact, if one assumes further that only the pile directly upstream from its neighbor causes the velocity in the subsequent pile to fluctuate, one is not limited to aligning piles along the x- axis. In other words, imagine the following configuration:

1 3

U∞ 2 4

Figure 3-2. Definition sketch for scenario of offset piles (Top View).

One could say that pile 2 “experiences” a velocity that is smaller than the velocity

“experienced” by pile 1 (pile 1 “experiences” a velocity of U∞) . In fact, this reduced velocity should be given directly by Schlichting’s equation. Likewise, one could also say that pile 3

“experiences” a velocity that is slightly smaller than the velocity “experienced” by pile 2, and that this reduced velocity should be given by Schlichting’s equations (equation 3.1). This of course assumes that the wakes from pile 1 and pile 2 do not significantly interact and that the piles are far enough apart to allow full development of the wake. It also assumes that the offset between piles 1 and 2 is slight enough so that pile 2 is still within pile 1’s wake zone. Therefore,

with a slight offset in the y-direction, velocity in a configuration similar to the configuration shown in Figure 2-2 can also be determined.

When pile wakes interact, flow in the wake region will be slightly different. Flow velocity will increase as a fluid moves between the two piles, and velocity will decrease directly behind each of the piles in the flow field.

A 1 U ∞ B x

Figure 3-3. Definition Sketch of Side-by-Side Piles (Top View).

Schlichting cites R. Gran Olsson who proposed the following equation for side-by-side piles:

2 U ⎛ s ⎞ s ⎛ y ⎞ u = ∞ ⎜ ⎟ cos⎜2π ⎟ (3-2) 8π 3 ⎝ l ⎠ x ⎝ s ⎠ where s is the centerline spacing between the piles and l is the mixing length. Olsson verified that this equation is valid for x/s > 4, and l/s equals 0.103 for s/d = 8 (Schlichting 1979).

Near Wake Region

These two formulae provide a good starting point for this study. The distance between most piles within a pile group on bridge piers is not large enough for the downstream pile to be in the “far-wake zone.” However, the question is how accurate are these equations in the near- wake region. If experimental studies can show that these equations are accurate to within say

10% then they may be useful in estimating the flow field in the near-wake region. Perhaps they can be modified to improve their accuracy in the near-wake region.

Regardless of whether a combination of Schlichting’s Equation, Olsson’s Equation, and

Morison’s Equation can be used to estimate the drag forces on pile groups, a study of the hydrodynamics in the vicinity of pile groups and the forces on the pile groups from a coastal engineering perspective would be useful. A robust dataset of the velocity at various Reynolds

Numbers for various pile configurations and a dataset of forces on various pile groups at various

Reynolds Numbers would be useful for improving knowledge of flow patterns and forcing within a pile group. The goals of this thesis are to measure the velocity field in the vicinity of various pile groups, measure the forces on various pile groups, and use this dataset to further understanding of fluid flow through a pile group.

Pile Configurations

Four different pile configurations were studied: a single pile, a row of three piles, two piles side-by-side, and a three-by-three matrix of piles. Spacing between piles was fixed at 3d where s is the centerline distance between the piles and d is the pile diameter (Figure 3-4). This spacing is typical of most pile groups on bridge piers (Spacing for typical bridge piers ranges from 3d to

5d).

A C

U∞ U∞

B D

3d 3d

U∞ U∞ 3d

Figure 3-4. Pile arrangements used in this study.

All experiments in this study were conducted in a steady flow. Measurements were made of the velocity field within each pile configuration . Total forces on each of the pile configurations were also made.

Particle Image Velocimetry

Problems with Traditional Measuring Techniques

Traditionally, the most common way to measure water velocity in a flume is to insert a probe and measure the velocity at a point. Several problems are inherent in this method. The first is that as soon as the probe is inserted, it disturbs the flow in the vicinity of the probe.

Downstream of the probe, after flow has fully developed once again, flow returns to its natural state, but in a bridge pier pile group, piles are typically spaced three to five diameters apart.

When measuring the water’s velocity around the pile cluster, flow disturbances cannot be tolerated.

The second problem with measuring flow with a probe is that the probe only measures velocity at one point. One of the goals of this project is to provide a general way to characterize flow downstream from each pile. Therefore, the full velocity field within the pile cluster needs to be understood. To fully capture the velocity field within a pile cluster using a probe, hundreds of measurements would have to be made. The probe would have to be inserted, a measurement made, then the probe moved, another measurement made, etc. Multiple probes would be impossible because of the flow disturbances caused by the probes’ insertion into the flow. Using one probe would take too long and it would be arduous. It would be almost impossible to complete a series of measurements in one sitting, and it would be difficult to recreate the conditions within the test flume (water height, water velocity, temperature within the lab, etc.) exactly for each series of measurements. It would be better if there was a method that would capture the entire flow field in one series of measurements.

Particle Image Velocimetry

Particle Image Velocimetry (PIV) allows a user to measure flow throughout a flow field at specified intervals in time. The technology for PIV has progressed steadily over the past fifteen years with advances in optics, lasers, electronics, video, and computer processing power.

According to Raffel (1998), PIV’s ability to measure an entire velocity field is unique. Except for Doppler global Velocimetry (DGV), which is only appropriate for high-speed airflows, all other techniques for velocity measurement only allow measurement at a single point.

PIV works by the user adding tiny reflective particles to the fluid and using a high-speed digital camera to capture the fluid flow. These particles must be illuminated in a plane at least twice within a short time interval. Each time light hits the fluid, and subsequently the particles, the particles reflect light. The idea is to figure out where the particles moved from one frame to the next. If the change in position of the particles and the time lapse between frames are known, the velocity can be determined for each image-pair. The sum of these individual velocities yields the complete velocity flow field.

A typical PIV setup looks like the following (Figure 3-5):

Figure 3-5. Typical PIV setup schematic drawing (Oshkai 2007).

For data analysis, the PIV images are divided into pairs. In other words, image 1 is compared with image 2, image 3 is compared with image 4, etc. Computers are not powerful enough yet to compare image 1 to image 2, image 2 to image 3, image 3 to image 4, etc. Each image is divided into small sub-areas called interrogation areas. Local displacement vectors for the image-pairs are determined using a statistical cross correlation. It is assumed that all particles in one interrogation area have moved homogeneously between the two illuminations.

Therefore if the time delay between light bursts is known, the velocity field can be computed.

This interrogation technique is repeated for all interrogation areas and for all image pairs (Raffel

1998).

Figure 3-6. Typical PIV image pair

Turner Fairbanks Highway Research Center PIV Flume

The Turner Fairbanks Highway Research Center (TFHRC) in McLean, VA has a PIV flume. The TFHRC setup resembles a typical PIV (Figure 3-7):

Figure 3-7. TFHRC PIV Flume in McLean, VA

Setup

The Plexiglas flume measures twelve inches across by twelve inches deep by sixteen feet long, and is connected to a Saftronics Rapidpak Converter Pump. A flap-gate is installed at the downstream end of the flume to control the water level. Upstream, an air conditioning filter mat, a honeycomb apparatus, and a trumpet entrance section are installed to ensure that a uniform flow exists in the flume. A depth sensor is installed upstream of the piles to further monitor water elevation and to ensure that flow depth is constant during the test.

Figure 3-8. Photograph of “trumpet” used to ensure uniform flow

A high-powered New Wave Research SoloPIV laser is installed to provide the light source for the experiments. A Megaplus Model ES 1.0 high-speed digital camera is used to capture the image snapshots.

Figure 3-9. SoloPIV laser

Figure 3-10. MegaPlus Camera used in the Experiments

Because the goal was to measure the flow in a horizontal plane through the water, and the cameras were mounted on the side of the PIV tank, a mirror was used so that the camera could capture an image looking “up” through the water column (Figure 3-11).

Figure 3-11. Photo of camera-mirror setup

Conduct-O-Fill spherical particles are added to the water to reflect the light during the experiments.

A Plexiglas top-rig was attached to the PIV tank so that piles could be inserted into the flow field. The top-rig apparatus consists of two Plexiglas plates with holes drilled at the appropriate spacing. This plate-on-plate design, combined with tight-fitting holes through the

Plexiglas, minimizes the amount of wobble that the piles will experience as the flow moves past them. Three-quarter inch Plexiglas piles are inserted into the rig in the desired configuration for each test (Figure 3-12). This pile configuration was checked to ensure that maximum blockage

recommendations for experiments in a flume were satisfied. Blockage within the flume is less than twenty percent – the general rule of thumb for flume-based experiments.

Figure 3-12. PIV rig setup used during experiments

Figure 3-13. PIV with laser on piles

PIV measurement

After the appropriate pile group had been installed, the test was initiated. All light in the lab was turned off to make sure that the particles in the water were only reflecting the laser’s

light. The camera’s time delay was adjusted so that errors in image pairs would be minimized.

This adjustment process was trial and error. For example, a time delay of 10ms might be tried for a certain configuration. A test correlation would be run, and if it produced nonsensical results, the time delay was adjusted until the errors were minimized. Nonsensical results mean that the velocity vectors produced by the test correlation displayed no discernable pattern.

Obviously velocity should look uniform upstream and downstream from the pile group. Within the cluster, velocity should still look relatively uniform with minor fluctuations (Figure 3-14).

After the appropriate time delay had been determined, the flow was measured for sixty seconds.

Figure 3-14. PIV output with the correct time delay. The large arrows represent “errors.” Time delay adjustments are completed until the number of errors is small.

Potential errors in PIV measurements

The idea behind adjusting the time delay is to minimize the number of large arrows, or

“error arrows” as seen in Figure 3-14. The reason the errors occur is because of the computation method used by the PIV software. The PIV divides the measurement area into boxes – each box measuring sixty-four pixels by sixty-four pixels, and the boxes overlap one another by seventy- five percent. Then, the cross-correlation program looks for the brightest pixel in each box in the first image and the brightest pixel in each box in the second image. Suppose the brightest pixel is on the edge of the box in the first image. It could potentially move out of the frame bound by the box, in the amount of time it took the laser to flash from the first to the second image. When the PIV looks for the brightest pixel in the second image then, it is looking at a different pixel – hence, the potential for large errors. If the time delay is correct, the pixel movements will be small enough from image to image so as to minimize the number of times that these errors occur.

The obvious question is, why not make the time delay as small as possible to remove all errors? The answer again is due to the PIV method. Each pixel on the screen represents a certain number of millimeters. A typical ratio between pixels and millimeters would be 247 pixels = 960mm. As far as the computer “knows,” for a particle to “move” at all, it needs then to be displaced at least 0.257mm (247/960). If this doesn’t happen, it will put the brightest pixel in the second image in the same location as the brightest pixel from the first image – therefore, no displacement is tracked, which also is obviously an error, referred to as a “small error.”

The goal then, is to get the largest time delay possible which minimizes the number of large errors, so that the number of small errors can also be minimized. Presently, there is no quantitative minimization technique used to determine when the number of small and large errors is at a minimum. Instead, visual inspection based on experience is used. Generally, an average pixel displacement of 4.00 – 6.00 pixels per image pair is a “good” time delay.

Verification with ADV Probe

To ensure that the PIV was recording the correct velocities, a Son-Tek Micro ADV velocity probe was inserted into the water to determine the velocity in the free stream. Results from the ADV readings were compared with a PIV reading with no piles. Both the PIV and the

ADV read at the same frequency – 15Hz. The ADV and the PIV are relatively close, with the error from the PIV compared to the ADV between 9.5% and 11% (Figure 3-15, Figure 3-16,

Figure 3-17, and Figure 3-18). The diameter of the ADV probe is small compared with the diameter of the piles; the ADV probe’s diameter was approximately 20% the diameter of the piles used in these experiments. Although the ADV probe did create a slight wake, the probe was placed far enough upstream from the piles so that all wake effects were sufficiently dissipated by the time the water had reached the piles. Velocity (cm/s)

First Velocity Second Velocity Third Velocity 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 16 18 20 Time (sec)

Figure 3-15. ADV Probe Measurements

Figure 3-16. Average Velocity Image in PIV at First Velocity

Figure 3-17. Average Velocity Image in PIV at Second Velocity

Figure 3-18. Average Velocity Image in PIV at Third Velocity

PIV Data Analysis

Data analysis with data obtained from the TFHRC PIV is complex. After the images have been taken, the piles and the sides of the tank are blacked out, or “masked,” to prevent the cross- correlation algorithm from calculating any velocity in these regions (Figure 3-19).

Figure 3-19. Masked PIV Image (9 Piles)

The cross-correlation is then run. After the correlation is run, a series of images are produced that look like this:

Figure 3-20. Example of a Correlation Image (no piles)

After the correlation is run, the images are checked for errors. First, the “large errors” are eliminated by checking each image for the case where one velocity vector is significantly larger than the surrounding velocity vectors. If this happens, the large vector is probably an error, and it is eliminated. Second, the images are checked for the case where one velocity vector points in one direction and all its neighbors are pointing another direction. If this happens, the velocity vector that is pointing in the wrong direction is eliminated. Next, any gaps in the velocity field are filled with interpolated values based on the neighboring vectors’ values. Finally, the entire velocity field is smoothed using a smoothing algorithm.

After the images have been checked for errors, they are averaged. Then, colors can be assigned to each velocity magnitude, and a contour map of velocity at each image-frame-step can be constructed. These images can be used to make a movie of the water velocity during the entire time-domain. Of particular interest to this project is the average velocity image

(Figure 3-21). After the velocity images have been completed, a black mask, which represents the pile, is inserted into each image to visualize the pile location.

Figure 3-21. Average Velocity Image Example.

Force Measurements

The two-dimensional force balance used in this investigation consisted of a metal carriage that moves freely laterally in the x-direction and vertically in the z-direction. Movement of the force balance is tracked via strain gauges that measure a voltage change based on the amount of displacement, and a linear relationship between strain and voltage is assumed.

TFHRC Force Balance Setup

The TFHRC is equipped with a custom-built force balance that was used to measure the forces directly on each of the pile configurations. The force balance is connected to an ELEKT-

AT Deck Force Analyzer DF2D amplifier and the amplifier is connected to a computer so that the computer can signal when the force balance should begin its measurement. The force-

balance is installed in a lift mechanism so that the piles can be lowered into position just above the flume bottom (Figure 3-22).

Figure 3-22. TFHRC Force Balance Setup

The force balance’s lift mechanism is attached to the side of the flume via an ELEKT-AT

LMP Control Unit. The control unit allows the lift-mechanism to move transversely along the tank, although for these experiments, the piles were fixed in the x-direction (the direction along the flume).

The Plexiglas Force Balance Flume is fourteen inches wide by twenty inches deep by twenty-seven feet long. A computer-controlled flapgate is installed at the downstream end of the flume to accurately control water level (Figure 3-23). In the upstream portion of the flume a trumpet entrance section, air conditioning filter fabric, and a honeycomb element are installed to ensure uniform flow. Water level in the force balance experiments was varied to ensure that the

amount of force on each pile configuration is large enough to be accurately read by the force balance.

Figure 3-23. Flapgate Setup in Force Balance Flume

Figure 3-24. Trumpet Setup in Force Balance Flume

A computer-controlled SonTek MICRO-ADV Robot is attached to the flume upstream of force apparatus (Figure 3-25).

Figure 3-25. SonTek MICRO-ADV Robot

This robot, like the ADV probe in the PIV flume, can be lowered into the water to verify the fluid velocity because the velocity used for the calculations and measured while the force is being measured is given by a Venturi flow meter. The robot is removed before forces are measured on the piles to ensure that flow approaching the piles is undisturbed.

Two ultrasonic depth gauges are installed in the Force Balance Flume – one upstream and one downstream from the force balance. They work in conjunction with the flapgate and the computer-controlled pump, to monitor depth within the flume to ensure that depth remains constant during the test. A top-rig mechanism, which is similar to the rig used in the PIV Flume, is attached to the force balance, and the same piles used in the PIV experiments were used for the force measurements.

Figure 3-26. View of force balance, piles, and flume looking downstream.

Measurements in the Force Balance Flume

Piles were set up in the appropriate configuration, and water was run through the force balance flume at several velocities continuously until the desired water level had stabilized.

Once the water level was stable, a sixty-second measurement was taken. Each experiment was repeated three times to ensure repeatability.

The force balance only measures the force on the pile group. Attempts were made to use the force balance to isolate the forces on each pile by adding piles one-by-one. For example, to determine the force on the third pile in the three-pile-in-a-line arrangement, force was first measured with just one pile, then two piles in a line, and finally on the third pile in a line.

Because total force is known in each instance, one only needs to subtract one result from the other to find the new pile’s individual contribution to total force.

The assumption behind this line of thinking is that the downstream pile does not affect the forces on the upstream pile. Upon analyzing the results from experiments in the Force Balance and PIV flumes, and reviewing more literature, it became clear that this assumption was invalid.

In other words, the velocity field downstream from a pile standing by itself in the free-stream is different than the velocity field downstream from the same pile if another pile is placed in the first pile’s wake. As a result, the Force Balance dataset obtained in this study can only be used for the pile configurations considered.

Although PIV measurements were only made at three Reynolds Numbers, measurements in the Force Balance Flume were made over a wide range of Reynolds Numbers. Capturing data in the PIV is time consuming – each dataset takes about 40 hours to get from the “capture” point to the final product because of post-processing time. Measurements in the force balance flume are much easier – one simply has to adjust the velocity, wait a couple of minutes, and a new reading is ready. Therefore, the number of PIV measurements was limited, whereas force measurement readings were made over a range of Reynolds Numbers.

Methods Summary

The purpose of this project is to measure the velocity field in the vicinity of a pile group and to determine the hydrodynamic drag force on piles within the pile group.

The PIV system was used to measure the velocity field in each pile group.

The force balance was used to measure the forces on the piles

CHAPTER 4 EXPERIMENTAL RESULTS

PIV Data

Velocity Fields

PIV average velocity color contour plots are shown in Figures 4-1 through 4-12 for the pile configurations and Reynolds Numbers investigated in this study.

Velocity Profiles from PIV data

To supplement the contour plots, horizontal velocity profiles normal to the approach flow were extracted and presented in Figure 4-13 through Figure 4-42. This data series shows the velocity profiles at 20mm spacings from the center of the first pile.

Vorticity Data

From fluid mechanics, vorticity is defined as the curl of the velocity vector. In other words: ω = ∇ ×U , where U is the velocity vector defined as U = ui + vj + wk and ω is the vorticity.

The velocity data was used to compute the vorticity for each image frame, and contour plots for vorticity are plotted for each frame. These frames are compiled into a series of animations that show how the vortices move downstream from each of the pile configurations.

Animations were limited to the first fifteen seconds of data to cut down on file size, but if necessary, animations could be made of the entire data series. One-minute time average images of vorticity magnitude are presented in Figure 4-43 through Figure 4-54.

Strouhal Number Comparison

Animations for one pile were used to verify the Strouhal Number that one should expect for the given datasets. Strouhal Number is defined as:

fd St = (4-1) U ∞ where f is the frequency of the vortex shedding, d is the pile diameter, and U is the upstream velocity. From published data, the Strouhal number versus Reynolds number plot should look like that shown in Figure 4-43. Given these values of Strouhal number for the range of Reynolds numbers in the PIV experiments, one can estimate what the vortex frequency should be. This can be compared with the actual shedding frequency seen in the animations. The comparison is shown in Figure 4-56.

Figure 4-1. Re = 5.13x103, 1 Pile Average Velocity Image

Figure 4-2. Re = 3.85x103, 1 Pile Average Velocity

Figure 4-3. Re = 2.57x103, 1 Pile Average Velocity

Figure 4-4. Re = 5.13x103, 2 Piles Average Velocity

Figure 4-5. Re = 3.85x103, 2 Piles Average Velocity

Figure 4-6. Re = 2.57x103, 2 Piles Average Velocity

Figure 4-7. Re = 5.13x103, 3 Piles Average Velocity

Figure 4-8. Re = 3.85x103, 3 Piles Average Velocity

Figure 4-9. Re = 2.57x103, 3 Piles Average Velocity

Figure 4-10. Re = 5.13x103, 9 Piles Average Velocity

Figure 4-11. Re = 3.85x103, 9 Piles Average Velocity

Figure 4-12. Re = 2.57x103, 9 Piles Average Velocity

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 30 35 V (cm/sec)

3 Figure 4-13. Velocity Profiles through the Center of the First Pile for Re = 5.13x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 30 35 40 V (cm/sec)

Figure 4-14. Velocity Profiles 20mm from the Center of the First Piles for Re = 5.13x103

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 30 35 40 V (cm/sec)

3 Figure 4-15. Velocity Profiles 40mm from the Center of the First Pile for Re = 5.13x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 30 35 V (cm/sec)

3 Figure 4-16. Velocity Profiles 60mm from the Center of the First Pile for Re = 5.13x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 30 35 V (cm/sec)

3 Figure 4-17. Velocity Profiles 80mm from the Center of the First Pile for Re = 5.13x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 30 35 V (cm/sec)

3 Figure 4-18. Velocity Profiles 100mm from the Center of the First Pile for Re = 5.13x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 30 35 V (cm/sec)

Figure 4-19. Velocity Profiles 120mm from the Center of the First Pile for Re = 5.13x103

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 30 35 V (cm/sec)

3 Figure 4-20. Velocity Profiles 140mm from the Center of the First Pile for Re = 5.13x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 30 35 V (cm/sec)

3 Figure 4-21. Velocity Profiles 160mm from the Center of the First Pile for Re = 5.13x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 30 35 V (cm/sec)

3 Figure 4-22. Velocity Profiles 180mm from the Center of the First Pile for Re = 5.13x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 V (cm/sec)

3 Figure 4-23. Velocity Profiles through the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 30 V (cm/sec)

3 Figure 4-24. Velocity Profiles 20mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 30 V (cm/sec)

3 Figure 4-25. Velocity Profiles 40mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 30 V (cm/sec)

3 Figure 4-26. Velocity Profiles 60mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 V (cm/sec)

3 Figure 4-27. Velocity Profiles 80mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 V (cm/sec)

3 Figure 4-28. Velocity Profiles 100mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -5 0 5 10 15 20 25 V (cm/sec)

3 Figure 4-29. Velocity Profiles 120mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 V (cm/sec)

3 Figure 4-30. Velocity Profiles 140mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 V (cm/sec)

3 Figure 4-31. Velocity Profiles 160mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 5 10 15 20 25 V (cm/sec)

3 Figure 4-32. Velocity Profiles 180mm from the Center of the First Pile for Re = 3.85x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -20 2 4 6 8 10121416 V (cm/sec)

3 Figure 4-33. Velocity Profiles through the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -4 -2 0 2 4 6 8 10 12 14 16 18 V (cm/sec)

3 Figure 4-34. Velocity Profiles 20mm from the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -2 0 2 4 6 8 10 12 14 16 18 V (cm/sec)

3 Figure 4-35. Velocity Profiles 40mm from the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -2 0 2 4 6 8 10 12 14 16 18 V (cm/sec)

3 Figure 4-36. Velocity Profiles 60mm from the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 V (cm/sec)

3 Figure 4-37. Velocity Profiles 80mm from the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 V (cm/sec)

3 Figure 4-38. Velocity Profiles 100mm from the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 -20 2 4 6 8 10121416 V (cm/sec)

3 Figure 4-39. Velocity Profiles 120mm from the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 V (cm/sec)

3 Figure 4-40. Velocity Profiles 140mm from the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 2 4 6 8 10 12 14 16 V (cm/sec)

3 Figure 4-41. Velocity Profiles 160mm from the Center of the First Pile for Re = 2.57x10

1_cyl 3_cyl 9_cyl Y (mm) 0 50 100 150 200 250 0 2 4 6 8 10121416 V (cm/sec)

3 Figure 4-42. Velocity Profiles 180mm from the Center of the First Pile for Re = 2.57x10

Figure 4-43. Re = 5.13x103 Average Vorticity for One Pile

Figure 4-44. Re = 3.85x103 Average Vorticity for One Pile

Figure 4-45. Re = 2.57x103 Average Vorticity for One Pile

Figure 4-46. Re = 5.13x103 Average Vorticity for Two Piles

Figure 4-47. Re = 3.85x103 Average Vorticity for Two Piles

Figure 4-48. Re = 2.57x103 Average Vorticity for Two Piles

Figure 4-49. Re = 5.13x103 Average Vorticity for Three Piles

Figure 4-50. Re = 3.85x103 Average Vorticity for Three Piles

Figure 4-51. Re = 2.57x103 Average Vorticity for Three Piles

Figure 4-52. Re = 5.13x103 Average Vorticity for Nine Piles

Figure 4-53. Re = 3.85x103 Average Vorticity for Nine Piles

Figure 4-54. Re = 2.57x103 Average Vorticity for Nine Piles

Figure 4-55. Published Strouhal Number Data (Sarpkaya 1981)

Measured (1/s) Frequency 012345 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Theoretical Frequency (1/s)

Figure 4-56. Strouhal Number Data from PIV dataset.

Measured Force Data

Results from the force balance measurements are presented in this section. Force balance measurements were for Reynolds numbers from 4.00x103 to 1.10x104.

One Pile

The first goal in measuring forces on piles is to compare these measurements with values from previous researchers. In the range of Reynolds Numbers used in these experiments, the drag coefficient should be about 1.0. For the range of Reynolds Numbers investigated in this study, the drag coefficient is approximately constant. Therefore the slope of the best-fit line of a

2 plot of Fx/h vs. ρdV , will equal the drag coefficient. This plot is shown in Figure 4-57.

Data y = x Linear Data Fit

/h y = 0.9783x x F R2 = 0.9352 0 0.5 1 1.5 2 2.5 3 3.5 00.511.522.533.5 0.5ρdV2

Figure 4-57. Drag Coefficient for 1 Pile

As can be seen in Figure 4-57, the slope of the best-fit line is approximately equal to 1.0.

This indicates that the force balance is accurately measuring forces in this range of Reynolds

Numbers.

Aligned Piles

After the one-pile experiment was run, piles were added behind the first pile to see what their effect would be on the total force on the pile group. First, a second pile was added behind the first pile, and then a third. Forces on the pile groups are presented in Figure 4-58. The yellow line represents the force on one pile, the red line the force on the two in-line piles, and the green line the force on the three in-line pile group.

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1 Pile 2 Aligned Piles 3 Aligned Piles Lowest PIV Reading Highest PIV Reading /h (System) x F 0.00 1.00 2.00 3.00 4.00 5.00 2.00E+03 4.00E+03 6.00E+03 8.00E+03 1.00E+04 1.20E+04 Re

Figure 4-58. Results for One Row of Piles

Side-by-Side Piles

An experiment was run on a two side-by-side pile arrangement to determine how flow between the two piles affected forcing on the pile group. The results are presented in Figure 4-

59:

1 Pile 2 Piles Side-by-Side Lowest PIV Reading Highest PIV Reading /h (System) x F 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 2.00E+03 4.00E+03 6.00E+03 8.00E+03 1.00E+04 1.20E+04 Re

Figure 4-59. Results for Two Side-by-Side Piles.

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Pile Group Configurations

Attempts were made to determine the force on individual piles in the pile group using the force balance. Therefore, a series of more complex pile arrangements were studied with the hope that subtracting the results of one experiment from another experiment would provide forces on an individual pile. The assumption with this method is that adding downstream piles to the group does not change the forcing on the upstream piles. Further analysis of PIV data, data from other researchers and analysis of results from this series of experiments shows that this assumption is not valid. That is, adding downstream piles changes the pressure distribution and flow around the upstream piles. Unfortunately then, it is not possible to determine the force on individual piles within the group from the data obtained in this study. The total force on the pile group configurations investigated are, however, valid. The configurations tested are shown in

Figure 4-60a and 4-60b..

U∞ U∞

(1.) (2.)

Figure 4-60a. First Two Complex Configurations Run in the Force Balance

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U∞ U∞

(3.) (4.)

Figure 4-60b. Second Two Complex Configurations Run in the Force Balance Flume

The total force on these pile group arrangements are presented in Figure 4-61. The purple line corresponds to the four-pile arrangement, the grey line to the six-pile arrangement, the blue line to the seven-pile arrangement, and the orange line to the nine-pile arrangement.

4 piles Fx/h (System) 6 piles 7 piles 9 piles Lowest PIV Reading Highest PIV Reading 0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 2.00E+03 4.00E+03 6.00E+03 8.00E+03 1.00E+04 1.20E+04 Re

Figure 4-61. Results for Complex Pile Arrangements

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Results Summary

PIV data was used to create the following:

• One-minute average velocity fields

• One minute average velocity profiles normal to the approach flow

• One minute average vorticity magnitude plots

• Animations of fifteen second averages of vorticity

• A comparison between observed vortex shedding frequency and that published in

the literature

Force Balance data was used to obtain total forces on the following pile arrangements:

• One pile

• Two in-line piles

• Three in-line piles

• Two side-by-side piles

• A series of more complex pile arrangements

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CHAPTER 5 DISCUSSION

PIV Data Analysis

Recall that the original motivation for this research was to find a way to use an expression similar to the expressions given by Schlichting and Olsson to find a way to reduce velocity in the near-wake region. The hope was that because drag force on a pile is given by the following equation,

1 F = ρC AU |U | (5-1) D 2 D accurate velocity measurements would yield a better way the approach velocities at downstream piles. The presumption was that if piles are aligned, the first pile will reduce velocity downstream from it. This reduced velocity, u’, would be the velocity to use in the drag equation to predict the force on the second pile. The second pile would reduce velocity even further in the wake region past it, and this would cause the third pile in the alignment to use the second reduced velocity, u’’, in its drag computation. This pattern would persist along the entire line of piles.

This method assumes that the pressure field around the first pile in the alignment and subsequent piles in the alignment are similar. Therefore, a similar drag coefficient could be used for each pile in the alignment – the standard drag coefficient graph that has been validated for decades!

Unfortunately, after running the PIV experiments, and looking at existing data, it is clear that this entire line of thinking does not work. Using a “reduced velocity” method for predicting

105

forces on downstream piles does not work because the downstream pile induces changes in the velocity field at the upstream piles.

Average Velocity Field Measurements

One pile arrangement

The one pile PIV arrangement behaved as expected. The presence of the pile induced a wake downstream from the pile. As the wake-region propagated downstream, the velocity in the wake region increased. PIV measurement windows were limited to 247mm by 247mm, and within this region, the downstream fluid velocity did not completely return to approach velocity, but if the window was increased, it is likely that in the far-wake of the one-pile arrangement, velocity would eventually return to the upstream value.

In the one-pile arrangement, the wake region is largest at lower Reynolds Numbers and smallest at higher Reynolds Numbers. At the highest Reynolds Number, the wake extended about 40mm (about 2 diameters) downstream from the pile, and at the lowest region, the extent of the wake was almost 60mm (about 3 diameters) downstream from the pile.

Recall that pile spacing was fixed at 3d where d is the diameter. Also recall that three- quarter inch plexiglass piles were used in the experiment. If another pile is placed 3d from the first pile, it would lie directly in the wake in all cases. In the best-case scenario – where the wake is the smallest (at the highest Reynolds Numbers) the front face of the pile still penetrates the first pile’s wake. In the worst case (small Reynolds Numbers) the entire pile would lie within the first pile’s wake.

At higher Reynolds Numbers, it seems that the front face of a hypothetical second pile would experience a negative velocity if the spacing was fixed at 3d. At lower Reynolds

Numbers, it is likely that the entire hypothetical second pile would experience a negative

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velocity. A pile in a negative velocity field likely has a much different pressure gradient surrounding it than a pile in a positive velocity field.

Velocities within the dead region are similar for all Reynolds Numbers. At lower

Reynolds Numbers, the negative velocity within the wake is not “more negative” than they would be at higher Reynolds Numbers.

Three pile arrangement

The three-pile arrangement demonstrates that the situation is even more different than originally anticipated from the one-pile measurements. With the three pile arrangement, the wake from the first pile seems to encompass the second pile for all Reynolds Numbers. In other words, the second pile induces changes in the wake from the wake that would have existed had the second pile not been present.

The wake behind the second pile is very small for the range of Reynolds Numbers considered. Wake behind the second pile does not seem to be a function of upstream velocity because the extent of the wake behind the second pile is similar for each Reynolds Number.

Velocities leading into the third pile are similarly reduced at each Reynolds Number.

Recall that what was expected was that the force on the first pile in the arrangement would have some value. The force then on the second pile in the arrangement would have some value that was lower than the force on the first pile. The force on the third pile would be even less, and so-on. This force reduction would be explained by the reduction in velocity.

The three-pile arrangement shows that this is not the case. The force on the first pile does, indeed have some value. The force on the second pile is clearly lower, but what was not expected is that the PIV measurements seem to indicate that the force on the second pile is actually negative because the velocity field surrounding the second pile is negative. The force on

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the third pile goes back to a positive force, but, which from these velocity measurements appears to be less than the force on the first pile, but still greater than the force on the second pile.

Additionally, the wake behind the third pile is almost non-existent at all Reynolds

Numbers. It is almost as if the wake beyond the first pile engulfs the second pile and the third completely. The wake beyond the first pile looks different than it would if the other two piles were not there, but there does not appear to be three independent sets of wakes as was originally anticipated. Instead, it is as if the wake from the first pile extends itself to beyond the third pile because of the presence of the two extra piles.

Nine pile arrangement

The nine pile arrangement reinforces the observations made with the three pile arrangement. In the nine pile configuration, if one only looked at the wake regions, it would appear that the three rows of piles act almost independent of each other. The wake from the top three piles in the PIV image seem to act independently from those in the second and bottom rows of the PIV image. The wake behind the first pile in each row encompasses the entire second pile, and the second pile seems to be experiencing a negative flow velocity. At the third pile in each row, the flow returns to a positive value –indicating a positive force on the third pile.

A closer look at the nine pile arrangement does, however, reveal that there is some flow interaction between the pile rows. This is due to the flow being contracted as it passes through the space between the piles. Fluid velocity in these regions is higher than the free stream velocity.

The wake regions in the nine pile arrangement exhibit almost the same patterns as the wake regions in the three in-line pile arrangement in that it almost looks like the wake from the front pile is just extending itself further downstream because of the presence of the other two

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piles. The nine-pile configuration seems to confirm the fact that the three piles do not exhibit independent wake characteristics.

Two Pile Arrangement

The two-pile arrangement seems to support the hypothesis that the wakes from the rows of piles in the nine-pile configuration act independently from one another. The piles’ wakes in the two-pile arrangement appear to interfere from the front face of the piles to about 90mm downstream from the piles for all three Reynolds Numbers. In this region, velocity is significantly higher than the free-stream velocity – as was seen between the piles in the nine-pile configuration. After this 90mm zone, the wakes seem to merge to form a single wake.

Demorphing PIV Data

There is a potential source of error in the PIV data that could show up with the nine pile configuration. Recall that the method in which data was collected with the PIV involved a camera snapshot bounced off of a mirror. The camera lens is curved, so light leaves the camera as shown in Figure 5-1.

As the light leaves the camera, it scatters outward according to the curvature of the lens.

When the light hits the mirror, it scatters again, and skews the location of the light even further.

In PIV jargon, this phenomenon is known as “morphing.”

The possibility for morphing in these experiments was examined, and determined to not be a significant source of error in these experiments. The evidence for this comes from the velocity fields obtained without any piles (Figures 5-2 – 5-4).

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Camera Light Sheet

Mirror

Figure 5-1. Schematic Drawing of Light Bouncing Off Mirror

Figure 5-2. Average Velocity Image in PIV at First Velocity

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Figure 5-3. Average Velocity Image in PIV at Second Velocity

Figure 5-4. Average Velocity Image in PIV at Third Velocity

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Because the tank in which these experiments were run had relatively parallel side walls, the velocity field across the horizontal cross section of the tank should be relatively uniform. If morphing was an issue, the velocities at the top of the image would be significantly different than the velocities at the bottom of the image. As can be seen from these figures, the velocity at different horizontal cuts across the image do not vary significantly. At most, the velocities at the top of the image vary from the bottom velocity by 0.2cm/sec. At worst, at the lowest Reynolds

Number, this represents a possible error of 1.8%.

Velocity Profile Measurements

Velocity profile measurements back up the assessment from the PIV average velocity images. At first glance at the average velocity measurements, one still might hold out hope at finding a function that would describe the average velocities within the wake regions. This would not, however, solve the problem associated with the impact of the downstream pile on the flow field around the upstream pile (and therefore the drag force on the upstream pile). As more piles are added to the row the wake region is simply extended further downstream.

Vorticity Measurements

Vorticity was computed from the velocity field data and the results were used to validate observed Strouhal Numbers against published vortex shedding frequency results. Observed shedding frequencies in these tests were consistently higher than published data, but only slightly. At higher Reynolds Numbers, the observed vortex shedding frequency was much closer to published data. Sarpkaya and Issaccson (1981) say that at lower Reynolds Numbers, Strouhal

Number varies within a wider range (Figure 5-5). Computations for observed vs. published frequencies assumed a Strouhal Number of 0.2, but using a value slightly different than 0.2 would allow the observed shedding frequencies to fall directly on the y=x line.

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Figure 5-5. Strouhal Number vs. Reynolds Number from Sarpkaya and Issacsson (1981)

Force Balance Data Analysis

One Pile Arrangement

The one pile force balance experiment indicates that the force balance is measuring the correct forces because the drag coefficient measured with the force balance is approximately equal to published values. Previous studies have shown that for the range of Reynolds Number in this study, the drag coefficient is approximately 1.0. The measured drag coefficient with the force balance was approximately 0.97 – a 3% difference from expected results.

Two Side by Side Pile Arrangement

Drag forces on the two side by side pile arrangement seem to indicate that the PIV hypothesis that stipulates that piles spaced at 3 diameters apart behave independently from one another is not 100% correct. The force on the two-pile arrangement is slightly greater than

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double that of the force on a single pile. If one assumes that the force is evenly distributed between the two piles, one would back-calculate a drag coefficient for each pile in the two-pile configuration of 1.13. This is slightly higher than the expected drag coefficient of 1.0, but not much higher.

This seems to indicate that the piles act almost independently from one another, but there is probably some interaction between them which caused this 13% jump in the drag coefficient.

This is most likely due to the higher velocities between the piles. The inner edges of each of these two piles experiences a different velocity than the velocity on the edge of a single pile.

This probably changes the pressure field around each of the piles in the two-pile configuration, thereby causing a slight increase in the drag force.

Three In-Line Pile Arrangement

As expected, the force on the three-pile arrangement is not triple the force on a single pile. Based on PIV data, it would not be unreasonable to assume that the first pile in the line takes the brunt of the force, but it is impossible to say how much of the force impacts the first pile because as the PIV data shows, the second pile induces changes in the wake regions of the first pile. Attempts were made to subtract a single pile force from the two pile total force, and to subtract the two-pile force from the three pile force. This method would provide a way of isolating the forces on the individual piles. Again, this assumes that the downstream piles do not induce a pressure field change in the upstream piles, which the PIV data shows is not the case.

However, when this analysis was completed, the results do agree with the PIV data. As shown in Figure 5-6, if one employs the subtraction method, one will find that the computed force on the second pile in the row is less than the computed force on the third pile. Although counter-intuitive, it does agree with the PIV data, which indicates that the force on the second pile is almost zero.

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System 1st Pile 2nd Pile 3rd Pile

System Avg 1st Pile Avg 2nd Pile Avg 3rd Pile Avg /h x F 012345 2.00E+03 4.00E+03 6.00E+03 8.00E+03 1.00E+04 1.20E+04 Re

Figure 5-6. Deduced Drag Forces Based on Measurements on a Three Inline Pile Arrangement

The problem is that much of the PIV data would also indicate that the force on the second pile should be negative, and computations using the force balance subtraction method do not show this. It is possible and even probable that 1) the force on the second pile is negative and 2) that the force on the first pile is increased by the presence of the second pile.

Force Decrease at High Reynolds Numbers in the Three-Pile Configuration

Also of note with the three-pile configuration is that at the highest Reynolds Number, the total force on the pile group seems to decrease. This result was consistent for all experiments – and again, each experiment was repeated three times. This is also very counterintuitive because one would think that increasing the velocity should increase the total force on the pile group.

The reason for the decrease in force as Reynolds Number increases is most likely due to a change in the flow separation point around the piles. At the lower Reynolds Number, it is

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possible that the flow separates at a certain angle when there are three piles in line with one another. As the velocity increases, the separation point may change to the point where it results in a higher pressure in the wake region and thus a reduced pressure drag component.

The tail-off of forces under the three pile configuration may also be due to experimental error. The force balance has four different ranges on which it can run zero to 0.5N, 1N, 2N, and

5N. Ranges for experiments were selected so that the most precise results would be obtained.

For example, if it was known that most of the force readings would be between 0.3N and 1.1N, the 1N range was selected. At the highest Reynolds Number, the force might be out of range –

1.5N for example. When the experiments were run, a mass was attached to the force balance when the forces approached the point where they were pushed out of range so that it would push the values back into range. Then, during post-processing, this force was added back to the total force at that Reynolds Number.

Complex Pile Arrangements Including Nine Pile Arrangement

As expected, the more complex pile arrangements exhibit a higher total force when more piles are added to the array. The original concept for this test sequence was to obtain the forces on the individual piles in the group with as few tests as possible with a single force transducer.

As additional piles were added to the group the additional force was thought to be that on the added piles. This assumes that the added piles do not impact the forces on the existing upstream piles, which was later proven not to be the case. While not providing sufficient data to determine the forces on the individual piles within the group it does give the total force on the various groups.

Interestingly, the piles in the more complex arrangement exhibit the same force reduction with increased Reynolds Number observed with the three-piles-in-line arrangement. The consistency of the data between the data sets would seem to rule out experimental error because

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the experiments for the nine-pile arrangement were all conducted at the 5N range, and the total force on the pile group was never “out of range” during the experiments. As pointed out earlier this could be due to a shift in the flow separation point on the leading piles impacting the pressure distribution on these piles. A study that includes the measurement of the pressure distribution around the piles would be helpful in explaining the reduction of force with increasing Reynolds Number.

Statistical Analysis of Force and PIV Data

A spectral analysis was undertaken for the in-line pile arrangements to further analyze both the force and the PIV data. After discovering that the force on the second pile in the three in-line pile arrangement must be negative, the hope was to draw some correlation between the vortex shedding frequency and the frequency of forcing on the second (and third) piles. A spectral analysis allows for isolation of frequencies within the dataset so that the dominant frequencies can be identified. If the forces on the in-line piles are due to the vortex shedding, and not the steady-state velocity in the flow domain, then the dominant velocity fluctuation frequency should approximately match the dominant force fluctuation frequency, at a given Reynolds Number.

Additionally, the force and velocity time series were analyzed to determine if values used throughout this paper (such as averages) are statistically meaningful. All force and velocity measurements were “de-meaned.” In other words, the average force and velocity over the time series was computed and subtracted from the signal so that only the fluctuations were analyzed.

If the average is truly a meaningful statistic, then the de-meaned velocity or force signal should fluctuate around zero, and the spectrum should show equal energies throughout the oscillating frequencies. If the average is not as statistically meaningful as originally thought, then the goal is two-fold:

1.) Explain why the average is not necessarily the best statistic of the flow to use

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2.) Discuss other possible methods for analyzing the flow around the piles.

Because the PIV measures velocity everywhere within the flow field, it would be possible to extract the flow spectra everywhere throughout the field. Within the flow window, a few key points that would isolate the vortex effects were thoroughly studied. First, when looking at a single-pile arrangement, the flow was sampled just before the water would have hit a second pile

(had a second pile been present) – approximately 35mm from the back edge of the single pile. In the three-pile arrangement, points were selected as labeled in Figure 5-7:

“Point A1”

“Point B” “Point D”

“Point A” “Point C”

Figure 5-7. Labeling Scheme for PIV Spectral Analysis.

Only the five flow conditions that allowed for comparison between PIV data and force data were thoroughly analyzed. There is a slight discrepancy between force measurements and velocity measurements (for example, the 5.13x103 velocity experiments are compared with the

4.76x103 force balance experiments, etc) because the idea to perform a spectral analysis came

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long after experiments for this thesis were completed. However, the Reynolds Numbers are close enough to allow for an accurate comparison between the measurements.

No-Pile PIV experiments

Before any pile configurations were analyzed, an analysis was conducted to see if there were any sloshing modes in the undisturbed flow. This was possible with the PIV because tests were run with nothing in the flume to confirm a steady flow. Figures 5-8 through 5-13 are time- series and spectral results of flow conditions in roughly the center of the PIV window.

As can be seen with these figures, there is a significant sloshing mode present in the PIV flume at a frequency of ~4.2Hz. Because there are no piles in the flow-field, this is certainly due to experimental conditions within the flume. Additionally, there are other spikes in the flow that are dependent on Reynolds Number. At Re = 5.13x103 there is a spike at ~1.5Hz; at Re =

3.85x103 there is a spike at ~1.25Hz; and at Re = 2.57x103 there is a spike at ~1.5Hz and ~5.5Hz.

-1 Velocity (cm/sec) Velocity

-2

-3

-4 0 5 10 15 Time (sec)

Figure 5-8. 0-Pile Time-Series in Middle of PIV Window, Re = 5.13x103

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Spectrum for 0 Piles Velocity in Middle of PIV Window Re = 5.13e3 6

3 Spectral Density 2

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-9. 0-Pile Spectrum in Middle of PIV Window, Re = 5.13x103 y 3

-1 Velocity (cm/sec) Velocity

-2

-3

-4 0 5 10 15 Time (sec)

Figure 5-10. 0-Pile Time Series in Middle of PIV Window, Re = 3.85x103

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Spectrum for 0 Piles Velocity in Middle of PIV Window Re = 3.85e3 5

4.5

3.5

2.5

2 Spectral Density 1.5

0.5

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-11. 0-Pile Spectrum in Middle of PIV Window, Re = 3.85x103

De-Meaned Velocity vs. Time for 0 Piles at in Middle of PIV Window, Re = 2.57e3 3

Velocity (cm/sec) Velocity -1

-2

-3 0 5 10 15 Time (sec)

Figure 5-12. 0-Pile Time Series in Middle of PIV Window, Re = 2.57x103

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Spectrum for 0 Piles Velocity in Middle of PIV Window Re = 2.57e3 4

3.5

2.5

1.5 Spectral Density

0.5

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-13. 0-Pile Time Series in Middle of PIV Window, Re = 2.57x103

At first, it is unclear what causes this ~4.2Hz spike at all three Reynolds Numbers.

However, if the tank dimensions are analyzed, the fundamental frequencies can be computed for the experimental flume. Because the flume is 12 inches wide, its fundamental frequencies are f = 2.009Hz, 4.1993Hz, 6.229Hz, 8.3986Hz, etc. for n = 1,2,3,4,etc. respectively. The spike of

~4.2Hz corresponds almost exactly to the n = 2 resonating fundamental frequency for the flume in which the experiments were run! Even though there were multiple methods used to ensure that flow was steady in the flume such as the flow straighteners, the air conditioning filter, and the entrance trumpet, there still seems to be some non-uniformity in the flow which is causing some lateral disturbances.

The second flow disturbance present in these time-series – the disturbance between 1.0Hz and 1.5Hz – depends on Reynolds Number. This disturbance does not match any of the fundamental frequencies for this flume, so resonating flume conditions cannot be its cause. The

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only other thing that could cause disturbances in the flow-field during this experiment is the pump that is pumping water through the flume. The pump works at different capacities to generate the different flow speeds for different Reynolds Numbers. At 100% capacity, the pump will run at a certain frequency – as is the case at the highest Reynolds Number; likewise at the lowest Reynolds Number, when the pump is only running at 25% capacity, it will have another frequency. Because this flow fluctuation depends on Reynolds Number, it is not unreasonable to assume that the pump’s frequency is responsible for it.

The final disturbance in the flow, the 5.5Hz disturbance, occurs only at the lowest

Reynolds Number. It is unknown what causes this disturbance as it does not match any fundamental frequencies that results from the geometry of the flume, nor does it appear at any of the other Reynolds Numbers. Further investigation would be useful to determine the causes of this flow fluctuation.

Single-Pile Experiments

Re ~ 5x103

PIV and force balance time-series and spectral results for the single-pile experiments at Re

~ 5x103 are presented in Figures 5-15 through 5-18. The plot of forces shows that taking the average force over the time series is statistically valid, although there are some spikes in the spectral density, particularly at ~ 3.5Hz. This 3.5Hz spike almost matches a similar spike in the velocity data at ~ 3.2Hz. However, the wave-nature of this spike appears to be minimal because the spectral-density of the spike, which is on the order of 10-4, is so small.

At the right-hand side of the force spectrum, the spectral density appears to be spiking, indicating a strong random signal or a frequency that is beyond the force balance’s measurement range (10Hz). Zooming into a five-second window of the time series (Figure 5-14), clearly exposes the saw-toothed nature of the signal.

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De-Meaned Velocity vs. Time for 1 Pile at Point A1, Re = 5.13e3 15

Velocity (cm/sec) Velocity -5

-10

-15 0 5 10 15 Time (sec)

Figure 5-14. De-Meaned Velocity vs. Time for 1 Pile at Point A1, Re = 5.13x103

Spectrum for 1 Pile Velocity at Point A1, Re = 5.13e3 14

6 Spectral Density

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-15. Velocity Spectrum for 1 Pile at Point A1, Re = 5.13x103

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0.1

0.05

Force (N) -0.05

-0.1

-0.15 0 10 20 30 40 50 60 Time (sec)

Figure 5-16. De-Meaned Force vs. Time for 1 Pile, Re = 4.72x103

-4 x 10 Spectrum for 1 Pile, Re = 5.13e3

3 Spectral Density 2

1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (Hz)

Figure 5-17. Force Spectrum for 1 Pile, Re = 4.72x103

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De-Meaned Force vs. Time for 1 Pile, Re = 5.13e3 0.2

0.15

0.1

0.05

0 Force (N) -0.05

-0.1

-0.15

-0.2 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 Time (sec)

Figure 5-18. Zoomed Force Time Series, Re = 5.13x103

Re ~ 4x103

At the middle Reynolds Number, the ~4.5Hz spike due to the resonance of the flume is still present in the signal. However, there does not appear to be any other significant spikes in the velocity signal. The wave-like nature of the force signal is also minimal as evidenced by the small magnitudes of the spectral density function, which is again on the order of 10-4. Results for single-pile at Re ~ 4.0 can be found from Figure 5-19 through Figure 5-22.

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De-Meaned Velocity vs . Time for 1 Pile at Point A1, Re = 3.85e3 10

-2 Velocity (cm/sec) Velocity -4

-6

-8

-10 0 5 10 15 Time (sec)

Figure 5-19. De-Meaned Velocity vs. Time for 1 Pile at Point A1, Re = 3.85x103 py, 25

10 Spectral Density

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-20. Velocity Spectrum for 1 Pile, Re = 3.85x103

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0.08

0.06

0.04

0.02

-0.02 Force (N)

-0.04

-0.06

-0.08

-0.1 0 10 20 30 40 50 60 Time (sec)

Figure 5-21. De-Meaned Force vs. Time for 1 Pile, Re = 3.83x103

-4 x 10 Spectrum for 1 Pile, Re = 3.85e3 2.4

2.2

1.8

1.6

1.4

1.2 Spectral Density 1

0.8

0.6

0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (Hz)

Figure 5-22. Force Spectrum for 1 Pile, Re = 3.83x103

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Three-Pile Experiments

As indicated in Figure 5-7, the three-pile arrangements were sampled at four different points. Because vortex movies were made of the flow around all pile configurations, and the goal was to isolate the vortex effects on the pile groups, the points that were chosen were purposely chosen where vortex effects would be seen. The goal was to isolate the fundamental vortex frequencies and to see if the vortex frequencies matched the measured forcing frequencies on the pile groups at the different Reynolds Numbers. Both lateral (u-direction) and transverse

(v-direction) velocities were studied in this analysis.

Re ~ 5x103

Velocity and forcing spectral results are presented from Figure 5-23 through Figure 5-25:

-4 x 10 Spectrum for 3 Piles, Re = 4.76e3 7

4 Spectral Density 3

1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (Hz)

Figure 5-23. Force Spectrum for 3 Piles, Re = 4.76x103

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30 Point A Point B 25 Point C Point D

15 Spectral Density 10

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-24. U-Velocity Spectrum for 3 Piles, Re = 5.13x103 yp 35 Point A Point B 30 Point C Point D 25

15 Spectral Density

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-25. V-Velocity Spectrum for 3 Piles, Re =5.13x103

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As seen from these figures, it looks as though the effect of the vortices on the forcing on the pile groups is small. The largest spikes in both the u-velocity and v-velocity spectra occur

~1.75Hz. In the corresponding forcing spectrum, there is a small spike around this frequency, but it is by-far the smallest spike on the chart. There are additional spikes on the v-velocity spectra at ~4.5Hz, which may correspond to a mid-range spike on the forcing spectra, or may be explained by the resonating oscillations of the experiment. Because the velocity spectra do not match the forcing spectra, it is reasonable to conclude that the vortices play a small role in the overall forcing on the pile group at this Reynolds Number.

Re ~ 4x103

Similarly, spectral results for Re ~ 4x103 are presented from Figure 5-26 through Figure 5-

28:

-4 x 10 Spectrum for 3 Piles, Re = 3.85e3

3 Spectral Density 2

1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (Hz)

Figure 5-26. Force Spectrum for 3 Piles, Re = 3.85x103

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14 Point A Point B 12 Point C Point D 10

6 Spectral Density

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-27. U-Velocity Spectrum for 3 Piles, Re = 3.85x103 V Velocity Spectrum for Re 3.85e3 25 Point A Point B Point C 20 Point D

10 Spectral Density

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-28. V-Velocity Spectrum for Re = 3.85x103

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Results at this Reynolds Number are very similar to results at the higher Reynolds

Number. There is a dual-spike in the v-velocity spectra, and the larger of the two spikes does not seem to coincide with any peaks on the forcing spectrum, but it does correspond to a similar spike in the u-velocity spectra. The second spike in the v-velocity spectra does match a spike in the forcing spectra, but again, from the zero-pile analysis, it is likely that a portion of this spike is due to experimental resonance. Because the largest spike in velocity does not correspond to any spike in forcing (in fact, it coincides with a minimum at this Reynolds Number!), this seems to indicate that once again, the vortices have a small effect on the overall forcing on the pile group.

Re ~ 3x103

Finally, spectral results at the lowest Reynolds Number are presented from Figure 5-29 through Figure 5-31.

-4 x 10 Spectrum for 3 Piles, Re = 2.86e3

3 Spectral Density 2

1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (Hz)

Figure 5-29. Force Spectrum for 3 Piles, Re = 2.86x103

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U-Velocity Spectrum for Re = 2.57e3 2.5 Point A Point B Point C 2 Point D

1.5

1 Spectral Density

0.5

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-30. U-Velocity Spectrum for 3 Piles, Re = 2.57x103 V Velocity Spectrum for Re 2.57e3 10 Point A 9 Point B Point C 8 Point D 7

4 Spectral Density 3

0 0 1 2 3 4 5 6 7 8 Frequency (Hz)

Figure 5-31. V-Velocity Spectrum for 3 Piles, Re = 2.57x103

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Results at this Reynolds Number look much different than at the higher two Reynolds

Number. There is a clear triple spike present in the forcing signal, and two of these peaks undoubtedly correspond to peaks present in the velocity spectra. At ~4.25Hz, there is a peak in the v-velocity spectrum and force spectrum, and at ~1.0Hz, there is a peak in the u-velocity spectrum and the force spectrum. There is another peak at ~2.5Hz in the force spectrum that does not however seem to coincide with any spike in the velocity spectrum. Although part of the

~4.25Hz spike in the velocity signal could probably be explained by the same resonant condition present in all the experiments, this signal is unique in that the spectral density of this signal is higher than all the other experiments.

Results at this Reynolds Number indicate that at this velocity, vortex effects have the greatest net-effect on forcing relative to average forcing on the pile group. If the total force at each spike on the force spectrum is computed from the root-mean-square of the amplitude, the

~1.0Hz force is 0.022N and the ~4.5Hz signal is 0.021N. Average forcing on the pile group at this Reynolds Number is ~ 0.139N; the force then due to vortices on this pile configuration is about 15% of the total force on the pile group, which is not insignificant. This explains why the velocity spectra seem to match the force spectrum so closely.

Further investigation to determine the net-effects of vortex forcing on piles at even lower

Reynolds Numbers would be valuable. Would the magnitude of the forcing on a downstream pile, which is on the order of 0.02N, remain the same for a lower Reynolds Number? If so, at the lower Reynolds Number, does it represent a greater percentage of the total force on the pile group?

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CHAPTER 6 FUTURE WORK

Additional Pressure Field Measurements

It would be beneficial to conduct drag force tests that include pressure measurements on, at least, some of the piles in the group in the TFHRC flume. Such tests would not be difficult to perform. A small hole would need to be drilled in each of the piles, and connected to a pressure transducer by small tubes. The pile could then be rotated to a finite number of positions over

360 degrees to yield the pressure distribution around the pile.

Additional Force Balance Measurements

With only one force balance and the present pile group setup only the total force on the pile group can be measured in the longitudinal direction. It would, however, be possible to construct a pile support system where the forces on only one of the piles in monitored. The other piles in the group could be placed in different configurations around the monitored pile so as to yield the forces on piles at all the locations within the group.

Other Drag Force Measurements

It would be beneficial to conduct experiments on other configurations of pile groups. The most complex group studied in this thesis was a three-by-three matrix of piles. It would be interesting to see what would happen if more columns of piles were added. For example, suppose instead of a configuration where there were three piles in a line, what would happen if there were four or five piles along the same line? Would the pattern of a large zero velocity zone behind the odd-numbered piles continue down the entire line? Would velocity consistently be larger on the even-numbered piles in the line?

From the experiments in this thesis, it seems as though this pattern would persist. The wake behind the first pile would engulf the second pile, the wake behind the third pile would

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engulf the fourth pile, and so-on. Measurements to back up this hypothesis would be useful so that a more comprehensive method for determining forces on pile groups of any configuration could be developed.

The addition of flow skew angles would add another dimension to the study. There would obviously be much greater wake interaction when the flow is skewed to the alignment of the piles. This situation is, however, important from an application point of view since some level of skew is almost always present in practice.

The addition of transverse force measurements would also greatly enhance this study.

Although the present TFHRC force balance setup does not allow for transverse force measurements, it would be possible to design a new force balance that could measure transverse forces. Load cells could also be used instead of the force balance to determine the forces on all three planes in three dimensions. It is expected that at higher flow velocities, the transverse forcing on the piles due to the oscillating nature of the vortices shedding from the piles can become significant.

Inertial and Wave Force Measurements

This thesis is aimed at studying the drag forces on piles within pile groups as a precursor to determining the force on pile groups subjected to waves. Wave forces on a single pile are given by the Morison Equation, which says that total force equals the sum of the inertial and drag forces on the pile.

The first step in understanding the wave forcing on pile groups is to measure the inertial force. An experiment is needed where piles are attached to load cells in a wave tank, and wave forces are measured directly. First, this experiment would help to validate that the drag forcing data that was obtained in this thesis under steady flow conditions can be utilized to predict the drag component of the Morison Equation under unsteady flow conditions. Secondly, this

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experiment would give a value for the inertial force on the piles within the pile group. Since the inertial force and the drag force on a pile in a wave field are out of phase, the two components can be separated.

Measuring wave forces on a pile group has its own inherent difficulties. At first glance, it would appear that the first piles in line would experience the greatest wave force. Sarpkaya and

Issacsson (1981) say that this is not necessarily the case; internal piles in the group may experience wave forces greater than the piles on the group’s front-face. There has already been work done that shows that this is due to constructive interference of waves. As waves propagate past the first couple of columns of piles in a group, they are scattered, and they can interfere with one another. If these interfering waves meet one another, and interfere constructively with one another, their amplitudes are magnified, and this magnification can cause a greater wave force on the internal piles. Once inertial forces are fully understood, an analysis needs to be conducted where constructive interference patterns of these waves are studied, so that one can predict when constructive interference will occur.

Large Scale Testing

All of the aforementioned studies involve small-scale laboratory testing of pile group configurations. After a reliable method for determining wave forcing on piles within groups has been found in the lab, further study needs to be conducted to determine how this relates to piles in the field. Reynolds Numbers in the lab are limited; the highest Reynolds Number possible in the TFHRC flume is on the order of 104. Under field conditions, piles will be subject to average

Reynolds Numbers two orders of magnitude higher. It has already been shown that drag coefficient varies considerably at higher Reynolds Numbers. As the Reynolds Number increases, the pressure gradient around piles in steady flow decreases, and then it increases again.

It would not be unreasonable to imagine that similar things happen during wave action – i.e., as

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Reynolds Number approaches the Reynolds Number that will be expected in the field, the behavior of the drag and inertial forcing components of wave forcing will also change.

Larger-scale tests are then needed to verify the lab results. Ideally, new bridge piers that are built could include an instrumentation package whereby the forces on the piles subject to wave action are studied.

Future Work Summary

Ultimately, coastal/ocean engineers need to be able to compute design current and wave induced forces on bridge sub and superstructures. When pile groups are present the forces on the individual piles as well as the resultant force on the group must be estimated. At present even the forces on a pile group in steady flow is not well understood. Even for steady flows there are many parameters involved including pile spacing, pile arrangement, pile group orientation to the flow, etc. that need to be investigated in the laboratory. The addition of waves increases the complexity of the problem by adding additional water and wave parameters (i.e. the addition of water depth, and wave height, period and direction).

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BIOGRAPHICAL SKETCH

Raphael Crowley has always had a dual-interest in both fluids and structures. In 2004,

Crowley graduated from Bucknell University with a BS in civil engineering. Although the focus of his coursework at Bucknell was in structural engineering, all of his research was in hydraulics.

While at Bucknell, under the tutelage of Dr. Richard Crago, Crowley began his research in hydrology. In the summer of 2002, Crowley assisted Crago with a project comparing the NRCS

Curve Number method for computing runoff with the TOPMODEL method. Results were presented at Bucknell University’s annual Kalman research symposium.

In summer 2003, Crowley worked again with Crago on a project to validate the applicability of the advection aridity approach for predicting evaporation and to determine a new method for finding kB-1 as a function of leaf area index. This project was followed up by Nikki

Hervol, and in 2004, two papers were accepted for publication in The Journal of Hydrology.

Crowley stumbled into Coastal Engineering in 2004 when he accepted a job at M.G.

McLaren, P.C. as a marine engineer. After a year with McLaren, Crowley began graduate school at the University of Florida to pursue his Ph.D. in coastal engineering. Under the tutelage of D.

Max Sheppard, Crowley’s research focuses on coastal-structural interactions. Particular interest is paid to research surrounding the 2004-2005 hurricane-related bridge collapses. In addition to this thesis, Crowley assisted in research projects to determine the wave loading on bridge decks, drag and lift forces on bridge decks under steady flow conditions, scour below bridge decks during inundation under steady flow conditions (pressure flow scenario), and shear forces below a bridge deck during inundation under steady flow conditions. Crowley hopes to graduate with his M.S. in coastal and oceanographic engineering in May 2008 and finish his Ph.D. by

December 2010.

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